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TITLE: Two matrix proofs
QUESTION [0 upvotes]: linear algebra problem I'm having some trouble wrapping my head around:
Given two square matrices $A,B$ with dimensions $n\times n$ and that $A=I-AB$ :
I've already proved with relative ease that $A$ is invertible and that $AB=BA$
Now I'm being asked to prove two more arguments:
Prove that if $B$ is symmetric then $A$ is symmetric as well
Prove that $B^3=0$ if and only if $A=I-B+B^2$
For the first argument I think one of the properties of symmetric matrices - the product of two symmetric matrices is also symmetric iff the two matrices commute - is helpful here. As for the second argument I really don't know where to start. Can someone shed some light on this please? Thanks!
REPLY [1 votes]: By definition,
$$
A=I-AB
$$
next, plug $A=I-AB$ into the right hand side of this equation:
$$
A=I-AB=I-\overbrace{(I-AB)}^{\equiv A}B
$$
and simplify:
$$
A=I-B+AB^2
$$
next, repeat the same trick: replace $A=I-AB$ into the right hand side of this equation:
$$
A=I-B+AB^2=I-B+\overbrace{(I-AB)}^{\equiv A}B^2
$$
and simplify:
$$
A=I-B+B^2-AB^3
$$
This means that
$$
AB^3=I-B+B^2-A
$$
Thus, $B^3=0$ iff the r.h.s$=0$ (remember that $A$ is invertible, as you already know)
REPLY [1 votes]: $A=I-AB$ implies that $A(I+B)=I$, thus $A$ is the inverse of $I+B$. If $B$ is symmetric, $B^t=B$, thus $(A(I+B))^t=I^t=(I^t+B^t)A^t=(I+B)A^t=I$. We deduce that $A^t$ is also the inverse of $I+B$ thus $A^t=A$. Since the inverse of a matrix is unique.
, $(I+B)(I-B+B^2)=I+B^3$ thus $B^3=0$ iff $I-B+B^2$ is the inverse of $I+B$, this is equivalent to saying that $B^3=0$ iff $I-B+B^2=A$ since the inverse of a matrix is unique if it exists done.
| 150,562
|
TITLE: $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$
QUESTION [1 upvotes]: I'm trying to show that
$T := \mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ for some $d \in \mathbb{Z}$.
Now I know that as $T$ is a tensor product, there exists a bilinear map $t : \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \rightarrow T$ such that for all Abelian groups $A$ and all bilinear maps $b : \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \rightarrow A$, there exists a unique $\tilde{b} : T \rightarrow A$ such that $b = \tilde{b} \circ t$. This is by definition of the tensor product.
I first note that for $d := \text{gcd}(m,n)$, the map
$$b :\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/d\mathbb{Z}, (x \mod m, y \mod n) \mapsto xy \mod d$$
is a well-defined bilinear map. This then gives us our unique $\tilde{b}$.
Now if it were true that $(\mathbb{Z}/d\mathbb{Z}, b)$ is also a tensor product, we would obtain a map $\tilde{t}$ such that $\tilde{b} = \tilde{t}^{-1}$, hence giving a group isomorphism $T \tilde{\rightarrow} \mathbb{Z}/d\mathbb{Z}$.
To prove this, I would have to show that for every Abelian group $A$ and every bilinear map $a : \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z} \rightarrow A$, there exists a unique map $\tilde{t}: \mathbb{Z}/d\mathbb{Z} \rightarrow A$ such that $a = \tilde{t} \circ b$.
Unfortunately I have no idea why that would be the case.
REPLY [2 votes]: If you like a more algebraic approach, consider the map
$$
f : \mathbb Z \to \mathbb Z / m \mathbb Z \otimes_{\mathbb Z} \mathbb Z / n \mathbb Z,\; a \mapsto a \otimes 1.
$$
This is clearly surjective, so we must have
$$
\mathbb Z / m \mathbb Z \otimes_{\mathbb Z} \mathbb Z / n \mathbb Z \simeq \mathbb Z / \ker f.
$$
Now all that is left is to show that $\ker f$ is of the form $d \mathbb Z$. In particular if $a \in (m,n)$ then there are $k,\ell$ such that $a = km + \ell n$. Then
$$
a \otimes 1 = km \otimes 1 + 1 \otimes \ell n = 0 \otimes 1 + 1 \otimes 0 = 0.
$$
This shows $(m,n) \subset \ker f$.
The converse is also true, so we are done.
Big edit
My "proof" essentially sidesteps using the universal property of the tensor product, and instead pushes it to the last step which I skip. I have tried to make this step rigorous but fail to do this in a nice way. So let us instead use your method.
Suppose everything in your post. In particular we have the map $a : \Bbb Z / m \Bbb Z \times \Bbb Z / n \Bbb Z \to A$. We must show that there exists a unique $\Bbb Z$-linear map $a' : \Bbb Z / d \Bbb Z \to A$ such that $a'\circ b = a$.
The tough part (I guess) is to find the map $a'$ and to recognize that $d = \gcd(m,n)$.
So let $d = \gcd(m,n)$ and define the map
$$
a' : \Bbb Z / d \Bbb Z \to A,\; x \mapsto a(x,1).
$$
One then shows that this map is well-defined, $\Bbb Z$-linear, satisfies $a'\circ b = a$ and unique.
Assume first that $a'$ is well-defined.
The $\Bbb Z$-linearity follows immediately from the $\Bbb Z$-bilinearity of $a$.
One checks that $a'\circ b(x,y) = a'(xy) = a(xy, 1) = y \cdot a(x,1) = a(x,y)$. Uniqueness can also easily be checked.
So what remains to be shown is well-definedness. Suppose $x \equiv y \mod d$, then $x - y = z \cdot d$ some $d$-multiple. Since $d = \gcd(m,n)$ there are $k,\ell$ integers such that $d = km + \ell n$. Now
$$
a'(x) - a'(y) = a(x,1) - a(y,1) = a(x - y,1)\\ = a(zd,1) = a(zkm + z \ell n,1) = a(zkm,1) + a(1,z \ell n) = a(0,1) + a(1,0) = 0.
$$
So $a$ is well-defined and we are done.
| 55,285
|
TITLE: Find general formula for the terms
QUESTION [5 upvotes]: Find a general formula for the terms of the sequence
$${a_n}=\left\{ \frac{11}{7},\frac{107}{49},\frac{659}{343},\frac{4883}{2401},\frac{33371}{16807},\frac{234569}{117649},\dots \right\}$$
I don't know how to approach this question as it is not arithmetic or geometric. I know the denominator is geometric increasing by a factor of $7$ but I can't find what the numerator should be for the general formula for the terms. Anyone know what it is?
REPLY [7 votes]: The general formula for an (infinite) sequence of (e. g. real) numbers from the finite number $n$ of its (first) members is in principle impossible, as the next (not listed) $(n+1)^\mathrm{th}$ member may be an arbitrary number, and there is still a formula for expressing $a_1, \dots, a_n, a_{n+1},$ e. g. as a polynomial of order $n$:
$$a_k = \sum_{i=0}^nb_ik^i,\quad k = 1, \dots,n+1$$
The process for finding coefficients $b_0, \dots, b_n$ is straightforward enough.
In other words, if someone will find the formula for your "sequence", there is still the infinity number of other formulas, giving different sequences, but all of them producing your "sequence", i. e. $$\frac{11}{7},\frac{107}{49},\frac{659}{343},\frac{4883}{2401},\frac{33371}{16807},\frac{234569}{117649}.$$
Note:
It means that all psychological tests of type
What is the next number of the sequence $1, 2, 3, 4, 5?$
are in principle meaningless ones, because you may tell "$1762$", and then show to surprised psychologist a formula supporting the correctness of your answer:
$$a_k = {439\over30}k^5-{439\over 2}k^4+{7463\over 6}k^3-{6585\over 2}k^2+{60158\over 15}k-1756$$
If he/she will not trust you, launch SageMath, which will produce accurate, non-rounded results, and write commands
sage: var("k")
sage: a(k) = (439/30)*k^5 - (439/2)*k^4+(7463/6)*k^3-(6585/2)*k^2+(60158/15)*k-1756
sage: a(1), a(2), a(3), a(4), a(5), a(6), a(7), a(8)
to obtain the result
(1, 2, 3, 4, 5, 1762, 10543, 36884)
(and to give the psychologist two more members for free).
Note 2:
It doesn't mean that there is not a simpler formula - including a recurrent one or other "recipe" - for the same (finite) sequence.
For example, there is so simple one for the rather not so trivial sequence
$$\color{blue}{1, 11, 21, 1211, 111221}$$
that even 6-7 year-old child is able to write down the next element ($\color{red}{312211}$) if you tell it the rule, or - perhaps - even without telling it.
No, you have no chance to discover this simple rule (supposing your age is $10$+). Don't waste your time. It's a good advice, believe me.
(Googling for it is a much better approach.)
| 143,231
|
\begin{document}
\title[Modular forms with poles on hyperplane arrangements]{Modular forms with poles on hyperplane arrangements}
\author{Haowu Wang}
\address{Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Korea}
\email{haowu.wangmath@gmail.com}
\author{Brandon Williams}
\address{Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany}
\email{brandon.williams@matha.rwth-aachen.de}
\subjclass[2020]{11F55, 32S22, 17B22, 11F50, 11F46}
\date{\today}
\keywords{Modular forms on symmetric domains, Orthogonal groups, Unitary groups, Hyperplane arrangements, Looijenga compactification, Root systems, Jacobi forms, Theta blocks conjecture}
\begin{abstract}
We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct $8$ free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the $2U\oplus D_{11}$ lattice, which admits a free algebra on $14$ meromorphic generators. Along the way, we determine minimal systems of generators for non-free algebras of orthogonal modular forms for $26$ reducible root lattices and prove the modularity of formal Fourier--Jacobi series associated to them. By exploiting an identity between weight one singular additive and multiplicative lifts on $2U\oplus D_{11}$, we prove that the additive lift of any (possibly weak) theta block of positive weight and $q$-order one is a Borcherds product; the special case of holomorphic theta blocks of one elliptic variable is the theta block conjecture of Gritsenko, Poor and Yuen.
\end{abstract}
\maketitle
\begin{small}
\tableofcontents
\end{small}
\addtocontents{toc}{\setcounter{tocdepth}{1}}
\section{Introduction}
\subsection{Modular forms with poles on hyperplane arrangements}
Geometric invariant theory provides a means of constructing and compactifying many interesting moduli spaces. On the other hand, global Torelli theorems often yield identifications of moduli spaces with locally symmetric varieties under the Hodge-theoretic period map. It is natural to compare the GIT-compactifications of these moduli spaces with the Baily--Borel compactifications of the related arithmetic quotients. The moduli space of polarized $\mathrm{K3}$ surfaces of a fixed degree can be identified with the quotient of a symmetric domain of type IV and dimension $19$ by an arithmetic group, and this identification extends to the respective compactifications. Similar identifications also hold for the moduli spaces of Enriques surfaces and Del Pezzo surfaces.
However, for many interesting geometric objects, the image of the moduli space under the period map is not the full quotient of a Hermitian symmetric domain but rather the complement of a hyperplane arrangement.
In 2003, Looijenga \cite{Loo03a, Loo03b} constructed a compactification for the complements of certain hyperplane arrangements in arithmetic quotients of complex balls and type IV symmetric domains. The Looijenga compactification is, roughly speaking, an interpolation between the Baily--Borel and toroidal compactifications, and it coincides with the Baily--Borel compactification if the hyperplane arrangement is empty. For a number of moduli spaces, it has been proved that the GIT compactifications are identified with the Looijenga compactifications via an extension of the period map; for example, the moduli spaces of quartic curves \cite{Kon00}, genus four curves \cite{Kon02}, rational elliptic surfaces \cite{HL02}, cubic surfaces \cite{ACT02}, cubic threefolds \cite{ACT11, LS07} and cubic fourfolds \cite{Loo09, Laz10}.
Let $\cD$ be a Hermitian symmetric domain and let $\Gamma$ be a congruence subgroup acting on $\cD$. The Baily--Borel compactification of $\cD / \Gamma$ can be understood as the $\mathrm{Proj}$ of the algebra of holomorphic modular forms on $\cD$ for $\Gamma$ \cite{BB66}. Let $\mathcal{H}$ be a $\Gamma$-invariant arrangement of hyperplanes on $\cD$. Assume that $\mathcal{H}$ satisfies the \emph{Looijenga condition} which guarantees in particular that a generalized Koecher's principle holds; that is, the algebra $M_*^!(\Gamma)$ of modular forms on $\cD$ for $\Gamma$ with poles contained in $\mathcal{H}$ is generated by forms of positive weight. Looijenga proved that $M_*^!(\Gamma)$ is finitely generated and its $\mathrm{Proj}$ characterizes the Looijenga compactification of the complement $(\cD - \mathcal{H}) / \Gamma$.
Holomorphic modular forms on symmetric domains have a very rich theory. The study of (holomorphic) modular forms on orthogonal groups has important applications to infinite-dimensional Lie algebras and birational geometry; for example, the classification of generalized Kac--Moody algebras \cite{Bor95, GN98, Sch06}, the proof that moduli spaces of polarized $\mathrm{K3}$ surfaces of degree larger than $122$ are of general type \cite{GHS07}, and the proof of the finiteness of orthogonal modular varieties not of general type \cite{Ma18}. Modular forms with singularities on hyperplane arrangements, by contrast, have attracted less attention. There are only a few explicit structure theorems for these algebras in the literature. In this paper we investigate the automorphic side of the Looijenga compactification, and especially the construction of free algebras of meromorphic modular forms for which these compactifications are simple weighted projective spaces.
In \cite{Wan21a} the first named author found necessary and sufficient conditions for an algebra of modular forms on $\Orth(l, 2)$ to be free. These conditions rely on the Jacobian of a set of potential generators having simple zeros exactly on the mirrors of reflections in the modular group. By Bruinier's converse theorem \cite{Bru02, Bru14} such a Jacobian must be a Borcherds product \cite{Bor98}. The necessary condition yields an explicit classification of free algebras of orthogonal modular forms \cite{Wan21a}. Using the sufficient part of the criterion, we constructed a number of free algebras of orthogonal modular forms \cite{Wan21RNT, WW20b, WW20c}.
In \cite{WW21a} we extended this approach to modular forms on complex balls attached to unitary groups of signature $(l,1)$. In this paper we further extend this argument to modular forms with singularities on hyperplane arrangements. Our first main theorem is
\begin{theorem}\label{MTH1}
Let $\cD_l$ be a symmetric domain of type IV and dimension $l\geq 3$ or a complex ball of dimension $l\geq 2$. Let $\Gamma$ be a congruence subgroup of $\Orth(l,2)$ or $\U(l,1)$ acting on $\cD_l$. Let $\cH$ be a $\Gamma$-invariant arrangement of hyperplanes satisfying the Looijenga condition. Then the algebra of modular forms on $\cD$ for $\Gamma$ with poles supported on $\mathcal{H}$ is freely generated by $l+1$ forms if and only if the Jacobian $J$ of the $l+1$ potential generators vanishes with multiplicity $d_\sigma-1$ on mirrors of reflections $\sigma$ in $\Gamma$ which are not contained in $\mathcal{H}$, and the other zeros and poles of $J$ are contained in $\mathcal{H}$, where $d_\sigma$ is the order of $\sigma$.
\end{theorem}
By applying the Jacobian criterion to symmetric domains of type IV attached to root lattices, we obtain the structure of the following algebras:
\begin{theorem}\label{MTH2}
Let $L$ be any lattice in the following three families of root lattices:
\begin{align*}
&\text{$A$-type:}& & \left(\bigoplus_{j=1}^t A_{m_j}\right)\oplus A_m, \quad t\geq 0, \quad m\geq 1, \quad (m+1)+\sum_{j=1}^t (m_j+1) \leq 11; \\
&\text{$AD$-type:}& & \left(\bigoplus_{j=1}^t A_{m_j}\right)\oplus D_m, \quad t\geq 0, \quad m\geq 4, \quad m + \sum_{j=1}^t (m_j+1) \leq 11; \\
&\text{$AE$-type:}& & E_6,\quad A_1 \oplus E_6, \quad A_2\oplus E_6, \quad E_7,\quad A_1\oplus E_7.
\end{align*}
Let $U$ be the even unimodular lattice of signature $(1,1)$. Let $\widetilde{\Orth}^+(2U\oplus L)$ denote the subgroup of $\Orth(2U\oplus L)$ which respects the symmetric domain and acts trivially on the discriminant group $L'/L$. Then there exists an arrangement of hyperplanes $\mathcal{H}$ such that the ring of modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ with poles supported on $\mathcal{H}$ is a polynomial algebra.
\end{theorem}
If we write $L = L_0 \oplus L_1$ with $L_1 = A_m, D_m, E_6, E_7$, then the hyperplane arrangement $\cH$ is a union $\cH_{L, 0} \cup \cH_{L, 1}$, where $\cH_{L, 0}$ consists of the orbits of hyperplanes $v^{\perp}$ for vectors $v\in L_0'$ of minimal norm in the dual of each component of $L_0$, and where $\cH_{L, 1}$ consists of the orbits of hyperplanes $(r+u)^{\perp}$, where $u\in L'$ are vectors of minimal norm in their cosets in $L'/L$ that satisfy $u^2 > 2$, and $r\in U$ with $(r,r)=-2$. This is made more precise in Theorem \ref{th:2precise}. There are $147$ hyperplane arrangements in total.
When $L = A_m$ for $1 \leq m \leq 7$ or $L = D_m$ for $4 \leq m \leq 8$ or $L = E_6, E_7$, the arrangement $\mathcal{H}$ above is empty. The corresponding free algebras of holomorphic modular forms have been constructed in \cite{Igu62, FH00, DK03, Kri05, Vin10, Vin18, WW20a}. By \cite{Wan21a}, these $14$ algebras and the algebra associated to $E_8$ computed in \cite{HU14} are the only free algebras of holomorphic modular forms for groups of type $\widetilde{\Orth}^+(2U\oplus M)$. The free algebra associated to $L=A_1\oplus A_2$ was determined in \cite{Nag21} using the period map between the moduli space of $U\oplus E_7\oplus E_6$-polarised $\mathrm{K3}$ surfaces and the arithmetic quotient attached to $2U\oplus A_1\oplus A_2$. The case $A_1\oplus A_1$ is related to Hermitian modular forms of degree two over $\QQ(\sqrt{-1})$. The remaining $131$ free algebras of modular forms with poles on (nonempty) arrangements seem to be new, and we expect them to also have interpretations of moduli spaces of lattice polarised $\mathrm{K3}$ surfaces. We remark that Hermitian modular forms of degree two and Siegel modular forms of degree two have been treated by a similar method in our previous papers \cite{WW21c, WW21d}.
Vinberg and Shvartsman \cite{VS17} showed that the ring of holomorphic modular forms on symmetric domains of type IV and dimension $l>10$ is never free. However, Theorem \ref{MTH2} includes a number of free algebras of meromorphic modular forms in dimension $l > 10$. In the most extreme case, the algebra of meromorphic modular forms associated to $L=D_{11}$ will turn out to be freely generated in weights $1$, $4$, $4$, $6$, $6$, $8$, $8$, $10$, $10$, $12$, $12$, $14$, $16$ and $18$; the associated variety has dimension $13$.
Some of the lattices in Theorem \ref{MTH2} have complex multiplication over $\QQ(\sqrt{-1})$ or $\QQ(\sqrt{-3})$. If in addition the restriction $\cH_{\U}$ of $\cH$ to the complex ball satisfies the Looijenga condition, then the associated algebra of unitary modular forms whose poles are contained in $\cH_{\U}$ is also free. This is described in Theorem \ref{th:algebras-unitary}. For example, $2U\oplus D_{10}$ has complex multiplication over $\QQ(\sqrt{-1})$, and the ring of meromorphic modular forms on the $6$-dimensional complex ball attached to $D_{10}$ is freely generated in weights $2$, $4$, $8$, $8$, $12$, $12$, $16$. This modular group is an example of a finite-covolume reflection group acting on complex hyperbolic space of dimension $6$ with quotient birational to $\mathbb{CP}^6$.
Let us sketch the idea of the proof of Theorem \ref{MTH2}. To apply Theorem \ref{MTH1}, we need to construct the potential generators, verify that their Jacobian is nonzero, and that it has the zero divisor prescribed by Theorem \ref{MTH1}. In general, there seems to be no rule to characterize the weights of the generators; however, when the underlying lattice is related to a root system, we find that the weights of the generators can be predicted in terms of invariants of the root system, using the theory of Jacobi forms. Jacobi forms are connected to modular forms on orthogonal groups by the Fourier--Jacobi expansion (cf. \cite{EZ85, Gri94}). Wirthm\"uller \cite{Wir92} proved that for any irreducible root system $R$ other than $E_8$, the ring of weak Jacobi forms associated to its root lattice that are invariant under the Weyl group is a polynomial algebra, and the weights and indices of its generators are natural invariants of $R$. We found in \cite{WW20a} that the weights of the generators of a free algebra of holomorphic modular forms related to a root system are simply $k_i + 12t_i$, where $(k_i, t_i)$ are the weights and indices of the generators of the ring of Weyl-invariant weak Jacobi forms. The weights of the generators of the rings of meromorphic modular forms in Theorem \ref{MTH2} are more complicated (cf. Theorem \ref{th:2precise}) and are controlled by exceptional generators related to theta blocks (cf. \cite{GSZ19}). The first part $\cH_{L, 0}$ of the hyperplane arrangement corresponds to the divisors of these theta blocks. The generators are constructed in terms of Borcherds' additive and multiplicative singular theta lifts \cite{Bor95, Bor98} and we will be able to read their algebraic independence off of their leading Fourier--Jacobi coefficients. To compute the divisor of their Jacobian, we prove the existence of a preimage of the Jacobian under Borcherds' lift. We do not construct their preimages directly; instead, we prove their existence in a uniform way by an argument similar to that used by the first named author \cite{Wan19} in the classification of reflective modular forms.
It was proved in \cite{HU14} that the ring of holomorphic modular forms for $\widetilde{\Orth}^+(2U\oplus E_8)$ is also free. However, we know from \cite{Wan21b} that the ring of Weyl-invariant weak Jacobi forms for $E_8$ is not a polynomial algebra. For this reason, the proof of Theorem \ref{MTH2} does not apply to this case. For $9\leq m\leq 11$ we have $2U\oplus D_m \cong 2U\oplus E_8 \oplus D_{m-8}$, but it is not clear to us how the root lattice structure of the latter model is related to the algebra structure. The restrictions in Theorem \ref{MTH2} are natural; if the condition ‘‘$\leq 11$" in Theorem \ref{MTH2} fails to hold, then the Looijenga condition never holds; and if we define the arrangement in other ways then it seems that either the Looijenga condition fails or the algebra of meromorphic modular forms is non-free.
\subsection{The theta block conjecture}
The generators of ``abelian type" (cf. Theorem \ref{th:2precise}) of algebras of meromorphic modular forms are constructed as rational expressions in the Gritsenko lifts of basic Jacobi forms. In order to control the divisor that results, we will prove and then use at several points that fact that the (possibly singular) Gritsenko lift of any lattice-valued pure theta block of $q$-order one is a Borcherds product. This result is of independent interest: the case of holomorphic theta blocks associated to lattices of rank one is the \emph{Theta Block Conjecture} of Gritsenko--Poor--Yuen \cite{GPY15}, and it is closely related to the question of which modular forms are simultaneously Borcherds products and Gritsenko lifts.
The theory of theta blocks is developed in the paper \cite{GSZ19} of Gritsenko--Skoruppa--Zagier. The (pure) theta block associated to a function $f: \NN \to \NN$ with finite support is defined as
\begin{equation}
\Theta_f(\tau,z)=\eta(\tau)^{f(0)} \prod_{a=1}^\infty (\vartheta(\tau,a z)/\eta(\tau))^{f(a)}, \quad (\tau, z) \in \HH \times \CC,
\end{equation}
where $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n)$ is the Dedekind eta function and $\vartheta$ is the odd Jacobi theta function
\begin{equation}
\vartheta(\tau,z)=q^{1/8}(e^{\pi iz}-e^{-\pi i z})\prod_{n=1}^\infty (1-q^ne^{2\pi i z})(1-q^ne^{-2\pi iz})(1-q^n), \quad q=e^{2\pi i\tau}.
\end{equation}
The function $\Theta_f$ is a weak Jacobi form (with character) of weight $f(0)/2$ in the sense of Eichler--Zagier \cite{EZ85}. Gritsenko--Poor--Yuen conjectured that if $\Theta_f$ is a holomorphic Jacobi form of integral weight and has $q$-order one, i.e. $$f(0)+2\sum_{a=1}^\infty f(a)=24,$$ then its Gritsenko lift \cite{Gri94} is a Borcherds product. This has been proved in \cite{GPY15, Gri18, GW20, DW20} for the eight infinite families of theta blocks of $q$-order one (built in \cite{GSZ19}) related to root systems, and in particular for all theta blocks of weight at least $4$. However, there are many theta blocks of $q$-order one in weights two and three which do not belong to these families. Surprisingly, the proof of the theta block conjecture in general becomes much easier when we extend it to the (singular) Gritsenko lifts of theta blocks which are merely weak Jacobi forms. We will prove that all such Gritsenko lifts are (meromorphic) Borcherds products. This follows from identities for singular theta lifts associated to the root lattices $D_m$:
\begin{theorem}\label{MTH3}
As a meromorphic modular form of weight $12-m$ on $\widetilde{\Orth}^+(2U\oplus D_m)$, the Borcherds additive lift of the generalized theta block
\begin{equation}
\vartheta_{D_m}(\tau, \mathfrak{z}) =\eta(\tau)^{24-3m} \prod_{j=1}^m \vartheta(\tau, z_j), \quad 1\leq m \leq 11
\end{equation}
is the Borcherds multiplicative lift of the weight zero weak Jacobi form $-(\vartheta_{D_m}| T_{-}(2) ) / \vartheta_{D_m}$, where $T_{-}(2)$ is the index-raising Hecke operator on Jacobi forms. As a result, the singular additive lift of any generalized theta block of positive weight and $q$-order one is a Borcherds product. In particular, the Gritsenko--Yuen--Poor conjecture is true.
\end{theorem}
The inputs into Borcherds' singular additive lifts \cite{Bor95, Bor98} are Jacobi forms of positive weight. When the inputs are holomorphic Jacobi forms, the singular additive lift coincides with the Gritsenko lift. When $m\leq 8$, the form $\vartheta_{D_m}$ above is a holomorphic Jacobi form of index $D_m$ so its Gritsenko lift is a holomorphic modular form, and in these cases Theorem \ref{MTH3} was proved in \cite{GPY15}. When $m>8$, $\vartheta_{D_m}$ is no longer a holomorphic Jacobi form and its additive lift will have singularities along $\cH_{D_m,1}$ defined in Theorem \ref{MTH2} (note $\cH_{D_m,0}=\emptyset$).
The Borcherds product in Theorem \ref{MTH3} has only simple zeros outside of the hyperplane arrangement $\cH_{D_m,1}$, and these simple zeros are determined by the zeros of the theta block $\vartheta_{D_m}$. This allows us to reduce the proof of the theorem to the Koecher principle on the Looijenga compactification.
\subsection{Non-free algebras of holomorphic modular forms}
Theorem \ref{MTH2} also includes a class of generators of ``Jacobi type", whose leading Fourier--Jacobi coefficients are $\Delta^{t}\phi_{k,t}$ where $\Delta=\eta^{24}$ is the cusp form of weight $12$ on $\SL_2(\ZZ)$, and $\phi_{k,t}$ are the generators of weight $k$ and index $t$ of the ring of Weyl invariant weak Jacobi forms. This motivates us to study the rings of holomorphic modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ satisfying the following property:
\vspace{3mm}
\textit{Let $J_{k,L,t}^{\w,\widetilde{\Orth}(L)}$ denote the space of weak Jacobi forms of weight $k$ and index $t$ associated to $L$ which are invariant under $\widetilde{\Orth}(L)$. For any $\phi_{k,t}\in J_{k,L,t}^{\w,\widetilde{\Orth}(L)}$, there exists a holomorphic modular form of weight $12t+k$ for $\widetilde{\Orth}^+(2U\oplus L)$ whose leading Fourier--Jacobi coefficient is $\Delta^{t}\phi_{k,t}$.}
\vspace{3mm}
Following \cite{Aok00} we can estimate the dimensions of spaces of orthogonal modular forms in terms of Jacobi forms:
$$
\dim M_k(\widetilde{\Orth}^+(2U\oplus L)) \leq \sum_{t=0}^\infty \dim J_{k-12t,L,t}^{\w,\widetilde{\Orth}(L)}.
$$
The above property means that this inequality is actually an equality in every weight. This property also implies the modularity of formal Fourier--Jacobi series, which has applications to the modularity of certain generating series which appear in arithmetic geometry (see \cite{BR15}). We classify all root lattices satisfying this property and we also determine the corresponding rings of modular forms in the last main theorem:
\begin{theorem}\label{MTH4}
Let $L$ be a direct sum of irreducible root lattices not of type $E_8$. Assume that $L$ satisfies the $\mathrm{Norm}_2$ condition, i.e. $\delta_L \leq 2$, where
$$
\delta_L := \max\{ \min\{\latt{y,y}: y\in L + x \} : x \in L' \}.
$$
Then the algebra of holomorphic modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ is minimally generated by forms of weights $4$, $6$ and $12t+k$, where the pairs $(k, t)$ are the weights and indices of the generators of the ring of $\widetilde{\Orth}(L)$-invariant weak Jacobi forms.
\end{theorem}
There are $40$ root lattices that satisfy the conditions of Theorem \ref{MTH4}. Fourteen of them are irreducible and they correspond to the free algebras of holomorphic modular forms which were included in Theorem \ref{MTH2}. The $26$ reducible root lattices in Theorem \ref{MTH4} yield non-free algebras of modular forms. Some of these non-free algebras have a lot of generators, and a complicated structure which seems hard to determine by other methods. For example, the algebra of holomorphic modular forms for $\widetilde{\Orth}^+(2U\oplus A_2\oplus E_6)$ has Krull dimension $11$ but is minimally generated by $33$ generators of weights 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 21, 22, 24. The construction of generators of Jacobi type for lattices of $AE$-type in Theorem \ref{MTH2} is completed by Theorem \ref{MTH4}.
\subsection{The structure of this paper} In \S \ref{sec:Looijenga} we review the theory of meromorphic modular forms on type IV symmetric domains related to the Looijenga compactification, and we establish the modular Jacobian approach (i.e. Theorem \ref{MTH1}). In \S \ref{sec:Jacobiforms} we review Jacobi forms of lattice index and the Gritsenko and Borcherds lifts. \S \ref{sec:theta block conjecture} contains the proof of Theorem \ref{MTH3} and therefore the theta block conjecture. The heart of the paper is \S \ref{sec:type IV} where we prove Theorem \ref{MTH2}, with the exception of some lattices related to $E_6$ and $E_7$. In \S \ref{sec:non-free} we prove Theorem \ref{MTH4} and complete the proof of Theorem \ref{MTH2}. In \S \ref{sec:ball quotients} we review Looijenga's compactifications for unitary groups and we use Theorem \ref{MTH2} to construct eight free algebras of meromorphic unitary modular forms, including one such algebra on a six-dimensional complex ball. In the appendix (cf. \S \ref{appendix}), we formulate the weights of generators of the algebras of modular forms described in Theorems \ref{MTH2} and \ref{MTH4}.
\addtocontents{toc}{\setcounter{tocdepth}{2}}
\section{Meromorphic modular forms on orthogonal groups}\label{sec:Looijenga}
In this section we review the Looijenga compactification on symmetric domains of type IV and establish the modular Jacobian approach for meromorphic modular forms whose poles are contained in an arrangement of hyperplanes.
\subsection{The Looijenga compactification}\label{subsec:Looijenga}
Let $M$ be an even lattice of signature $(l,2)$ for some $l\geq 3$ with bilinear form $(-,-)$. We equip the complex vector space $V(M):=M\otimes \CC$ with the $\QQ$-structure defined by $M$ and the non-degenerate symmetric bilinear form induced by $(-,-)$. The corresponding orthogonal group $\Orth(V(M))\cong\Orth(l,2)$ is therefore an algebraic group defined over $\QQ$. The set of complex lines $\mathcal{Z} \in \PP(V(M))$ satisfying $( \mathcal{Z}, \mathcal{Z} ) = 0$ and $( \mathcal{Z}, \overline{\mathcal{Z}} ) < 0$ has two conjugate connected components. We fix one component once and for all; it is a symmetric domain of Cartan type IV, denoted $\cD(M)$.
Let $\Orth^+(M)$ be the subgroup of $\Orth(l,2)$ that preserves $M$ and $\mathcal{D}(M)$. The subgroup of $\Orth^+(M)$ which acts trivially on the discriminant group $M'/M$ is the \emph{discriminant kernel}, $$\widetilde{\Orth}^+(M) = \{\gamma \in \Orth^+(M): \; \gamma x - x \in M \; \text{for all} \; x \in M'\}.$$
The symmetric domain $\mathcal{D}(M)$ also determines a set of oriented real planes in $V(M)$ after identifying $\mathcal{Z} = X+iY$ with $\frac{i}{2} \mathcal{Z} \wedge \mathcal{\overline{Z}} = X \wedge Y$. For $\mu \in M$ let $[\mu]$ denote the space $\RR\mu$. For any rational isotropic line $I \in \PP(V_{\mathbb{Q}}(M))$, define the negative cone $$C_I = \{I \wedge [\mu]: \; \mu \in I^{\perp}, \; (\mu, \mu) < 0\} \cap \overline{\mathcal{D}(M)} \subseteq \mathbb{P}(\wedge^2 V_{\mathbb{R}}(M)),$$ and for any rational isotropic plane $J \subseteq \PP(V_{\mathbb{Q}}(M))$, define the isotropic half-line $$C_J = [\wedge^2 J] \cap \overline{\mathcal{D}(M)} \subseteq \mathbb{P}(\wedge^2 V_{\mathbb{R}}(M)).$$ Note that the sets $C_I$, $C_J$ are pairwise disjoint and that $I \subseteq J$ if and only if $C_J \subseteq \overline{C_I}$. We define $$C_0 = \{0\}, \quad C_{J, +} = C_J \cup \{0\}, \quad C_{I, +} = C_I \cup \bigcup_{I \subseteq J} C_J \cup \{0\}.$$ Then $C_0$, $C_{J, +}$ and $C_{I, +}$ are the $0$-, $1$- and $2$-dimensional faces of a rational polyhedral fan $\mathcal{C}(M) \subseteq \wedge^2 V_{\mathbb{R}}(M)$ called the \emph{conical locus} $\mathcal{C}(M)$ of $M$ (see \cite[Definition 2.1]{Loo03b}).
For any vector $v$ of positive norm in $M'$, the associated hyperplane (or rational quadratic divisor)
$$
\cD_v(M) := v^\perp \cap \cD(M) = \{ [\mathcal{Z}] \in \cD(M): (\mathcal{Z}, v) = 0 \}
$$
is the type IV symmetric domain attached to the signature $(l-1,2)$ lattice $M_v:=v^\perp \cap M$. For any finite-index subgroup $\Gamma$ of $\Orth^+(M)$, a ($\Gamma$-invariant) hyperplane arrangement $\cH$ is defined to be a finite union of $\Gamma$-orbits of hyperplanes $\cD_v(M)$. An arrangement $\cH$ is said to satisfy \textit{the Looijenga condition} (cf. Corollary 7.5 of \cite{Loo03b}) if every one-dimensional intersection of hyperplanes in $\cH$ (taken in the ambient space $V(M)$) is positive definite.
Let $\cD^\circ$ denote the arrangement complement $\cD(M) - \cH$. Looijenga \cite{Loo03b} defined a compactification $\widehat{X^{\circ}}$ of $X^\circ:=\cD^\circ / \Gamma$ by successively blowing up nonempty intersections of members of $\cH$ modulo $\Gamma$ and then blowing down the final blow-up. On the level of sets, $\widehat{X^\circ}:=\widehat{\cD^\circ} / \Gamma$, where $\widehat{\cD^\circ}$ is the disjoint union
\begin{align*}
\widehat{\cD^\circ} = \cD^\circ \sqcup \bigsqcup_{L\in \mathrm{PO}(\cH)} \pi_L \cD^\circ \sqcup \bigsqcup_{\sigma \in \sum(\cH)} \pi_{V_\sigma} \cD^\circ.
\end{align*}
The notation above is as follows. The first union is taken over the poset $\mathrm{PO}(\cH)$ of nonempty intersections inside of hyperplanes in $\cH$ within $\cD(M)$. In the second union, note that any hyperplane $H \in \cH$ containing $I$ defines a hyperplane $\{I \wedge [\mu]: \; \mu \in H\}$ in the facet $C_{I, +}$; that as $H$ runs through $\cH$ this family is locally finite; and therefore $\cH$ determines a decomposition of the conical locus $\mathcal{C}(M)$ into locally rational cones. The collection of these cones is labelled $\sum(\cH)$. For any $\sigma \in \sum(\cH)$ contained in the facet $C_{I, +}$ but not itself a face of the conical locus, its \emph{support space} is the span $V_{\sigma} \subseteq I^{\perp}$ of vectors $\mu$ for which $I \wedge [\mu] \subseteq \sigma$; and for any one-dimensional face $\sigma = C_{J, +} \in \sum(\cH)$, $$V_{\sigma} = J^{\perp} \cap \bigcap_{\substack{H \in \cH \\ J \subseteq H}} H.$$ Finally, for any vector subspace $W \subseteq V(M)$, $\pi_W$ is the canonical projection $$\pi_W : \Big( \PP(V(M)) - \PP(W) \Big) \longrightarrow \PP(V(M) / W).$$ This union carries a $\Gamma$-equivariant topology defined in \cite{Loo03b}. When $\mathcal{H}$ is empty, $\widehat{\cD^{\circ}} = \widehat{\cD}$ is the union of $\cD(M)$ with the entire conical locus and it carries the usual Satake--Baily--Borel topology, and therefore $\widehat{X^{\circ}}$ is exactly the Baily--Borel compactification.
The set $\cD^\circ$ is open and dense in $\widehat{\cD^\circ}$. When we pass to the $\Gamma$-orbit space, the boundary of the compactification $\widehat{X^\circ}$ of $X^\circ$ is decomposed into finitely many components. The Looijenga condition on $\cH$ guarantees that these boundary components have codimension at least two.
It is convenient to rephrase Looijenga's condition on $\cH$ in terms of intersections of rational quadratic divisors (within $\cD(M)$). Note that an intersection $\cD_{v_1}(M)\cap \cdots \cap \cD_{v_t}(M)$ is nonempty and of dimension $m\geq 0$ if and only if $\mathbb{Z}v_1 + ... + \mathbb{Z}v_t$ is a positive-definite lattice of rank $l-m$. It is not hard to show that the Looijenga condition is equivalent to the following condition.
\begin{condition}\label{condition}
For every nonempty intersection $\mathcal{I} = \cD(M) \cap \cD_{v_1}(M)\cap \cdots \cap \cD_{v_t}(M)$ of hyperplanes in $\cH$, the following hold:
\begin{enumerate}
\item $\mathcal{I}$ is not discrete.
\item If $\mathcal{I}$ is of dimension $2$, then the lattice $M\cap v_1^\perp \cap \cdots \cap v_t^\perp$ does not contain an isotropic plane.
\item If $\mathcal{I}$ is of dimension $1$, then the lattice $M\cap v_1^\perp \cap \cdots \cap v_t^\perp$ does not contain an isotropic vector.
\end{enumerate}
\end{condition}
\subsection{Orthogonal modular forms}
The Baily--Borel compactification can be realized as the $\mathrm{Proj}$ of the algebra of holomorphic modular forms, and the Looijenga compactification has a similar characterization. In the latter case, the corresponding modular forms are defined as meromorphic sections of $\widehat{\mathcal{L}}$ which are regular on $X^\circ$, where $\widehat{\mathcal{L}}$ is the ample line bundle on $\widehat{X^\circ}$ which has the same restriction to $X^\circ$ as the automorphic line bundle on $X$, and the Looijenga compactification is again the Proj of the graded algebra of modular forms. We will explain this more explicitly below.
Let us define the affine cone over $\cD(M)$ as
$$\cA(M)=\{ \mathcal{Z} \in V(M): [\mathcal{Z}] \in \cD(M) \}.$$
\begin{definition}
Let $k$ be an integer. A \emph{modular form} of weight $k$ and character $\chi\in \mathrm{Char}(\Gamma)$ with poles on $\cH$ is a meromorphic function $F: \mathcal{A}(M)\to \CC$ which is holomorphic away from $\cH$ and satisfies
\begin{align*}
F(t\mathcal{Z})&=t^{-k}F(\mathcal{Z}), \quad \forall t \in \CC^\times,\\
F(g\mathcal{Z})&=\chi(g)F(\mathcal{Z}), \quad \forall g\in \Gamma.
\end{align*}
\end{definition}
All modular forms with poles on $\cH$ of integral weight and trivial character form a graded algebra over $\CC$ and we denote it by
$$
M_*^!(\Gamma)=\bigoplus_{k = -\infty}^{\infty} M_k^!(\Gamma).
$$
When $\cH$ is empty, this algebra reduces to the usual algebra of holomorphic modular forms
$$
M_*(\Gamma)=\bigoplus_{k = 0}^\infty M_k(\Gamma).
$$
It is possible that $\cH$ is defined by the zero of a holomorphic modular form (see \cite[\S 3.2 and \S 5.2]{Loo03b} for criteria). However, this will never happen when $\cH$ satisfies the Looijenga condition. In fact, the Looijenga condition on $\cH$ implies that every modular form with poles on $\cH$ defines a regular section of a non-negative tensor power of $\widehat{\mathcal{L}}$. Equivalently, a modular form of non-positive weight with poles on $\cH$ must be constant. Furthermore, Looijenga proved in \cite[Corollary 7.5]{Loo03b} that the algebra $M_*^!(\Gamma)$ is finitely generated by forms of positive weights and its $\mathrm{Proj}$ gives the Looijenga compactification $\widehat{X^\circ}$.
As mentioned above, the Looijenga condition implies a form of Koecher's principle for modular forms with poles on $\cH$. Due to its importance, we give a new proof using the pullback trick.
\begin{lemma}\label{lem:Koecher}
Assume that $\cH$ satisfies Condition \ref{condition}. Let $F$ be a nonzero modular form of weight $k$ with poles on $\cH$. Let $c_F$ denote the maximal multiplicity of poles of $F$. Then $k\geq c_F$. In particular, if $k=0$ then $F$ is constant.
\end{lemma}
\begin{proof}
Assume that $F$ has poles of multiplicity $c_F$ on the hyperplane $\cD_v(M)$ in $\cH$. The restrictions of members of $\cH$ to $\cD_v(M)$ define a hyperplane arrangement $\cH_v$ which also satisfies Condition \ref{condition}. The quasi pullback of $F$ to $\cD_v(M)$ is a modular form of weight $k-c_F$ with poles supported on $\cH_v$. This allows us to prove the lemma by induction of the dimension of the symmetric domain $\cD(M)$. When $k=0$, $c_F$ must be zero, which yields that $F$ is holomorphic and then must be constant by the usual Koecher's principle.
\end{proof}
In this paper the hyperplane arrangements will always be finite unions of Heegner divisors. Let $\gamma \in M'/M$ and $a$ be a positive rational number. The (primitive) Heegner divisor of discriminant $(a,\gamma)$ is
$$
H(a,\gamma)= \bigcup_{\substack{v \in M + \gamma \\ v \; \text{primitive in} \; M' \\ (v,v) = 2a}} \mathcal{D}_v(M),
$$
and it is locally finite and $\widetilde{\Orth}^+(M)$-invariant and therefore descends to an (irreducible) analytic divisor on the modular variety $\cD(M) / \widetilde{\Orth}^+(M)$. Note that $\mathcal{D}_v(M) = \mathcal{D}_{-v}(M)$ implies $H(a, \gamma) = H(a, -\gamma)$.
The lemma below describes the intersections in $\cD(M)$ of members of $\widehat{H}(a, \gamma)$ and therefore is useful to verify the Looijenga condition \ref{condition} for arrangements of hyperplanes, where $\widehat{H}(a, \gamma)$ is the non-primitive Heegner divisor defined as
$$
\widehat{H}(a,\gamma)= \bigcup_{\substack{v \in M + \gamma \\ (v,v) = 2a}} \mathcal{D}_v(M),
$$
which is a finite union of some primitive Heegner divisors defined above.
\begin{lemma}\label{lem:intersection}
Let $u, v \in M+\gamma$ such that $\cD_u(M), \cD_v(M) \in \widehat{H}(a, \gamma)$.
\begin{enumerate}
\item If $a\leq \frac{1}{4}$, then $\cD_u(M)\cap\cD_v(M)=\emptyset $.
\item Suppose that $\cD_u(M)\cap\cD_v(M) \neq \emptyset $. If $\frac{1}{4}< a \leq \frac{1}{2}$, then $\widehat{H}(a, \gamma) \cap \cD_v(M)$, viewed as a non-primitive Heegner divisor in $\cD(M_v) = \cD_v(M)$, is
\begin{equation*}
\widehat{H}(1 - 1/(4a), u') = \bigcup_{\substack{x \in M_v + u' \\ (x,x)/2 = 1-1/(4a) }} \cD_x(M_v).
\end{equation*}
\end{enumerate}
Here $M_v$ is the orthogonal complement of $v$ in $M$, and $u'$ is the orthogonal projection of $u$ to the dual lattice $M_v'$.
\end{lemma}
\begin{proof}
(1) $\cD_u(M)\cap\cD_v(M)$ is nonempty if and only if the sublattice generated by $u$ and $v$ is positive definite. Since $(u, u) = (v, v) = 2a$, this holds if and only if $|(u, v)| < 2a$. On the other hand, $u - v \in M$ implies
$$
(u,v) = (v,v) + (u-v, v) \in 2a + \ZZ,
$$
which forces $|(u,v)| \geq 2a$ if $a \le 1/4$.
(2) Let $w\in M+\gamma$ with $(w,w)=2a$. As in (1), the intersection $\cD(M_v)\cap w^\perp $ is nonempty if and only if $|(v, w)| < 2a$, which occurs if and only if $(v, w)=2a-1$ under the assumption $1/4 < a \le 1/2$. When $(v, w)=2a-1$, the projection of $w$ to $M_v'$ is
$$
w'= w -\frac{(w,v)}{(v,v)}v = w + \left(\frac{1}{2a}-1\right)v,
$$
and it satisfies $(w',w')=2(1-\frac{1}{4a})$ and $w'-u'\in M_v$. This proves our claim.
\end{proof}
\begin{corollary}\label{cor:intersection} Define $a_0 = 0$ and $a_k = 1 / (4 - 4a_{k-1})$ for $k \ge 1$. Suppose that $\cH$ is a finite collection of distinct non-primitive Heegner divisors $\widehat{H}(a, \gamma)$ with $a<1/2$, of which $b_k$ have norm $a_{k-1} < a \le a_k$. If $$\sum_k k \cdot b_k < l - 2,$$ then $\cH$ satisfies the Looijenga condition \ref{condition}.
\end{corollary}
Recall that $M$ has signature $(l, 2)$. Only the values $a_1 = 1/4$, $a_2 = 1/3$, $a_3 = 3/8$, $a_4 = 2/5$ and $a_5 = 5/12$ will be needed in the paper.
\begin{proof}
Let $\widehat{H}(a,\gamma)$ be a Heegner divisor contained in $\cH$ satisfying $a_{k-1}<a\leq a_k$. By Lemma \ref{lem:intersection}, the intersection between $\widehat{H}(a,\gamma)$ and one of its hyperplanes is a Heegner divisor $\widehat{H}(a',-)$ satisfying $a_{k-2}<a'\leq a_{k-1}$. By repeating this process, we find that all nonempty intersections of hyperplanes in $\widehat{H}(a,\gamma)$ have codimension at most $k$. It follows that the nonempty intersections of hyperplanes in $\cH$ have codimension at most $\sum_k kb_k$, and therefore dimension at least $3$. Thus $\cH$ satisfies the Looijenga condition \ref{condition}.
\end{proof}
\subsection{Reflections and the Jacobian}
In \cite{Wan21a} the first named author introduced the modular Jacobian criterion to decide when the ring $M_*(\Gamma)$ of holomorphic modular forms is a polynomial algebra. This states that a set of modular forms freely generates $M_*(\Gamma)$ if and only if their Jacobian vanishes exactly with multiplicity one on hyperplanes fixed by reflections in $\Gamma$. In this section we extend this approach to the ring $M_*^!(\Gamma)$ of modular forms with poles on a hyperplane arrangement.
Let $M$ be an even lattice of signature $(l, 2)$ with $l \ge 3$ and let $\Gamma \le \Orth^+(M)$ be a finite-index subgroup.
Let $r$ be a non-isotropic vector of $M$. The \textit{reflection} associated to $r$ is defined as
$$
\sigma_r: v \mapsto \frac{2(v,r)}{(r,r)}r \in \Orth(M\otimes \QQ).
$$
A reflection $\sigma_r$ preserves $\cD(M)$ if and only if its spinor norm is $+1$, i.e. $(r,r)>0$. Therefore, we need only consider reflections associated to vectors of positive norm. The set of fixed points of $\sigma_r$ on $\cD(M)$ is the hyperplane $\cD_r(M)$, and is called the \emph{mirror} of $\sigma_r$. A vector $r\in M'$ of positive norm is called \textit{reflective} if its reflection $\sigma_r$ preserves the lattice $M$. By definition, a primitive vector $r\in M$ of norm $(r,r)=2d$ is reflective if and only if $\Div(r)=d$ or $2d$, where $\Div(r)$ denotes the positive generator of the ideal $\{ (r, v): v\in M \}$. It follows that a primitive vector $x \in M'$ is reflective if and only if there exists a positive integer $d$ such that $(x,x)=2/d$ and $\ord(x)=d$ or $d/2$, where $\ord(x)$ is the order of $x$ in $M'/M$. Every vector of norm $2$ in $M$ is reflective, and the corresponding reflection acts trivially on $M'/M$; conversely, the discriminant kernel only contains this type of reflection.
The Jacobian of modular forms is a generalization of the Rankin--Cohen--Ibukiyama differential operator for Siegel modular forms introduced in \cite{AI05}. For $0\leq j \leq l$, let $F_j$ be a modular form of weight $k_j$ and character $\chi_j$ for $\Gamma$ with poles on an arrangement $\cH$ of hyperplanes. View these $F_j$ as modular forms on the tube domain $c^\perp / c$, where $c$ is a primitive isotropic vector of $M$, and let $z_1, ..., z_l$ be coordinates on the tube domain. The Jacobian of these $l+1$ meromorphic modular forms in these coordinates is
$$
J(F_0, ..., F_{l+1}) = \mathrm{det} \begin{pmatrix} k_0 F_0 & k_1 F_1 & \cdots & k_l F_l \\ \partial_{z_1} F_0 & \partial_{z_1} F_1 & \cdots & \partial_{z_1} F_l \\ \vdots & \vdots & \ddots & \vdots \\ \partial_{z_l} F_0 & \partial_{z_l} F_1 & \cdots & \partial_{z_l} F_l \end{pmatrix},
$$
and it defines a meromorphic modular form of weight $k_J$ and character $\chi_J$ for $\Gamma$, holomorphic away from $\cH$, where
$$
k_J=l + \sum_{j=0}^l k_j, \quad \chi_J = \mathrm{det} \otimes \bigotimes_{j=0}^l \chi_j.
$$
Moreover, the Jacobian is not identically zero if and only if these $l+1$ forms are algebraically independent over $\CC$.
The Jacobian satisfies the product rule in each component and every modular form with poles on $\cH$ can be written as a quotient $f = g/h$, where $g, h$ are holomorphic. Using $$J(g/h, f_1,...,f_l) = \frac{1}{h^2} \Big( h \cdot J(g, f_1,...,f_l) - g \cdot J(h, f_1,...,f_l) \Big)$$ and the analogous equations in the other components, we can reduce the claims above to the Jacobians of holomorphic modular forms, where they are already known (cf. \cite[Theorem 2.5]{Wan21a}).
Assume that the characters of all $F_j$ are trivial, so their Jacobian has the determinant character. Let $\cD_r(M)$ be the mirror of a reflection $\sigma_r\in \Gamma$ which is not contained in $\cH$. From $\det(\sigma_r)=-1$, we see that the Jacobian must vanish on $\cD_r(M)$.
We will prove some necessary conditions for $M_*^!(\Gamma)$ to be a free algebra. The following theorem is an analogue of \cite[Theorem 3.5]{Wan21a} and the proof is essentially the same.
\begin{theorem}\label{th:freeJacobian}
Let $\cH$ be a hyperplane arrangement satisfying the Looijenga condition. Suppose that the ring of modular forms for $\Gamma$ with poles on $\cH$ is freely generated by $F_j$ for $0\leq j \leq l$.
\begin{enumerate}
\item The group $\Gamma$ is generated by reflections.
\item The Jacobian $J:=J(F_0,...,F_l)$ is a nonzero meromorphic modular form of character $\det$ for $\Gamma$ which satisfies the conditions:
\begin{enumerate}
\item $J$ vanishes with multiplicity one on all mirrors of reflections in $\Gamma$ which are not contained in $\cH$;
\item all other zeros and poles of $J$ are contained in $\cH$. \end{enumerate}
\item Let $\{ \Gamma \pi_1, ..., \Gamma \pi_s\}$ be the distinct $\Gamma$-equivalence classes of mirrors of reflections in $\Gamma$ which are not contained in $\cH$. Let $P$ be the polynomial expression of $J^2$ in terms of the generators $F_j$. Then $P$ decomposes into a product of $s$ irreducible polynomials, and each irreducible factor is a modular form for $\Gamma$ which vanishes with multiplicity two on some $\Gamma\pi_i$ and whose other zeros and poles are contained in $\cH$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) We use the notations in \S \ref{subsec:Looijenga}. Let $\cD^\circ=\cD - \cH$ and $X^\circ = \cD^\circ / \Gamma$. Recall that the Looijenga compactification $\widehat{X^\circ}$ is defined as $\mathrm{Proj}(M_*^!(\Gamma))$. Let $Y^\circ=\cA^\circ / \Gamma$ be the affine cone over $X^\circ$ and $\widehat{Y^\circ}$ be the affine span of $\widehat{X^\circ}$ defined by the maximal spectrum of $M_*^!(\Gamma)$. The freeness of $M_*^!(\Gamma)$ implies that $\widehat{X^\circ}$ is a weighted projective space and that $\widehat{Y^\circ}$ is simply the affine space $\CC^{l+1}$. We known from \cite{Loo03b} that the boundary of $\widehat{X^\circ} - X^\circ$ has only finitely many components and each component has codimension at least two in $\widehat{X^\circ}$. Therefore, a similar property holds for $\widehat{Y^\circ}-Y^\circ$. It follows that $Y^\circ$ is smooth and simply connected. Applying a theorem of Armstrong \cite{Arm68} (cf. \cite[Theorem 3.2]{Wan21a}) to $Y^\circ$, we find that $\Gamma$ is generated by elements having fixed points. Then a theorem of Gottschling \cite{Got69a} (cf. \cite[Theorem 3.1]{Wan21a}) implies that $\Gamma$ is generated by reflections.
(2) Let $k_i$ denote the weight of $F_i$ for $0\leq i \leq l$. Then $$[z] \mapsto [F_0(z),...,F_{l}(z)]$$ identifies $\widehat{X^\circ}$ with $\PP(k_0,...,k_l)$ and further induces a holomorphic map
$$
\phi: \cA^\circ \to Y^\circ = \cA^\circ / \Gamma \hookrightarrow \CC^{l+1}\setminus \{0\}, \quad z \mapsto (F_0(z),...,F_{l}(z)).
$$
If $x\in \cA^\circ$ has trivial stabilizer $\Gamma_x$, then $\phi$ is biholomorphic on a neighborhood of $x$ since the action of $\Gamma$ on $\cA^\circ$ is properly discontinuous, so the Jacobian $J$ (which can be regarded as the usual Jacobian determinant of $\phi$) is nonvanishing there. Then Gottschling's theorem implies that $J$ is nonzero on $\cD^\circ$ away from mirrors of reflections in $\Gamma$. From the character of $J$, we conclude that the restriction of $J$ to the complement $\cD^\circ$ vanishes precisely on mirrors of reflections in $\Gamma$. The fact that the multiplicity of the zeros is one follows from the fact that the reflections have order two.
(3) The proof of this is similar to that of \cite[Theorem 3.5 (3)]{Wan21a}. We only need to compare the decomposition of the zero divisor of $J$ on $\cD^\circ$ and the decomposition of the zero locus of $P$ in $\PP(k_0,...,k_l)$.
\end{proof}
At the end of this section, we establish the modular Jacobian criterion, which will be used to construct free algebras of meromorphic modular forms later.
\begin{theorem}\label{th:Jacobiancriterion}
Let $\cH$ be an arrangement of hyperplanes satisfying the Looijenga condition. Suppose that there exist $l+1$ algebraically independent modular forms with poles on $\cH$ such that the restriction of their Jacobian to the arrangement complement $\cD^\circ$ vanishes precisely on mirrors of reflections in $\Gamma$ with multiplicity one. Then the algebra of modular forms with poles on $\cH$ is freely generated by these $l+1$ forms.
\end{theorem}
\begin{proof}
The proof is similar to that of \cite[Theorem 5.1]{Wan21a}. For the reader's convenience we give a short proof. Let $f_i$ ($1\leq i\leq l+1$) be $l+1$ modular forms of weight $k_i$ with poles on $\cH$ whose Jacobian $J$ satisfies the conditions of the theorem.
Suppose that $M_*^!(\Gamma)$ is not generated by $f_i$. Let $f_{l+2} \in M_{k_{l+2}}^!(\Gamma)$ be a modular form of minimal weight which does not lie in $\CC[f_1,...,f_{l+1}]$. For $1\leq t \leq l+2$ we define $$J_{t} = J(f_1,..., \hat f_t, ..., f_{l+2})$$ as the Jacobian of the $l+1$ modular forms $f_i$ omitting $f_t$, such that $J=J_{l+2}$. It is clear that $g_t := J_t/J \in M_*^!(\Gamma)$.
The identity
$$
0 = \mathrm{det} \begin{psmallmatrix} k_1 f_1 & k_2 f_2 & \cdots & k_{l+2} f_{l+2} \\ k_1 f_1 & k_2 f_2 & \cdots & k_{l+2} f_{l+2} \\ \nabla f_1 & \nabla f_2 & ... & \nabla f_{l+2} \end{psmallmatrix} = \sum_{t=1}^{l+2} (-1)^t k_t f_t J_t = J\cdot \Big( \sum_{t=1}^{l+2} (-1)^t k_t f_t g_t \Big)
$$
and the equality $g_{l+2}=1$ yield
$$
(-1)^{l+1}k_{l+2}f_{l+2}= \sum_{t=1}^{l+1}(-1)^t k_t f_t g_t.
$$
Each $g_t$ has weight strictly less than that of $f_{l+2}$. The construction of $f_{l+2}$ implies that $g_t \in \CC[f_1,...,f_{l+1}]$ and then $f_{l+2} \in \CC[f_1,...,f_{l+1}]$, which contradicts our assumption.
\end{proof}
\section{Jacobi forms and singular theta lifts}\label{sec:Jacobiforms}
In this section we review some results from the theories of Jacobi forms of lattice index and Borcherds' singular theta lifts, which are necessary to prove the main theorems in the introduction.
\subsection{Jacobi forms of lattice index}\label{sec:Jacobi} In 1985 Eichler and Zagier introduced the theory of Jacobi forms in their monograph \cite{EZ85}. These are holomorphic functions in two variables $(\tau,z)\in \HH \times \CC$ which are modular in $\tau$ and quasi-periodic in $z$. Jacobi forms of lattice index were defined in \cite{Gri88} by replacing $z$ with a vector of variables associated with a positive definite lattice. As a bridge between different types of modular forms, Jacobi forms have many applications in mathematics and physics. Let $L$ be an even integral positive definite lattice of rank $\rk(L)$ with bilinear form $\latt{-,-}$ and dual lattice $L'$.
\begin{definition}\label{def:JFs}
Let $k\in \ZZ$ and $t\in \NN$. A \emph{nearly holomorphic Jacobi form} of weight $k$ and index $t$ associated to $L$ is a holomorphic function $\varphi : \HH \times (L \otimes \CC) \rightarrow \CC$ which satisfies
\begin{align*}
\varphi \left( \frac{a\tau +b}{c\tau + d},\frac{\mathfrak{z}}{c\tau + d} \right) &= (c\tau + d)^k \exp\left( t\pi i \frac{c\latt{\mathfrak{z},\mathfrak{z}}}{c \tau + d}\right) \varphi ( \tau, \mathfrak{z} ), \quad \left( \begin{array}{cc}
a & b \\
c & d
\end{array} \right) \in \SL_2(\ZZ),\\
\varphi (\tau, \mathfrak{z}+ x \tau + y)&= \exp\left(-t\pi i ( \latt{x,x}\tau +2\latt{x,\mathfrak{z}} )\right) \varphi ( \tau, \mathfrak{z} ), \quad x,y\in L,
\end{align*}
and has a Fourier expansion of the form
\begin{equation*}
\varphi ( \tau, \mathfrak{z} )= \sum_{ n \gg -\infty}\sum_{ \ell \in L'}f(n,\ell)q^n \zeta^\ell, \quad q=e^{2\pi i\tau}, \; \zeta^\ell = e^{2\pi i \latt{\ell, \mathfrak{z}}}.
\end{equation*}
If $f(n,\ell) = 0$ whenever $n <0$ (resp. $2nt - \latt{\ell,\ell} <0$), then $\varphi$ is called a \textit{weak} (resp. \textit{holomorphic}) Jacobi form. We denote the vector spaces of nearly holomorphic, weak and holomorphic Jacobi forms of weight $k$ and index $t$ respectively by
$$
J_{k,L,t}^{!} \supset J_{k,L,t}^{\w} \supset J_{k,L,t}.
$$
\end{definition}
Jacobi forms of index $0$ are independent of the lattice variable $\mathfrak{z}$ and are therefore classical modular forms on $\SL_2(\ZZ)$. In general, the quasi-periodicity and the transformation under $\begin{psmallmatrix} -1 & 0 \\ 0 & -1 \end{psmallmatrix} \in \SL_2(\ZZ)$ imply the following constraints on the coefficients.
\begin{lemma}\label{lem:periodic}
\noindent
\begin{enumerate}
\item Let $t\geq 1$. The Fourier coefficients of $\varphi \in J_{k,L,t}^!$ satisfy
$$
f(n_1,\ell_1)=f(n_2,\ell_2) \quad \text{if} \quad 2n_1t - \latt{\ell_1,\ell_1} = 2n_2t - \latt{\ell_2,\ell_2} \; \text{and} \; \ell_1 - \ell_2 \in tL.
$$
\item The Fourier coefficients of the $q^0$-term of $\varphi \in J_{k,L,1}^{\w}$ satisfy
$$
f(0,\ell) \neq 0 \quad \Rightarrow \quad \latt{\ell,\ell} \leq \latt{\ell_1,\ell_1} \; \text{for all $\ell_1 \in \ell + L$}.
$$
\item The Fourier coefficients of $\varphi \in J_{k,L,t}^!$ satisfy
$$
f(n,\ell)=(-1)^k f(n,-\ell).
$$
\end{enumerate}
\end{lemma}
Let $J_{*,L,t}^{\w}$ (resp. $J_{*,L,t}$) denote the spaces of weak (resp. holomorphic) Jacobi forms of fixed index $t$ and arbitrary weight associated to $L$. Both are free graded modules of rank $|L'/L|$ over the graded ring of modular forms
$$
M_*(\SL_2(\ZZ)):=\bigoplus_{k=0}^\infty M_k(\SL_2(\ZZ))=\CC[E_4,E_6].
$$
We will need the following raising index Hecke operators of Jacobi forms.
\begin{lemma}[see Corollary 1 of \cite{Gri88}]\label{lem:Hecke}
Let $\varphi \in J_{k,L,t}^{!}$. For any positive integer $m$, we have
\begin{equation}\label{T(m)}
(\varphi \lvert T_{-}(m))(\tau,
\mathfrak{z}):=m^{-1}\sum_{\substack{ad=m,a>0\\ 0\leq b <d}}a^k \varphi
\left(\frac{a\tau+b}{d},a\mathfrak{z}\right) \in J_{k,L,mt}^{!},
\end{equation}
and the Fourier coefficients of
$\varphi \lvert T_{-}(m)$ are given by the formula
$$
f_m(n,\ell)=\sum_{\substack{a\in \NN\\ a \mid (n,\ell,m)}}a^{k-1} f
\left( \frac{nm}{a^2},\frac{\ell}{a}\right),
$$
where $a\mid(n,\ell,m)$ means that $a\mid (n,m)$
and $a^{-1}\ell\in L'$.
\end{lemma}
In \cite{Gri18} Gritsenko proved the following useful identities for Jacobi forms of weight $0$. These are variants of identities of Borcherds \cite[Theorem 10.5]{Bor98} that hold for vector-valued modular forms.
\begin{lemma}\label{Lem:q^0-term}
Every nearly holomorphic Jacobi form of weight $0$ and index $1$ with Fourier expansion
$$
\phi(\tau,\mathfrak{z})=\sum_{n\in\ZZ}\sum_{\ell\in L'}f(n,\ell)q^n\zeta^\ell \in J_{0,L,1}^!
$$
satisfies the identity
\begin{equation}\label{eq:q^0-term}
C:= \frac{1}{24}\sum_{\ell\in L'}f(0,\ell)-\sum_{n<0}\sum_{\ell\in L'}f(n,\ell)\sigma_1(-n)=\frac{1}{2\rk(L)} \sum_{\ell\in L'}f(0,\ell)\latt{\ell,\ell}
\end{equation}
and the identity
\begin{equation}\label{eq:vectorsystem}
\sum_{\ell\in L'}f(0,\ell)\latt{\ell,\mathfrak{z}}^2=2C\latt{\mathfrak{z},\mathfrak{z}},
\end{equation}
where $\sigma_1(m)$ is the sum of the positive divisors of $m \in \mathbb{N}$.
\end{lemma}
\subsection{Additive lifts and Borcherds products}
In \cite{Bor95, Bor98} Borcherds introduced the singular theta lift to construct modular forms on type IV symmetric domains with specified divisors. We will quickly review this theory from the point of view of Jacobi forms.
\subsubsection{Modular forms for the Weil representation}
Let $\mathrm{Mp}_2(\mathbb{Z})$ be the metaplectic group, consisting of pairs $A = (A, \phi_A)$, where $A = \begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in \mathrm{SL}_2(\mathbb{Z})$ and $\phi_A$ is a holomorphic square root of $\tau \mapsto c \tau + d$ on $\mathbb{H}$, with the standard generators $T = (\begin{psmallmatrix} 1 & 1 \\ 0 & 1 \end{psmallmatrix}, 1)$ and $S = (\begin{psmallmatrix} 0 & -1 \\ 1 & 0 \end{psmallmatrix}, \sqrt{\tau})$. Let $M$ be an even lattice with bilinear form $(-,-)$ and quadratic form $Q(-)$. The \emph{Weil representation} $\rho_M: \mathrm{Mp}_2(\mathbb{Z})\to \GL (\mathbb{C}[M'/M])$ is defined by $$\rho_M(T) e_x = \mathbf{e}(-Q(x)) e_x \quad \text{and} \quad \rho_M(S) e_x = \frac{\mathbf{e}(\mathrm{sig}(M) / 8)}{\sqrt{|M'/M|}} \sum_{y \in M'/M} \mathbf{e}((x,y )) e_y,$$
where $e_\gamma$, $\gamma\in M'/M$ is the standard basis of the group ring $\CC[M'/M]$, and $\mathbf{e}(z)=e^{2\pi i z}$.
A \emph{nearly holomorphic modular form} of weight $k \in \frac{1}{2}\mathbb{Z}$ for the Weil representation $\rho_M$ is a holomorphic function $f : \mathbb{H} \rightarrow \mathbb{C}[M'/M]$ which satisfies $$f(A \cdot \tau) = \phi_A(\tau)^{2k} \rho_M(A) f(\tau), \quad \text{for all $(A, \phi_A) \in \mathrm{Mp}_2(\mathbb{Z})$}$$ and is meromorphic at the cusp $\infty$, that is, its Fourier expansion $$f(\tau) = \sum_{x \in M'/M} \sum_{n \in \mathbb{Z} - Q(x)} c(n, x) q^n e_x$$
has only finitely many negative exponents.
We denote the space of such forms by $M_k^!(\rho_M)$.
We define the \textit{principal part} of $f$ by the sum
$$
\sum_{x \in M'/M} \sum_{n < 0} c(n, x) q^n e_x.
$$
We call $f$ a \textit{holomorphic} modular form if $c(n,x)=0$ whenever $n<0$, i.e. its principal part is zero. We denote the space of holomorphic modular forms of weight $k$ for $\rho_M$ by $M_k(\rho_M)$.
If $M$ splits two hyperbolic planes, then modular forms for the Weil representation can be identified with Jacobi forms. Let $U$ be a hyperbolic plane, i.e. the unique even unimodular lattice of signature $(1,1)$. We fix the basis of $U$ as
\begin{equation}\label{eq:basis of U}
U=\ZZ e + \ZZ f, \quad (e,e)=(f,f)=0, \quad (e,f)=-1.
\end{equation}
Let $U_1$ be a copy of $U$ with a similar basis $e_1, f_1$. As in Section \ref{sec:Jacobi}, let $L$ be an even positive definite lattice with bilinear form $\latt{-,-}$. Then $M:=U_1\oplus U\oplus L$ is an even lattice of signature $(2+\rk(L),2)$ and $M'/M=L'/L$. Define the theta functions
$$
\Theta_{L, \gamma}(\tau,\mathfrak{z}) = \sum_{\ell \in L + \gamma} \exp\left(\pi i \latt{\ell,\ell}\tau + 2\pi i\latt{\ell, \mathfrak{z}}\right), \quad \gamma \in L'/L.
$$
The \emph{theta decomposition} yields an isomorphism between $M_{k-\rk(L)/2}^!(\rho_L)$ and $J_{k,L,1}^!$ :
$$
f(\tau)=\sum_{\gamma \in L'/L} f_\gamma(\tau) e_\gamma \mapsto \sum_{\gamma \in L'/L} f_\gamma(\tau) \Theta_{L,\gamma}(\tau, \mathfrak{z}).
$$
For a nearly holomorphic Jacobi form $\varphi \in J_{k,L,1}^!$, the Fourier coefficients $f(n,\ell)q^n\zeta^\ell$ satisfying $2n-\latt{\ell,\ell}<0$ are called the \textit{singular Fourier coefficients}. The singular coefficients of $\varphi$ are precisely the coefficients which appear in the principal part at $\infty$ of the vector-valued modular form $f$. Therefore, the theta decomposition defines an isomorphism between the spaces of holomorphic forms $M_{k-\rk(L)/2}(\rho_L)$ and $J_{k,L,1}$. It follows in particular that the minimum possible weight of a non-constant holomorphic Jacobi form for $L$ is $\rk(L)/2$. This is called the \textit{singular weight}; since the associated vector-valued form $f$ must be constant, a singular-weight Jacobi form has its Fourier series supported on exponents $(n, \ell)$ of hyperbolic norm $2n - \latt{\ell, \ell} = 0$.
\subsubsection{Borcherds' singular additive lift}
When $M$ is of signature $(l,2)$ and $f \in M_{\kappa}^!(\rho_M)$ has weight $\kappa = k+1-l/2$ with integral $k\geq 1$, Borcherds' (singular) theta lift of $f$ is a meromorphic modular form of weight $k$ for $\widetilde{\Orth}^+(M)$ whose only singularities are poles of order $k$ along the hyperplanes $\cD_v(M)$ satisfying (see \cite[Theorem 14.3]{Bor98} for full details)
$$
\sum_{n=1}^\infty c(-n^2Q(v),nv)\neq 0, \quad \text{where $v\in M'$ is primitive.}
$$
This construction is called the \textit{Borcherds singular additive lift}. Note that the singular additive lift is holomorphic if and only if the principal part of the input is zero.
Suppose as before that $M = U_1 \oplus U \oplus L$ with $U = \ZZ e + \ZZ f$ and $U_1 = \ZZ e_1 + \ZZ f_1$. Around the one-dimensional cusp associated to the isotropic plane $\ZZ e_1 + \ZZ e$, $\cD(M)$ can be realized as the tube domain
$$
\cH(L):= \{Z=(\tau,\mathfrak{z},\omega)\in \HH\times (L\otimes\CC)\times \HH:
(\im Z,\im Z)<0\},
$$
where $(\im Z,\im Z)=-2\im \tau \im \omega +
\latt{\im \mathfrak{z},\im \mathfrak{z}}$.
Let $F$ be a holomorphic modular form of weight $k$ and trivial character for $\widetilde{\Orth}^+(2U\oplus L)$. The Fourier expansion of $F$ on $\cH(L)$ has the shape
$$
F(\tau,\mathfrak{z},\omega) = \sum_{n=0}^\infty \sum_{m=0}^\infty \sum_{\ell \in L'} f(n,\ell,m)q^n\zeta^\ell\xi^m, \quad q=e^{2\pi i\tau}, \; \zeta^\ell =e^{2\pi i\latt{\ell,\mathfrak{z}}},\; \xi=e^{2\pi i\omega}.
$$
The holomorphy of $F$ on the boundary of $\cH(L)$ in $\cD(M)$ forces $f(n,\ell,m)=0$ if $2nm-\latt{\ell,\ell}<0$. This series can be reorganized as the \emph{Fourier--Jacobi expansion}
\begin{equation}\label{eq:FJdef}
F(\tau, \mathfrak{z}, \omega) = \sum_{m = 0}^\infty f_m(\tau, \mathfrak{z})\xi^m, \quad \xi=e^{2\pi i \omega },
\end{equation}
where each $f_m$ is a holomorphic Jacobi form of weight $k$ and index $m$ associated to $L$. If $F$ is non-constant, then the weight must satisfy $k\geq \rk(L)/2$ since this is true for each $f_m$. In particular, the minimum possible weight of a non-constant modular form on a lattice of signature $(l,2)$ is $l/2-1$ (as before we assume $l\geq 3$), which is again called the \textit{singular weight}.
Through the theta decomposition, we can realize nearly-holomorphic Jacobi forms of weight $k$ and index $1$ for $L$ as the inputs into the weight $k$ singular additive lift. Combining \cite[Theorem 7.1, Theorem 9.3]{Bor95} and \cite[Theorem 14.3]{Bor98} yields a simple expression for the Fourier--Jacobi expansion of Borcherds' singular additive lifts in terms of the input Jacobi form.
\begin{theorem}[\cite{Bor95, Bor98}]\label{th:additive}
Suppose that $k\geq 1$ is integral and that $\varphi_k \in J_{k,L,1}^!$ has Fourier expansion
$$
\varphi_k(\tau,\mathfrak{z})=\sum_{ n \in \ZZ}\sum_{ \ell \in L'}f(n,\ell)q^n \zeta^\ell.
$$
Then the series
$$
\Grit(\varphi_k)(\tau,\mathfrak{z},\omega) = \sum_{m=0}^\infty (\varphi_k | T_{-}(m)) (\tau, \mathfrak{z}) \cdot \xi^m
$$
defines a meromorphic modular form of weight $k$ and trivial character for $\widetilde{\Orth}^+(2U\oplus L)$, where the Hecke operator $T_{-}(0)$ is
\begin{align*}
(\varphi_k | T_{-}(0)) (\tau, \mathfrak{z}) &= -\frac{f(0,0)B_k}{2k} + \sum_{(n,\ell)>0} \sum_{d | (n,\ell)} d^{k-1} f(0,\ell/d) q^n \zeta^\ell.
\end{align*}
Here, $B_n$ are the Bernoulli numbers defined by $\sum_{n=0}^\infty B_n t^k / n! = t/ (e^t -1)$, and $(n,\ell)>0$ means that either $n>0$ or $n=0$ and $\ell >0$. In particular, when the input has trivial $q^0$-term the zeroth Fourier--Jacobi coefficient $\varphi_k | T_{-}(0)$ is zero.
\end{theorem}
The image $\varphi_k | T_{-}(0)$ can be expressed in closed form in terms of the Weierstrass zeta function, $$\zeta(\tau, z) = \frac{1}{z} + \sum_{\substack{w \in \ZZ \tau + \ZZ \\ w \neq 0}} \left( \frac{1}{z - w} + \frac{1}{w} + \frac{z}{w^2} \right) = \frac{1}{z} + \frac{1}{2} \sum_{\substack{w \in \ZZ\tau + \ZZ \\ w \neq 0}} \left( \frac{1}{z - w} + \frac{1}{z + w} + \frac{2z}{w^2} \right).$$ Using the well-known partial fractions decomposition, $$\lim_{N \rightarrow \infty} \sum_{n = -N}^N \frac{1}{z + n} = \pi i - \frac{2\pi i}{1 - e^{2\pi i z}}, \quad z \notin \ZZ,$$ and applying Eisenstein summation, we find the Fourier series for $\zeta$:
\begin{align*} \zeta(\tau, z) =& \frac{\pi^2}{3} E_2(\tau) z + \frac{1}{2} \lim_{M \rightarrow \infty} \sum_{m = -M}^M \lim_{N \rightarrow \infty} \sum_{n=-N}^N \left( -\frac{1}{-z + m\tau + n} + \frac{1}{z + m \tau + n} \right) \\
=& \frac{\pi^2}{3} E_2(\tau) z + \pi i \sum_{m = -\infty}^{\infty} \left( \frac{1}{1 - q^m \zeta^{-1}} - \frac{1}{1 - q^m \zeta} \right) \\
=& \frac{\pi^2}{3} E_2(\tau) z + \pi i \left( \frac{1}{1 - \zeta^{-1}} - \frac{1}{ 1 - \zeta} \right) \\
& + \pi i \sum_{m=1}^{\infty} \left( \frac{1}{1 - q^m \zeta^{-1}} - \frac{1}{1 - q^m \zeta} + \frac{q^m \zeta^{-1}}{1 - q^m \zeta^{-1}} - \frac{q^m \zeta}{1 - q^m \zeta} \right) \\
=& \frac{\pi^2}{3} E_2(\tau) z - \pi i - 2\pi i \sum_{n=1}^{\infty} \zeta^n - 2\pi i \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} q^{mn} (\zeta^n - \zeta^{-n}), \; 0 < |\zeta| < |q|,
\end{align*}
where $\zeta$ also denotes $e^{2\pi i z}$. When $k = 1$, the nonexistence of (nonzero) modular forms of weight $2$ forces the identity $$\sum_{\ell > 0} f(0, \ell) \latt{\ell, \mathfrak{z}} \equiv 0,$$ which, together with $f(0, \ell) = -f(0, -\ell)$, implies
\begin{align*}
\sum_{(n, \ell) > 0} \sum_{d | (n, \ell)} f(0, \ell / d) q^n \zeta^{\ell} &= \sum_{\ell > 0} f(0, \ell) \left( -\frac{1}{2\pi i} \zeta(\tau, \latt{\ell, \mathfrak{z}}) - \frac{\pi i}{6} E_2(\tau) \latt{\ell, \mathfrak{z}} \right) \\
&= -\frac{1}{2\pi i} \sum_{\ell > 0} f(0,\ell) \zeta(\tau, \latt{\ell, \mathfrak{z}}).
\end{align*}
For $k \ge 2$, a similar argument yields
\begin{align*}
\sum_{(n, \ell) > 0} \sum_{d | (n, \ell)} f(0, \ell/d) q^n \zeta^{\ell} &= \delta_k f(0, 0) \sum_{n = 1}^{\infty} \sigma_{k - 1}(n)q^n - \frac{1}{(2\pi i)^k} \sum_{\ell > 0} f(0, \ell) \zeta^{(k-1)}(\tau, \latt{\ell, \mathfrak{z}}) \\
&= -\frac{f(0, 0)B_k \delta_k}{2k} (E_k(\tau) - 1) + \frac{1}{(2\pi i)^k} \sum_{\ell > 0} f(0, \ell) \wp^{(k-2)}(\tau, \latt{\ell, \mathfrak{z}}),
\end{align*}
where $\delta_k = 1$ if $k$ is even and $\delta_k = 0$ if $k$ is odd; where $E_k(\tau)=1+O(q)$ is the normalized Eisenstein series of weight $k$ on $\SL_2(\ZZ)$; where $\wp(\tau, z) = -\partial_z \zeta(\tau, z)$ is the Weierstrass elliptic function; and where $\wp^{(k)} = \partial_z^k \wp(\tau, z)$.
Altogether, we have the formula
\begin{equation}
(\varphi_k | T_{-}(0))(\tau, \mathfrak{z}) = -\delta_k f(0, 0) \cdot \frac{B_k}{2k}E_k(\tau)- \frac{1}{(2\pi i)^k} \sum_{\ell > 0} f(0, \ell) \zeta^{(k-1)}(\tau, \latt{\ell, \mathfrak{z}}).
\end{equation}
When $\varphi_k$ is a holomorphic Jacobi form, $\Grit(\varphi_k)$ is also holomorphic and is exactly the Gritsenko lift constructed in \cite{Gri88, Gri94}, which is a generalization of the Saito--Kurokawa lift or Maass lift.
\subsubsection{Borcherds products}
When the nearly holomorphic modular form $f$ for $\rho_M$ is of weight $1-l/2$, Borcherds discovered that the modified exponential of the singular theta lift of $f$ gives a meromorphic modular form of weight $c(0,0)/2$ and character for $\widetilde{\Orth}^+(M)$, which has an infinite product expansion and all of whose zeros or poles lie on hyperplanes $\cD_r(M)$, each with multiplicity
$$
\sum_{n\in \NN} c(-n^2Q(r),n r), \quad \text{where $r\in M'$ is primitive.}
$$
This remarkable modular form is called a \textit{Borcherds product} denoted $\Borch(f)$. The following expression for Borcherds products in terms of Jacobi forms when $M = 2U \oplus L$ is due to Gritsenko and Nikulin \cite{GN98, Gri18}.
\begin{theorem}[{\cite[Theorem 4.2]{Gri18}}]\label{th:product}
Let $\phi$ be a nearly holomorphic Jacobi form of weight $0$ and index $1$ for $L$ with Fourier expansion
\[
\phi(\tau,\mathfrak{z})=\sum_{n\in \ZZ}\sum_{\ell\in L'}f(n,\ell)q^n\zeta^\ell\in J_{0,L,1}^!
\]
satisfying $f(n,\ell)\in \ZZ$ for all $2n-\latt{\ell,\ell}\leq 0$. There is a meromorphic modular form of weight
$f(0,0)/2$ and character $\chi$ with respect to
$\widetilde{\Orth}^+(2U\oplus L)$, given by the series
\begin{equation}\label{eq:JacobiLift}
\Borch(\phi)(\tau,\mathfrak{z},\omega)=
\biggl(\Theta_{f(0,\ast)}
(\tau,\mathfrak{z})\cdot \xi^C \biggr)
\exp \left(-\sum_{m=1}^\infty (\phi | T_{-}(m)) (\tau, \mathfrak{z}) \cdot \xi^m \right),
\end{equation}
convergent in an open subset of $\cH(L)$, where $C$ is defined by \eqref{eq:q^0-term} and where
\begin{equation*}\label{FJtheta}
\Theta_{f(0,\ast)}(\tau,\mathfrak{z})
=\eta(\tau)^{f(0,0)}\prod_{\ell >0}
\biggl(\frac{\vartheta(\tau,\latt{\ell,\mathfrak{z}})}{\eta(\tau)}
\biggr)^{f(0,\ell)}
\end{equation*}
is a generalized theta quotient.
The character $\chi$ is induced by the character of the above theta block
and by the relation $\chi(V)=(-1)^D$, where
$V\colon (\tau,\mathfrak{z}, \omega) \mapsto (\omega,\mathfrak{z},\tau)$,
and $D=\sum_{n<0}\sigma_0(-n) f(n,0)$, here $\sigma_0(x)$ denotes the number of positive divisors of a positive integer $x$.
\end{theorem}
\begin{remark}\label{rem:divisor}
We will rephrase some properties of the zeros and poles of additive lifts and Borcherds products in the context of Jacobi forms. Write vectors in $M = 2U\oplus L$ in the form $v = (a, b, \ell, c, d)$ with $\ell \in L$ and $a, b, c, d \in \ZZ$, such that $(v, v) = -(ad+bc)+\latt{\ell, \ell}$.
(1) The Eichler criterion (see e.g. \cite[Proposition 3.3]{GHS09}) states that if $v_1$ and $v_2$ are primitive vectors of $M'$ that have the same norm and satisfy $v_1-v_2\in M$, then there exists $g\in \widetilde{\Orth}^+(M)$ such that $g(v_1)=v_2$.
Let $v$ be a primitive vector of positive norm in $M'$. Then there exists a vector $(0,n,\ell,1,0)\in 2U
\oplus L'$ such that $(v,v)=-2n+\latt{\ell,\ell}$ and
$v- (0,n,\ell,1,0)\in M$. Therefore, the Heegner divisor $\cH(\frac{1}{2}(v,v),v)$ is exactly the $\widetilde{\Orth}^+(M)$-orbit of $\cD_{(0,n,\ell,1,0)}(M)$. Its restriction to the tube domain $\cH(L)$ is $$\cD_{(0, n, \ell, 1, 0)} \cap \cH(L) = \{Z = (\tau, \mathfrak{z}, \omega): \; \tau - \latt{\ell, \mathfrak{z}} + n\omega = 0\}.$$
(2) In Theorem \ref{th:additive}, the singular additive lift $\Grit(\varphi_k)$ has poles of order $k$ along $\cD_{(0,n,\ell,1,0)}(M)$ if and only if $2n-\latt{\ell,\ell}<0$ and
$$
\delta(n,\ell):= \sum_{d=1}^\infty f(d^2 n, d\ell)\neq 0.
$$
(3) In Theorem \ref{th:product}, the zeros or poles of the Borcherds product $\Borch(\phi)$ lie on $\cD_{(0,n,\ell,1,0)}(M)$ with multiplicity $\delta(n,\ell)$.
(4) By Lemma \ref{lem:periodic}, the singular Fourier coefficients of
$$
\varphi_k = \sum_{n\geq n_0}\sum_{\ell \in L'} f(n,\ell) q^n \zeta^\ell \in J_{k,L,1}^!
$$
are represented by
$$
f(n,\ell), \quad n_0 \leq n \leq \widehat{\delta}_L, \; \ell \in L',\; 2n-\latt{\ell,\ell}<0,
$$
where $\widehat{\delta}_L$ is the largest integer less than $\delta_L/2$ and as in the introduction
$$
\delta_L := \max\{ \min\{\latt{y,y}: y\in L + x \} : x \in L' \}.
$$
(5) Let $\ell$ be a nonzero vector of $L'$. If $\varphi_k \in J_{k,L,1}^{\w}$ has trivial $q^0$-term (i.e. $\varphi_k = O(q)$) and vanishes on the divisor
$$
\{(\tau, \mathfrak{z})\in \HH\times (L\otimes\CC): \latt{\ell,\mathfrak{z}} \in \ZZ\tau+\ZZ\},
$$
then $\Grit(\varphi_k)$ vanishes on $\cD_{(0,0,\ell,1,0)}(M)$.
\end{remark}
\subsection{Weyl-invariant weak Jacobi forms}\label{subsec:Weyl}
In this subsection we review some known results about the algebras of Jacobi forms, which will be used to prove the main theorems.
Let $G$ be a subgroup of $\Orth(L)$. A Jacobi form $\varphi$ for $L$ is called \textit{$G$-invariant} if it satisfies
$$
\varphi(\tau, \sigma(\mathfrak{z})) = \varphi(\tau, \mathfrak{z}), \quad \text{for all $\sigma\in G$}.
$$
All $G$-invariant weak Jacobi forms of integral weight and integral index for $L$ form a bigraded algebra over $\CC$
$$
J_{*,L,*}^{\w, G} = \bigoplus_{t\in \NN} J_{*,L,t}^{\w,G}, \quad \text{where} \quad J_{*,L,t}^{\w,G}=\bigoplus_{k\in\ZZ} J_{k,L,t}^{\w,G}.
$$
It is conjectured that the algebra $J_{*,L,*}^{\w,G}$ is always finitely generated. The most important result in this direction is due to Wirthm\"uller.
Let $R$ be an irreducible root system (cf. \cite{Bou60}) of rank $\rk(R)$, with Weyl group $W(R)$. The root lattice $L_R$ is the lattice generated by the roots of $R$ together with the standard scalar product, rescaled by two if that lattice is odd. When $R$ is not of type $E_8$, Wirthm\"uller \cite{Wir92} showed that $J_{*,L_R,*}^{\w,W(R)}$ is a polynomial algebra over $\CC[E_4,E_6]$ generated by $\rk(R)+1$ weak Jacobi forms. The weights and indices of the generators are invariants of the root system $R$. More precisely, we know the following.
\begin{enumerate}
\item There is always a generator of weight $0$ and index $1$.
\item The other indices are the coefficients of the dual of the highest coroot of $R$, written as a linear combination of the simple roots of $R$.
\item The negative weights of the other generators are the degrees of the generators of the ring of $W(R)$-invariant polynomials, or equivalently, the exponents of $W(R)$ increased by $1$.
\end{enumerate}
It was proved in \cite{Wan21b} that $J_{*,E_8,*}^{\w,W(E_8)}$ is not a polynomial algebra, and we refer to \cite{KW21} for an explicit description of its structure. An automorphic proof of Wirthm\"uller's theorem was given in \cite{Wan21c}.
We provide some information about the generators of $J_{*,R,*}^{\w,W(R)}$. We formulate the weights $k_j$ and indices $m_j$ of generators in Table \ref{Tab:Jacobi} below. We note that $A_3=D_3$ and
$$
W(B_n) = \Orth(nA_1), \quad W(G_2) = \Orth(A_2), \quad W(F_4) = \Orth(D_4), \quad W(C_n) = \Orth(D_n) \; \text{if $n\neq 4$}.
$$
The $W(A_1)$-invariant weak Jacobi forms are actually classical weak Jacobi forms of even weight introduced by Eichler--Zagier \cite{EZ85}.
Explicit constructions of generators of type $A_n$, $B_n$ and $D_4$ were first obtained in \cite{Ber00a, Ber00b}. The generators of type $C_n$ and $D_n$ were obtained in \cite{AG20}. The generators of type $E_6$ and $E_7$ were constructed in \cite{Sak19}, and the generators of type $F_4$ were constructed in \cite{Adl20}.
We fix some notation for the generators to avoid confusion later.
\begin{Notation}\label{notation}
\noindent
\begin{enumerate}
\item When $R=A_n, E_6, E_7$, there are no distinct generators of the same weight and index. Thus we use $\phi_{k,R,t}$ to stand for the generator of weight $k$ and index $t$ associated to $W(R)$.
\item Since $W(C_n)$ is generated by $W(D_n)$ and the reflection which changes the sign of any fixed coordinate, we can choose the generators of $J_{*,D_n,*}^{\w,W(D_n)}$ in the following way:
\begin{itemize}
\item[(a)] Index one: $\phi_{0,D_n,1}$, $\phi_{-2,D_n,1}$, $\phi_{-4,D_n,1}$, which are invariant under $W(C_n)$.
\item[(b)] Index two: $\phi_{-2k,D_n,2}$ for $3\leq k \leq n-1$, which are invariant under $W(C_n)$.
\item[(c)] Index one: $\psi_{-n,D_n,1}$, which is invariant under $W(D_n)$, but anti-invariant under the above reflection.
\end{itemize}
The forms $\phi_{-,D_n,-}$ together with $\psi_{-n,D_n,1}^2$ form a system of generators of $J_{*,D_n,*}^{\w,W(C_n)}$.
\end{enumerate}
\end{Notation}
\begin{table}[ht]
\caption{Weights $k_j$ and indices $m_j$ of generators of $J_{*,L_R,*}^{\w,W(R)}$ ($A_n: n\geq 1$, $D_n: n\geq 4$, $B_n: n\geq 2$, $C_n: n\geq 3$)}\label{Tab:Jacobi}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|c|}
\hline
R & L_R & (k_j,m_j) \\
\hline
A_n & A_n& (0,1), (-s,1) : 2\leq s\leq n+1\\
\hline
D_n & D_n & (0,1), (-2,1), (-4,1), (-n,1), (-2s,2) : 3\leq s \leq n-1 \\
\hline
E_6 & E_6 & (0,1), (-2,1), (-5,1), (-6,2), (-8,2), (-9,2), (-12,
3) \\
\hline
E_7 & E_7 & (0,1), (-2,1), (-6,2), (-8,2), (-10,2), (-12,
3), (-14,3), (-18,4) \\
\hline
B_n & n A_1 & (-2s,1) : 0\leq s \leq n \\
\hline
C_n & D_n & (0,1), (-2,1), (-4,1), (-2s,1) : 3\leq s \leq n \\
\hline
G_2 & A_2 & (0,1), (-2,1), (-6,2) \\
\hline
F_4 & D_4 & (0,1), (-2,1), (-6,2), (-8,2), (-12,3)\\
\hline
\end{array}
\]
\end{table}
\begin{remark}\label{rem:sum of Jacobi forms}
When $L$ is a direct sum of irreducible root lattices, we can determine the algebra of weak Jacobi forms for $L$ using \cite[Theorem 2.4]{WW21c}. Let $L=\bigoplus_{j=1}^n R_j$ and $G=\bigotimes_{j=1}^n W(R_j)$. As a free module over $M_*(\SL_2(\ZZ))$, the space of weak Jacobi forms $J_{*,L,t}^{\w, G}$ of given index $t$ is generated by the tensor products of generators of these $J_{*,R_j,t}^{\w,W(R_j)}$. In other words, there is an isomorphism of graded $\CC[E_4,E_6]$-modules
$$
J_{*,R_1\oplus \cdots \oplus R_n, t}^{\w, W(R_1)\otimes \cdots \otimes W(R_n)} \cong J_{*,R_1,t}^{\w, W(R_1)} \otimes \cdots \otimes J_{*,R_n,t}^{\w, W(R_n)}.
$$
\end{remark}
\section{A proof of the theta block conjecture}\label{sec:theta block conjecture}
In this section we prove Theorem \ref{MTH3} and prove the theta block conjecture as a corollary. As we mentioned in the introduction, Gritsenko, Skoruppa and Zagier \cite{GSZ19} developed the theory of theta blocks to construct holomorphic Jacobi forms of low weight for $A_1$. We recall their construction and its generalization to Jacobi forms of lattice index. The Dedekind eta function
$$
\eta(\tau)= q^{\frac{1}{24}}\prod_{n=1}^\infty (1-q^n), \quad q=e^{2\pi i\tau}
$$
is a modular form of weight $1/2$ on $\SL_2(\ZZ)$ with a multiplier system of order $24$ denoted $v_\eta$. The odd Jacobi theta function
$$
\vartheta(\tau,z)=q^{\frac{1}{8}}(e^{\pi iz}-e^{-\pi i z})\prod_{n=1}^\infty (1-q^ne^{2\pi i z})(1-q^ne^{-2\pi iz})(1-q^n), \quad z\in \CC
$$
is a holomorphic Jacobi form of weight $1/2$ and index $1/2$ for $A_1$ with a multiplier system of order $8$ (see \cite{GN98}). More precisely, $\vartheta$ satisfies the transformation laws
\begin{align*}
\vartheta (\tau, z+ x \tau + y)&= (-1)^{x+y} \exp(- \pi i ( x^2 \tau +2xz )) \vartheta ( \tau, z ), \quad x,y \in \ZZ,\\
\vartheta \left( \frac{a\tau +b}{c\tau + d},\frac{z}{c\tau + d} \right) &= \upsilon_{\eta}^3 (A) \sqrt{c\tau + d} \exp\left(\frac{\pi i c z^2}{c \tau + d} \right) \vartheta ( \tau, z ), \quad A=\left( \begin{array}{cc}
a & b \\
c & d
\end{array} \right) \in \SL_2(\ZZ).
\end{align*}
A (pure) \textit{theta block} is a holomorphic function of the form
$$
\Theta_f(\tau,z)=\eta(\tau)^{f(0)} \prod_{a=1}^\infty (\vartheta(\tau,a z)/\eta(\tau))^{f(a)}, \quad (\tau, z) \in \HH \times \CC,
$$
where $f: \NN \to \NN$ is a function with finite support. From the modular properties of $\eta$ and $\vartheta$ we see that $\Theta_f$ defines a weak Jacobi form of weight $f(0)/2$ and index $m_f$ for $A_1$, with multiplier system $v_\eta^{d_f}$ and leading term $q^{d_f/24}$ in its Fourier expansion, where
$$
m_f=\frac{1}{2}\sum_{a=1}^\infty a^2 f(a), \quad d_f=f(0)+2\sum_{a=1}^\infty f(a).
$$
The number $d_f/24$ is called the \textit{$q$-order} of $\Theta_f$. This is called a \emph{holomorphic theta block} if it is holomorphic as a Jacobi form, i.e. its singular Fourier coefficients vanish. For example, the theta block $\eta^{-6}\vartheta^4\vartheta_2^3\vartheta_3^2\vartheta_4$ gives an explicit construction of the Jacobi Eisenstein series of weight $2$ and index $25$, where $\vartheta_a:=\vartheta(\tau,a z)$. In \cite{GSZ19} Gritsenko, Skoruppa and Zagier associated to a root system $R$ an infinite family of holomorphic theta blocks of weight $\rk(R)/2$. These infinite families are closely related to the famous Macdonald identities (see \cite[Theorem 10.1, Theorem 10.6]{GSZ19}). Some of them also appear as the leading Fourier--Jacobi coefficients of holomorphic Borcherds products of singular weight which vanish precisely on mirrors of reflections (see \cite{DW20}).
Siegel paramodular forms of level $N$ are Siegel modular forms for the paramodular group of degree two and level $N$,
\[
K(N)=\begin{pmatrix}
\ast & N\ast & \ast & \ast \\ \ast & \ast & \ast & \ast/N \\ \ast & N\ast & \ast & \ast \\ N\ast & N\ast & N\ast &\ast
\end{pmatrix}\cap \Sp_4(\QQ), \quad \text{ all } \ast\in \ZZ.
\]
Paramodular forms can be identified with modular forms for $\widetilde{\Orth}^+(2U\oplus A_1(N))$ (see \cite{GN98}), so we can construct Siegel paramodular forms using Gritsenko lifts and Borcherds products. In \cite{GPY15} Gritsenko, Poor and Yuen formulated the \textit{theta block conjecture} which characterizes Siegel paramodular forms which are simultaneously Borcherds products and Gritsenko lifts.
\begin{conjecture}[Conjecture 8.1 in \cite{GPY15}]\label{conj:theta}
Suppose that the $q$-order one theta block $\Theta_f$ is a holomorphic Jacobi form of integral weight $k$ and integral index $N$ for $A_1$. Then as a Siegel paramodular form of weight $k$ and level $N$ the Gritsenko lift of $\Theta_f$ is a Borcherds product. More precisely,
\[
\Grit(\Theta_f)=\Borch\left( - \frac{\Theta_f|T_{-}(2)}{\Theta_f} \right).
\]
\end{conjecture}
This conjecture has been proved for all (eight) infinite families of holomorphic theta blocks of $q$-order one which are related to root systems (see \cite{GPY15, Gri18, GW20, DW20}). Every theta block of $q$-order one has weight $k\leq 11$ and can be written as
\begin{equation}\label{eq:theta q-order 1}
\Theta_{\mathbf{a}}(\tau, z) := \eta^{3k-12} \prod_{j=1}^{12-k}\vartheta_{a_j}, \quad \vartheta_{a_j}:=\vartheta(\tau,a_j z).
\end{equation}
When $k\geq 4$, every theta block of $q$-order one is a holomorphic Jacobi form, and it comes from the infinite family $\prod_{j=1}^8 \vartheta_{a_j}$ associated to the root system $8A_1$ and its quasi pullbacks, in which case Conjecture \ref{conj:theta} was proved. However, when $k=2$ or $3$, the theta block \eqref{eq:theta q-order 1} is usually not a holomorphic Jacobi form, and there do exist holomorphic theta blocks of $q$-order one which do not belong to any of the infinite families associated to root systems. In these cases, Conjecture \ref{conj:theta} has remained open. We remark that there are no holomorphic Jacobi forms of weight $1$ for $A_1$ (see \cite{Sko84}) so the theta block \eqref{eq:theta q-order 1} is never holomorphic when $k=1$.
It is in fact easier to prove the generalization of Conjecture \ref{conj:theta} to meromorphic modular forms on higher-dimensional type IV domains. We first define theta blocks associated to an even positive definite lattice $L$ with bilinear form $\latt{-,-}$. For any finite set $\mathbf{s}=\{s_j\in L' : 1\leq j \leq d \}$, the function
\begin{equation}
\Theta_{\mathbf{s}}(\tau,\mathfrak{z}) = \eta(\tau)^{24-3d} \prod_{j=1}^d \vartheta(\tau, \latt{s_j, \mathfrak{z}}), \quad \mathfrak{z}\in L\otimes\CC
\end{equation}
is called a \textit{theta block of $q$-order one associated to $L$} if it defines a weak Jacobi form of index $1$ for $L$. This holds if and only if $\mathbf{s}$ satisfies the identity
$$
\sum_{j=1}^d \latt{s_j, \mathfrak{z}}^2 = \latt{\mathfrak{z},\mathfrak{z}},
$$
or equivalently, the map
$$
\iota_{\mathbf{s}}: \quad \ell \mapsto (\latt{s_1, \ell}, ..., \latt{s_d, \ell})
$$
defines an embedding of $L$ into the odd unimodular lattice $\ZZ^d$ and therefore into its maximal even sublattice $D_d$.
Note that there exist lattices which cannot be embedded into $\ZZ^d$ for any $d$; however, for a given lattice $L$ there exists a positive integer $m$ such that $L(m)$ can be embedded into some $\ZZ^d$ (see \cite{CS89}). For example, there is no embedding from $E_6$ into any $\ZZ^d$, but $E_6(2)$ is a sublattice of $\ZZ^7$. Therefore, for any given lattice, one can construct theta blocks of $q$-order one and of sufficiently large index.
We first show that the generalized theta block conjecture is true for the $q$-order one theta blocks associated to $D_n$. Let $(\varepsilon_1,...,\varepsilon_n)$ denote the standard orthogonal basis of $\RR^n$. We fix the model
\begin{equation}\label{model of D_n}
D_n=\left\{ (x_i)_{i=1}^n \in \ZZ^n : \sum_{i=1}^n x_i \in 2\ZZ \right\}
\end{equation}
and write $\mathfrak{z} \in D_n \otimes \CC$ in coordinates $\mathfrak{z} = (z_1,...,z_n)$.
\begin{theorem}\label{th:theta D_n}
For $1\leq n \leq 11$, as meromorphic modular forms of weight $12-n$ for $\widetilde{\Orth}^+(2U\oplus D_n)$, the following identity holds:
\begin{equation}
\Grit(\vartheta_{D_n}) = \Borch\left( - \frac{\vartheta_{D_n}|T_{-}(2)}{\vartheta_{D_n}} \right),
\end{equation}
where the theta block $\vartheta_{D_n}$ is
\begin{equation}\label{eq:theta block D_n}
\vartheta_{D_n}(\tau, \mathfrak{z}) = \eta(\tau)^{24-3n}\prod_{j=1}^n \vartheta(\tau, \latt{\varepsilon_j,\mathfrak{z}}) =\eta(\tau)^{24-3n} \prod_{j=1}^n \vartheta(\tau, z_j).
\end{equation}
\end{theorem}
When $n\leq 8$, the theta block $\vartheta_{D_n}$ is a holomorphic Jacobi form, and this was proved in \cite[Theorem 8.2]{GPY15}. We only need to consider the remaining cases $n=9$, $10$, $11$.
\begin{proof}
Let $\cH_n$ be the full Heegner divisor of discriminant $n/4-2$: $$\cH_n = \bigcup_{\substack{v \in M_n' \\ (v, v) = n/4 - 2}} \cD_v(M_n)$$ in $M_n = 2U \oplus D_n$. Since
$$
\delta_{D_n}:=\max\{ \min\{ \latt{v,v} : v \in D_n + \gamma \}: \gamma \in D_n' \} = n/4,
$$
Remark \ref{rem:divisor} (4) implies that representatives of all singular Fourier coefficients of $\vartheta_{D_n}$ appear in the $q^1$-term of its Fourier expansion. (Note that $\vartheta_{D_n}= O(q)$.) Therefore, the only possible singularities of $\Grit(\vartheta_{D_n})$ lie on the arrangement $\cH_n$.
Write $\Psi_{D_n} := -\vartheta_{D_n} | T_{-}(2) / \vartheta_{D_n}$. Since $\vartheta(\tau, z)$ vanishes, if $\tau \in \mathbb{H}$ is fixed, with simple zeros exactly when $z \in \ZZ \tau + \ZZ$, it follows that $\vartheta_{D_n}(\tau, \mathfrak{z})$ vanishes exactly when one of the coordinates $z_j$ is a lattice vector. This implies that $\Psi_{D_n}$ is holomorphic and therefore a weak Jacobi form of weight $0$ and index $1$ for $D_n$. By Lemma \ref{lem:Hecke} and the infinite product expansion of $\vartheta_{D_n}$, the input $\Psi_{D_n}$ has integral Fourier expansion beginning $$\Psi_{D_n}(\tau, \mathfrak{z}) = \sum_{j=1}^n (e^{2\pi i z_j} + e^{-2\pi i z_j}) + 2 (12 - n) + O(q).$$ From this we see that the Borcherds product $\mathrm{Borch}(\Psi_{D_n})$ has weight $12-n$ and simple zeros on the irreducible Heegner divisor $H(1/2,\varepsilon_1)$, and that its other zeros and poles are contained in $\cH_n$.
Remark \ref{rem:divisor} (5) implies that $\Grit(\vartheta_{D_n})$ vanishes on the divisor $H(1/2,\varepsilon_1)$. It follows that the quotient $\mathrm{Grit}(\vartheta_{D_n}) / \mathrm{Borch}(\Psi_{D_n})$ has weight zero and is holomorphic away from $\cH_n$.
We will show in Lemma \ref{Lem:KoecherD_n} below that $\cH_n$ satisfies the Looijenga condition. By Koecher's principle, i.e. Lemma \ref{lem:Koecher}, $\mathrm{Grit}(\vartheta_{D_n}) / \mathrm{Borch}(\Psi_{D_n})$ is a constant. By Theorem \ref{th:additive} and Theorem \ref{th:product}, the leading Fourier--Jacobi coefficients of these forms are both $\vartheta_{D_n}\cdot \xi$, which implies that the constant is $1$.
\end{proof}
\begin{lemma}\label{Lem:KoecherD_n}
\noindent
\begin{enumerate}
\item Every intersection of two distinct hyperplanes in $\cH_9$ is empty.
\item Every nonempty intersection of hyperplanes in $\cH_{10}$ has dimension $11$ or $10$.
\item Every nonempty intersection of hyperplanes in $\cH_{11}$ has dimension $12$, $11$ or $10$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Since $\cH_9 = \cH(1/8, \sum_{j=1}^9\varepsilon_j/2)$, the claim follows from Lemma \ref{lem:intersection} (1).
(2) Write $\cH_{10}=\cH(1/4,v_1)\cup \cH(1/4,v_2)$ where $v_1, v_2 \in D_{10}'/D_{10}$ are the cosets containing non-integral vectors. More precisely, $v_1=\sum_{j=1}^{10}\varepsilon_j/2$ and $v_2=v_1-\varepsilon_1$. By Lemma \ref{lem:intersection} (1), the intersection of two different hyperplanes in the same irreducible component is empty, so nonempty intersections only arise from the intersection of one hyperplane in $\cH(1/4,v_1)$ and one hyperplane in $\cH(1/4,v_2)$. This yields the claim. Since $2U\oplus D_{10} \cong 2U\oplus E_8\oplus 2A_1$ we see that the $10$-dimensional intersections are all symmetric spaces associated to the lattice $2U\oplus E_8$.
(3) We have $\cH_{11}=H(3/8, v)$ with $v=\sum_{j=1}^{11}\varepsilon_j/2$. By Lemma \ref{lem:intersection} (2), the restriction of $\cH(3/8, v)$ to a hyperplane in $\cH(3/8, v)$ is a Heegner divisor of type $\cH(1/3,-)$. Similarly, the restriction of $\cH(1/3, -)$ to a hyperplane in $\cH(1/3, -)$ is a Heegner divisor of type $\cH(1/4,-)$. The intersection of two different hyperplanes in $\cH(1/4,-)$ is empty. In particular, all nonempty intersections have dimension at least $10$. From $2U\oplus D_{11} \cong 2U\oplus E_8\oplus A_3$ we see that the $11$-dimensional (resp. $10$-dimensional) intersections correspond to the lattice $2U\oplus E_8\oplus A_1$ (resp. $2U\oplus E_8$).
\end{proof}
\begin{remark}
By calculating the $q^1$-term of $\Psi_{D_n}$, we find the explicit divisor:
$$
\Div(\Borch{\Psi_{D_n}}) = H(1/2,\varepsilon_1) - (12-n)\cH_n, \quad n=9,10,11.
$$
Since $\Grit(\vartheta_{D_n})$ can have at worst poles of order $12-n$ along the components of $\cH_n$, one can obtain a more direct proof that the quotient $\Grit(\vartheta_{D_n}) / \Borch(\Psi_{D_n})$ is holomorphic everywhere and therefore constant. Our first proof is more satisfying: for one thing, it does not require explicit computation of $\Psi_{D_n}$, and for another it shows that both $\Grit(\vartheta_{D_n})$ and $\Borch(\Psi_{D_n})$ lie in the space of modular forms of weight $12-n$ for $\widetilde{\Orth}^+(2U\oplus D_n)$ with poles on the arrangement $\cH_n$, and the equality is guaranteed by the divisor of the theta block and the Looijenga condition on $\cH_n$. We will see in the next section that this vector space is one-dimensional.
\end{remark}
\begin{corollary}\label{cor:theta}
Let $\Theta_{\mathbf{s}}$ be a theta block of positive weight and $q$-order one associated to $L$. Then we have the identity
$$
\Grit(\Theta_{\mathbf{s}}) = \Borch\left( - \frac{\Theta_{\mathbf{s}}|T_{-}(2)}{\Theta_{\mathbf{s}}} \right).
$$
\end{corollary}
\begin{proof}
The definition of $\Theta_{\mathbf{s}}$ yields an embedding $\iota_\mathbf{s}$ of $L$ into $D_n$, where $n\geq \rk(L)$ is the number of elements of the set $\mathbf{s}$. Then $\Theta_{\mathbf{s}}$ is the pullback of $\vartheta_{D_n}$ along $\iota_{\mathbf{s}}$. Moreover, $\Grit(\Theta_{\mathbf{s}})$ is the pullback of $\Grit(\vartheta_{D_n})$ along the induced embedding $\iota_{\mathbf{s}}: 2U\oplus L \hookrightarrow 2U\oplus D_n$. It is also known that the pullback of a Borcherds product is again a Borcherds product (see \cite{Ma19}). By Theorem \ref{th:theta D_n}, we find that $\Grit(\Theta_{\mathbf{s}})$ is a Borcherds product. The expression $- \left(\Theta_{\mathbf{s}}|T_{-}(2)\right) / \Theta_{\mathbf{s}}$ for the input into the Borcherds product can be determined by comparing the Fourier--Jacobi expansions of singular additive lifts and Borcherds products in Theorem \ref{th:additive} and Theorem \ref{th:product}.
\end{proof}
\begin{corollary}
The theta block conjecture \ref{conj:theta} is true.
\end{corollary}
\begin{proof}
Every holomorphic theta block of weight $k$ and $q$-order one is of the form $\Theta_{\mathbf{s}}$ for a subset $\mathbf{s} \subseteq A_1(N)'$, for some $N \in \mathbb{N}$.
\end{proof}
The above two corollaries also imply the ``if" part of \cite[Conjecture 4.10]{GW20} which is a generalization of the theta block conjecture to holomorphic modular forms on higher-rank orthogonal groups. The ``only if" part remains open.
\section{Free algebras of meromorphic modular forms on type IV symmetric domains}\label{sec:type IV}
In this section we prove Theorem \ref{MTH2}. We first state a more detailed form of Theorem \ref{MTH2}, then prove the existence of the potential modular Jacobians, and finally construct the generators.
\subsection{A precise statement of Theorem \ref{MTH2} and the outline of its proof}
\subsubsection{The construction of the hyperplane arrangement $\cH_L$}
Let $L$ be a lattice in the following three families of root lattices:
\begin{equation}\label{eq:lattices}
\begin{aligned}
&\text{$A$-type:}& & \left(\bigoplus_{j=1}^t A_{m_j}\right)\oplus A_m, \quad t\geq 0, \quad m\geq 1, \quad (m+1)+\sum_{j=1}^t (m_j+1) \leq 11; \\
&\text{$AD$-type:}& & \left(\bigoplus_{j=1}^t A_{m_j}\right)\oplus D_m, \quad t\geq 0, \quad m\geq 4, \quad m + \sum_{j=1}^t (m_j+1) \leq 11; \\
&\text{$AE$-type:}& & E_6, \quad A_1 \oplus E_6, \quad A_2\oplus E_6, \quad E_7,\quad A_1\oplus E_7.
\end{aligned}
\end{equation}
Corresponding to the above decomposition, we write $L = L_0 \oplus L_1$, where $L_1$ is $A_m$, $D_m$ or $E_6$ or $E_7$. When $t = 0$, we take $L_0 = \{0\}$.
\begin{remark}
Different decompositions may yield equivalent lattices, e.g. $A_2 \oplus A_1 \cong A_1 \oplus A_2$; however, we will view them as distinct as long as the $L_1$ are distinct, because we associate different hyperplane arrangements to them. In this sense, there are $97$ $A$-type lattices and $45$ $AD$-lattices, and therefore $147$ lattices altogether. There are only two repeats in the sense that both the lattices and the hyperplane arrangements are the same (and even in these cases, the decomposition of $\cH_L$ into $\cH_{L, 0}$ and $\cH_{L, 1}$ and therefore the classification below of the generators into abelian and Jacobi type generators is distinct):
\begin{align*}
2U \oplus (A_1 \oplus A_3) \oplus D_4 &= 2U \oplus 3A_1 \oplus D_5,\\
2U \oplus A_3 \oplus D_6 &= 2U \oplus 2A_1 \oplus D_7.
\end{align*}
\end{remark}
We list the lattices of $A$-type and $AD$-type in Table \ref{Tab:MTH2}.
\begin{table}[ht]
\caption{Lattices of $A$-type and $AD$-type in the families \eqref{eq:lattices}}\label{Tab:MTH2}
\renewcommand\arraystretch{1.0}
\noindent\[
\begin{array}{|c|c|}
\hline
& \\ [-4mm]
L_1 & L_0 \\ \hline
& \\ [-4mm]
A_1 & 0,\; A_1,\; 2A_1,\; 3A_1,\; 4A_1,\; A_1\oplus A_2,\; A_1\oplus A_3,\; A_1\oplus A_4,\; A_1\oplus A_5,\; A_1\oplus A_6, \\
& A_1\oplus 2A_2,\; A_1\oplus A_2\oplus A_3,\; 2A_1\oplus A_2,\; 2A_1\oplus A_3,\; 2A_1\oplus A_4,\; 3A_1\oplus A_2,\; A_2,\; 2A_2,\\
& 3A_2,\; A_2\oplus A_3,\; A_2\oplus A_4,\; A_2\oplus A_5,\; A_3,\; 2A_3,\; A_3\oplus A_4,\; A_4,\; A_5,\; A_6,\; A_7,\; A_8 \\ \hline
& \\ [-4mm]
A_2 & 0,\; A_1,\; 2A_1,\; 3A_1,\; 4A_1,\; A_1\oplus A_2,\; A_1\oplus A_3,\; A_1\oplus A_4,\; A_1\oplus A_5,\; A_1\oplus 2A_2,\\
& 2A_1\oplus A_2,\; 2A_1\oplus A_3,\; A_2,\; 2A_2,\; A_2\oplus A_3,\; A_2\oplus A_4,\; A_3,\; 2A_3,\; A_4,\; A_5,\; A_6,\; A_7 \\ \hline
& \\ [-4mm]
A_3,\; D_4 & 0,\; A_1,\; 2A_1,\; 3A_1,\; A_1\oplus A_2,\; A_1\oplus A_3,\; A_1\oplus A_4,\\
& 2A_1\oplus A_2,\; A_2,\; 2A_2,\; A_2\oplus A_3,\; A_3,\; A_4,\; A_5,\; A_6 \\ \hline
& \\ [-4mm]
A_4,\; D_5 & 0,\; A_1,\; 2A_1,\; 3A_1,\; A_1\oplus A_2,\; A_1\oplus A_3,\; A_2,\; 2A_2,\; A_3,\; A_4,\; A_5\\ \hline
& \\ [-4mm]
A_5,\; D_6 & 0,\; A_1,\; 2A_1,\; A_1\oplus A_2,\; A_2,\; A_3,\; A_4 \\ \hline
& \\ [-4mm]
A_6,\; D_7 & 0,\; A_1,\; 2A_1,\; A_2,\; A_3 \\ \hline
& \\ [-4mm]
A_7,\; D_8 & 0,\; A_1,\; A_2 \\ \hline
& \\ [-4mm]
A_8,\; D_9 & 0,\; A_1 \\ \hline
& \\ [-4mm]
A_9,\; D_{10} & 0 \\ \hline
& \\ [-4mm]
A_{10},\; D_{11} & 0 \\
\hline
\end{array}
\]
\end{table}
We will construct a hyperplane arrangement $\cH_L$ for every $L$. View $A_n$ as a sublattice of $D_{n+1}$:
$$
A_n = \left\{ (x_i)_{i=1}^{n+1} \in \ZZ^{n+1}: \sum_{i=1}^{n+1} x_i =0 \right\}.
$$
As before, let $\varepsilon_1,..., \varepsilon_{n+1}$ be the standard basis of $\RR^{n+1}$. We fix the following coordinates
\begin{equation}\label{model of A_n}
e_j=\varepsilon_{j+1} - \varepsilon_j, \quad 1\leq j \leq n; \quad \mathfrak{z}=\sum_{j=1}^n z_j e_j \in A_n\otimes \CC.
\end{equation}
Up to sign, the nonzero vectors of minimal norm in $A_n'$ are
\begin{equation}\label{minimal vectors}
u_j:=\varepsilon_j - \frac{1}{n+1}\sum_{i=1}^{n+1} \varepsilon_i, \quad 1\leq j\leq n+1.
\end{equation}
These vectors have norm $\latt{u_j,u_j}=n/(n+1)$ and they lie in the same coset of $A_n'/A_n$.
The hyperplane arrangement $\cH_L$ consists of two parts. The first part is the union of Heegner divisors associated to the minimal vectors in the dual of every irreducible component $A_{m_j}$ of the direct summand $L_0=\oplus_{j=1}^t A_{m_j}$:
\begin{equation}\label{eq:H_0}
\cH_{L,0} = \bigcup_{j=1}^t H\left( \frac{m_j}{2(m_j+1)}, v_j \right), \quad \text{$v_j$ is a minimal norm vector in $A_{m_j}'$.}
\end{equation}
By Remark \ref{rem:divisor} (1), the above Heegner divisor associated to $v_j$ is the $\widetilde{\Orth}^+(2U\oplus L)$-orbit of the hyperplane $\cD_{(0,0,v_j,1,0)}$.
By convention, if $t=0$ and therefore $L_0 = \{0\}$, then $\cH_{L,0}$ is empty.
We introduce the second part of $\cH_L$. For a class $\gamma$ of $L'/L$, we define the minimal norm of $\gamma$ as
\begin{equation}\label{eq:delta_gamma}
\delta_\gamma = \min\{ \latt{v,v} : v \in L + \gamma \}.
\end{equation}
As before we define the minimal norm of $L$ as
\begin{equation}\label{eq:delta_L}
\delta_L = \max\{ \delta_\gamma: \gamma \in L'/L \}.
\end{equation}
\begin{lemma}\label{lem:norm of L}
All lattices $L$ in the three families \eqref{eq:lattices} satisfy $\delta_L < 3$.
\end{lemma}
\begin{proof} Direct computation.
\end{proof}
The second part of $\cH_L$ is defined as the following finite union of Heegner divisors:
\begin{equation}\label{eq:H_1}
\cH_{L,1} = \bigcup_{\substack{ \gamma \in L'/L\\ \delta_\gamma > 2}} H\left( \frac{\delta_\gamma}{2}-1, \gamma \right).
\end{equation}
By Remark \ref{rem:divisor} (1), the above Heegner divisor associated to $\gamma$ is the $\widetilde{\Orth}^+(2U\oplus L)$-orbit of the hyperplane $\cD_{(0,1,v,1,0)}$, where $v\in L'$ is a vector of minimal norm $\delta_\gamma$ in the class $\gamma + L$.
In particular, if $\delta_L \leq 2$ then $\cH_{L,1}$ is empty.
The $\widetilde{\Orth}^+(2U\oplus L)$-invariant hyperplane arrangement on $\cD(2U\oplus L)$ is then defined as
\begin{equation}
\cH_L=\cH_{L,0}\cup \cH_{L,1}.
\end{equation}
When $L$ is an irreducible root lattice (i.e. $L_0=\{0\}$) satisfying $\delta_L \leq 2$, then the arrangement $\cH_L$ is empty. The lattices satisfying this are precisely $A_m$ for $m\leq 7$, $D_m$ for $4\leq m\leq 8$, $E_6$ and $E_7$.
\begin{lemma}
For all lattices $L$ in the three families \eqref{eq:lattices}, the arrangement $\cH_L$ satisfies the Looijenga condition \ref{condition}.
\end{lemma}
\begin{proof}
We use induction on the rank of $L$; when $L$ has rank one, it is the $A_1$ root lattice and $\cH_L$ is empty.
In general, by Lemma \ref{lem:norm of L}, the arrangement $\cH_L$ is a finite union of Heegner divisors of the form $H(a,-)$ with $a< 1/2$. Every $n$-dimensional intersection of hyperplanes including at least one in $\cH_{L,0}$ is a symmetric domain attached to the lattice $2U \oplus M$ for another $M$ appearing in \eqref{eq:lattices}, and that every $(n-1)$-dimensional intersection of hyperplanes contained in it is contained in the arrangement $\cH_M$ associated to $M$, so in this case the claim follows by induction.
Therefore, we only need to consider intersections of hyperplanes from $\cH_{L,1}$. Enumerating the minimal norm vectors in each $L'$ shows that the divisors $\cH_{L, 1}$ satisfy Corollary \ref{cor:intersection} and therefore the Looijenga condition \ref{condition}.
\end{proof}
\subsubsection{A more precise version of Theorem \ref{MTH2}}
\begin{theorem}\label{th:2precise}
Let $L=L_0\oplus L_1$ be one of the $147$ lattices in \eqref{eq:lattices}. Then the ring of modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ with poles supported on $\cH_L$ is the polynomial algebra on $\rk(L)+3$ generators. The generators fall naturally into three groups:
\begin{enumerate}
\item Eisenstein type: there are two generators of weights $4$ and $6$ whose zeroth Fourier--Jacobi coefficients are respectively the $\SL_2(\ZZ)$ Eisenstein series $E_4$ and $E_6$.
\item Abelian type: for each component $A_{m_j}$ of $L_0=\bigoplus_{j=1}^t A_{m_j}$, there are $m_j$ generators of weights
$$
m_j + 1, \quad m_j+1-i \quad \text{for $2\leq i\leq m_j$}.
$$
\item Jacobi type: there are $\rk(L_1)+1$ generators associated to $L_1$ whose weights are given by
$$
k_i + t_i\left( 12 - \sum_{j=1}^t (m_j+1) \right), \quad 0\leq i \leq \rk(L_1)
$$
where the pairs $(k_i,t_i)$ are the weights and indices of the free generators of $W(L_1)$-invariant weak Jacobi forms described in \S \ref{subsec:Weyl}.
\end{enumerate}
\end{theorem}
The grouping is based on the leading Fourier--Jacobi coefficient, which is either an Eisenstein series, an abelian function associated to $L_0$, or a Jacobi form associated to $L_1$. In more detail:
(1) The zeroth Fourier--Jacobi coefficient of a modular form that is holomorphic away from $\cH_{L, 1}$ is a (holomorphic) modular form in $\CC[E_4, E_6]$. The generators of Eisenstein type will turn out to have their poles supported on $\cH_{L, 1}$.
(2) The generators of abelian type are related to theta blocks associated to root systems of type $A$. It follows from Wirthm\"uller's theorem \cite{Wir92} that the $\CC[E_4, E_6]$-module of weak Jacobi forms of index $1$ for $A_n$ is generated by forms of weights $0$, $-2$, $-3$, ..., $-(n+1)$. The minimal weight generator can be constructed as a theta block:
\begin{equation}\label{eq:theta-blockA_n}
\phi_{-(n+1),A_n,1}(\tau, \mathfrak{z})= \prod_{s=1}^{n+1} \frac{\vartheta(\tau, \latt{u_s, \mathfrak{z}})}{\eta(\tau)^3} = -\frac{\vartheta(\tau,z_1)\vartheta(\tau,z_n)}{\eta(\tau)^6} \prod_{s=1}^{n-1} \frac{\vartheta(\tau, z_s - z_{s+1})}{\eta(\tau)^3},
\end{equation}
where $u_s$ are minimal norm vectors of $A_n'$ defined in \eqref{minimal vectors} and we use the model of $A_n$ fixed in \eqref{model of A_n}. Let $\Delta=\eta^{24}$. Then
$$
\vartheta_{A_n}:=\Delta \phi_{-(n+1),A_n,1}
$$
is a theta block of weight $11-n$ and $q$-order one associated to $A_n$, and it is the pullback of the theta block $\vartheta_{D_{n+1}}$. Corollary \ref{cor:theta} implies that its singular additive lift is also a Borcherds product:
\begin{lemma}\label{lem:theta A_n}
When $n\leq 10$, as meromorphic modular forms of weight $11-n$ for $\widetilde{\Orth}^+(2U\oplus A_n)$,
$$
\Grit(\vartheta_{A_n}) = \Borch\left( - \frac{\vartheta_{A_n}|T_{-}(2)}{\vartheta_{A_n}} \right).
$$
\end{lemma}
For each component $A_{m_j}$ of $L_0$, there will be $m_j$ generators of abelian type whose zeroth Fourier--Jacobi coefficients are the abelian functions
\begin{equation}\label{eq:generators of type (2)}
\phi_{-i,A_{m_j},1} / \phi_{-(m_j+1),A_{m_j},1}, \quad i=0, 2,3, ..., m_j.
\end{equation}
These can be understood as meromorphic Jacobi forms of weight $(m_j + 1) - i$ and index $0$ associated to $L$ which are constant in the components of $L$ other than $A_{m_j}$.
(3) The generators of Jacobi type are motivated by the results of \cite{WW20a}. For the $14$ lattices $L$ with $\cH_L=\emptyset$, we proved in \cite{WW20a} that the ring of holomorphic modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ is freely generated by $\rk(L)+3$ modular forms whose leading Fourier--Jacobi coefficients are respectively
$$
E_4,\quad E_6,\quad \Delta^{s} f_{L,s} \cdot \xi^{s}, \quad \text{where} \; \xi=e^{2\pi i \omega},
$$
where $f_{L,s}$ runs through the generators of index $s$ of Wirthm\"uller's ring of $W(L)$-invariant weak Jacobi forms. In the present case, we expect to find $\rk(L_1)+1$ generators associated to the direct summand $L_1$ for which the leading terms in the Fourier--Jacobi expansions are
\begin{equation}\label{eq:generators of type (3)}
\left(\bigotimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1}\right)^{t_i} \otimes (\Delta^{t_i}\phi_{k_i,L_1,t_i})\cdot \xi^{t_i}
\end{equation}
and, if $L_1 = D_m$, $m \ge 4$, the additional form beginning with
\begin{equation}\label{eq:generators Dn(1)}
\left(\bigotimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1}\right)\otimes (\Delta \psi_{-m,D_m,1})\cdot \xi.
\end{equation}
We refer to Notation \ref{notation} for the description of these generators of Weyl invariant weak Jacobi forms.
This coefficient is a Jacobi form of weight $k_i+t_i(12-\sum_{j=1}^t(m_j+1))$ and index $t_i$ associated to $L=L_0\oplus L_1$.
Note that the expected generators are all of positive weight. Every generator of index $t_i$ and weight $k_i$ of the ring of $W(L_1)$-invariant weak Jacobi forms satisfies $k_i / t_i \geq -K_{L_1}$, where $K_{L_1}$ is $m+1$ for $L_1=A_m$, $m$ for $L_1=D_m$, and $5$ for $L_1=E_6, E_7$. Thus the generators of Jacobi type have positive weight, because of the constraints $$(m+1) + \sum_{j=1}^t (m_j + 1) \le 11$$ for lattices of $A$-type and $$m + \sum_{j=1}^t (m_j + 1) \le 11$$ for lattices of $AD$-type.
\begin{remark}
If a lattice $L$ in the family of $A$-type has $$K:=m+1+\sum_{j=1}^t (m_j+1)\geq 12,$$ then the hyperplane arrangement $\cH_L$ does \emph{not} satisfy the Looijenga condition. Indeed, if $\theta_L$ is the theta block of $q$-order one defined as $$\theta_L := \Delta \cdot \left( \bigotimes_{j=1}^t \phi_{-(m_j + 1), A_{m_j}, 1} \right) \otimes \phi_{-(m+1), A_m, 1},$$ then the Borcherds lift of $-2(\theta_L | T_{-}(2)) / \theta_L$ is a meromorphic modular form of non-positive weight $24-2K$ which is holomorphic away from $\cH_{L, 1}$, violating Koecher's principle.
Therefore, Theorem \ref{th:2precise} cannot be extended to the lattices satisfying $K\geq 12$. A similar result holds for the family of $AD$-type lattices.
\end{remark}
\subsubsection{The sketch of the proof}
To prove Theorem \ref{th:2precise}, we apply the modular Jacobian criterion of Theorem \ref{th:Jacobiancriterion}. We sketch the main steps of the proof below:
\vspace{3mm}
\begin{itemize}
\item[(I)] We construct a Borcherds product $\Phi_L$ for $\widetilde{\Orth}^+(2U\oplus L)$ satisfying the following conditions:
\begin{itemize}
\item[(i)] The weight of $\Phi_L$ equals the sum of the weights of the $\rk(L)+3$ generators prescribed in Theorem \ref{th:2precise} plus $\rk(L)+2$.
\item[(ii)] The function $\Phi_L$ vanishes with multiplicity one on hyperplanes associated to vectors of norm $2$ in $2U\oplus L$ which are not contained in the arrangement $\cH_L$.
\item[(iii)] All other zeros and poles of $\Phi_L$ are contained in $\cH_L$.
\end{itemize}
\item[(II)] We construct the $\rk(L)+3$ generators which have weights given in Theorem \ref{th:2precise} and the leading Fourier--Jacobi coefficients described above. By the discussion above, we conclude from the algebraic independence of generators of Jacobi forms that the leading Fourier--Jacobi coefficients of these generators of orthogonal type are algebraically independent over $\CC$. Therefore, these generators of orthogonal type are algebraically independent, which implies that their Jacobian $J_L$ is not identically zero.
\item[(III)] The quotient $J_L / \Phi_L$ defines a modular form of weight $0$ which is holomorphic away from $\cH_L$, and is therefore constant by Lemma \ref{lem:Koecher}.
\item[(IV)] Theorem \ref{th:2precise} follows by applying Theorem \ref{th:Jacobiancriterion}.
\end{itemize}
\vspace{3mm}
Step (I) is Theorem \ref{th:existence}. For Step (II), we construct the generators of Eisenstein type in Lemma \ref{lem: type (1)}, the generators of abelian type in Lemma \ref{lem: type (2) AD} and Lemma \ref{lem: type (2) AE}, and the generators of Jacobi type in \S\ref{subsubsec: generators of type (3)} for the families of $A$-type and $AD$-type. Finally, we find the generators of Jacobi type for the family of $AE$-type in \S \ref{sec:non-free}.
Steps (III) and (IV) then follow immediately.
\subsection{Preimage of the modular Jacobian under the Borcherds lift} For each lattice $L$ in \eqref{eq:lattices}, we will prove the existence of a Borcherds product $\Phi_L$ which will turn out to be (up to a scalar) the Jacobian of any set of generators of the algebra of meromorphic modular forms.
\begin{theorem}\label{th:existence}
For each lattice $L$ in the families \eqref{eq:lattices}, there exists a Borcherds product $\Phi_L$ satisfying conditions $(i)$, $(ii)$, $(iii)$ of Step $(I)$. More precisely, $\Phi_L$ is the Borcherds multiplicative lift of a nearly holomorphic Jacobi form of weight $0$ and index $1$ associated to $L$ whose Fourier expansion has the form
\begin{equation}\label{eq:Jacobian q^0}
\phi_L(\tau, \mathfrak{z}) = q^{-1} + 2k + \sum_{\substack{ r\in L\\ \latt{r,r}=2}} \zeta^r + \sum_{j=1}^t c_{m_j} \sum_{u\in \mathcal{U}_j}(\zeta^u + \zeta^{-u}) + O(q),
\end{equation}
where $\mathcal{U}_j$ is the set of minimal norm vectors in the dual of the component $A_{m_j}$ of $L_0$ defined in \eqref{minimal vectors}. The multiplicities $c_{m_j}$ are determined by the formulae
\begin{equation}
c_1 = 2(h-2), \quad c_{n} = h - (n+1) \quad \text{if $n>1$},
\end{equation}
where $h$ is the Coxeter number of the root lattice $L_1$.
The weight $k$ of $\Phi_L$ is determined by
\begin{equation}
k = 12 (h+1) - \frac{1}{2} \rk(L_1)h - \sum_{j=1}^t \left(h-1 - \frac{1}{2}m_j \right)(m_j+1).
\end{equation}
\end{theorem}
The proof of the theorem follows essentially from the following Lemma.
\begin{lemma}\label{lem:Jacobi}
Let $L$ be an even positive definite lattice with bilinear form $\latt{-,-}$.
For every nonzero $\gamma \in L'/L$, we define the basic orbit associated to $\gamma$ as
$$
\orb(\gamma)= \sum_{\substack{v\in L+\gamma \\ \latt{v,v}=\delta_\gamma }} (\zeta^v + \zeta^{-v}), \quad \zeta^v=e^{2\pi i\latt{v,\mathfrak{z}}}, \quad \mathfrak{z} \in L\otimes\CC,
$$
where as before $\delta_\gamma = \min\{ \latt{v,v} : v\in L + \gamma \}$. The number of distinct basic orbits is denoted $N$.
\begin{enumerate}
\item The space $J_{2*,L,1}^{\w}$ of weak Jacobi forms of even weight and index $1$ associated to $L$ is a free module over $\CC[E_4, E_6]$ on $N+1$ generators. The $q^0$-term of any weak Jacobi form in $J_{2*,L,1}^{\w}$ is a $\CC$-linear combination of the basic orbits and the constant term, and the $q^0$-terms of any basis of the $\CC[E_4, E_6]$-module are linearly independent over $\CC$.
\item Let $L=\oplus_{i=1}^n L_i$. For every $L_i$, assume that
\begin{itemize}
\item[(i)] the free module $J_{*,L_i,1}^{\w}=\oplus_{k\in\ZZ}J_{k,L_i,1}^{\w}$ is generated by forms of non-positive weight;
\item[(ii)] $J_{*,L_i,1}^{\w}$ has no generator of weight $-1$;
\item[(iii)]$J_{*,L_i,1}^{\w}$ has exactly one generator each of weight $-2$ and weight $0$.
\end{itemize}
For $1\leq i\leq n-1$, choose any basic orbit $\orb(\gamma_i)$ for $L_i$. Then there exists a nearly holomorphic Jacobi form of weight $0$ and index $1$ associated to $L$ with rational Fourier expansion
$$
\phi_L = q^{-1} + \sum_{\substack{r\in L \\ \latt{r,r}=2}} \zeta^r + \sum_{j=1}^{n-1} c_j \orb(\gamma_j) + c_0 + O(q),
$$
and where all $c_j$ are rational numbers.
\end{enumerate}
\end{lemma}
For general $L$, $\phi_L$ will not be unique.
\begin{proof}
(1) This is standard result. (See also \cite[Proposition 2.2]{WW21c}.) Briefly, following the argument of \cite[Theorem 8.4]{EZ85}, $J_{2*,L,1}^{\w}$ can be shown to be a free module over $M_*(\SL_2(\ZZ))$ using only the fact that the weight of weak Jacobi forms is bounded from below. If we view Jacobi forms as modular forms for $\rho_L$, then Borcherds' obstruction principle \cite{Bor99} shows that there exist weak Jacobi forms of sufficiently large weight whose $q^0$-term is a constant or any single basic orbit. This implies that there are $N+1$ generators. The $q^0$-terms of any basis are linearly independent, as otherwise some $\CC[E_4, E_6]$-linear combination of the generators would be a weak Jacobi form whose $q^0$-term is zero and therefore is a product of $\Delta$ and a $\CC[E_4,E_6]$-linear combination of generators of lower weight, leading to a contradiction.
(2) Define two rational vector spaces:
\begin{align*}
V=& \QQ \oplus \bigoplus_{\substack{\gamma \in L' / L \\ \gamma \neq 0}} \QQ \cdot \orb(\gamma); \\
V_0=& \{ \text{$q^0$-terms of weak Jacobi forms of weight $0$ and index $1$ for $L$ with rational coefficients} \}.
\end{align*}
The space $V_0$ is naturally a subspace of $V$. By \cite{McG03}, all spaces of vector-valued modular forms for the representations $\rho_L$ have bases with rational Fourier coefficients, so the generators of $J^{\w}_{*, L_i, 1}$ can all be chosen to be rational. By assumption (i), the free module $J_{2*,L,1}^{\w}$ is generated by forms of non-positive weight, and therefore the codimension of $V_0$ in $V$ is the number of generators of weight $-2$ of $J_{2*,L,1}^{\w}$ which is in fact $n$ by assumptions (ii) and (iii). We claim that $\{1, \orb(\gamma_i), 1\leq i\leq n-1 \}$ is a basis of the quotient space $V/V_0$; in other words, that there is no nonzero weak Jacobi form of weight $0$ and index $1$ for $L$ whose $q^0$-term lies in the span of this basis. This follows from the vector system relation \eqref{eq:vectorsystem} satisfied by the $q^0$-term of any weak Jacobi form of weight $0$:
$$
\sum_{\ell \in L'} f(0,\ell) \latt{\ell, \mathfrak{z}}^2 = 2C \latt{\mathfrak{z},\mathfrak{z}}, \quad \mathfrak{z}\in L\otimes\CC,
$$
which is not satisfied by any combination of $1$ and $\orb(\gamma_i)$ (since these $\gamma_i$ does not span the whole space $L\otimes\RR$). Therefore, for any basic orbit $\orb(\gamma)$ related to $L$, there exists a weak Jacobi form of weight $0$ and index $1$ for $L$ with Fourier expansion
$$
\phi_{L,\gamma} = \orb(\gamma) + \sum_{i=1}^{n-1} a_i \orb(\gamma_i) + a_0,
$$
where $a_i \in \QQ$ for $0\leq i\leq n-1$. Since $J_{2*,L,1}^{\w}$ is generated in non-positive weights, part (1) implies that there exists a weak Jacobi form of weight $12$ whose $q^0$-term is $1$ and whose other Fourier coefficients are rational. We divide it by $\Delta$ and obtain a nearly holomorphic Jacobi form of weight $0$ and index $1$ for $L$ whose Fourier expansion begins
$$
q^{-1}+\sum_{r\in L, \latt{r,r}=2}\zeta^r + C(\zeta) + O(q), \; \; C(\zeta) \in V.
$$
By modifying it with the above $\phi_{L,\gamma}$, we obtain the desired Jacobi form $\phi_L$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:existence}]
The root lattice $L$ satisfies all assumptions in the above lemma by \S \ref{subsec:Weyl}, so we obtain the existence of a nearly holomorphic Jacobi form $\phi_L$ of weight $0$ and index $1$ for $L$ whose Fourier expansion has the form \eqref{eq:Jacobian q^0}. It remains to determine $k$ and $c_{m_j}$. The multiplicities $c_{m_j}$ can be computed by applying the identity \eqref{eq:vectorsystem} to every irreducible component of $L_0$, and $k$ is determined by the identity \eqref{eq:q^0-term}.
\end{proof}
\begin{remark}
For the lattices in \eqref{eq:lattices}, the Jacobi form $\phi_L$ from \eqref{eq:Jacobian q^0} is unique because $J_{-12,L,1}^{\w} = \{0\}$. Lemma \ref{lem:norm of L} and Remark \ref{rem:divisor} (4) show that the representative singular Fourier coefficients of $\phi_L$ only appear in its $q^{-1}$-term, $q^0$-term and $q^1$-term. The representative singular Fourier coefficients in $q^{-1}$-term correspond to mirrors of reflections in $\widetilde{\Orth}^+(2U\oplus L)$. The representative singular Fourier coefficients in $q^{0}$-term correspond to the first part $\cH_{L,0}$ of the arrangement $\cH_L$. The representative singular Fourier coefficients in $q^{1}$-term correspond to the second part $\cH_{L,1}$ of $\cH_L$.
In Theorem \ref{th:existence} we only showed that $\phi_L$ has rational Fourier coefficients. This is enough for our purposes: suppose $N \in \mathbb{N}$ is such that $N \phi_L$ has integral Fourier coefficients and is therefore a valid input in Borcherds' lift. We will simply replace step (III) of the proof of Theorem \ref{th:2precise} by showing that $J_L^N / \Borch(N\phi_L)$ is constant. From the divisor of $J_L^N$ we see \emph{a posteriori} that the singular coefficients of $\phi_L$ (in particular those that appear in the $q^1$-term) were in fact integers.
\end{remark}
\begin{remark}
Let us verify that the weight $k$ in Theorem \ref{th:existence} equals the weight of the Jacobian of generators predicted by Theorem \ref{th:2precise}. Wirthm\"uller \cite{Wir92} determined the weights and indices of generators of $W(R)$-invariant weak Jacobi forms in terms of invariants of $R$ (we refer to \cite[Theorem 2.2]{Wan21b} for a clear description). In particular, the sums of the weights $k_i$ and indices $t_i$ satisfy
$$
\sum k_i = -\left( 1 + \frac{h}{2} \right)\rk(R), \quad \sum t_i = h,
$$
where $h$ is the Coxeter number of $R$. Using these identities, it follows that $k - (\mathrm{rk}(R) + 2)$ is the sum of the weights of the generators in Theorem \ref{th:2precise}.
\end{remark}
\subsection{The construction of generators}\label{subsec:generators}
In this subsection we construct the generators required in Theorem \ref{th:2precise}.
\subsubsection{Generators of Eisenstein type}
\begin{lemma}\label{lem:E4E6}
Assume that the free $M_*(\SL_2(\ZZ))$-module $J_{2*,L,1}^{\w}$ is generated by forms of non-positive weight. Then there exist weak Jacobi forms with the following Fourier expansions:
\begin{align*}
E_{4,L} &= 1 +O(q) \in J_{4,L,1}^{\w},\\
E_{6,L} &= 1 +O(q) \in J_{6,L,1}^{\w}.
\end{align*}
Their singular additive lifts define meromorphic modular forms of weight $4$ and $6$ for $\widetilde{\Orth}^+(2U\oplus L)$ with Fourier--Jacobi expansions
\begin{align*}
\cE_{4,L} &:= 240 \Grit(E_{4,L}) = E_4 + O(\xi),\\
\cE_{6,L} &:= -504 \Grit(E_{6,L}) = E_6 + O(\xi),
\end{align*}
where $\xi=e^{2\pi i \omega}$ as before.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:Jacobi}, the $q^0$-terms of generators of $J_{2*,L,1}^{\w}$ are linearly independent, and the number of generators equals the number of basic orbits plus one. Therefore the forms $E_{4, L}$ and $E_{6, L}$ can be constructed as suitable $\CC[E_4,E_6]$-linear combinations of the generators. The second part of the lemma follows from Theorem \ref{th:additive}.
\end{proof}
The above form $E_{4,L}$ (resp. $E_{6,L}$) is unique if and only if $J_{-8,L,1}^{\w}$ (resp. $J_{-6,L,1}^{\w}$) is trivial.
The following gives the construction of generators of Eisenstein type.
\begin{lemma}\label{lem: type (1)}
For every lattice $L$ in \eqref{eq:lattices}, the possible poles of $\cE_{4,L}$ and $\cE_{6,L}$ are contained in $\cH_{L,1}$.
\end{lemma}
\begin{proof}
All singular Fourier coefficients of $E_{4,L}$ and $E_{6,L}$ must appear in their $q^1$-terms, and these coefficients correspond to divisors in $\cH_{L,1}$ under the singular additive lift.
\end{proof}
\subsubsection{Generators of abelian type}
The generators of abelian type will be constructed as quotients of singular additive lifts. For $n\geq 4$, the minimal weight generator of index $1$ for $J_{*,D_n,*}^{\w,W(D_n)}$ which is anti-invariant under changing an odd number of signs (e.g. the map $z_1 \mapsto -z_1$) can be constructed as the theta block
\begin{equation}
\psi_{-n, D_n,1}(\tau, \mathfrak{z}) = \frac{\vartheta_{D_n}(\tau, \mathfrak{z})}{\eta(\tau)^{24}} = \prod_{j=1}^n \frac{\vartheta(\tau,z_j)}{\eta(\tau)^3} \in J_{-n,D_n,1}^{\w, W(D_n)}.
\end{equation}
Here we have fixed the model of $D_n$ in \eqref{model of D_n} and $\vartheta_{D_n}$ is defined in \eqref{eq:theta block D_n}.
We recall the basic Jacobi forms fixed in Notation \ref{notation}. The generators of $W(A_n)$-invariant weak Jacobi forms are labelled $\phi_{-k,A_n,1}$ for $k=0$ or $2\leq k \leq n+1$. Note that $\phi_{-(n+1),A_n,1}$ is the theta block \eqref{eq:theta-blockA_n}.
\begin{lemma}\label{lem: type (2) AD}
Let $L=L_0\oplus L_1$ be a lattice in the family of $A$-type or $AD$-type with $L_0\neq \{0\}$. For any component $A_{m_s}$ of $L_0=\oplus_{j=1}^t A_{m_j}$, the associated $m_s$ generators can be constructed by
\begin{equation}
\begin{aligned}
&\frac{\Grit\left[\Delta \left(\bigotimes_{j=1}^{s-1}\phi_{-(m_j+1),A_{m_j},1}\right)\otimes \phi_{-k,A_{m_s},1} \otimes \left(\bigotimes_{j=s+1}^{t}\phi_{-(m_j+1),A_{m_j},1}\right)\otimes f_{L_1} \right]}{\Grit\left[\Delta\left(\bigotimes_{j=1}^{t}\phi_{-(m_j+1),A_{m_j},1}\right)\otimes f_{L_1} \right]} \\
=& \frac{\phi_{-k,A_{m_s},1}}{\phi_{-(m_s+1),A_{m_s},1}} + O(\xi), \quad \text{for $k=0$ or $2\leq k \leq m_s$}
\end{aligned}
\end{equation}
where $f_{L_1}=\phi_{-(m+1),A_m,1}$ if $L_1=A_m$, or $f_{L_1}=\psi_{-m, D_m,1}$ if $L_1=D_m$.
\end{lemma}
\begin{proof}
We write the above quotient as $\Grit(f)/\Grit(g)$ for convenience. By definition, $g$ is a theta block of positive weight and $q$-order one. By Corollary \ref{cor:theta}, $\Grit(g)$ is a Borcherds product. Moreover, on the complement $\cD(2U\oplus L) - \cH_{L} $ the additive lift $\Grit(g)$ vanishes precisely with multiplicity one along the Heegner divisor corresponding to the theta block $f_{L_1}$, which is either $H(m/2(m+1),u)$ for a minimal norm vector $u$ in the dual of $L_1=A_m$, or $H(1/2,\varepsilon_1)$ for the minimal vector $\varepsilon_1$ in the dual of $L_1=D_m$. By Remark \ref{rem:divisor} (5), $\Grit(f)$ also vanishes on this Heegner divisor. In addition, all poles of $\Grit(f)$ and $\Grit(g)$ are contained in $\cH_{L,1}$. Therefore, the above quotient is holomorphic away from $\cH_{L}$. Its leading Fourier--Jacobi coefficient is the quotient of the leading Fourier--Jacobi coefficients of $\Grit(f)$ and $\Grit(g)$.
\end{proof}
The generators of abelian type for the three $AE$-lattices with $L_0 \neq \{0\}$ must be constructed by a different argument. In this case, $\delta_L \leq 2$, so $\cH_{L, 1}$ is empty.
\begin{lemma}\label{lem: type (2) AE}
For the three lattices of $AE$-type with $L_0\neq \{0\}$, the generators of abelian type exist:
\begin{align*}
G_{2, A_1\oplus E_6} &= \Grit(\phi_{-2,A_1,1}\otimes E_{4,E_6}) = c_1 \frac{\phi_{0,A_1,1}}{\phi_{-2,A_1,1}} + O(\xi),\\
G_{2, A_1\oplus E_7} &= \Grit(\phi_{-2,A_1,1}\otimes E_{4,E_7}) = c_2 \frac{\phi_{0,A_1,1}}{\phi_{-2,A_1,1}} + O(\xi),\\
G_{1, A_2\oplus E_6} &= \Grit(\phi_{-3,A_2,1}\otimes E_{4,E_6}) = c_3 \frac{\phi_{-2,A_2,1}}{\phi_{-3,A_2,1}} + O(\xi),\\
G_{3, A_2\oplus E_6} &= \Grit(\phi_{-3,A_2,1}\otimes E_{6,E_6}) - G_{1, A_2\oplus E_6}^3 = c_4 \frac{\phi_{0, A_2,1}}{\phi_{-3,A_2,1}} + O(\xi),
\end{align*}
where $c_i$ are nonzero constants, and where $E_{4,E_6}$, $E_{6,E_6}$ and $E_{4,E_7}$ are defined in Lemma \ref{lem:E4E6}.
\end{lemma}
\begin{proof}
The above meromorphic modular forms are well defined and their poles are supported on $\cH_{L,0}$. We only need to show that they have the claimed zeroth Fourier--Jacobi coefficients. This follows from Theorem \ref{th:additive}, together with the identity (\cite{EZ85}, Theorem 3.6) $$\frac{\phi_{0, A_1, 1}}{\phi_{-2, A_1, 1}} = -\frac{3}{\pi^2} \wp$$ in the first two cases, and the identity (\cite{WW21c}, Lemma 4.1) \begin{align*} \frac{\phi_{-2, A_2, 1}(\tau, z_1, z_2)}{\phi_{-3, A_2, 1}(\tau, z_1, z_2)} &= \frac{1}{2\pi i} \left( -\frac{\vartheta'(\tau, z_1)}{\vartheta(\tau, z_1)} + \frac{\vartheta'(\tau, z_1-z_2)}{\vartheta(\tau, z_1-z_2)} + \frac{\vartheta'(\tau, z_2)}{\vartheta(\tau, z_2)} \right) \\ &= \frac{1}{2\pi i} \left( -\zeta(\tau, z_1) + \zeta(\tau, z_1-z_2) + \zeta(\tau, z_2) \right) \end{align*} in the third case. As for $G_{3, A_2 \oplus E_6}$, we can see that if the (unique) allowed pole in $\cH_{L, 0}$ is cut out locally by $z = 0$ then $G_{1, A_2 \oplus E_6}$ and $\Grit(\phi_{-3,A_2,1}\otimes E_{6,E_6})$ have Taylor expansions \begin{align*} G_{1, A_2 \oplus E_6}(\tau, \mathfrak{z}, \omega) &= (2\pi i z)^{-1} + O(z) \\ \Grit(\phi_{-3,A_2,1}\otimes E_{6,E_6})(\tau, \mathfrak{z}, \omega) &= (2\pi i z)^{-3} + O(z) \end{align*} and therefore that $G_{3, A_2 \oplus E_6}$ has at worst a simple pole. Its weight is also less than the singular weight for $2U \oplus A_2 \oplus E_6$, so to see that it indeed has a simple pole on its only allowed singularity it is enough to check that it does not vanish identically. This is clear from the leading Fourier--Jacobi coefficient.
\end{proof}
\subsubsection{Generators of Jacobi type}\label{subsubsec: generators of type (3)}
Finally, we construct the generators of Jacobi type. The construction is easiest for the family of $A$-type lattices:
\begin{lemma}
Let $L=L_0\oplus A_m$ be a lattice in the family of $A$-type. The $m+1$ generators of Jacobi type can be constructed as singular additive lifts:
\begin{equation}
\Grit\left(\Delta \left( \otimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1} \right) \otimes \phi_{-k,A_m,1} \right), \quad \text{for $k=0$ or $2\leq k \leq m+1$}.
\end{equation}
\end{lemma}
\begin{proof}
It follows immediately from Theorem \ref{th:additive} that these lifts are holomorphic away from $\cH_{L, 1}$ and have the necessary leading Fourier--Jacobi coefficients.
\end{proof}
Now we consider the family of $AD$-type lattices. For every index-one generator $f_{D_m}$ of the ring of $W(D_m)$-invariant weak Jacobi forms, the associated meromorphic generator is again a singular theta lift:
\begin{equation}
\Grit\left(\Delta \left( \otimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1} \right) \otimes f_{D_m} \right).
\end{equation}
Unlike the $L_1 = A_m$ case, the ring $J_{*,D_m,*}^{\w,W(D_m)}$ has generators of index $2$. The corresponding meromorphic generators can be constructed as polynomials in singular theta lifts, but to do this directly in all but the simplest cases apparently requires intricate identities among elliptic functions and Jacobi forms. We will construct only the lowest weight generators explicitly and prove the existence of the other generators indirectly from our work on modular forms for the lattice $2U\oplus D_8$ \cite{WW20a}.
\begin{lemma}\label{lem:L1}
Let $L=L_0\oplus D_m$ be a lattice in the family of $AD$-type. There exist weak Jacobi forms of weights $2$ and $4$ and lattice index $L$ with the following $q^0$-terms:
\begin{align}
f_{2,L} &= \sum_{j=1}^m (e^{2\pi i z_j} + e^{-2\pi i z_j}) -2m + O(q) \in J_{2,L,1}^{\w},\\
f_{4,L} &= \sum_{j=1}^m (e^{2\pi i z_j} + e^{-2\pi i z_j}) + O(q) \in J_{4,L,1}^{\w},
\end{align}
where $(z_1,...,z_m)$ is the coordinates of $D_m\otimes \CC$ fixed in \eqref{model of D_n}.
\end{lemma}
\begin{proof}
As in Notation \ref{notation}, let $\phi_{0,D_m,1}$, $\phi_{-2,D_m,1}$ and $\phi_{-4,D_m,1}$ be the index-one generators of the ring of $W(C_m)$-invariant Jacobi forms. Let $E_{4,L_0}$ and $E_{6,L_0}$ be the weak Jacobi forms defined in Lemma \ref{lem:E4E6} (which reduce to the usual Eisenstein series $E_4$ and $E_6$ when $L_0 = \{0\}$). Note that the $q^0$-terms of $\phi_{k,D_m,1}$ are linearly independent and involve only the constant term and the two basic orbits
$$
\sum_{j=1}^m (e^{2\pi i z_j} + e^{-2\pi i z_j}) \quad \text{and} \quad \sum_{v \in \{\pm 1/2\}^m} \prod_{j=1}^m e^{2\pi i v_j z_j},
$$
Therefore, there are $\CC$-linear combinations $f_{2, L}$ of $E_{4,L_0}\phi_{-2,D_m,1}$ and $E_{6,L_0}\phi_{-4,D_m,1}$ and $f_{4, L}$ of $E_{4,L_0}\phi_{0,D_m,1}$, $E_{6,L_0}\phi_{-2,D_m,1}$ and $E_4E_{4,L_0}\phi_{-4,D_m,1}$ with the desired $q^0$-terms.
\end{proof}
\begin{lemma}\label{lem:L2}
Let $L=L_0\oplus D_m$ be a lattice in the family of $AD$-type. There exist meromorphic modular forms of weights $2$ and $4$ for $\widetilde{\Orth}^+(2U\oplus L)$, with double poles on $H(1/2,\varepsilon_m)$ and whose other poles are contained in $\cH_{L,1}$, and whose Fourier--Jacobi expansions are
\begin{align}
\Psi_{2,L}(\tau, \mathfrak{z}, \omega) &=-\frac{3}{\pi^2} \sum_{j=1}^m \wp(\tau,z_j) + O(\xi),\\
\Psi_{4,L}(\tau, \mathfrak{z}, \omega) &= \frac{9}{\pi^4}\sum_{1\leq j_1<j_2\leq m} \wp(\tau,z_{j_1})\wp(\tau,z_{j_2})+ O(\xi).
\end{align}
\end{lemma}
\begin{proof}
The singular additive lifts of $f_{2,L}$ and $f_{4,L}$ have Fourier--Jacobi expansions beginning
\begin{align*}
F_{2,L} &= \frac{1}{(2\pi i)^2}\sum_{j=1}^m \wp(\tau,z_j) + O(\xi),\\
F_{4,L} &= \frac{1}{(2\pi i)^4} \sum_{j=1}^m \wp''(\tau,z_j) + O(\xi),
\end{align*}
and their poles are supported on $\cH_{L,1}\cup H(1/2,\varepsilon_m)$ with multiplicity $2$ and $4$ respectively. Since $\wp(\tau, z) = z^{-2} + O(z^2)$, the Taylor expansions of $F_{2, L}$ and $F_{4, L}$ about the divisor $z_m = 0$ which represents $H(1/2, \varepsilon_m)$ begin
\begin{align*} F_{2, L} &= (2\pi i z_m)^{-2} + f_{2, 2} + O(z_m^2), \\
F_{4, L} &= 6 \cdot (2\pi i z_m)^{-4} + f_{4, 4} + O(z_m^2),
\end{align*}
for some functions $f_{2, 2}$ and $f_{4, 4}$ which are holomorphic near $z_m = 0$.
Clearly we can take $\Psi_{2, L} = 12 F_{2, L}$. The construction \begin{align*}
\Psi_{4, L} = 72 F_{2,L}^2 - 12 F_{4,L} - 120 \Grit(E_{4,L}) = -\frac{36}{\pi^2}f_{2,2} z_m^{-2} + O(z_m^0)\end{align*} therefore has at most a double pole on $z_m = 0$. Using the Weierstrass differential equation $$\wp''(\tau, z) = 6 \wp(\tau, z)^2 - \frac{2}{3}\pi^4 E_4(\tau)$$ we obtain the first Fourier--Jacobi coefficient of $\Psi_{4, L}$.
\end{proof}
\begin{lemma}\label{lem:L3}
Let $L=L_0\oplus D_m$ be a lattice of $AD$-type and let $\kappa = 2(12-\sum_{j=1}^t (m_j+1))$. The generators of Jacobi type of weights $\kappa - 2m + 2$ and $\kappa - 2m + 4$ corresponding to $\phi_{-2(m-1),D_m,2}$ and $\phi_{-2(m-2),D_m,2}$ can be constructed as
\begin{align*}
\Phi_{\kappa-2m+2,L} &:= B_{\kappa/2 -m, L}^2 \Psi_{2,L} = \Delta^2 (\otimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1}^2)\otimes \phi_{-2(m-1),D_m,2}\cdot\xi^2 + O(\xi^3),\\
\Phi_{\kappa-2m+4,L} &:= B_{\kappa/2 -m, L}^2 \Psi_{4,L} = \Delta^2 (\otimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1}^2)\otimes \phi_{-2(m-2),D_m,2}\cdot\xi^2 + O(\xi^3),
\end{align*}
where
$$
B_{\kappa/2-m,L} = \Grit\left[\Delta \cdot \left(\bigotimes_{j=1}^t \phi_{-(m_j+1),A_{m_j},1}\right)\otimes \psi_{-m,D_m,1}\right].
$$
\end{lemma}
\begin{proof}
Recall from \cite{AG20} that the index two generators $\phi_{-2(m-1),D_m,2}$ and $\phi_{-2(m-2),D_m,2}$ of the ring of $W(D_m)$-invariant Jacobi forms can be constructed using the pullback from $mA_1$ to $D_m(2)$ as the symmetric sums
\begin{align*}
\phi_{-2(m-1),D_m,2} = \sum_{\text{sym}} \phi_{-2,A_1,1}^{\otimes^{m-1}} \otimes \phi_{0,A_1,1},\\
\phi_{-2(m-2),D_m,2} = \sum_{\text{sym}} \phi_{-2,A_1,1}^{\otimes^{m-2}} \otimes \phi_{0,A_1,1}^{\otimes^2},
\end{align*}
where $\sum_{\text{sym}} f(z_1,...,z_n) = \frac{1}{n!} \sum_{\sigma \in S_n} f(z_{\sigma(1)},...,z_{\sigma(n)}).$ Note also that the input form in the additive lift to $B_{\kappa/2-m,L}$ is a theta block of $q$-order one, and that $B_{\kappa/2-m,L}$ has simple zeros along $H(1/2,\varepsilon_m)$, cancelling the poles of $\Psi_{2, L}$ and $\Psi_{4, L}$ there. It follows that the forms $\Phi_{-,L}$ are holomorphic away from $\cH_{L,1}$. Their leading Fourier--Jacobi coefficients can be determined using
\[
\psi_{-m,D_m,1}^2(\tau,\mathfrak{z}) = \prod_{j=1}^m \phi_{-2,A_1,1}(\tau,z_j) \quad \text{and} \quad -\frac{3}{\pi^2}\wp(\tau,z) = \frac{\phi_{0,A_1,1}(\tau,z)}{\phi_{-2,A_1,1}(\tau,z)}.
\qedhere \]
\end{proof}
To construct the other generators of Jacobi type, we need the following result for $2U\oplus D_8$:
\begin{lemma}[see Theorem 5.2 and Corollary 4.3 of \cite{WW20a}]\label{lem:D8}
The ring of holomorphic modular forms for $\Orth^+(2U\oplus D_8)$ is freely generated by $11$ additive lifts of Jacobi Eisenstein series. Moreover, for each index two generator $\phi_{k,D_8,2}$ of the ring of $W(C_8)$-invariant Jacobi forms, there exists a modular form for $\Orth^+(2U\oplus D_8)$ of weight $24+k$ with Fourier--Jacobi expansion
$$
G_{24+k,D_8} = \Delta^2 \phi_{k, D_8,2}\cdot \xi^2 + O(\xi^3), \quad k=-6, -8, ..., -12, -14, -16,
$$
\end{lemma}
The $11$ Jacobi Eisenstein series can all be expressed as $\CC[E_4, E_6]$-linear combinations of the index-one generators $\phi_{0,D_8,1}$, $\phi_{-2,D_8,1}$ and $\phi_{-4,D_8,1}$ of the ring of $W(C_8)$-invariant Jacobi forms.
We first consider the cases $L = D_m$ with $9 \leq m \leq 11$. (In particular, $L_0 = \{0\}$.)
\begin{lemma}\label{lem:D11}
For $m=9,10,11$, there exist modular forms, holomorphic away from $\cH_{L,1}$, with Fourier--Jacobi expansion beginning
$$
G_{24+k, D_m} = \Delta^2 \phi_{k, D_m,2}\cdot \xi^2 + O(\xi^3), \quad k=-6, -8, ...,-2(m-2), -2(m-1).
$$
\end{lemma}
\begin{proof}
It is enough to prove the lemma for $L=D_{11}$, as the forms $G_{24+k, D_{10}}$ and $G_{24+k, D_{9}}$ can be constructed by restricting $G_{24+k, D_{11}}$. By Lemma \ref{lem:L3} we can construct $G_{4,D_{11}}=\Phi_{4,D_{11}}$ and $G_{6,D_{11}}=\Phi_{6,D_{11}}$.
For $k\geq -16$, we can write the form $G_{24+k,D_8}$ in Lemma \ref{lem:D8} uniquely as a polynomial in the additive lifts of $\CC[E_4, E_6]$-linear combinations of $\phi_{0, D_8, 1}$, $\phi_{-2, D_8, 1}$, $\phi_{-4, D_8, 1}$: $$G_{24 + k, D_8} = P \left( \mathrm{Grit}( E_4^{a_0} E_6^{b_0} \phi_{0, D_8, 1} + E_4^{a_2} E_6^{b_2} \phi_{-2, D_8, 1} + E_4^{a_4} E_6^{b_4} \phi_{-4, D_8, 1}), \; 4a_i + 6b_i -i = 24+k \right).$$ We lift this to a form $\widehat{G}_{24+k, D_{11}}$ by replacing all instances of $\phi_{-k, D_8, 1}$ with $\phi_{-k, D_{11}, 1}$:
$$\widehat{G}_{24 + k, D_8} = P \left( \mathrm{Grit}( E_4^{a_0} E_6^{b_0} \phi_{0, D_{11}, 1} + E_4^{a_2} E_6^{b_2} \phi_{-2, D_{11}, 1} + E_4^{a_4} E_6^{b_4} \phi_{-4, D_{11}, 1}), \; 4a_i + 6b_i -i = 24+k \right).$$
This has at worst poles on $\cH_{L, 1}$ because that is true for all of the Gritsenko lifts appearing in $P$, and its Fourier--Jacobi expansion begins
$$
\widehat{G}_{24+k,D_{11}} = \Delta^2 \varphi_{k, D_{11},2} \cdot \xi^2 + O(\xi^3),
$$
where $\varphi_{k,D_{11},2} \in J_{k,D_{11},2}^{\w,W(C_{11})}$ has pullback $\phi_{k, D_8, 2}$ to $D_8$. Therefore, $\varphi_{k,D_{11},2} - \phi_{k,D_{11},2}$ is a $\CC[E_4, E_6]$-linear combination of $\phi_{-18,D_{11},2}$, $\phi_{-20,D_{11},2}$ and $\psi_{-11,D_{11},1}^2$. By modifying $\widehat{G}_{24+k, D_8}$ by a polynomial expression in $\cE_{4,D_{11}}, \cE_{6,D_{11}}$ and the forms $\Phi_{4, D_{11}}$, $\Phi_{6, D_{11}}$ and $B_{1, D_{11}}^2$, we obtain the desired generator $G_{24+k,D_{11}}$.
\end{proof}
By now we have constructed all of the generators for the lattices in the $D_n$-tower
$$
D_4 \hookrightarrow D_5 \hookrightarrow D_6 \hookrightarrow D_7 \hookrightarrow D_8 \hookrightarrow D_9 \hookrightarrow D_{10} \hookrightarrow D_{11}.
$$
We now consider the lattices of $AD$-type with $L_0\neq \{0\}$. Recall that $L=L_0\oplus D_m$.
\textbf{Case I.} When $m=4$ or $5$, there are at most two generators of Jacobi type, corresponding to the index-two generators of $J_{*,D_m,2}^{\w,W(D_m)}$, and they have been constructed in Lemma \ref{lem:L3}.
\textbf{Case II.} When $m\geq 6$, we need only consider the following towers:
\begin{align*}
&A_1\oplus D_6 \hookrightarrow A_1\oplus D_7 \hookrightarrow A_1\oplus D_8 \hookrightarrow A_1\oplus D_9, \\
&A_2\oplus D_5 \hookrightarrow A_2\oplus D_6 \hookrightarrow A_2\oplus D_7 \hookrightarrow A_2\oplus D_8, \\
&A_3\oplus D_4 \hookrightarrow A_3\oplus D_5 \hookrightarrow A_3\oplus D_6 \hookrightarrow A_3\oplus D_7, \\
&2A_1\oplus D_4 \hookrightarrow 2A_1\oplus D_5 \hookrightarrow 2A_1\oplus D_6 \hookrightarrow 2A_1\oplus D_7, \\
&A_1\oplus A_2 \oplus D_3 \hookrightarrow A_1\oplus A_2 \oplus D_4 \hookrightarrow A_1\oplus A_2 \oplus D_5 \hookrightarrow A_1\oplus A_2 \oplus D_6,\\
&A_4\oplus D_3 \hookrightarrow A_4\oplus D_4 \hookrightarrow A_4\oplus D_5 \hookrightarrow A_4\oplus D_6.
\end{align*}
For each of the above towers, the leftmost lattice can be embedded into $D_8$, and therefore we obtain the Jacobi type generators corresponding to index-two Jacobi forms (i.e. $\phi_{k,D_m,2}$ for $k=-6, -8, ..., -2m$) as pullbacks of generators for $2U\oplus D_8$. These forms can again be expressed as polynomials in additive lifts whose inputs are $\CC[E_4,E_6]$-linear combinations of the basic weak Jacobi forms of type $L_0 \oplus D_m$. We can construct the desired generators for the other lattices in the tower by the same argument as Lemma \ref{lem:D11}. We remind the reader that the pullbacks of generators of type $D_8$ are not exactly the desired generators for the first lattice of the tower, but we can obtain the generators with prescribed leading Fourier--Jacobi coefficients by modifying these pullbacks in a simple way; for example, the pullback of $G_{10,D_8}$ to $A_1\oplus D_6$ yields the form
$$
\widehat{G}_{10,A_1\oplus D_6} = \Delta^2 (\phi_{-2,A_1,1}^2\otimes \phi_{-10,D_6,2} + 2 (\phi_{-2,A_1,1}\phi_{0,A_1,1})\otimes \psi_{-6,D_6,1}^2)\cdot \xi^2 + O(\xi^3),
$$
and the desired generator $G_{10, A_1 \oplus D_6}$ is then constructed as the modification
\begin{align*}
G_{10,A_1\oplus D_6} & = \widehat{G}_{10,A_1\oplus D_6} - 2\Grit(\Delta\phi_{-2,A_1,1}\otimes \psi_{-6,D_6,1})\Grit(\Delta\phi_{0,A_1,1}\otimes\psi_{-6,D_6,1})\\
& = (\Delta^2 \phi_{-2,A_1,1}^2\otimes \phi_{-10,D_6,2})\cdot \xi^2 + O(\xi^3).
\end{align*}
Theorem \ref{th:2precise} has been proved for the families of $A$-type and $AD$-type. The existence of the generators of Jacobi type for the four $AE$-lattices is harder to prove directly; our argument appears in the next section.
\begin{remark}
If $D_n$ or $E_n$ appears as a component of $L_0$, then $\cH_L$ does not satisfy the Looijenga condition. All such lattices have a sublattice of type $D_4 \oplus A_m$, so we may assume $L_0 = D_4$ and $L_1 = A_m$. Define $\cH_{L,0}=H(1/2,\varepsilon_1)$ and $\cH_L=\cH_{L,0}\cup \cH_{L,1}$. Then
the quotient
$$
\Grit(\Delta\phi_{-4,D_4,1}\otimes \phi_{-(m+1),A_m,1}) / \Grit(\Delta\psi_{-4,D_4,1}\otimes \phi_{-(m+1),A_m,1})
$$
is a non-constant modular form of weight $0$ with poles on $\cH_L$, violating Koecher's principle.
\end{remark}
\begin{remark}
When $L_1$ is not irreducible, the ring of $W(L_1)$-invariant weak Jacobi forms is not a polynomial algebra, so the algebra of modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ with poles on $\cH_L$ is also not free.
\end{remark}
\begin{remark}
We know from \cite{Wan21a} that the Jacobian of generators of a free algebra of holomorphic modular forms is a cusp form. However, the Jacobian of generators of a free algebra of meromorphic modular forms can be non-holomorphic. For example, when $L_0=A_2$ and $L_1=A_1$, by Theorem \ref{th:existence} the corresponding Jacobian has poles of order $1$ along $\cH_{L,0}$.
\end{remark}
\subsection{A free algebra of meromorphic modular forms for the full orthogonal group}
Theorem \ref{th:2precise} shows that the algebra of modular forms for $\widetilde{\Orth}^+(2U\oplus 5A_1)$ with poles on $\cH_{5A_1}=\cH_{5A_1,0}\cup\cH_{5A_1,1}$ is freely generated in weights $2$, $2$, $2$, $2$, $2$, $4$, $4$, $6$. The arrangement $\cH_{5A_1,0}$ is not invariant under $\Orth^+(2U\oplus 5A_1)$; however, $\cH_{5A_1,1}$ is invariant under $\Orth^+(2U\oplus 5A_1)$.
We can prove that the ring of meromorphic modular forms for $\Orth^+(2U\oplus 5A_1)$ with poles on the hyperplane arrangement $\cH_{5A_1,1}$ is a polynomial algebra without relations. This construction is motivated by the free algebras of holomorphic modular forms for the full orthogonal groups of lattices in the tower
\begin{align}\label{eq:nA1}
2U\oplus A_1 \hookrightarrow 2U\oplus 2A_1 \hookrightarrow 2U\oplus 3A_1 \hookrightarrow 2U\oplus 4A_1,
\end{align}
considered by Woitalla in \cite{Woi18}, who proved that the algebra $M_*(\Orth^+(2U\oplus 4A_1))$ at the top is freely generated in weights $4$, $4$, $6$, $6$, $8$, $10$, $12$. The bottom of the tower is the famous Igusa algebra \cite{Igu62} of Siegel modular forms of degree two and even weight which is freely generated in weights $4$, $6$, $10$, $12$. A different interpretation of this tower was given in \cite{WW20a} by associating it to the $B_n$ root system.
Let $5A_1$ be the lattice $\ZZ^5$ with the diagonal Gram matrix $\mathrm{diag}(2,2,2,2,2)$. Then $\cH_{5A_1,1}$ is the Heegner divisor $H(\frac{1}{4}, (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}))$. There is a Borcherds product $\widehat{\Phi}_{5A_1}$ of weight $59$ for the full orthogonal group $\Orth^+(2U\oplus 5A_1)$ whose divisor is
$$
H(1,0) + \sum_{\substack{ v \in 5A_1'/5A_1 \\ v^2 = \frac{1}{2}} } H\left(\frac{1}{4}, v\right) + \sum_{\substack{ u \in 5A_1' /5A_1 \\ u^2 = 1} } H\left(\frac{1}{2}, u\right) + 21 H\left( \frac{1}{4}, \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) \right).
$$
The first three types of divisors correspond to reflections in $\Orth^+(2U\oplus 5A_1)$. Let $\phi_{k,5A_1,1}$ be the generators of $J_{*,5A_1,*}^{\w, \Orth(nA_1)}$, where $k=0, -2, -4, -6, -8, -10$. The additive lifts of $\Delta\phi_{k,5A_1,1}$ together with the forms of weight $4$ and $6$ related to the Eisenstein series $E_4$ and $E_6$ give the generators. Their Jacobian $J$ defines a nonzero modular form of weight $59$. By comparing the divisors we find that $J/\widehat{\Phi}_{59}$ is constant. Then Theorem \ref{th:Jacobiancriterion} implies the following:
\begin{theorem}
The algebra of modular forms for $\Orth^+(2U\oplus 5A_1)$ which are holomorphic away from the Heegner divisor $H(\frac{1}{4}, (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}))$ is freely generated by singular additive lifts of weights $2$, $4$, $4$, $6$, $6$, $8$, $10$, $12$. \end{theorem}
In particular, the $2U \oplus nA_1$-tower \eqref{eq:nA1} of modular forms extends naturally to $n = 5$. Similarly, we obtain free rings of meromorphic modular forms for $\Orth^+(2U\oplus D_m)$ with poles on $\cH_{D_m,1}$ for $m=9$, $10$, $11$. In this case, we need only square the generator of weight $12-m$ related to the theta block $\psi_{-m,D_m,1}$.
\section{Algebras of holomorphic modular forms on reducible root lattices}\label{sec:non-free}
In \cite{WW20a} we used Wirthm\"uller's theorem and the existence of modular forms with certain leading Fourier--Jacobi coefficients to determine the ring structure of some modular forms on irreducible root lattices. The argument proves in particular that all formal Fourier--Jacobi series satisfying a suitable symmetry condition actually define modular forms, and was first used by Aoki \cite{Aok00} to give a new proof of Igusa's structure theorem for $\mathrm{Sp}_4(\mathbb{Z})$, corresponding to the root lattice $A_1$.
We will use a similar approach to determine the algebras of modular forms on the discriminant kernel of some reducible root lattices. Unlike \cite{WW20a}, the algebras appearing here are not free.
Let $M=2U\oplus L$ and $G$ be a subgroup of $\Orth(L)$ containing $\widetilde{\Orth}(L)$. We denote by $\Gamma$ the subgroup generated by $\widetilde{\Orth}^+(M)$ and $G$. For any modular form $F$ of weight $k$ and trivial character for $\Gamma$, we write its Fourier and Fourier--Jacobi expansions as follows:
\begin{align*}
F(\tau,\mathfrak{z},\omega)=\sum_{\substack{n,m\in \NN, \ell\in L' \\2nm-\latt{\ell,\ell}\geq 0}}f(n,\ell,m)q^n\zeta^\ell\xi^m
=\sum_{m=0}^{\infty}\phi_m(\tau,\mathfrak{z})\xi^m,
\end{align*}
where as before $q=\exp(2\pi i\tau)$, $\zeta^\ell=\exp(2\pi i \latt{\ell, \mathfrak{z}})$, $\xi=\exp(2\pi i \omega)$.
The coefficients $\phi_m$ are $G$-invariant holomorphic Jacobi forms of weight $k$ and index $m$ associated to $L$, i.e. $\phi_m\in J_{k,L,m}^{G}$. The invariance of $F$ under the involution $(\tau,\mathfrak{z},\omega) \mapsto (\omega, \mathfrak{z}, \tau)$ yields the symmetry relation
$$
f(n,\ell,m)=f(m,\ell,n),\quad \text{for all} \; (n,\ell,m)\in \NN \oplus L' \oplus \NN,
$$
and further implies the bound (see \cite[\S 3]{WW20a} for details)
\begin{equation}\label{eq:MF-JF}
\dim M_k(\Gamma) \leq \sum_{t=0}^{\infty} \dim J_{k-12t,L,t}^{\w ,G}.
\end{equation}
If there exists a positive constant $\delta<12$ such that
\begin{equation}
J_{k,L,m}^{\w,G}=\{0\} \quad \text{if} \quad k<-\delta m,
\end{equation}
then \eqref{eq:MF-JF} can be improved to the finite upper bound
\begin{equation}
\dim M_k(\Gamma) \leq \sum_{t=0}^{[\frac{k}{12-\delta}]} \dim J_{k-12t,L,t}^{\w ,G},
\end{equation}
where $[x]$ is the integer part of $x$.
\begin{definition}
The subgroup $\Gamma$ will be called \emph{nice} if the inequality \eqref{eq:MF-JF} degenerates into an equality for every weight $k\in\NN$.
\end{definition}
\begin{lemma}\label{lem:nice}
The group $\Gamma$ is nice if and only if, for any weak Jacobi form $\phi_m\in J_{k,L,m}^{\w,G}$, there exists a modular form of weight $k+12m$ for $\Gamma$ whose leading Fourier--Jacobi coefficient is $\Delta^m\phi_m \cdot \xi^m$.
\end{lemma}
This implies in particular that $\Delta^m \phi_m$ is holomorphic for all weak Jacobi forms $\phi_m \in J_{k, L, m}^{\w, G}$.
\begin{proof}
For $r\geq 0$ we define
$$
M_k(\Gamma)(\xi^r)=\{F\in M_k(\Gamma) : F=O(\xi^r)\} \quad \text{and} \quad J_{k,L,m}^{G}(q^r)=\{\phi \in J_{k,L,m}^{G}: \phi=O(q^r) \}
$$
For any $r\geq 0$ we have the following exact sequences
\begin{align*}
&0\longrightarrow M_k(\Gamma)(\xi^{r+1})\longrightarrow M_k(\Gamma)(\xi^r)\stackrel{P_r}\longrightarrow J_{k,L,r}^{G}(q^r), \\
&0\longrightarrow J_{k,L,r}^{G}(q^r)\stackrel{Q_r}\longrightarrow J_{k-12r,L,r}^{\w, G},
\end{align*}
where $P_r$ sends $F$ to its $r^{\text{th}}$ Fourier--Jacobi coefficient $\phi_r$, and $Q_r$ maps $\phi_r$ to $\phi_r / \Delta^r$. Taking dimensions shows that $\Gamma$ is nice if and only if, for any $r\geq 0$, the extended sequences
\begin{align*}
&0\longrightarrow M_k(\Gamma)(\xi^{r+1})\longrightarrow M_k(\Gamma)(\xi^r)\stackrel{P_r}\longrightarrow J_{k,L,r}^{G}(q^r)\longrightarrow 0 , \\
&0\longrightarrow J_{k,L,r}^{G}(q^r)\stackrel{Q_r}\longrightarrow J_{k-12r,L,r}^{\w, G} \longrightarrow 0,
\end{align*}
are exact, and the proof follows.
\end{proof}
A formal series of holomorphic Jacobi forms
$$
\Psi(Z)=\sum_{m=0}^{\infty} \psi_m \xi^m \in \prod_{m=0}^\infty J_{k,L,m}^{G}
$$
is called a \textit{formal Fourier--Jacobi series} of weight $k$ if it satisfies the symmetry condition
$$
f_m(n,\ell)=f_n(m,\ell), \quad \text{for all} \quad m,n\in \NN, \ell \in L',
$$
where $f_m(n,\ell)$ are Fourier coefficients of $\psi_m$. We label the space of formal FJ-series $FM_k(\Gamma)$.
\begin{lemma}\label{lem:modularity}
Assume that $\Gamma$ is nice.
\begin{enumerate}
\item Every formal Fourier--Jacobi series converges on the tube domain $\cH(L)$ and defines a modular form for $\Gamma$. In other words, $$FM_k(\Gamma)=M_k(\Gamma), \quad \text{for all $k\in \NN$}.$$
\item The lattice $L$ satisfies the $\mathrm{Norm}_2$ condition:
$$
\delta_L := \max\{ \min\{\latt{y,y}: y\in L + x \} : x \in L' \} \leq 2.
$$
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The map
$$
M_k(\Gamma) \to FM_k(\Gamma), \quad F\mapsto \text{Fourier--Jacobi expansion of $F$}
$$
is injective. The assumption that $\Gamma$ is nice is precisely the claim that both sides of this map have the same dimension.
(2) Suppose that $\delta_L > 2$. Then there exists a weak Jacobi form $\phi$ whose $q^0$-term contains the nonzero coefficients $\zeta^\ell$ with $\latt{\ell,\ell}>2$. In this case, $\Delta \phi$ is not a holomorphic Jacobi form, which contradicts Lemma \ref{lem:nice}.
\end{proof}
If enough Jacobi forms occur as the leading Fourier--Jacobi coeffcients of modular forms for $\Gamma$, then $\Gamma$ is automatically nice:
\begin{assumption}\label{assum}
We make the following assumptions:
\begin{enumerate}
\item The bigraded ring $J_{*,L,*}^{\w,G}$ is minimally generated over $\CC[E_4,E_6]$ by $N$ generators
$$
\phi_i\in J_{k_i,L,m_i}^{\w,G}, \;1\leq i \leq N.
$$
\item There exist modular forms $\cE_4 \in M_4(\Gamma)$ and $\cE_6\in M_6(\Gamma)$ whose zeroth Fourier--Jacobi coefficients are respectively the Eisenstein series $E_4$ and $E_6$.
\item For each $1\leq i\leq N$, there exists a modular form $\Phi_i\in M_{k_i+12m_i}(\Gamma)$ whose Fourier--Jacobi expansion has the form
$$\Phi_i = (\Delta^{m_i} \phi_i) \xi^{m_i} + O(\xi^{m_i+1}).$$
\end{enumerate}
\end{assumption}
\begin{theorem}\label{th:criterion}
If Assumption \ref{assum} holds, then $\Gamma$ is nice. Moreover, $M_*(\Gamma)$ is minimally generated over $\CC$ by $\cE_4$, $\cE_6$ and $\Phi_i$, $1\leq i \leq N$.
\end{theorem}
\begin{proof}
The proof is similar to that of \cite[Proposition 4.2]{WW20a}. Condition (3) immediately implies that $\Gamma$ is nice, so we only need to prove that it is generated by the claimed forms.
Let $r\geq 1$ and $F_r=\sum_{m=r}^{\infty} f_{m,r} \xi^m \in M_k(\Gamma)(\xi^r)$. For any $m\geq r$, $$f_{m,r}\in J_{k,L,m}^{G}(q^r)$$ implies $f_{r,r}/ \Delta^r\in J_{k-12r,L,r}^{\w,G}$. By Assumption \ref{assum} there exists a polynomial $$P_r\in \CC[E_4,E_6, \phi_i, 1\leq i \leq N]$$ such that $f_{r,r}=\Delta^r P_r(E_4,E_6,\phi_i)$. It follows that
\begin{align*}
&F_{r+1}:=F_r-P_r(\cE_4,\cE_6,\Phi_i)=\sum_{m=r+1}^{\infty} f_{m,r+1} \xi^m \in M_k(\Gamma)(\xi^{r+1}),\\
&f_{m,r+1}\in J_{k,L,m}^G(q^{r+1}), \quad \text{for all $m\geq r+1$}.
\end{align*}
The proof follows by induction over $r$, since $M_k(\Gamma)(\xi^r) = \{0\}$ for $r$ sufficiently large.
We now explain why the set of generators is minimal. Suppose $P$ is a polynomial expression in the generators whose leading Fourier--Jacobi coefficients have index less than some $t \in \mathbb{N}$, and that $P$ itself has leading Fourier--Jacobi coefficient of index $t$. Then this leading coefficient factors as a product of Jacobi forms of index less than $t$ (indeed, all Fourier--Jacobi coefficients of $P$ do). By minimality of the generators of $J_{*, L, *}^{\w, G}$, this leading coefficient cannot be a $\mathbb{C}[E_4,E_6]$-linear combination of the index $t$ weak Jacobi form generators. This implies that none of the generators of $M_*(\Gamma)$ can be written as a polynomial in the others, so the system of generators is minimal.
\end{proof}
Theorem \ref{th:criterion} allows us to determine algebras of holomorphic modular forms for nice groups $\Gamma$. We focus on the most interesting case when $L$ is a root lattice and $\Gamma=\widetilde{\Orth}^+(2U\oplus L)$ (i.e. $G=\widetilde{\Orth}(L)$). We first classify all root lattices such that $\widetilde{\Orth}^+(2U\oplus L)$ are nice.
\begin{enumerate}
\item The existence of $\cE_4\in M_4(\Gamma)$ forces the singular weight to be at most $4$, and therefore $\rk(L)\leq 8$.
\item By Lemma \ref{lem:modularity} (2), the lattice $L$ satisfies $\delta_L \leq 2$.
\item We know from \cite{HU14} that $M_*(\widetilde{\Orth}^+(2U\oplus E_8))$ is a free algebra. However, by \cite[\S 6]{Wan21b} the ring $J_{*,E_8,*}^{\w,W(E_8)}$ is not a polynomial algebra, and the group $\widetilde{\Orth}^+(2U\oplus E_8)$ is not nice because
$$
\dim M_{34}(\widetilde{\Orth}^+(2U\oplus E_8))=12 \quad \text{but} \quad \sum_{t=0}^\infty \dim J_{34-12t,E_8,t}^{\w,W(E_8)} = 13.
$$
\end{enumerate}
The values of $\delta_L$ for irreducible ADE root lattices of rank at most $8$ are listed in Table \ref{tab:delta}.
\begin{remark} Modularity of symmetric formal Fourier--Jacobi series, on the other hand, does not imply that $\Gamma$ is nice; indeed $\Gamma = \widetilde{\Orth}^+(2U\oplus E_8)$ is a counterexample. A symmetric formal Fourier--Jacobi series defines a modular form if and only if it converges, and in particular this condition is preserved by passage to a sublattice; and the convergence of symmetric formal Fourier--Jacobi series for the subgroup $\widetilde{\Orth}^+(2U\oplus D_8)$ was shown in \cite{WW20a}.
\end{remark}
\begin{table}[ht]
\caption{The values of $\delta_L$}\label{tab:delta}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
L & A_1 & A_2 & A_3 & A_4 & A_5 & A_6 & A_7 & A_8 & D_4 & D_5 & D_6 & D_7 & D_8 & E_6 & E_7 & E_8 \\
\hline
\delta_L & \frac{1}{2} & \frac{2}{3} & 1 & \frac{6}{5} & \frac{3}{2} & \frac{12}{7} & 2 & \frac{20}{9} & 1 & \frac{5}{4} & \frac{3}{2} & \frac{7}{4} & 2 & \frac{4}{3} & \frac{3}{2} & 0 \\
\hline
\end{array}
\]
\end{table}
\begin{lemma}\label{lem:lattices}
There are exactly $40$ root lattices satisfying the $\mathrm{Norm}_2$ condition whose irreducible components do not contain $E_8$. They are listed in Table \ref{tab:Norm_2 lattices}.
\end{lemma}
\begin{table}[ht]
\caption{Lattices not of type $E_8$ satisfying the $\mathrm{Norm}_2$ condition}\label{tab:Norm_2 lattices}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|}
\hline
\text{Type} & \text{Lattice} \\ \hline
A & A_1,\; 2A_1,\; 3A_1,\; 4A_1,\; A_2,\; A_3,\; A_4,\; A_5,\; A_6,\; A_7,\; 2A_1\oplus A_2,\; 2A_1\oplus A_3,\; A_1\oplus A_2,\\
& A_1\oplus 2A_2,\; A_1\oplus A_3,\; A_1\oplus A_4,\; A_1\oplus A_5,\; 2A_2,\; 3A_2,\; A_2\oplus A_3,\; A_2\oplus A_4,\; 2A_3 \\ \hline
D & D_4,\; D_5,\; D_6,\; D_7,\; D_8,\; 2D_4 \\ \hline
AD & A_1\oplus D_4,\; A_1\oplus D_5,\; A_1\oplus D_6,\; 2A_1\oplus D_4,\; A_2\oplus D_4,\; A_2\oplus D_5,\; A_3\oplus D_4 \\ \hline
E & E_6,\; E_7 \\ \hline
AE & A_1\oplus E_6,\; A_2\oplus E_6,\; A_1\oplus E_7 \\
\hline
\end{array}
\]
\end{table}
We can now derive a classification of nice groups of type $\widetilde{\Orth}^+(2U\oplus L)$ for root lattices $L$.
\begin{theorem}\label{th:non-free}
Let $L$ be a root lattice. Then $\widetilde{\Orth}^+(2U\oplus L)$ is nice if and only if it is one of the $40$ lattices in Table \ref{tab:Norm_2 lattices}. For any $L$ in Table \ref{tab:Norm_2 lattices}, the ring of modular forms for $\widetilde{\Orth}^+(2U\oplus L)$ is minimally generated in weights $4$, $6$ and $12m+k$, where the pairs $(k, m)$ are the weights and indices of the generators of the ring of $\widetilde{\Orth}(L)$-invariant weak Jacobi forms.
\end{theorem}
In the $14$ cases where $L$ is irreducible, this was proved in \cite{WW20a} and the corresponding ring of modular forms is a free algebra. The algebras in the remaining $26$ cases are not freely generated.
\begin{proof}
By the discussions above, if $\widetilde{\Orth}^+(2U\oplus L)$ is nice then $L$ is a lattice in Lemma \ref{lem:lattices}. To prove the theorem it is enough to verify that every lattice in Lemma \ref{lem:lattices} satisfies Assumption \ref{assum}. Let $L=\oplus_{j=1}^n L_j$ be the decomposition into irreducible root lattices. By Wirthm\"uller's theorem \cite{Wir92}, the ring of $J_{*,L_j,*}^{\w,W(L_j)}$ is a polynomial algebra for $1\leq j\leq n$. Then Remark \ref{rem:sum of Jacobi forms} and the fact that
$$
\widetilde{\Orth}(L) = \bigotimes_{j=1}^n \widetilde{\Orth}(L_j) = \bigotimes_{j=1}^n W(L_j).
$$
together imply that $J_{*,L,*}^{\w, \widetilde{\Orth}(L)}$ is finitely generated by tensor products of generators of $J_{*,L_j,*}^{\w,W(L_j)}$. Moreover, the set of these generators is minimal. Therefore condition (1) in Assumption \ref{assum} is satisfied. Condition (2) follows from Lemma \ref{lem:E4E6}, because $\delta_L\leq 2$ so $\cE_4$ and $\cE_6$ can be constructed as the additive lifts of holomorphic Jacobi forms $E_{4,L}$ and $E_{6,L}$. Only the last condition remains to be verified. When $L$ is of type $A$, as in the previous section, the generators $\phi_i$ of Jacobi forms all have index one, so the generators other than $\cE_4$ and $\cE_6$ can be constructed as the additive lifts $\Delta\phi_i$. For the other cases, the construction is given in the lemmas below.
\end{proof}
We first construct the generators for lattices of type $AD$.
\begin{lemma}
Let $L$ be a lattice of type $AD$ in Table \ref{tab:Norm_2 lattices}. Then Condition $(3)$ in Assumption \ref{assum} holds.
\end{lemma}
\begin{proof}
Let us write $L=A_n\oplus D_m$.
We first construct the orthogonal generators corresponding to index-one generators of Jacobi forms (i.e. $\phi_{k,A_n,1}\otimes \phi_{l,D_m,1}$) using additive lifts. When $m \le 5$, we construct the orthogonal generators corresponding to index-two generators of Jacobi forms (i.e. $(\phi_{k_1,A_n,1}\phi_{k_2,A_n,1})\otimes \phi_{l,D_m,2}$) as the generators of Jacobi type in Lemma \ref{lem:L3}. More precisely, the generators corresponding to $(\phi_{k_1,A_n,1}\phi_{k_2,A_n,1})\otimes \phi_{-2(m-1),D_m,2}$ for $m=4,5$ can be constructed as
$$
\Grit(\Delta\phi_{k_1,A_n,1}\otimes \psi_{-m,D_m,1}) \Grit(\Delta\phi_{k_2,A_n,1}\otimes \psi_{-m,D_m,1}) \Psi_{2,L},
$$
and the generators corresponding to $(\phi_{k_1,A_n,1}\phi_{k_2,A_n,1})\otimes \phi_{-6,D_5,2}$ can be constructed as
$$
\Grit(\Delta\phi_{k_1,A_n,1}\otimes \psi_{-5,D_5,1}) \Grit(\Delta\phi_{k_2,A_n,1}\otimes \psi_{-5,D_5,1}) \Psi_{4,L}.
$$
Recall that $\psi_{-m,D_m,1}$ is the theta block \eqref{eq:theta block D_n}, and $\Psi_{2,L}$ and $\Psi_{4,L}$ are meromorphic modular forms constructed in Lemma \ref{lem:L2}. This construction also works for $L=2A_1\oplus D_4$.
Note that these lattices $L$ can be embedded into $D_8$, so the orthogonal generators corresponding to index-two generators of Jacobi forms can be constructed as pullbacks of generators of type $D_8$. This also yields a construction for $A_1\oplus D_6$. These forms are necessarily holomorphic because $\delta_L\leq 2$ implies $\cH_{L,1}=\emptyset$.
\end{proof}
Among lattices of type $D$, the only new example is $2D_4$. Since $2D_4$ is a sublattice of $D_8$, we can construct the generators similarly to the above lemma. We omit the details.
\begin{lemma}
Let $L$ be a lattice of type $AE$ in Table \ref{tab:Norm_2 lattices}. Then Condition $(3)$ in Assumption \ref{assum} holds.
\end{lemma}
\begin{proof} We may assume that $L = A_2 \oplus E_6$ or $L = A_1 \oplus E_7$, as any other lattices of type $AE$ can be embedded into one of them and the generators in Condition $(3)$ can be obtained by restriction.
It is enough to prove $$\mathrm{dim}\, M_k(\Gamma) \ge \sum_{t=0}^{\infty} \mathrm{dim}\, J_{k - 12t, L, t}^{\w, G}$$ for all $k \le k_0 := \max\{12t_i - k_i\}$, where $(-k_i, t_i)$ are the weights and indices of the generators of $J_{*, L, *}^{\w, G}$, as the reverse inequality is automatically true. We have $k_0 = 24$ if $L = A_2 \oplus E_6$ and $k_0 = 30$ if $L = A_1 \oplus E_7$.
To estimate $\mathrm{dim}\, M_k(\Gamma)$, we use exactly the argument of sections 5.2 and 5.3 of \cite{WW20a}. For any lattice vector $v \in L \backslash \{0\}$, there is a natural ring homomorphism $$P_v : M_k(\Gamma) \longrightarrow M_k(\widetilde{\mathrm{O}}^+(2U\oplus A_1(v^2/2))) \cong M_k(K(v^2/2)),$$ the pullback map, to the space of Siegel paramodular forms of level $v^2/2$ (cf. Section \ref{sec:theta block conjecture}), which is compatible with the Gritsenko lift in the sense that $$P_v(\mathrm{Grit}(\phi)) = \mathrm{Grit}( P_v(\phi))$$ where $P_v$ also denotes the natural pullback of Jacobi forms.
For any modular forms $f_1,...,f_n \in M_k(\Gamma)$, and any vectors $v_1,...,v_n \in L \backslash \{0\}$, we have $$\{\text{linear relations among}\; f_1,...,f_n\} \subseteq \bigcap_{i=1}^n \{\text{linear relations among} \; P_{v_i}(f_1),...,P_{v_i}(f_n)\},$$ so we obtain a lower bound for $\mathrm{dim}\, M_k(\Gamma)$ by computing the Fourier series of paramodular Gritsenko lifts to sufficiently high precision. This is more efficient than working with $M_k(\Gamma)$ directly.
The computation was carried out in SageMath \cite{sagemath} using the package ``WeilRep'' available from \cite{Wil20}. The source code (including our choices of lattice vectors $v_i$) and the results of this computation are available as ancillary material on arXiv.
\end{proof}
\begin{corollary}\label{cor: AE jacobi_type}
The generators of Jacobi type required by Theorem \ref{th:2precise} exist for the lattices of $AE$-type.
\end{corollary}
\begin{proof} Since $\Gamma$ is nice, the existence follows from Lemma \ref{lem:nice}.
\end{proof}
\begin{remark} The Hilbert--Poincar\'e series of $M_*(\widetilde{\Orth}^+(2U\oplus L))$ is $$\sum_{k=0}^{\infty} \mathrm{dim}\, M_k(\widetilde{\Orth}^+(2U\oplus L)) x^k = \sum_{t=0}^{\infty} \sum_{k=0}^{\infty} \mathrm{dim}\, J_{k - 12t, L, t}^{\w, \widetilde{\Orth}(L)} x^k = \sum_{t=0}^{\infty} \sum_{k \in \mathbb{Z}} \mathrm{dim}\, J_{k,L,t}^{\w, \widetilde{\Orth}(L)} x^{k + 12t}.$$ Write $L = L_1 \oplus \cdots \oplus L_n$ where each $L_i$ is irreducible. Then each $L_i$ yields the Hilbert--Poincar\'e series $$F_{L_i}(x, y) = \sum_{t=0}^{\infty} \sum_{k \in \mathbb{Z}} \mathrm{dim}\, J_{k, L_i, t}^{\w, \widetilde{\Orth}(L)} x^k y^t = \frac{1}{(1 - x^4)(1 - x^6) \prod_j (1 - x^{k_j} y^{m_j})}$$ where $(k_j, m_j)$ are the weights and indices of the generators of $J_{*,L_i,*}^{\w,W(L_i)}$ in Table \ref{Tab:Jacobi}. In view of the isomorphism of $\mathbb{C}[E_4,E_6]$-modules $$J_{*, L, t}^{\w, \widetilde{\Orth}(L)} = \bigotimes_{i=1}^n J_{*, L_i,t}^{\w, W(L_i)}$$ it follows that, if we write $$F_{L_i}(x, y) = \frac{1}{(1 - x^4)(1 - x^6)} \sum_{t=0}^{\infty} f_{L_i, t}(x) y^t, \; \; f_{L_i, t} \in \mathbb{C}[x, x^{-1}]$$ then $$F_L(x, y) := \sum_{t=0}^{\infty} \sum_{k \in \mathbb{Z}} \mathrm{dim}\, J_{k, L, t}^{\w,\widetilde{\Orth}(L)} x^k y^t = \frac{1}{(1 - x^4)(1 - x^6)} \sum_{t=0}^{\infty} \Big( \prod_{i=1}^n f_{L_i, t}(x) \Big) y^t$$ and therefore $$
\sum_{k=0}^{\infty} \mathrm{dim}\, M_k(\widetilde{\Orth}^+(2U\oplus L)) x^k = F_L(x, x^{12}).
$$
These series are computed explicitly in \S \ref{sec:tables_hol}.
\end{remark}
We can also consider rescalings of the root lattices.
\begin{theorem}
Let $L$ be a direct sum of rescaled irreducible root lattices, none of which are $E_8$ or $E_8(2)$, and suppose $L$ satisfies the $\mathrm{Norm}_2$ condition. Let $W(L)$ be the direct product of the Weyl groups of all components of $L$. Then the group $\Gamma$ generated by $\widetilde{\Orth}^+(2U\oplus L)$ and $W(L)$ is nice. Moreover, the ring $M_*(\Gamma)$ is minimally generated in weights $4$, $6$ and $12m+k$, where the pairs $(k, m)$ are the weights and indices of the generators of $J_{*,L,*}^{\w, W(L)}$.
\end{theorem}
This applies to the following lattices which do not appear in Table \ref{tab:Norm_2 lattices}:
\begin{align*}
&A_1(2)& &A_1(3)& &A_1(4)& &A_2(2)& \\
&A_2(3)& &A_3(2)& &D_4(2)& &2A_1(2)& \\
&A_1\oplus A_1(2)& &A_2\oplus A_1(2)& &A_3\oplus A_1(2)& &D_4\oplus A_1(2)&\\
&2A_1\oplus A_1(2)& &A_1\oplus A_1(3)& &A_1\oplus A_2(2)& &A_2\oplus A_2(2)&
\end{align*}
The proof is essentially the same as Theorem \ref{th:non-free} so we omit it.
Finally, we consider two examples related to Siegel paramodular forms, which are not covered by the above theorem (because we remove the Weyl invariance). Let $\phi_{0,1}$, $\phi_{-2,1}$ and $\phi_{-1,2}$ be the generators of $J_{*,A_1,*}^{\w}$ as determined by Eichler--Zagier \cite{EZ85}.
\begin{example}
Let $L=A_1(2)$ and $\Gamma=\widetilde{\Orth}(L)$. We have
$$
J_{*,A_1(2),*}^{\w}= \CC[E_4,E_6][\phi_{0,1}^2, \phi_{0,1}\phi_{-2,1}, \phi_{-2,1}^2, \phi_{-1,2}],
$$
where all generators are of index one when viewed as Jacobi forms for $A_1(2)$. We conclude from Theorem \ref{th:criterion} that $M_*(\Gamma)$ is generated by the weight $4, 6, 8, 10, 11, 12$ Gritsenko lifts of the Eisenstein series $E_{4, 2}, E_{6, 2}$ and $\Delta \phi$, $\phi \in \{\phi_{0,1}^2, \phi_{0,1}\phi_{-2,1}, \phi_{-2,1}^2, \phi_{-1,2}\}$. This corresponds to the ring of paramodular forms of degree $2$ and level $2$ which was determined in \cite{IO97}.
\end{example}
\begin{example}
Let $L=A_1(3)$ and $\Gamma=\widetilde{\Orth}(L)$. We have
$$
J_{*,A_1(3),*}^{\w}= \CC[E_4,E_6][\phi_{0,1}^3, \phi_{0,1}^2\phi_{-2,1}, \phi_{0,1}\phi_{-2,1}^2, \phi_{-2,1}^3, \phi_{0,1}\phi_{-1,2}, \phi_{-2,1}\phi_{-1,2}].
$$
All generators are of index one as Jacobi forms for $A_1(3)$. We conclude from Theorem \ref{th:criterion} that $M_*(\Gamma)$ is generated by the weight $4, 6, 6, 8, 9, 10, 11, 12$ Gritsenko lifts of the Eisenstein series $E_{4, 3}, E_{6, 3}$ and $\Delta \phi$, where $\phi$ is one of the above generators of $J_{*, A_1(3), *}^{\w}$. This corresponds to the ring of paramodular forms of degree $2$ and level $3$ which was determined in \cite{Der02}.
\end{example}
\section{Meromorphic modular forms on complex balls}\label{sec:ball quotients}
In this section we establish the modular Jacobian approach for meromorphic modular forms on complex balls attached to unitary groups of signature $(l,1)$ whose poles are supported on hyperplane arrangements. We also apply this criterion to root lattices with complex multiplication to construct free algebras of unitary modular forms with poles on hyperplane arrangements.
\subsection{The Looijenga compactification of ball quotients}
Following \cite[\S 2]{WW21a} we define modular forms with respect to unitary groups of Hermitian lattices. Let $d$ be a square-free negative integer. Then $\F=\QQ(\sqrt{d})$ is an imaginary quadratic field with ring of integers $\mathcal{O}_\F$. Let $D_\F$ denote the discriminant of $\F$, such that $$
\mathcal{O}_\F=\ZZ+\ZZ\cdot \zeta, \; \text{where} \; \zeta=(D_{\F}+\sqrt{D_\F})/2.
$$
Let $\cD_\F^{-1}$ denote the inverse different
$$
\cD_\F^{-1} = \{x \in \F: \; \mathrm{Tr}_{\F / \QQ}(xy) \in \mathbb{Z} \; \text{for all} \; y \in \mathcal{O}_\F\} = \frac{1}{\sqrt{D_{\F}}} \mathcal{O}_\F.
$$
A \emph{Hermitian lattice} $M$ is a free $\mathcal{O}_{\F}$-module equipped with a non-degenerate form
$$
h(-,-): M \times M \longrightarrow \F
$$
which is linear in the first component and conjugate-linear in the second.
We call $M$ \emph{even} if $h(x,x)\in \ZZ$ for all $x\in M$. We define the dual of $M$ by
$$
M' = \{x \in M \otimes_{\mathcal{O}_{\F}} \F: \; h(x,y) \in \cD_\F^{-1} \;\text{for all} \; y \in M\}.
$$
We view a Hermitian lattice $M$ as a usual $\ZZ$-lattice (denoted $M_\ZZ$) equipped with the following bilinear form induced by $h(-,-)$
$$
(-, -) := \mathrm{Tr}_{\F/\QQ} h(-,-) : M \otimes M \longrightarrow \mathbb{Q}.
$$
Under this identification, $M$ is even if and only if $M_\ZZ$ is even, and the $\mathcal{O}_\F$-dual $M'$ coincides with the $\ZZ$-dual $M_\ZZ'$.
Now assume that $M$ is an even Hermitian lattice of signature $(l, 1)$ with $l\geq 2$. We equip the complex vector space $V_{\U}(M) := M \otimes_{\mathcal{O}_{\F}} \CC$ with the $\QQ$-structure defined by $M$ and the sesquilinear form induced by $h(-,-)$. Then the unitary group $\U(V_{\U}(M))\cong \U(l,1)$ is an algebraic group defined over $\QQ$. The Hermitian symmetric domain attached to $M$ is the Grassmannian of negative-definite lines
$$
\cD_{\U}(M) = \{[z] \in \PP(V_{\U}(M)): \; h(z, z) < 0\},
$$ on which $\U(l,1)$ acts by multiplication, and it is biholomorphic to the complex ball of dimension $l$. The quotients $\cD_{\U}(M) / \Gamma$ by arithmetic subgroups $\Gamma \le \U(l,1)$ are usually called \emph{ball quotients}.
Let $\cA_{\U}(M)$ be the principal $\CC^\times$-bundle
$$
\cA_{\U}(M) = \{z \in V_{\U}(M): \; [z] \in \cD_{\U}(M)\}.
$$
Modular groups $\Gamma$ will be finite-index subgroups of
$$
\U(M) := \{\gamma \in \U(l,1):\; \gamma M = M\}.
$$
The most important example is the \emph{discriminant kernel},
$$
\widetilde{\U}(M) = \{\gamma \in \U(M): \; \gamma x - x \in M \; \text{for all} \; x \in M'\}.
$$
For any vector $v\in M'$ satisfying $h(v,v)>0$, the associated hyperplane $v^\perp$ is the set of vectors of $\cD_{\U}(M)$ orthogonal to $v$ and it defines a complex ball of dimension $l-1$. Following Looijenga's work \cite{Loo03a}, an arrangement $\cH_{\U}$ of hyperplanes is a finite union of $\Gamma$-orbits of hyperplanes in $\cD_{\U}(M)$. We say that $\cH_{\U}$ satisfies the \emph{Looijenga condition} if every one-dimensional intersection in $V_{\U}(M)$ of hyperplanes from $\cH_{\U}$ is positive definite. This condition guarantees that the analogue of Koecher's principle holds for the meromorphic modular forms defined below.
\begin{definition}
Let $k\in\ZZ$ and $\Gamma$ be a finite-index subgroup of $\U(M)$. A modular form for $\Gamma$ of weight $k$ and character $\chi$ with poles on $\cH_{\U}$ is a meromorphic function $F: \cA_{\U} \to \CC$ which is holomorphic away from $\cH_{\U}$ and satisfies
\begin{align*}
F(tz)&=t^{-k}F(z), \quad \text{for all $t\in \CC^\times$},\\
F(\gamma z)&=\chi(\gamma) F(z), \quad \text{for all $\gamma\in \Gamma$.}
\end{align*}
\end{definition}
Assume that $\cH_{\U}$ satisfies the Looijenga condition. By \cite[Corollary 5.8]{Loo03a}, the ring $M_*^!(\Gamma)$ of modular forms for $\Gamma$ of integral weight and trivial character with poles on $\cH_{\U}$ is finitely generated over $\CC$ by forms of positive weight. Moreover, the $\mathrm{Proj}$ of $M_*^!(\Gamma)$ gives the Looijenga compactification of the complement of $\cH_{\U}$ in $\cD_{\U}(M)$ with respect to $\Gamma$. It is similar to the orthogonal case that the boundary components of the Looijenga compactification have codimension at least two. When the arrangement $\cH_{\U}$ is empty, the Looijenga compactification coincides with the Baily--Borel compactification of $\cD_{\U}(M) / \Gamma$ which is obtained by adding finitely many $0$-dimensional cusps.
\subsection{The Jacobian of unitary modular forms}
Reflections are automorphisms of finite order whose fixed point set has codimension one. For any primitive vector $r\in M$ satisfying $h(r,r)> 0$ and $\alpha \in \mathcal{O}_\F^\times \setminus \{1\}$, the reflection associated to $r$ and $\alpha$ is
$$
\sigma_{r, \alpha}: x \mapsto x -(1-\alpha) \frac{ h(x, r)}{h(r,r)} r .
$$
The hyperplane $r^\perp$ is the fixed point set of $\sigma_{r,\alpha}$ and is called the \textit{mirror} of $\sigma_{r, \alpha}$. Note that $\sigma_{r, \alpha} \sigma_{r, \beta}$ equals $\sigma_{r, \alpha \beta}$ or the identity when $\alpha \beta = 1$, and therefore $$
\mathrm{ord}(\sigma_{r, \alpha}) = \mathrm{ord}(\alpha) \in \{2, 3, 4, 6\},
$$
where $\mathrm{ord}(\alpha)$ is the order of $\alpha$ in $\mathcal{O}_\F^{\times}$. We refer to \cite[\S 2.3]{WW21a} for a full description of the reflections contained in $\U(M)$ and $\widetilde{\U}(M)$.
For $1 \leq j \leq l+1$, let $F_j$ be a modular form for $\Gamma$ of weight $k_j$ and trivial character with poles on $\cH_{\U}$. View $F_j$ as meromorphic functions defined on the affine cone $\cA_{\U}(M) \subseteq \CC^{l, 1}$. With respect to the natural coordinates $(z_1,...,z_{l+1})$ on $\CC^{l, 1}$, the Jacobian determinant of $(F_1,...,F_{l+1})$ is defined in the usual way:
$$
J_{\U}:=J_{\U}(F_1, ...,F_{l+1})=\frac{\partial (F_1, F_2, ..., F_{l+1})}{\partial (z_1, z_2, ..., z_{l+1})}.
$$
Similar to \cite[Theorem 3.1]{WW21a}, the Jacobian $J_{\U}$ satisfies the following properties.
\begin{enumerate}
\item $J_{\U}$ is a modular form for $\Gamma$ of weight $l+1 + \sum_{j=1}^{l+1}k_j$ and character $\det^{-1}$ with poles on $\cH_{\U}$, where $\det$ is the determinant character.
\item $J_{\U}$ is not identically zero if and only if these $F_j$ are algebraically independent over $\CC$.
\item Let $r\in M$ and $\alpha \in \mathcal{O}_\F^\times \setminus \{1\}$. If $\sigma_{r,\alpha}\in \Gamma$ and $r^\perp$ is not contained in $\cH_{\U}$, then the vanishing order of $J_{\U}$ on $r^\perp$ satisfies
$$
\mathrm{ord}(J_{\U}, r^\perp) \equiv -1 \, \mathrm{mod}\, \ord(\alpha).
$$
\item If we view $F_j$ as functions of $\tau, z_1,...,z_{l-1}$ on the Siegel domain $\cH_{\U}$ attached to a zero-dimensional cusp (i.e. a rational isotropic line), then the Jacobian takes the form
$$
\left\lvert \begin{array}{cccc}
k_1F_1 & k_2F_2 & \cdots & k_{l+1}F_{l+1} \\
\partial_\tau F_1 & \partial_\tau F_2 & \cdots & \partial_\tau F_{l+1} \\
\partial_{z_1} F_1 & \partial_{z_1} F_2 & \cdots & \partial_{z_1} F_{l+1} \\
\vdots & \vdots & \ddots & \vdots \\
\partial_{z_{l-1}} F_{1} & \partial_{z_{l-1}} F_{2} & \cdots & \partial_{z_{l-1}} F_{l+1}
\end{array} \right\rvert.
$$
\end{enumerate}
As analogues of Theorem \ref{th:freeJacobian}, Theorem \ref{th:Jacobiancriterion} and \cite[Theorem 3.3, Theorem 3.4]{WW21a}, we will describe the link between the Jacobian and free algebras of meromorphic modular forms.
\begin{theorem}\label{th:unitaryfreeJacobian}
Let $\cH_{\U}$ be an arrangement of hyperplanes satisfying the Looijenga condition. Suppose that the ring of unitary modular forms for $\Gamma$ with poles on $\cH_{\U}$ is freely generated by forms $F_j$ of weight $k_j$ for $1\leq j \leq l+1$. Then:
\begin{enumerate}
\item $\Gamma$ is generated by reflections.
\item $J_{\U}=J_{\U}(F_1,...,F_{l+1})$ is a nonzero modular form for $\Gamma$ of weight $l+1 + \sum_{j=1}^{l+1}k_j$ and character $\det^{-1}$ which satisfies:
\begin{enumerate}
\item If a mirror $r^\perp$ of reflections in $\Gamma$ is not contained in $\cH_{\U}$, then $J_{\U}$ vanishes on $r^\perp$ with multiplicity
$$
\mathrm{ord}(J_{\U}, r^{\perp}) = -1 + \max\{\mathrm{ord}(\alpha): \; \sigma_{r, \alpha} \in \Gamma\}.
$$
\item The other zeros and poles of $J_{\U}$ are contained in $\cH_{\U}$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
This theorem can be proved in a similar way to Theorem \ref{th:freeJacobian} and \cite[Theorem 3.3]{WW21a}. The essential fact is that the boundary components of the Loojenga compactification of the arrangement complement $(\cD_{\U}(M)- \cH_{\U}) / \Gamma$ have codimension at least two.
\end{proof}
\begin{theorem}\label{th:unitaryJacobiancriterion}
Let $\cH_{\U}$ be an arrangement of hyperplanes satisfying the Looijenga condition. Suppose that there exist $l+1$ algebraically independent modular forms of trivial character for $\Gamma$ with poles on $\cH_{\U}$, and suppose the restriction of their Jacobian to the complement $\cD_{\U}(M) - \cH_{\U}$ vanishes precisely on all mirrors of reflections in $\Gamma$ with multiplicity $m-1$, where $m$ is the maximal order of the reflections in $\Gamma$ through that mirror. Then these forms freely generate the ring of unitary modular forms for $\Gamma$ with poles on $\cH_{\U}$. In particular, $\Gamma$ is generated by reflections.
\end{theorem}
\begin{proof}
The proof is an adjustment to the proofs of Theorem \ref{th:Jacobiancriterion} and \cite[Theorem 3.4]{WW21a}.
\end{proof}
\subsection{Free algebras of meromorphic modular forms on complex balls}
Let $M$ be an even Hermitian lattice of signature $(l, 1)$. There are natural embeddings $$\U(M) \hookrightarrow \Orth^+(M_{\ZZ}) \quad \text{and} \quad \widetilde{\U}(M) \hookrightarrow \widetilde{\Orth}^+(M_{\ZZ}),$$ and modular forms for $\Orth^+(M_{\ZZ})$ can be pulled back to modular forms for $\U(M)$ by restricting to those lines in $\cD(M)$ which are preserved by the complex structure on $M$; for more detail see \cite[\S 2.4]{WW21a}. In \cite[\S 4]{WW21a} we used the relationship between the Jacobians of orthogonal and unitary modular forms to show that free algebras of modular forms for $\Gamma \le \Orth^+(M_{\ZZ})$ often yield free algebras of modular forms for $\Gamma \cap \U(M)$. This result naturally extends to modular forms with poles:
\begin{theorem}\label{th:twins}
Let $M$ be an even Hermitian lattice of signature $(l,1)$ over $\F=\QQ(\sqrt{-1})$ or $\QQ(\sqrt{-3})$ with $l\geq 2$. Let $\cH$ be a hyperplane arrangement on $\cD(M_{\ZZ})$ satisfying the Looijenga condition. Let $\cH_{\U}$ denote the hyperplane arrangement obtained as the restriction of $\cH$ to $\cD_{\U}(M)$, and assume that $\cH_{\U}$ also satisfies the Looijenga condition.
Suppose that the ring of modular forms for $\widetilde{\Orth}^+(M_{\ZZ})$ with poles on $\cH$ is freely generated by $2l+1$ orthogonal modular forms, $l$ of whose restrictions to $\cD_{\U}(M)$ are identically zero. Then the ring of modular forms for $\widetilde{\U}(M)$ with poles on $\cH_{\U}$ is freely generated by the restrictions to $\cD_{\U}(M)$ of the remaining $l+1$ generators of orthogonal type.
\end{theorem}
\begin{proof}
For simplicity we label $\cD^\circ=\cD(M_{\ZZ}) - \cH$ and $\cD_{\U}^\circ=\cD_{\U}(M) - \cH_{\U}$.
Let $F_j$, $0 \le j \le 2l$, be the $2l+1$ generators for $M_*(\widetilde{\Orth}^+(M_{\ZZ}))$, and let $f_j$ be their restrictions to $\cD_{\U}(M)$. Without loss of generality, suppose $f_j = 0$ for $j \geq l+1$. Let $J_{\Orth}$ be the Jacobian of $F_0,...,F_{2l}$, and let $J_{\U}$ be the Jacobian of $f_0,...,f_l$. Note that $J_{\Orth}$ has its zero locus supported on Heegner divisors, all of which intersect $\cD_{\U}(M)$ transversally, so its restriction $\widehat{J}_{\Orth}$ to $\cD_{\U}(M)$ does not vanish identically.
By \cite[Proposition 4.1]{WW21a} and its natural analogue to modular forms with poles, there is a meromorphic unitary modular form $g$, holomorphic away from $\cH_{\U}$, such that $\widehat{J}_{\Orth}=g J_{\U}$. It follows that $J_{\U}$ is also not identically zero. According to \cite[Lemma 2.2]{WW21a}, the restrictions of mirrors of reflections in $\widetilde{\Orth}^+(M_{\ZZ})$ to $\cD_{\U}(M)$ are exactly the mirrors of reflections in $\widetilde{\U}(M)$. By Theorem \ref{th:2precise}, on the complement $\cD^\circ$, the orthogonal Jacobian $J_{\Orth}$ vanishes precisely with multiplicity one on mirrors of reflections in $\widetilde{\Orth}^+(M_{\ZZ})$. As a factor of $\widehat{J}_{\Orth}$, the unitary Jacobian $J_{\U}$ also vanishes only on mirrors of reflections in $\widetilde{\U}(M)$ on $\cD_{\U}^\circ$.
By considering its character, we see that $J_{\U}$ must vanish on every mirror of a reflection in $\widetilde{\U}(M)$ which is not contained in $\cH_{\U}$. Therefore $J_{\U}$ vanishes precisely on the mirrors of reflections in $\widetilde{\U}(M)$ in $\cD_{\U}^\circ$. To apply Theorem \ref{th:unitaryJacobiancriterion}, we must show that the order of vanishing of $J_{\U}$ on any mirror is exactly $1$ when $d=-1$ and $2$ when $d=-3$. This is true because the restriction $\widehat{J}_{\Orth}$ of $J_{\Orth}$ vanishes with multiplicity at most $$N := |\mathcal{O}_\F^\times / \{\pm 1\}| = \begin{cases}2 : & d = -1; \\ 3: & d = -3; \end{cases},$$ so its factor $J_{\U}$ vanishes to at most that multiplicity; and because $\ord(J_{\U}) \equiv -1$ mod $N$.
\end{proof}
Note that the Looijenga condition for $\cH$ does not imply the Looijenga condition for $\cH_{\U}$ in general. Two examples where this fails are $L = A_1 \oplus A_1$ and $L = 3A_1 \oplus A_1$, viewed as Gaussian lattices, where $\cH_L$ is defined in Theorem \ref{th:2precise}.
The above proof also shows that, up to constant multiple, $J_{\U} = \widehat{J}_{\Orth}^{1/2}$ if $d=-1$ and $J_{\U} = \widehat{J}_{\Orth}^{2/3}$ if $d=-3$.
As in \cite[Theorem 4.2]{WW21a}, the above theorem holds when we replace $\widetilde{\Orth}^+(M_{\ZZ})$ and $\widetilde{\U}(M)$ with $\Orth(M_{\ZZ})$ and $\U(M)$ and the proof is similar. We cannot extend this theorem to other discriminants, because when $d \notin \{-1, -3\}$ the restriction of a reflection in $\Orth^+(M_{\ZZ})$ is not necessarily a reflection in $\U(M)$.
The following lemma is a convenient way to prove that certain modular forms vanish along $\cD_{\U}(M)$.
\begin{lemma}\label{lem:zero}
Let $M=H\oplus L$ be an even Hermitian lattice of signature $(l,1)$ over $\F = \QQ(\sqrt{d})$ split by a complex line $H$ satisfying $H_{\ZZ}=U$. We write $L=\oplus_{j=1}^n L_j$. Let $\Gamma$ denote the subgroup generated by $\widetilde{\Orth}^+(M_{\ZZ})$ and these $\Orth((L_j)_{\ZZ})$.
Suppose that $F$ is a nonzero meromorphic modular form of integral weight $k$ and trivial character for $\Gamma$. Let $f$ denote the restriction of $F$ to $\cD_{\U}(M)$.
\begin{enumerate}
\item When $d=-1$, if $k$ is not a multiple of $4$, then $f$ is identically zero.
\item When $d=-3$, if $k$ is not a multiple of $6$, then $f$ is identically zero.
\end{enumerate}
\end{lemma}
\begin{proof}
For any unit $\alpha \in \mathcal{O}_\F^\times$, the map $\alpha_{L_j}: z \mapsto \alpha\cdot z$ lies in $\U(L_j)$, and $\alpha_H : z \mapsto \alpha\cdot z$ lies in $\U(H) = \widetilde{\U}(H)$. Therefore $\Gamma \cap \U(M)$ contains the automorphism $\alpha_M : z \mapsto \alpha\cdot z$, which can be viewed as the composition of $\alpha_H$ and these $\alpha_{L_j}$. Suppose $\alpha$ has order $a$. Then
$$
f(z)=f(\alpha \cdot z)=\alpha^{-k}f(z),
$$
which implies that $a|k$ if $f$ is nonzero.
\end{proof}
Some of the root lattices in Theorem \ref{th:2precise} have complex multiplication over $\QQ(\sqrt{-1})$ or $\QQ(\sqrt{-3})$. We obtain the following result by applying Theorem \ref{th:twins} to them.
\begin{theorem}\label{th:algebras-unitary}
\noindent
Let $M$ be a Gaussian lattice whose associated $\ZZ$-lattice is $M_{\ZZ}=2U\oplus L$, where $L$ is $2A_1\oplus D_4$, $2A_1\oplus D_6$ or $D_{10}$, or an Eisenstein lattice whose associated $\ZZ$-lattice is $M_{\ZZ}=2U\oplus L$, where $L$ is $A_2\oplus A_2$, $2A_2\oplus A_2$, $A_2\oplus D_4$, $2A_2\oplus D_4$ or $A_2\oplus E_6$. Let $\cH_L$ be the hyperplane arrangement for $M_{\ZZ}$ defined in Theorem \ref{th:2precise} and let $\cH_{\U}$ be the restriction of $\cH_L$ to $\cD_{\U}(M)$. Then $\cH_{\U}$ satisfies the Looijenga condition and the ring of modular forms for $\widetilde{\U}(M)$ with poles on $\cH_{\U}$ is freely generated.
\end{theorem}
The weights of the generators of the twin free algebras $M_*^!(\widetilde{\Orth}^+(2U \oplus L))$ and $M_*^!(\widetilde{\U}(M))$ are listed in Tables \ref{tab:-1} and \ref{tab:-3} below.
\begin{proof}
It is easy to verify that the Looijenga condition is satisfied for all $\cH_{\U}$ by Lemma \ref{lem:intersection} and its unitary analogue. To apply Theorem \ref{th:twins} we need to argue that there are $l$ generators of orthogonal type whose restrictions to $\cD_{\U}(M)$ are identically zero. (Recall that the signature of $M_{\ZZ}$ is $(2l,2)$, so $l = \rk(L) / 2 - 1$.)
We will prove the existence of these generators for two of the harder cases; the other cases are similar and we omit the details.
(1) $L=2A_1\oplus D_6$ as a Gaussian lattice. As explained in Notation \ref{notation}, we can choose the generators of the ring of $W(D_6)$-invariant weak Jacobi forms as follows: (i) $\phi_{k,D_6,2}$ are invariant under $\Orth(D_6)$ for $k=-6, -8, -10$; (ii) $\phi_{k,D_6,1}$ are invariant under $\Orth(D_6)$ for $k=0, -2, -4$. (iii) $\psi_{-6,D_6,1}$ is invariant under $W(D_6)$ and anti-invariant under the odd sign change. Therefore, we can choose the generators of the ring of modular forms for $\widetilde{\Orth}^+(M_{\ZZ})$ with poles on $\cH_L$ in the following way.
\begin{itemize}
\item[(a)] The two generators of Eisenstein type are invariant under $\Orth(2A_1)\otimes\Orth(D_6)$. By Lemma \ref{lem:zero}, the generator of Eisenstein type of weight $6$ restricts to zero.
\item[(b)] There are two generators of abelian type, both of weight $2$. Up to scalar there is a unique linear combination of them which is invariant under $\Orth(2A_1)\otimes\Orth(D_6)$, and whose restriction must be zero by Lemma \ref{lem:zero}.
\item[(c)] There are six generators of Jacobi type related to the six basic $\Orth(D_6)$-invariant weak Jacobi forms. They have weights $8$, $6$, $4$, $10$, $8$, $6$. By Lemma \ref{lem:zero}, the restrictions of three generators of weight $6$, $10$ and $6$ are zero.
\end{itemize}
We have found 5 generators of $M^!_*(\widetilde{\Orth}^+(M_{\ZZ}))$ whose restrictions are identically zero, so we can apply Theorem \ref{th:twins}.
\vspace{3mm}
(2) $L=2A_2\oplus D_4$ as an Eisenstein lattice. This case is more subtle. Recall that the ring of $W(A_2)$-invariant weak Jacobi forms has generators $\phi_{0,A_2,1}$, $\phi_{-2,A_2,1}$ and $\phi_{-3,A_2,1}$. Note that $\Orth(A_2)$ is generated by $W(A_2)$ and the sign change $\mathfrak{z}\mapsto -\mathfrak{z}$, so the even weight generators are invariant under $\Orth(A_2)$, but the odd weight generator is anti-invariant under the sign change. The $W(D_4)$-invariant weak Jacobi forms have five generators: $\phi_{0,D_4,1}$, $\phi_{-2,D_4,1}$, $\phi_{-4,D_4,1}$, $\psi_{-4,D_4,1}$ and $\phi_{-6,D_4,2}$. The full orthogonal group of $D_4$ is the Weyl group of the $F_4$ root system and its ring of weak Jacobi forms is generated in weights and indices $(0,1)$, $(-2,1)$, $(-6,2)$, $(-8,2)$, $(-12,3)$; we can take $\phi_{0,D_4,1}$, $\phi_{-2,D_4,1}$ and $\phi_{-6,D_4,2}$ to be $\Orth(D_4)$-invariant. Similar to the above case, we can choose the generators of the ring of modular forms for $\widetilde{\Orth}^+(M_{\ZZ})$ with poles on $\cH_L$ in the following way.
\begin{itemize}
\item[(a)] The two generators of Eisenstein type are invariant under $\Orth(2A_2)\otimes\Orth(D_4)$. Lemma \ref{lem:zero} shows that the weight $4$ generator restricts to zero.
\item[(b)] There are four generators of abelian type of weights $1,1,3,3$. The two generators of weight $1$ are labelled $F_1$ and $G_1$, and they are mapped into each other by the element $\sigma \in \Orth(L)$ which swaps the two copies of $A_2$. Then $F_1^2+G_1^2$ and $F_1^2G_1^2$ are meromorphic modular forms of weight $2$ and $4$, invariant under $\Orth(2A_2)\otimes\Orth(D_4)$, whose restrictions have to be zero by Lemma \ref{lem:zero}. It follows that the restrictions of $F_1$ and $G_1$ themselves are identically zero.
\item[(c)] There are five generators of Jacobi type, corresponding to the five Jacobi forms $\phi_{0,D_4,1}$, $\phi_{-2,D_4,1}$, $\phi_{-4,D_4,1}$, $\psi_{-4,D_4,1}$ and $\phi_{-6,D_4,2}$, and we label them $F_{6}$, $F_{4}$, $F_{2}$, $G_{2}$ and $G_{6}$ respectively, where the subscript indicates the weight. Our choice of the generators guarantees that $F_6^2$, $F_4^2$ and $G_6$ are invariant under $\Orth(2A_2)\otimes\Orth(D_4)$, which implies that the restriction of $F_4$ is zero. By \cite[\S 4.1]{Adl20}, the basic $\Orth(D_4)$-invariant weak Jacobi form of weight $-8$ and index $2$ can be expressed as $\phi_{-4,D_4,1}^2 -\psi_{-4,D_4,1}^2$ if we choose $\phi_{-4,D_4,1}$ and $\psi_{-4,D_4,1}$ in a suitable way. Then $F_2^2 - G_2^2$ is a modular form of weight $4$, which is invariant under $\Orth(2A_2)\otimes\Orth(D_4)$, and therefore restricts to zero. It follows that the restriction of one of $F_2+G_2$ or $F_2-G_2$ is identically zero.
\end{itemize}
As before, having found $5$ generators of $M^!_*(\widetilde{\Orth}^+(M_{\ZZ}))$ whose restrictions are identically zero, we can now apply Theorem \ref{th:twins}.
\end{proof}
\begin{table}[ht]
\caption{Free algebras of modular forms for $\widetilde{\U}(M)$ with poles on $\cH_{\U}$ over $\QQ(\sqrt{-1})$}\label{tab:-1}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|c|}
\hline
L & \text{weights of generators of $M_*^!(\widetilde{\Orth}^+(M_\ZZ))$} & \text{weights of generators of $M_*^!(\widetilde{\U}(M))$} \\
\hline
2A_1\oplus D_4 & 2, 2, 4, 4, 4, 6, 6, 8, 10 & 2, 4, 4, 4, 8 \\
\hline
2A_1\oplus D_6 & 2, 2, 2, 4, 4, 6, 6, 6, 8, 8, 10 & 2, 2, 4, 4, 8, 8\\
\hline
D_{10} & 2, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 16, 18 & 2, 4, 8, 8, 12, 12, 16\\
\hline
\end{array}
\]
\end{table}
\begin{table}[ht]
\caption{Free algebras of modular forms for $\widetilde{\U}(M)$ with poles on $\cH_{\U}$ over $\QQ(\sqrt{-3})$}\label{tab:-3}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|c|}
\hline
L & \text{weights of generators of $M_*^!(\widetilde{\Orth}^+(M_\ZZ))$} & \text{weights of generators of $M_*^!(\widetilde{\U}(M))$} \\
\hline
A_2\oplus A_2 & 1, 3, 4, 6, 6, 7, 9 & 3, 6, 6, 9 \\
\hline
2A_2\oplus A_2 & 1, 1, 3, 3, 3, 4, 4, 6, 6 & 3, 3, 3, 6, 6 \\
\hline
A_2\oplus D_4 & 1, 3, 4, 5, 5, 6, 7, 9, 12 & 3, 5, 6, 9, 12 \\
\hline
2A_2\oplus D_4 & 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 6 & 2, 3, 3, 6, 6, 6 \\
\hline
A_2\oplus E_6 & 1, 3, 4, 4, 6, 7, 9, 9, 10, 12, 15 & 3, 6, 9, 9, 12, 15\\
\hline
\end{array}
\]
\end{table}
\begin{remark}
As a direct consequence of Theorem \ref{th:algebras-unitary}, the modular groups $\widetilde{\U}(M)$ related to these free algebras are generated by reflections. Thus they provide explicit examples of finite-covolume reflection groups acting on complex hyperbolic spaces (see \cite{All00a}).
\end{remark}
\bigskip
\noindent
\textbf{Acknowledgements}
H. Wang thanks Max Planck Institute for Mathematics (MPIM Bonn) for its hospitality where part of the work was done, and thanks Zhiwei Zheng for helpful discussions on the Looijenga compactification. H. Wang was supported by the Institute for Basic Science (IBS-R003-D1). B. Williams thanks Cris Poor and David Yuen for interesting discussions, especially related to Section \ref{sec:non-free}.
\addtocontents{toc}{\setcounter{tocdepth}{1}}
\section{Appendix: Tables}\label{appendix}
\subsection{Weights of generators of free algebras of meromorphic modular forms}
On the following pages we list the weights of generators for the $147$ free algebras of meromorphic modular forms on the discriminant kernels of $2U \oplus L$, which $L=L_0\oplus L_1$ are root lattices in the three families \eqref{eq:lattices}. The allowed poles lie on the hyperplane arrangement $\cH_L = \cH_{L, 0} \cup \cH_{L, 1}$ defined in Section \ref{sec:type IV}. Generators of abelian type are never holomorphic, because they have poles on $\cH_{L, 0}$; while generators of Eisenstein and Jacobi type may be holomorphic or have poles only on the part $\cH_{L, 1}$. The tables list the weights of generators of Eisenstein, abelian, and Jacobi type, and the weight of their Jacobian.
We also list in Table \ref{tab:predict} some algebras we predict to be freely generated as well as the weights, but for which we have been unable to construct appropriate generators. The ring structure of $M_*(\Orth^+(2U \oplus E_8))$ was shown by \cite{HU14} to be the free algebra on generators of weights $4$, $10$, $12$, $16$, $18$, $22$, $24$, $28$, $30$, $36$, $42$ and it is very exceptional. It does not fit into the pattern of generators for the other root lattices so we omit it and other potential interesting algebras of meromorphic modular forms on root lattices containing it as a component.
\vspace{5mm}
\noindent\[
\renewcommand{\arraystretch}{1.0}
\begin{array}{|c|c|c|c|c|c|}
\hline
L_0 & L_1 & \text{Eisenstein} & \text{Abelian} & \text{Jacobi} & \text{wt}\, J \\
\hline
0 & A_1 & 4, 6 & - & 10, 12 & 35\\
A_1 & A_1 & 4, 6 & 2 & 8, 10 & 34 \\
2A_1 & A_1 & 4, 6 & 2, 2 & 6, 8 & 33 \\
3A_1 & A_1 & 4, 6 & 2, 2, 2 & 4, 6 & 32 \\
4A_1 & A_1 & 4, 6 & 2, 2, 2, 2 & 2, 4 & 31\\
A_1 \oplus A_2 & A_1 & 4, 6 & 1, 2, 3 & 5, 7 & 34 \\
A_1 \oplus A_3 & A_1 & 4, 6 & 1, 2, 2, 4 & 4, 6 & 36 \\
A_1 \oplus A_4 & A_1 & 4, 6 & 1, 2, 2, 3, 5 & 3, 5 & 39 \\
\hline
\end{array}
\]
\clearpage
\noindent\[
\renewcommand{\arraystretch}{1.0}
\begin{array}{|c|c|c|c|c|c|}
\hline
L_0 & L_1 & \text{Eisenstein} & \text{Abelian} & \text{Jacobi} & \text{wt}\, J \\
\hline
A_1 \oplus A_5 & A_1 & 4, 6 & 1, 2, 2, 3, 4, 6 & 2, 4 & 43 \\
A_1\oplus A_6 & A_1 & 4, 6 & 1, 2, 2, 3, 4, 5, 7 & 1, 3 & 48 \\
A_1 \oplus 2A_2 & A_1 & 4, 6 & 1, 1, 2, 3, 3 & 2, 4 & 34 \\
A_1 \oplus A_2 \oplus A_3 & A_1 & 4, 6 & 1, 1, 2, 2, 3, 4 & 1, 3 & 36 \\
2A_1 \oplus A_2 & A_1 & 4, 6 & 1, 2, 2, 3 & 3, 5 & 33 \\
2A_1 \oplus A_3 & A_1 & 4, 6 & 1, 2, 2, 2, 4 & 2, 4 & 35 \\
2A_1 \oplus A_4 & A_1 & 4, 6 & 1, 2, 2, 2, 3, 5 & 1, 3 & 38 \\
3A_1 \oplus A_2 & A_1 & 4, 6 & 1, 2, 2, 2, 3 & 1, 3 & 32 \\
A_2 & A_1 & 4, 6 & 1, 3 & 7, 9 & 35 \\
2A_2 & A_1 & 4, 6 & 1, 1, 3, 3 & 4, 6 & 35 \\
3A_2 & A_1 & 4, 6 & 1, 1, 1, 3, 3, 3 & 1, 3 & 35 \\
A_2 \oplus A_3 & A_1 & 4, 6 & 1, 1, 2, 3, 4 & 3, 5 & 37 \\
A_2 \oplus A_4 & A_1 & 4, 6 & 1, 1, 2, 3, 3, 5 & 2, 4 & 40 \\
A_2 \oplus A_5 & A_1 & 4, 6 & 1, 1, 2, 3, 3, 4, 6 & 1, 3 & 44 \\
A_3 & A_1 & 4, 6 & 1, 2, 4 & 6, 8 & 37 \\
2A_3 & A_1 & 4, 6 & 1, 1, 2, 2, 4, 4 & 2, 4 & 39 \\
A_3 \oplus A_4 & A_1 & 4, 6 & 1, 1, 2, 2, 3, 4, 5 & 1, 3 & 42 \\
A_4 & A_1 & 4, 6 & 1, 2, 3, 5 & 5, 7 & 40 \\
A_5 & A_1 & 4, 6 & 1, 2, 3, 4, 6 & 4, 6 & 44 \\
A_6 & A_1 & 4, 6 & 1, 2, 3, 4, 5, 7 & 3, 5 & 49 \\
A_7 & A_1 & 4, 6 & 1, 2, 3, 4, 5, 6, 8 & 2, 4 & 55 \\
A_8 & A_1 & 4, 6 & 1, 2, 3, 4, 5, 6, 7, 9 & 1, 3 & 62 \\
0 & A_2 & 4, 6 & - & 9, 10, 12 & 45\\
A_1 & A_2 & 4, 6 & 2 & 7, 8, 10 & 42 \\
2A_1 & A_2 & 4, 6 & 2, 2 & 5, 6, 8 & 39 \\
3A_1 & A_2 & 4, 6 & 2, 2, 2 & 3, 4, 6 & 36 \\
4A_1 & A_2 & 4, 6 & 2, 2, 2, 2 & 1, 2, 4 & 33 \\
A_1 \oplus A_2 & A_2 & 4, 6 & 1, 2, 3 & 4, 5, 7 & 39 \\
A_1 \oplus A_3 & A_2 & 4, 6 & 1, 2, 2, 4 & 3, 4, 6 & 40 \\
A_1 \oplus A_4 & A_2 & 4, 6 & 1, 2, 2, 3, 5 & 2, 3, 5 & 42 \\
A_1 \oplus A_5 & A_2 & 4, 6 & 1, 2, 2, 3, 4, 6 & 1, 2, 4 & 45 \\
A_1 \oplus 2A_2 & A_2 & 4, 6 & 1, 1, 2, 3, 3 & 1, 2, 4 & 36 \\
2A_1 \oplus A_2 & A_2 & 4, 6 & 1, 2, 2, 3 & 2, 3, 5 & 36 \\
2A_1 \oplus A_3 & A_2 & 4, 6 & 1, 2, 2, 2, 4 & 1, 2, 4 & 37 \\
A_2 & A_2 & 4, 6 & 1, 3 & 6, 7, 9 & 42 \\
2A_2 & A_2 & 4, 6 & 1, 1, 3, 3 & 3, 4, 6 & 39 \\
A_2 \oplus A_3 & A_2 & 4, 6 & 1, 1, 2, 3, 4 & 2, 3, 5 & 40 \\
A_2 \oplus A_4 & A_2 & 4, 6 & 1, 1, 2, 3, 3, 5 & 1, 2, 4 & 42 \\
A_3 & A_2 & 4, 6 & 1, 2, 4 & 5, 6, 8 & 43 \\
2A_3 & A_2 & 4, 6 & 1, 1, 2, 2, 4, 4 & 1, 2, 4 & 41 \\
A_4 & A_2 & 4, 6 & 1, 2, 3, 5 & 4, 5, 7 & 45 \\
A_5 & A_2 & 4, 6 & 1, 2, 3, 4, 6 & 3, 4, 6 & 48 \\
A_6 & A_2 & 4, 6 & 1, 2, 3, 4, 5, 7 & 2, 3, 5 & 52 \\
A_7 & A_2 & 4, 6 & 1, 2, 3, 4, 5, 6, 8 & 1, 2, 4 & 57 \\
0 & A_3 & 4, 6 & - & 8, 9, 10, 12 & 54 \\
A_1 & A_3 & 4, 6 & 2 & 6, 7, 8, 10 & 49 \\
2A_1 & A_3 & 4, 6 & 2, 2 & 4, 5, 6, 8 & 44 \\
\hline
\end{array}
\]
\clearpage
\noindent\[
\renewcommand{\arraystretch}{1.0}
\begin{array}{|c|c|c|c|c|c|}
\hline
L_0 & L_1 & \text{Eisenstein} & \text{Abelian} & \text{Jacobi} & \text{wt}\, J \\
\hline
3A_1 & A_3 & 4, 6 & 2, 2, 2 & 2, 3, 4, 6 & 39 \\
A_1 \oplus A_2 & A_3 & 4, 6 & 1, 2, 3 & 3, 4, 5, 7 & 43 \\
A_1 \oplus A_3 & A_3 & 4, 6 & 1, 2, 2, 4 & 2, 3, 4, 6 & 43 \\
A_1 \oplus A_4 & A_3 & 4, 6 & 1, 2, 2, 3, 5 & 1, 2, 3, 5 & 44 \\
2A_1 \oplus A_2 & A_3 & 4, 6 & 1, 2, 2, 3 & 1, 2, 3, 5 & 38 \\
A_2 & A_3 & 4, 6 & 1, 3 & 5, 6, 7, 9 & 48 \\
2A_2 & A_3 & 4, 6 & 1, 1, 3, 3 & 2, 3, 4, 6 & 42 \\
A_2 \oplus A_3 & A_3 & 4, 6 & 1, 1, 2, 3, 4 & 1, 2, 3, 5 & 42 \\
A_3 & A_3 & 4, 6 & 1, 2, 4 & 4, 5, 6, 8 & 48 \\
A_4 & A_3 & 4, 6 & 1, 2, 3, 5 & 3, 4, 5, 7 & 49 \\
A_5 & A_3 & 4, 6 & 1, 2, 3, 4, 6 & 2, 3, 4, 6 & 51 \\
A_6 & A_3 & 4, 6 & 1, 2, 3, 4, 5, 7 & 1, 2, 3, 5 & 54 \\
0 & A_4 & 4, 6 & - & 7, 8, 9, 10, 12 & 62 \\
A_1 & A_4 & 4,6 & 2 & 5, 6, 7, 8, 10 & 55 \\
2A_1 & A_4 & 4, 6 & 2, 2 & 3, 4, 5, 6, 8 & 48 \\
3A_1 & A_4 & 4,6 & 2, 2, 2 & 1, 2, 3, 4, 6 & 41 \\
A_1 \oplus A_2 & A_4 & 4, 6 & 1, 2, 3 & 2, 3, 4, 5, 7 & 46 \\
A_1 \oplus A_3 & A_4 & 4, 6 & 1, 2, 2, 4 & 1, 2, 3, 4, 6 & 45 \\
A_2 & A_4 & 4, 6 & 1, 3 & 4, 5, 6, 7, 9 & 53 \\
2A_2 & A_4 & 4, 6 & 1, 1, 3, 3 & 1, 2, 3, 4, 6 & 44 \\
A_3 & A_4 & 4, 6 & 1, 2, 4 & 3, 4, 5, 6, 8 & 52 \\
A_4 & A_4 & 4, 6 & 1, 2, 3, 5 & 2, 3, 4, 5, 7 & 52 \\
A_5 & A_4 & 4, 6 & 1, 2, 3, 4, 6 & 1, 2, 3, 4, 6 & 53 \\
0 & A_5 & 4, 6 & - & 6, 7, 8, 9, 10, 12 & 69 \\
A_1 & A_5 & 4, 6 & 2 & 4, 5, 6, 7, 8, 10 & 60 \\
2A_1 & A_5 & 4, 6 & 2, 2 & 2, 3, 4, 5, 6, 8 & 51 \\
A_1 \oplus A_2 & A_5 & 4, 6 & 1, 2, 3 & 1, 2, 3, 4, 5, 7 & 48 \\
A_2 & A_5 & 4, 6 & 1, 3 & 3, 4, 5, 6, 7, 9 & 57 \\
A_3 & A_5 & 4, 6 & 1, 2, 4 & 2, 3, 4, 5, 6, 8 & 55 \\
A_4 & A_5 & 4, 6 & 1, 2, 3, 5 & 1, 2, 3, 4, 5, 7 & 54 \\
0 & A_6 & 4, 6 & - & 5, 6, 7, 8, 9, 10, 12 & 75 \\
A_1 & A_6 & 4, 6 & 2 & 3, 4, 5, 6, 7, 8, 10 & 64 \\
2A_1 & A_6 & 4, 6 & 2, 2 & 1, 2, 3, 4, 5, 6, 8 & 53 \\
A_2 & A_6 & 4, 6 & 1, 3 & 2, 3, 4, 5, 6, 7, 9 & 60 \\
A_3 & A_6 & 4, 6 & 1, 2, 4 & 1, 2, 3, 4, 5, 6, 8 & 57 \\
0 & A_7 & 4,6 & - & 4, 5, 6, 7, 8, 9, 10, 12 & 80 \\
A_1 & A_7 & 4, 6 & 2 & 2, 3, 4, 5, 6, 7, 8, 10 & 67 \\
A_2 & A_7 & 4,6 & 1, 3 & 1, 2, 3, 4, 5, 6, 7, 9 & 62 \\
0 & A_8 & 4, 6 & - & 3, 4, 5, 6, 7, 8, 9, 10, 12 & 84 \\
A_1 & A_8 & 4, 6 & 2 & 1, 2, 3, 4, 5, 6, 7, 8, 10 & 69 \\
0 & A_9 & 4, 6 & - & 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 & 87 \\
0 & A_{10} & 4,6 & - & 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 & 89 \\
\hline
\hline
0 & D_4 & 4, 6 & - & 8, 8, 10, 12, 18 & 72 \\
A_1 &D_4 & 4, 6 & 2 & 6, 6, 8, 10, 14 & 63 \\
2A_1 & D_4 & 4, 6 & 2, 2 & 4, 4, 6, 8, 10 & 54 \\
3A_1 & D_4 & 4, 6 & 2, 2, 2 & 2, 2, 4, 6, 6 & 45 \\
\hline
\end{array}
\]
\clearpage
\noindent\[
\renewcommand{\arraystretch}{1.0}
\begin{array}{|c|c|c|c|c|c|}
\hline
L_0 & L_1 & \text{Eisenstein} & \text{Abelian} & \text{Jacobi} & \text{wt}\, J \\
\hline
A_1 \oplus A_2 & D_4 & 4, 6 & 1, 2, 3 & 3, 3, 5, 7, 8 & 51 \\
A_1 \oplus A_3 & D_4 & 4, 6 & 1, 2, 2, 4 & 2, 2, 4, 6, 6 & 49 \\
A_1 \oplus A_4 & D_4 & 4, 6 & 1, 2, 2, 3, 5 & 1, 1, 3, 4, 5 & 48 \\
2A_1\oplus A_2 & D_4 & 4, 6 & 1, 2, 2, 3 & 1, 1, 3, 4, 5 & 42 \\
A_2 & D_4 & 4, 6 & 1, 3 & 5, 5, 7, 9, 12 & 60 \\
2A_2 & D_4 & 4, 6 & 1, 1, 3, 3 & 2, 2, 4, 6, 6 & 48 \\
A_2 \oplus A_3 & D_4 & 4, 6 & 1, 1, 2, 3, 4 & 1, 1, 3, 4, 5 & 46 \\
A_3 & D_4 & 4, 6 & 1, 2, 4 & 4, 4, 6, 8, 10 & 58 \\
A_4 & D_4 & 4, 6 & 1, 2, 3, 5 & 3, 3, 5, 7, 8 & 57 \\
A_5 & D_4 & 4, 6 & 1, 2, 3, 4, 6 & 2, 2, 4, 6, 6 & 57 \\
A_6 & D_4 & 4, 6 & 1, 2, 3, 4, 5, 7 & 1, 1, 3, 4, 5 & 58 \\
0 & D_5 & 4, 6 & - & 7, 8, 10, 12, 16, 18 & 88 \\
A_1 & D_5 & 4, 6 & 2 & 5, 6, 8, 10, 12, 14 & 75 \\
2A_1& D_5 & 4, 6 & 2, 2 & 3, 4, 6, 8, 8, 10 & 62 \\
3A_1 & D_5 & 4, 6 & 2, 2, 2 & 1, 2, 4, 4, 6, 6 & 49 \\
A_1 \oplus A_2 & D_5 & 4, 6 & 1, 2, 3 & 2, 3, 5, 6, 7, 8 & 57 \\
A_1 \oplus A_3 & D_5 & 4, 6 & 1, 2, 2, 4 & 1, 2, 4, 4, 6, 6 & 53 \\
A_2 & D_5 & 4, 6 & 1, 3 & 4, 5, 7, 9, 10, 12 & 70 \\
2A_2 & D_5 & 4, 6 & 1, 1, 3, 3 & 1, 2, 4, 4, 6, 6 & 52 \\
A_3 & D_5 & 4, 6 & 1, 2, 4 & 3, 4, 6, 8, 8, 10 & 66 \\
A_4 & D_5 & 4, 6 & 1, 2, 3, 5 & 2, 3, 5, 6, 7, 8 & 63 \\
A_5 & D_5 & 4, 6 & 1, 2, 3, 4, 6 & 1, 2, 4, 4, 6, 6 & 61 \\
0 & D_6 & 4, 6 & - & 6, 8, 10, 12, 14, 16, 18 & 102 \\
A_1 & D_6 & 4, 6 & 2 & 4, 6, 8, 10, 10, 12, 14 & 85 \\
2A_1 & D_6 & 4, 6 & 2, 2 & 2, 4, 6, 6, 8, 8, 10 & 68 \\
A_1 \oplus A_2 & D_6 & 4, 6 & 1, 2, 3 & 1, 3, 4, 5, 5, 6, 8 & 59 \\
A_2 & D_6 & 4, 6 & 1, 3 & 3, 5, 7, 8, 9, 10, 12 & 78 \\
A_3 & D_6 & 4, 6 & 1, 2, 4 & 2, 4, 6, 6, 8, 8, 10 & 72 \\
A_4 & D_6 & 4, 6 & 1, 2, 3, 5 & 1, 3, 4, 5, 5, 6, 8 & 65 \\
0 & D_7 & 4, 6 & - & 5, 8, 10, 12, 12, 14, 16, 18 & 114 \\
A_1 & D_7 & 4, 6 & 2 & 3, 6, 8, 8, 10, 10, 12, 14 & 93 \\
2A_1 & D_7 & 4, 6 & 2, 2 & 1, 4, 4, 6, 6, 8, 8, 10 & 72 \\
A_2 & D_7 & 4, 6 & 1, 3 & 2, 5, 6, 7, 8, 9, 10, 12 & 84 \\
A_3 & D_7 & 4, 6 & 1, 2, 4 & 1, 4, 4, 6, 6, 8, 8, 10 & 68 \\
0 & D_8 & 4, 6 & - & 4, 8, 10, 10, 12, 12, 14, 16, 18 & 124 \\
A_1 & D_8 & 4, 6 & 2 & 2, 6, 6, 8, 8, 10, 10, 12, 14 & 99 \\
A_2 & D_8 & 4, 6 & 1, 3 & 1, 4, 5, 6, 7, 8, 9, 10, 12 & 88 \\
0 & D_9 & 4, 6 & - & 3, 8, 8, 10, 10, 12, 12, 14, 16, 18 & 132 \\
A_1 & D_9 & 4, 6 & 2 & 1, 4, 6, 6, 8, 8, 10, 10, 12, 14 & 103 \\
0 & D_{10} & 4, 6 & - & 2, 6, 8, 8, 10, 10, 12, 12, 14, 16, 18 & 138 \\
0 & D_{11}& 4, 6 & - & 1, 4, 6, 8, 8, 10, 10, 12, 12, 14, 16, 18 & 142 \\
\hline
\hline
0 & E_6 & 4, 6 & - & 7, 10, 12, 15, 16, 18, 24 & 120 \\
A_1 & E_6 & 4, 6 & 2 & 5, 8, 10, 11, 12, 14, 18 & 99 \\
A_2 & E_6 & 4, 6 & 1, 3 & 4, 7, 9, 9, 10, 12, 18 & 93 \\
0 & E_7 & 4, 6 & - & 10, 12, 14, 16, 18, 22, 24, 30 & 165 \\
A_1 & E_7 & 4, 6 & 2 & 8, 10, 10, 12, 14, 16, 18, 22 & 132 \\
\hline
\end{array}
\]
\begin{table}[ht]
\caption{\textbf{Predicted} free algebras of meromorphic modular forms for discriminant kernels of lattices $2U \oplus L_0 \oplus L_1$ of type $AE$.}\label{tab:predict}
\renewcommand\arraystretch{1.0}
\noindent\[
\begin{array}{|c|c|c|c|c|c|}
\hline
L_0 & L_1 & \text{Eisenstein} & \text{Abelian} & \text{Jacobi} & \text{wt}\, J \\
\hline
2A_1 & E_6 & 4, 6 & 2, 2 & 3, 6, 7, 8, 8, 10, 12 & 78 \\
3A_1 & E_6 & 4, 6 & 2, 2, 2 & 1, 3, 4, 4, 6, 6, 6 & 57 \\
A_1 \oplus A_2 & E_6 & 4, 6 & 1, 2, 3 & 2, 5, 5, 6, 7, 8, 9 & 69 \\
A_1 \oplus A_3 & E_6 & 4, 6 & 1, 2, 2, 4 & 1, 3, 4, 4, 6, 6, 6 & 61 \\
2A_2 & E_6 & 4, 6 & 1, 1, 3, 3 & 1, 3, 4, 4, 6, 6, 6 & 60 \\
A_3 & E_6 & 4, 6 & 1, 2, 4 & 3, 6, 7, 8, 8, 10, 12 & 82 \\
A_4 & E_6 & 4, 6 & 1, 2, 3, 5 & 2, 5, 5, 6, 7, 8, 9 & 75 \\
A_5 & E_6 & 4, 6 & 1, 2, 3, 4, 6 & 1, 3, 4, 4, 6, 6, 6 & 69 \\
2A_1 & E_7 & 4, 6 & 2, 2 & 6, 6, 8, 8, 10, 10, 12, 14 & 99 \\
3A_1 & E_7 & 4, 6 & 2, 2, 2 & 2, 4, 4, 4, 6, 6, 6, 6 & 66 \\
A_1 \oplus A_2 & E_7 & 4, 6 & 1, 2, 3 & 4, 5, 6, 7, 7, 8, 9, 10 & 84 \\
A_1 \oplus A_3 & E_7 & 4, 6 & 1, 2, 2, 4 & 2, 4, 4, 4, 6, 6, 6, 6 & 70 \\
A_2 & E_7 & 4, 6 & 1, 3 & 7, 8, 9, 10, 12, 13, 15, 18 & 117 \\
2 A_2 & E_7 & 4, 6 & 1, 1, 3, 3 & 2, 4, 4, 4, 6, 6, 6, 6 & 69 \\
A_3 & E_7 & 4, 6 & 1, 2, 4 & 6, 6, 8, 8, 10, 10, 12, 14 & 103 \\
A_4 & E_7 & 4, 6 & 1, 2, 3, 5 & 4, 5, 6, 7, 7, 8, 9, 10 & 90 \\
A_5 & E_7 & 4, 6 & 1, 2, 3, 4, 6 & 2, 4, 4, 4, 6, 6, 6, 6 & 78 \\
\hline
\end{array}
\]
\end{table}
\subsection{Non-free algebras of holomorphic modular forms related to reducible root lattices}\label{sec:tables_hol}
In the following we describe the $26$ non-free algebras of holomorphic modular forms determined in \S \ref{sec:non-free}, associated to the discriminant kernel of $2U+R$ for a reducible root lattice $R$, by describing the weights of a minimal system of generators and the dimensions of the spaces of modular forms. The notation $k^n$ in the weights of the generators means that there are $n$ generators of weight $k$. The rational function associated to each root system $R$ is the Hilbert--Poincar\'e series $$\mathrm{Hilb}\, M_*(\widetilde{\Orth}^+(2U \oplus R)) = \sum_{k=0}^{\infty} \mathrm{dim}\, M_k(\widetilde{\Orth}^+(2U \oplus R)) t^k,$$ from which the dimensions can be extracted easily.
\vspace{3mm}
\begin{small}
\begin{enumerate}
\item $R=2A_1$: weights of generators: 4, 6, 8, 10, 10, 12;
$$
\frac{1 + t^{10}}{(1 - t^4)(1-t^6)(1-t^8)(1-t^{10})(1-t^{12})}
$$
\item $R=3A_1$: weights of generators: $4, 6^2, 8^3, 10^3, 12$;
$$
\frac{1 + 2t^8 + 2t^{10} + t^{18}}{(1 - t^4)(1-t^6)^2(1-t^8)(1-t^{10})(1-t^{12})}
$$
\item $R=4A_1$: weights of generators: $4^2, 6^5, 8^6, 10^4, 12$;
$$
\frac{1 + 3t^6 + 5t^8 + 3t^{10} + 3t^{14} + 5t^{16} + 3t^{18} + t^{24}}{(1 - t^4)^2(1-t^6)^2 (1-t^8)(1-t^{10})(1-t^{12})}
$$
\item $R=A_1\oplus A_2$: weights of generators: $4, 6, 7, 8, 9, 10, 10, 12$;
$$
\frac{1+ t^{10} - t^{17} - t^{19}}{(1 - t^4)(1-t^6)(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
$$
\item $R=2A_1\oplus A_2$: weights of generators: $4, 5, 6^2, 7^2, 8^3, 9, 10^3, 12$;
$$
\frac{1 + t^5 + t^7 + 2t^8 + 3t^{10} + t^{12} - 2t^{17} + t^{18} - 2t^{19} - t^{22} - t^{24} - t^{25} - 2t^{27} - t^{29}}{(1-t^4)(1 - t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
$$
\item $R=2A_2$: weights of generators: $4, 6^2, 7^2, 8, 9^2, 10^2, 12$;
$$
\frac{1 + t^4 + t^7 + t^8 + t^9 + t^{10} + t^{11} + t^{12} + t^{13} + t^{14} + t^{15} +t^{18} + t^{22}}{(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
$$
\item $R=3A_2$: weights of generators: $3, 4^4, 5^3, 6^5, 7^6, 8^3, 9^3, 10^3, 12$;
\begin{align*}
&( 1 + 3t^4 + 3t^5 + 3t^6 + 5t^7 + 5t^8 + 8t^9 + 8t^{10} + 9t^{11} + 12t^{12} + 12t^{13}\\
&+ 15t^{14} + 16t^{15} + 16t^{16} +14t^{17} + 16t^{18} + 16t^{19} + 15t^{20} + 12t^{21}\\
&+ 12t^{22} + 9t^{23} + 8t^{24} + 8t^{25} + 5t^{26} + 5t^{27} + 3t^{28} + 3t^{29} + 3t^{30} + t^{34} ) \\
&/ (1 - t^3)(1-t^4)(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})
\end{align*}
\item $R=A_1\oplus A_3$: weights of generators: $4, 6, 6, 7, 8, 8, 9, 10, 10, 12$;
$$
\frac{1 - t^2 + t^6 - t^{15}}{(1-t^2)(1-t^4)(1 - t^6)(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
$$
\item $R=2A_1\oplus A_3$: weights of generators: $4^2, 5, 6^4, 7^2, 8^4, 9, 10^3, 12$;
\begin{align*}
&(1 + t^5 + 2t^6 + t^7 + 3t^8 + 3t^{10} + t^{12} +t^{14} - t^{15} + 2t^{16} - 2t^{17}\\
&+ t^{18} - 2t^{19} - t^{21} -t^{22} - 2t^{23} - t^{24} - 3t^{25} - 2t^{27} - t^{29}) \\
&/(1-t^4)^2(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})
\end{align*}
\item $R=A_2\oplus A_3$: weights of generators: $4, 5, 6^3, 7^2, 8^2, 9^2, 10^2, 12$;
\begin{align*}
\frac{1 + t^5 + t^6 + t^7 + t^8 + t^9 + 2t^{10} + t^{11} + t^{12} + t^{13} + t^{14} + t^{15} + t^{16} + t^{18} + t^{20} +t^{21}}{(1 - t^6)(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
\end{align*}
\item $R=2A_3$: weights of generators: $4^2, 5^2, 6^4, 7^2, 8^3, 9^2, 10^2, 12$;
\begin{align*}
&(1 + 2t^5 + 2t^6 + t^7 + 2t^8 + t^9 + 3t^{10} + 2t^{11} +2t^{12} + 2t^{13} \\
&+ 2t^{14} + 2t^{15} + 3t^{16} + t^{17} + 2t^{18} + t^{19} + 2t^{20} + 2t^{21} +t^{26}) \\
&/(1-t^4)^2(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})
\end{align*}
\item $R=A_1\oplus A_4$: weights of generators: $4, 5, 6, 6, 7, 7, 8, 8, 9, 10, 12$;
$$
\frac{1+t^7+t^8+t^{10}-t^{13}-t^{15}-t^{17}-t^{19}}{(1-t^4)(1 - t^5)(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
$$
\item $R=A_2\oplus A_4$: weights of generators: $4^2, 5^2, 6^3, 7^3, 8^2, 9^2, 10^2, 12$;
\begin{align*}
&(1 + t^4 + t^5 + t^6 + 2t^7 + 2t^8 + 2t^9 + 2t^{10} + 2t^{11} + 2t^{12} + 2t^{13}\\
&+ 2t^{14} + 2t^{15} + 2t^{16} + t^{17} + 2t^{18} + t^{19} + t^{20} + t^{21} + t^{22}) \\
&/ (1-t^4)(1-t^5)(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})
\end{align*}
\item $R=A_1\oplus A_5$: weights of generators: $4^2, 5, 6^3, 7^2, 8^2, 9, 10^2, 12$;
\begin{align*}
\frac{1 + t^6 + t^7 + t^8 + t^{10} - t^{11} - t^{13} -t^{15} - t^{17} - t^{19}}{(1 - t^4)^2(1-t^5)(1-t^6)^2(1-t^7)(1-t^8)(1-t^9)(1-t^{10})(1-t^{12})}
\end{align*}
\item $R=A_1\oplus 2A_2$: weights of generators: $4^2, 5^2, 6^3, 7^4, 8^3, 9^2, 10^3, 12$;
\begin{align*}
&(1 + t^{4} + 2t^{5} + t^{6} + 3t^{7} + 3t^{8} + 3t^{9} + 5t^{10} + 3t^{11} + 5t^{12} + 4t^{13} + 5t^{14} + 6t^{15} + 5t^{16}\\
& + 4t^{17} + 6t^{18} + 4t^{19} + 5t^{20} + 4t^{21} + 3t^{22} + 3t^{23} + t^{24} + 2t^{25} + t^{26} + t^{28} + t^{30})\\
& / (1 - t^4)(1 - t^6)^2(1 - t^7)(1 - t^8)(1 - t^9)(1 - t^{10})(1 - t^{12})
\end{align*}
\item $R=A_1\oplus D_4$: weights of generators: $4, 6^3, 8^3, 10^2, 12, 14, 16, 18$;
\begin{align*}
\frac{1 + 2t^6 + t^8 + t^{10} + 3t^{12} + t^{14} + 2t^{16} + 3t^{18} + t^{20} + t^{22} + t^{24}}{(1 - t^4)(1-t^6)(1-t^8)^2 (1-t^{10})(1-t^{12})(1-t^{14})(1-t^{18})}
\end{align*}
\item $R=2A_1\oplus D_4$: weights of generators: $4^3, 6^6, 8^5, 10^4, 12^3, 14^3, 16^2, 18$;
\begin{align*}
&(1 - t^{2} + 2t^{4} + 3t^{6} + t^{8} + 5t^{10} + 7t^{12} + 4t^{14} + 10t^{16} + 8t^{18} \\
&+ 7t^{20} + 9t^{22} + 8t^{24} + 6t^{26} + 6t^{28} + 4t^{30} + 3t^{32} + t^{34} + t^{36}) \\
&/(1-t^2)(1-t^4)(1-t^6)(1-t^8)^2(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{18})
\end{align*}
\item $R=A_2\oplus D_4$: weights of generators: $4, 5^2, 6^3, 7, 8^3, 9, 10^2, 12^2, 13, 14, 15, 16, 18$;
\begin{align*}
&(1 + t^{5} + t^{6} + t^{7} + t^{8} + t^{9} + 2t^{10} + 3t^{12} + 2t^{14} + t^{15} + 2t^{16} + t^{17} + 2t^{19} - t^{20} + t^{21} - t^{22}\\
&- t^{23} - t^{24} - t^{25} - t^{27} - t^{29} - 2t^{30} - 2t^{32} - t^{33} - 2t^{34} - t^{35} - 2t^{36} - t^{37} - t^{38} - t^{39}) \\
& / (1 - t^4)(1-t^5)(1-t^6)^2(1-t^8)^2(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{18})
\end{align*}
\item $R=A_3\oplus D_4$: weights of generators: $4^3, 5^2, 6^4, 7, 8^4, 9, 10^3,11, 12^3, 13, 14^2, 15, 16, 18$;
\begin{align*}
&(1 - t^{3} + t^{4} + t^{5} + 3t^{6} + 2t^{8} - 2t^{9} + 4t^{10} - t^{11} + 8t^{12} - 3t^{13} + 6t^{14} - 6t^{15} + 8t^{16} - 4t^{17}\\
&+ 9t^{18} - 6t^{19} + 7t^{20} - 7t^{21} + 7t^{22} - 7t^{23} + 9t^{24} - 7t^{25} + 7t^{26} - 9t^{27} + 5t^{28} - 9t^{29}\\
&+ 3t^{30} - 7t^{31} + 3t^{32} - 5t^{33} + t^{34} - 5t^{35} + t^{36} - 2t^{37} + t^{38} - 2t^{39} - 2t^{41} - t^{42} - t^{43}) \\
&/ (1-t^3)(1-t^4)^2(1-t^5)(1-t^6)(1-t^8)^2 (1-t^{10})(1-t^{12})(1-t^{14})(1-t^{18})
\end{align*}
\item $R=2D_4$: weights of generators: $4^5, 6^5, 8^5, 10^8, 12^6, 14^6, 16^2, 18^2$;
\begin{align*}
&(1 + 2t^{4} + 3t^{6} + 5t^{8} + 8t^{10} + 11t^{12} + 10t^{14} + 12t^{16} + 10t^{18} + 13t^{20} + 9t^{22} + 10t^{24} + 4t^{26} - 4t^{28} \\
& - 10t^{30} - 9t^{32} - 13t^{34} - 10t^{36} - 12t^{38} - 10t^{40} - 11t^{42} - 8t^{44} - 5t^{46} - 3t^{48} - 2t^{50} - t^{54}) \\
& /(1-t^4)^3(1-t^6)^2 (1-t^8)^2 (1-t^{10})^2 (1 - t^{12})(1-t^{14})(1-t^{18})
\end{align*}
\item $R=A_1\oplus D_5$: weights of generators: $4, 5, 6^2, 7, 8^2, 10^2, 12^2, 14^2, 16^2, 18$;
\begin{align*}
\frac{1 + t^6 + t^7 + t^8 + t^{10} + 2t^{12} + 3t^{14} + 2t^{16} + 2t^{18} + 2t^{20} + t^{22} + t^{24} + t^{26}}{(1 - t^4)(1-t^5)(1-t^6)(1-t^8)(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{16})(1-t^{18})}
\end{align*}
\item $R=A_2\oplus D_5$: weights of generators: $4^2, 5^2, 6^2, 7^2, 8^2, 9, 10^3, 11, 12^3, 13^2, 14^2, 15, 16^2, 18$;
\begin{align*}
&(1 - t^{3} + t^{4} + t^{5} + t^{6} + t^{7} + t^{8} + t^{9} + t^{10} + 2t^{11} + 3t^{12} + 2t^{13} + 3t^{14} \\
& + t^{15} + 3t^{16} + t^{17} + 4t^{18} + 2t^{19} + 4t^{20} + 2t^{21} + 4t^{22} + 2t^{23} + 4t^{24} \\
&+ 2t^{25} + 3t^{26} + t^{27} + 2t^{28} + t^{29} + 2t^{30} + 2t^{31} + t^{32} + t^{33} + t^{34} + t^{35}) \\ & /(1-t^3)(1-t^4)(1-t^5)(1-t^6)(1-t^8)(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{16})(1-t^{18})
\end{align*}
\item $R=A_1\oplus D_6$: weights of generators: $4^2, 6^3, 8^2, 10^3, 12^3, 14^3, 16^2, 18$;
\begin{align*}
\frac{1+t^4+t^6+2t^8+2t^{10}+4t^{12}+4t^{14}+4t^{16}+3t^{18}+3t^{20}+2t^{22}+2t^{24}+t^{26}+t^{28}}{(1-t^4)(1-t^6)^2(1-t^8)(1-t^{10})^2(1-t^{12})(1-t^{14})(1-t^{16})(1-t^{18})}
\end{align*}
\item $R=A_1\oplus E_6$: weights of generators: $4, 5, 6, 7, 8, 10^2, 11, 12^2, 13, 14^2, 15, 16^2, 18^2, 20, 22, 24$;
\begin{align*}
&(1 + t^{5} + t^{7} + t^{8} + 2t^{10} + 2t^{12} + 2t^{13} + 2t^{14} + 2t^{15} + 2t^{16} + 2t^{17} + 3t^{18} + 2t^{19} + 4t^{20} + 2t^{21} + 4t^{22} \\
& + 2t^{23} + 3t^{24} + 2t^{25} + 3t^{26} + 2t^{27} + 3t^{28} + t^{29} + 3t^{30} + t^{31} + 2t^{32} + t^{33} + t^{34} + t^{35} + t^{36} + t^{38} + t^{40}) \\
& / (1-t^4)(1-t^6)(1-t^{10})(1-t^{11})(1-t^{12})(1-t^{14})(1-t^{15})(1-t^{16})(1-t^{18})(1-t^{24})
\end{align*}
\item $R=A_2\oplus E_6$: weights of generators: $4^2$, 5, 6, $7^2$, 8, 9, $10^3$, 11, $12^3$, $13^2$, $14^2$, $15^2$, $16^3$, 17, $18^3$, 19, 20, 21, 22, 24;
\begin{align*}
&(1 + t^{4} + t^{5} + 2t^{7} + 2t^{8} + 2t^{9} + 4t^{10} + 3t^{11} + 6t^{12} + 6t^{13} + 6t^{14} + 8t^{15} + 9t^{16} + 11t^{17}\\
& + 12t^{18} + 13t^{19} + 16t^{20} + 16t^{21} + 19t^{22} + 18t^{23} + 21t^{24} + 23t^{25} + 23t^{26} + 24t^{27} \\
& + 27t^{28} + 26t^{29} + 29t^{30}+ 28t^{31} + 29t^{32} + 28t^{33} + 30t^{34} + 28t^{35} + 28t^{36} + 28t^{37} \\
& + 28t^{38} + 25t^{39} + 26t^{40} + 22t^{41} + 22t^{42} + 21t^{43} + 18t^{44} + 16t^{45} + 16t^{46} + 13t^{47} \\
&+ 11t^{48} + 10t^{49} + 9t^{50} + 7t^{51} + 6t^{52} + 4t^{53} + 3t^{54} + 3t^{55} + 2t^{56} + t^{57} + t^{58} + t^{59}) \\
& / (1-t^4)(1-t^6)(1-t^9)(1-t^{10})(1-t^{11})(1-t^{12})(1-t^{14})(1-t^{15})(1-t^{16})(1-t^{18})(1-t^{24})
\end{align*}
\item $R=A_1\oplus E_7$: weights of generators: $4, 6, 8, 10^3, 12^3, 14^3, 16^3, 18^3,20^2, 22^3, 24^2, 26, 28, 30$;
\begin{align*}
\frac{1 - t^2 + t^8 + t^{10} + t^{16} + t^{18} + t^{20} + t^{26} + t^{28} + t^{36}}{(1 - t^2)(1-t^4)(1-t^6)(1-t^{10})(1-t^{12})(1-t^{14})(1-t^{16})(1-t^{18})(1-t^{22})(1-t^{24})(1-t^{30})}
\end{align*}
\end{enumerate}
\end{small}
\bibliographystyle{plainnat}
\bibliofont
\bibliography{refs}
\end{document}
| 160,088
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Tripod Blue Sapphire Earringbluboho
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The three points on our blue sapphires which are guardians of innocence, bestowers of truth, and promoters of good health. Also known as the "traveler's protector," a blue sapphire brings gifts of fulfillment, joy, and worldliness.
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| 177,240
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With a keen eye for detail, a devotion to client satisfaction and several years’ experience in various aspects of property transactions, Shakira has quickly proven an asset to WGC.
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| 6,572
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TITLE: For a connected curve $X$ over a field, is $H^0(X,\mathcal{O}_X)$ necessarily a field?
QUESTION [2 upvotes]: Let $X$ be a connected projective finite type noetherian scheme of dimension 1 over a field $k$.
When can $H^0(X,\mathcal{O}_X)$ fail to be a field?
If it is a field $k'$, then is $X$ also a curve over $k'$? Must $X$ be geometrically connected over $k'$?
Note that I don't assume $X$ to be smooth, regular, irreducible, or reduced, or even geometrically connected. Nor do I assume that $k$ is algebraically closed.
Certainly $H^0(X,\mathcal{O}_X)$ is a $k$-algebra. If $k$ is algebraically closed, then $H^0(X,\mathcal{O}_X)$is just $k$, hence a field.
What are some useful keywords/references to look up?
REPLY [1 votes]: Here are some relevant facts. Lets start with $X$ connected projective noetherian over $k$, with structure morphism $f : X\rightarrow\text{Spec }k$.
Firstly, $\text{Spec }H^0(X,\mathcal{O}_X) = \underline{Spec}_k f_*\mathcal{O}_X$, and hence the Stein factorization of $f$ gives us:
$$X\stackrel{f'}{\rightarrow}\text{Spec }H^0(X,\mathcal{O}_X)\stackrel{g}{\rightarrow}\text{Spec k}$$
with $f'$ having geometrically connected fibers and $g$ being the normalization map of $\text{Spec }k$ inside $X$, hence finite. In particular, $f'$ is surjective since $f$ was. This already tells us quite a lot.
Assume $X$ integral. Then $H^0(X,\mathcal{O}_X)$ is an integral $k$-algebra of finite dimension (since $X$ is projective), and hence is a field, and by the Stein factorization, we see that $X$ is a geometrically connected curve over $H^0(X,\mathcal{O}_X)$.
Assume $X$ not necessarily integral, but at least reduced. Then $H^0(X,\mathcal{O}_X)$ is a reduced finite $k$-algebra, hence a product of fields, but $f'$ is surjective and $X$ is connected, so again $H^0(X,\mathcal{O}_X)$ must be a single field.
Lastly, if $X$ is not reduced, then at least $A := H^0(X,\mathcal{O}_X)$ is an Artinian $k$-algebra, which must be local since $f'$ is surjective and $X$ connected. Let $m$ be the maximal ideal of $A$, and let $k' := A/m$, so $k'$ is a finite field extension of $k$. By the fiberwise criteria of flatness, $f'$ is flat, and so $X/A$ becomes a deformation of the geometrically connected curve $X_{k'} := X\times_A A/m$. In general without additional assumptions on $X$ or $X_{k'}$, this leads into the realm of deformation theory. However, a simple case arises when $X_{k'}$ has no (nontrivial) infinitesimal deformations - for example, if $X_{k'} = \mathbb{P}^1_{k'}$, in which case Mohan's example is the only possibility: ie, $X = \mathbb{P}^1_{k'}\times_{k'} A$.
| 211,403
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\begin{document}
\title{Transient Chaos in Strongly Monotone Dynamical Systems\thanks{Supported by NSF of China No.11825106, 11971232 and 12090012.}}
\setlength{\baselineskip}{16pt}
\author {
Jinxiang Yao
\\[2mm]
School of Mathematical Sciences\\
University of Science and Technology of China\\
Hefei, Anhui, 230026, P. R. China\\[3pt]
jxyao@mail.ustc.edu.cn
}
\date{}
\maketitle
\begin{abstract}
For strongly monotone dynamical systems, it is shown that the principal Lyapunov exponent $\lambda_{1}(x)\leq0$ holds on a prevalent set in the measure-theoretic sense. The nonexistence of observable chaos is thus obtained as a by-product. Together with the existing works, we point out that transient chaos is ubiquitous in strongly monotone dynamical systems, which is a form of unobservable chaos introduced by Young [Commun. Pure Appl. Math. 66(2013)].
\par
\textbf{Keywords}: Observable chaos; Transient chaos; Lyapunov exponents; Sharpened dynamics alternative; Improved prevalent dynamics; $C^1$-robustness; Monotone dynamical systems.
\end{abstract}
\section{Introduction}
\emph{Observable chaos} (see e.g., Young
\cite[Section 1, p.1441]{Y13},\cite[Section 4]{Y13'},\cite[Section 3]{Y18}) indicates that the principal Lyapunov exponent $\lambda_{1}(x)>0$ holds on a positive Lebesgue measure set, which implies that the instability persists for all future times and occur on a set large enough to be observable. As Young \cite{Y13} pointed out that, the presence of a horseshoe does not imply the system has observable chaos. Such unobservable complicated dynamics are characterized as \emph{transient chaos} in Young \cite[Section 1, p.1442]{Y13},\cite[Section 4]{Y13'}.
Transient chaos is ubiquitous in fluid, chemical, biological, and engineering systems. See also \cite{ASY97,LT11,T15} (and references therein) etc. for more details of the theory and applications of transient chaos.
The ground-breaking work by Hirsch \cite{H85} and the extended work by Pol\'{a}\v{c}ik \cite{P89} and Smith and Thieme \cite{ST91} showed that, generic precompact orbit of a strongly monotone semiflow converges to equilibria (see also \cite{HS05}). Here, {\it generic} describes the properties hold residually i.e., on a countable intersection of open dense subsets in a Baire space. For strongly monotone discrete-time dynamical systems, Pol\'{a}\v{c}ik and Tere\v{s}\v{c}\'{a}k \cite{PT92} first proved the {\it generic convergence to cycles} occurs provided that the mapping $F$ is of class $C^{1,\alpha}$ (i.e., $F$ is a $C^1$-map with a locally $\alpha$-H\"{o}lder derivative $DF$, $\alpha\in (0,1]$). Motivated by Tere\v{s}\v{c}\'{a}k \cite{T94}, Wang and Yao \cite{WY20-1,WY21-2} extended the result to $C^1$-smooth strongly monotone discrete dynamical systems. Recently, generic Poincar\'{e}-Bendixson Theorem was obtained by Feng et al. \cite{FWWpreprint,FWW17} for smooth flows strongly monotone with respect to $2$-cone. In the measure-theoretic sense, Enciso, Hirsch and Smith \cite{EHS08} investigated the prevalent behavior of strongly monotone semiflows and proved that the set of points that converge to equilibria is prevalent. Here, {\it Prevalent} describes properties of interest occurs for ``almost surely" in an infinite-dimensional space from a probabilistic or measure-theoretic perspective. It is a natural generalization to separable Banach spaces of the notion of the full Lebesgue measure for Euclidean spaces (see definition in Section \ref{S:Not-pre}). In particular, on $\mathbb{R}^d$, it is equivalent to the notion of ``Lebesgue almost everywhere". Wang et al. \cite{WYZ20-2} proved the set of points that converge to periodic orbits is prevalent for discrete-time systems. Very recently, Wang et al. \cite{WYZ22} further studied the prevalent behavior of flows with invariant k-cones and obtained an almost sure Poincar\'e-Bendixson theorem.
Although a lot of generic and prevalent asymptotic behavior theory have been built in monotone systems, one can not expect that monotone systems can only have simply dynamics. Smale \cite{S76} first smartly constructed a profound example to show that essentially arbitrary dynamics, including positive entropy, horseshoes, chaos in the sense of Li-Yorke and/or Devaney can be found in the codimension-1 invariant manifolds in monotone systems, even if in low-dimensional systems (see also this example in Smith \cite[Chapter 4, p.71-72]{S95}). For the investigation of complicated dynamics in monotone dynamical systems, one may also refer to \cite{DP98,S95,S97,S88,WJ01,WX10} (and references therein) etc. for more details.
In the present paper, we shall focus on and prove the nonexistence of observable chaos for $C^1$-smooth strongly monotone dynamical systems, as well as its $C^1$-robustness for $C^1$-perturbed systems. Together with the existing works on the presence of complexity in monotone systems, this implies the presence of transient chaos in monotone systems. That is to say, if the chaotic behavior exists, it is so unstable as to be unobservable. To be more precise, we formulate some standing hypotheses:
\vskip 2mm
\noindent \textbf{(H1)} $(X,C)$ is a strongly ordered separable Banach Space.
\vskip 1mm
\noindent \textbf{(H2)} $F_{0}:X \to X$ is a compact ${C}^{1}$-map, such that for any $x\in X$, the Fr\'echet derivative $DF_{0}(x)$
\quad\ is a strongly positive operator, i.e., $DF_{0}(x)v\gg0$ whenever $v>0$.
\newtheorem*{thma}{\textnormal{\textbf{Theorem A}}}
\begin{thma}[Nonexistence of observable chaos for mappings]
\emph{Assume that \textnormal{(H1)-(H2)} hold. Assume also $F_{0}$ is pointwise dissipative with an attractor $A$. Then the set
$$S_{0}:=\{x\in X:\ \lambda_{1}(x,F_{0})\leq 0\},$$
is prevalent in $X$. In particular, if $X=\mathbb{R}^{d}$, then $F_{0}$ cannot have observable chaos.}
\end{thma}
Motivated by Theorem A, we further consider $C^1$-perturbations of the $C^1$-smooth mapping $F_{0}$, and obtain the nonexistence of observable chaos for the $C^1$-perturbed systems.
More precisely, we present an additional standing hypothesis:
\vskip 3mm
\noindent \textbf{(H3)} Let $J=[-\epsilon_{0},\epsilon_{0}]\subset\mathbb{R}$, and $F:J\times X\to X; (\epsilon,x)\mapsto F_{\epsilon}(x)$ is a compact $C^{1}$-map, i.e., $DF(\epsilon,x)$ continuously depends on $(\epsilon,x)\in J\times X$.
\vskip 3mm
The following theorem reveals that the nonexistence of observable chaos of $F_0$ is robust under the $C^1$-perturbation.
\newtheorem*{thmb}{\textnormal{\textbf{Theorem B}}}
\begin{thmb}[$C^1$-robustness for nonexistence of observable chaos]
\emph{Assume that \textnormal{(H1)-(H3)} hold. Assume also $F_{0}$ is pointwise dissipative with an attractor $A$. Let $B_{1}\supset A$ be an open ball such that
\begin{equation}\label{E:C1-close}
\sup\{\Vert F_{\epsilon}x-F_{0}x\Vert +\Vert DF_{\epsilon}(x)-DF_{0}(x)\Vert : \epsilon\in J,\,x\in B_{1}\}
\end{equation}
sufficiently small. Then there exists a closed bounded set $M$ (${\rm Int}M\supset A$) and an integer $q>0$ such that, for any $|\epsilon|$ sufficiently small,}
\textnormal{(i)}. \emph{$F_{\ep}^{q}(M)\subset M$; and}
\textnormal{(ii)}. \emph{The set}
$$S_{\ep}:=\{x\in M:\ \lambda_{1}(x,F_{\epsilon})\leq 0\},$$
\emph{is prevalent in $M$. In particular, if $X=\mathbb{R}^{d}$, then the set $S_{\ep}$ is of full Lebesgue measure in the compact domain $M$, and $F_{\ep}^{q}|_M$ cannot have observable chaos.}
\end{thmb}
For a semiflow $\phi$, we present the following standing hypothesis:
\vskip 2mm
\noindent $\textbf{(H2)}^{\prime}$ $\phi:\RR\times X\to X$ is a $C^1$-smooth strongly monotone semiflow with compact orbit closures. For some fixed $t_0>0$, the Fr\'echet derivative $D\phi_{t_0}(x)$ is compact for any $x\in X$ and $D\phi_{t_0}(e)$ is strongly positive for any equilibrium $e\in X$.
\newtheorem*{thmc}{\textnormal{\textbf{Theorem C}}}
\begin{thmc}[Nonexistence of observable chaos for semiflows]
\emph{Assume that \textnormal{(H1)} and $\textnormal{(H2)}^{\prime}$ hold. Then the set
$$S_{\phi}:=\{x\in X:\ \lambda_{1}(x,\phi)\leq 0\},$$
is prevalent in $X$. In particular, if $X=\mathbb{R}^{d}$, then $\phi$ cannot have observable chaos.}
\end{thmc}
Theorem A-C concludes that observable chaos cannot exist in $C^1$-smooth strongly monotone systems, as well as for $C^1$-perturbed systems. This implies that the aforementioned unobservable complicated dynamics in strongly monotone systems are indeed transient chaos.
It deserves to point out that the existing works exhibited that any attractor for monotone systems cannot be chaotic in the sense of topological transitivity (see e.g., \cite{H84,H85attractor,H19}). Here, Theorem A-C provide us a new point of view in measure-theoretic sense to understand such highly unstable complicated dynamics in monotone dynamical systems, which is a widespread phenomenon in such systems. Meanwhile, it also reveals an interaction relationship between the generic (prevalent) asymptotic behavior (in the whole space) and arbitrary dynamics (in the codimension-1 invariant manifolds) in monotone systems.
Our approach is motivated by the \emph{Sharpened $C^1$-dynamics alternative} (Lemma \ref{P:im-da}) and its $C^1$-robustness (Lemma \ref{P:im-da-perturb}) in our recent work \cite{WY21-2}, which is a critical insight for the inherent structure of discrete-time strongly monotone systems. By appealing to the Sharpened $C^1$-dynamics alternative and its $C^1$-robustness and a useful tool of upper (lower) $\omega$-unstable sets introduced by Tak\'{a}\v{c} \cite{Ta92}, we obtain the \emph{Improved prevalent dynamics theorem} (Theorem \ref{T:im-pre}) and its $C^1$-robustness (Theorem \ref{T:im-pre-per}). Together with a formula for the spectral radius of an operator (Lemma \ref{L:spe-radi-norm}), we utilize the improved prevalent dynamics theorem to accomplish our approach.
This paper is organized as follows. In Section \ref{S:Not-pre}, we agree on some notations, give relevant definitions and preliminary results. In Section \ref{S:im-pre}, we establish the improved prevalent dynamics theorem (Theorem \ref{T:im-pre}) and its $C^1$-robustness (Theorem \ref{T:im-pre-per}), which turn out to be crucial in the proof our main results (Theorem A-C). Section \ref{S:tran-chaos} is devoted to the proof of the main results.
\section{Notations and Preliminary results}\label{S:Not-pre}
Let $(X,\|\cdot\|)$ be a Banach space. A cone $C$ is a closed convex subset of $X$ such that $\lambda C\subset C$ for all $\lambda>0$ and $C\cap(-C)=\{0\}$. $(X,C)$ is said to be a strongly ordered Banach space if $C$ has nonempty interior ${\rm Int}C$. For $x,y\in X$, we write $x\leq y$ if $y-x\in C$, $x<y$ if $y-x\in C\setminus\{0\}$, $x\ll y$ if $y-x\in{\rm Int}C$. The reversed signs are used in the usual way. A subset $J^{\prime}\subset X$ is called a \emph{simply ordered, open arc} if there is an increasing homeomorphism $h$ from an open interval $I\subset\mathbb{R}$ onto $J^{\prime}$ ($h$ is \emph{increasing} if $\xi_{1}<\xi_{2}$ implies $h\xi_{1}\ll h\xi_{2}$). A mapping $h:X\to X$ is \emph{monotone} (\emph{strongly monotone}), if $x\leq y$ ($x<y$) implies $hx\leq hy$ ($hx\ll hy$). Similarly, a semiflow $\phi:\RR\times X\to X$ is \emph{monotone} (\emph{strongly monotone}), if $x\leq y$ ($x<y$) implies $\phi_{t}(x)\leq \phi_{t}(y)$ ($\phi_{t}(x)\ll \phi_{t}(y)$) for all $t\geq0$ ($t>0$).
In this paper, we sometimes also need to deal with arguments for another solid cone $C_1(\subset C)$. Hence, for the sake of no confusion, we write $\leq_1, <_1, \ll_1$ as the corresponding order relation induced by the cone $C_1$ throughout the paper.
For a continuous map $h:X\to X$, the orbit of $x\in X$ is denoted by $O(x,h)$=$\{h^{n}x:n\geq0\} $. The $\omega$-limit set of $x\in X$ is $\omega(x,h)=\mathop{\bigcap}\limits_{k\geq0}\overline{\{h^{n}x:n\geq k\}}$. We say that $h$ is $\om$-compact in a subset $Y$ of $X$, if $O(x,h)$ is relatively compact for each $x\in Y$ and $\mathop{\bigcup}\limits_{x\in Y}\om(x,h)$ is relative compact. Given any $x\in X$, we define the \emph{upper} and \emph{lower $\omega$-limit sets} of $x$ by
$$\omega_{+}(x,h):=\bigcap_{\substack{u\in X\\u\gg x}}\overline{\bigcup_{\substack{y\in X\\ u\geq y>x}}\omega(y,h)} \quad and\quad
\omega_{-}(x,h):=\bigcap_{\substack{u\in X\\u\ll x}}\overline{\bigcup_{\substack{y\in X\\ u\leq y<x}}\omega(y,h)},$$
respectively. If $h$ is $\omega$-compact in some neighbourhood of $x$ in $X$, $\omega_{+}(x,h)$ (resp. $\omega_{-}(x,h)$) is non-empty and compact (see Tak\'{a}\v{c} \cite[Proposition 3.1]{Ta92}). Write
$$\mathcal{U}_{+}(h)=\{x\in X:\omega_{+}(x,h)\neq \omega(x,h)\} \quad and\quad \mathcal{U}_{-}(h)=\{x\in X:\omega_{-}(x,h)\neq \omega(x,h)\}$$
as the \emph{upper $\omega$-unstable set} and \emph{lower $\omega$-unstable set} in $X$, respectively. We denote the set $\mathcal{U}_2(h)=\mathcal{U}_{+}(h)\cap\mathcal{U}_{-}(h)$.
A point $x\in X$ is called a \emph{periodic point of $h$}, if $h^{p}x=x$ for some integer $p\geq1$. $p$ is then a \emph{period} of $x$. Moreover, if $h^{l}x\neq x$ for $l=1,2,\cdots,p-1$, $x$ is said to be $p$-periodic. $p$ is the \emph{minimal period} of $x$. Particularly, if $p=1$, $x$ is a fixed point of $h$. $K$ is a \emph{cycle} if $K=O(x,h)$ for some periodic point $x$. For a $C^1$-smooth map $h$ and $x\in X$, we define $$\lambda_{1}(x,h)={\limsup_{n \to +\infty}\frac{log\|Dh^{n}(x)\|}{n}}$$
as the principal \emph{Lyapunov exponent of x} (with respect to $h$). We say a cycle $K=O(x,h)$ (of minimal period $p$) is \emph{linearly stable} if $r_{\sigma}(Dh^p(x))\leq1$, where $r_{\sigma}(Dh^p(x))$ is the spectral radius of $Dh^p(x)$ ($x$ is also said to be a linearly stable $p$-periodic point of $h$). It is equivalent to $\lambda_{1}(x,h)\leq0$ (see e.g., Hess and Pol\'{a}\v{c}ik \cite[p.1316-1317]{PH93}). Particularly, if $p=1$, $x$ is called a linearly stable fixed point of $h$. Let $B\subset X$. $k$ is said to be a \emph{stable period} for the restriction ${h|}_{B}$ if there is a linearly stable $k$-periodic point $x$ of $h$ such that the orbit $O(x,h)\subset B$. If $B=X$, we simply say that $k$ is a stable period of $h$. For brevity, we hereafter say $\omega(x,h)$ is a linearly stable cycle (of minimal period $p$), if $\omega(x,h)$ is a linearly stable cycle (of minimal period $p$) of $h$.
A continuous map $h:X\to X$ is \emph{pointwise dissipative} if there is a bounded subset $B\subset X$ such that $B$ attracts each point of $X$. An invariant set $A$ is called an \emph{attractor} of $h$ if $A$ is the maximal compact invariant set which attracts each bounded subset of $X$. If $h:X\to X$ is compact and pointwise dissipative, then there is a connected attractor $A$ of $h$ (see e.g., \cite[Theorem 2.4.7]{Ha88}).
For a semiflow $\phi:\RR\times X\to X$, the orbit of $x\in X$ is $O(x,\phi)$=$\{\phi_{t}x:t\geq0\}$. The $\omega$-limit set of $x\in X$ is $\omega(x,\phi)=\mathop{\bigcap}\limits_{s\geq0}\overline{\{\phi_{t}x:t\geq s\}}$. A point $e\in X$ is called an \emph{equilibrium} if $\phi_{t}(e)=e$ for any $t\geq0$. We call a semiflow $\phi$ is $C^1$-smooth if the Fr\'echet derivative $D\phi_{t}(x)$ with respect to the state variable $x$ exists for each $x\in X$ and $t>0$, and $x\mapsto D\phi_{t}(x)$ is continuous. For a $C^1$-smooth semiflow $\phi$ and $x\in X$, we define the principal \emph{Lyapunov exponent of x} (with respect to $\phi$) as
$$\lambda_{1}(x,\phi)={\limsup_{n \to +\infty}\frac{log\|D\phi_{t}(x)\|}{t}}.$$
An equilibrium $e$ is called \emph{linearly stable}, if $\lambda_{1}(e,\phi)\leq0$.
Let $M$ be a compact domain of $\mathbb{R}^{d}$ or a finite-dimensional Riemannian manifold and $h: M\to M$ be a ${C}^{1}$-map. We call that $h$ has \emph{observable chaos} if $\lambda_{1}(\cdot,h)>0$ holds on a positive Lebesgue measure set (see e.g., Young
\cite[Section 1, p.1441]{Y13},\cite[Section 4]{Y13'},\cite[Section 3]{Y18}). Similarly, a $C^1$-smooth semiflow $\phi:\RR\times M\to M$ is said to have \emph{observable chaos} if $\lambda_{1}(\cdot,\phi)>0$ holds on a positive Lebesgue measure set. We say that a $C^1$-smooth semiflow $\phi_t$ (resp. mapping $h$) has \emph{transient chaos}, if $\phi_t$ (resp. $h$) has horseshoes but no observable chaos.
In what follows, we introduce the definition and some significant properties of \emph{prevalence} and \emph{shyness}. A Borel subset $W\subset X$ is called \emph{shy} if there exists a nonzero compactly supported Borel measure $\mu$ on $X$ such that $\mu(W+x)=0$ for every $x\in X$. A Borel subset $W\subset X$ is \emph{prevalent (in $X$)} if its complement $X\backslash W$ is shy. Given a Borel subset $V\subset X$, we say that a Borel subset $W\subset X$ is \emph{prevalent in} $V$ if $V\setminus W$ is shy.
Shyness has the following fundamental properties (\cite{HSY92,HK10}):
(i) Every Borel subset of a shy set is shy;
(ii) Every translation of a shy set is shy;
(iii) No nonempty open set is shy;
(iv) Every countable union of shy sets is shy;
(v) In finite-dimensional spaces, a Borel set $W$ is shy if and only if it has Lebesgue measure
\quad \ \ \,zero.
Now, we present a sufficient condition which guarantees a Borel subset $W\subset X$ to be shy.
\begin{prop}\label{P:Borel-suffi-condi}
Let $W\subset X$ be a Borel subset and assume that there exists $v\gg0$ such that $L\cap W$ is countable for every straight line $L$ parallel to $v$. Then $W$ must be shy in $X$.
\end{prop}
\begin{proof}
See Enciso, Hirsch and Smith \cite[Lemma 1]{EHS08}.
\end{proof}
The finite-dimensional space spanned by $v$ is also called a \emph{probe}, which is a very useful tool to show prevalence (see e.g., \cite{HSY92}). In the following, we give a structure proposition of $\mathcal{U}_{-}$ ($\mathcal{U}_{+}$, resp.), which is crucial in our proof of prevalence in Section \ref{S:im-pre}.
\begin{prop}\label{P:unsta-count}
Let $D\subset X$ be an open subset and $h:D\to D$ be strongly monotone. Assume that the set
$$D_0=\{x\in D:\ O(x,h)\text{ is relatively compact }\}$$
is dense in $D$. Let $J^{\prime}\subset D_0$ be a simply ordered, open arc. Then the set $J^{\prime}_{-}=J^{\prime}\cap\mathcal{U}_{-}(h)$ is at most countable. A corresponding result holds for $\mathcal{U}_{+}(h)$.
\end{prop}
\begin{proof}
See \cite[Corollary 3.4]{Ta92}.
\end{proof}
\begin{lem}\label{L:cycle-fixed pt}
Let $h:X\to X$ be a $C^1$-map. If $z$ is a linearly stable $k$-periodic point of $h^q$, then $z$ is a linearly stable periodic point of $h$ of minimal period at most $kq$.
\end{lem}
\begin{proof}
See the claim in the proof of \cite[Corollary 2.5]{WY21-2}.
\end{proof}
\section{Improved prevalent dynamics/and its $C^1$-robustness for $C^1$-perturbed Systems}\label{S:im-pre}
In this section, we will focus on the proof of the prevalence of ``convergence to cycles whose minimal periods are uniformly bounded'' (Theorem \ref{T:im-pre}), as well as its $C^1$-robustness for $C^1$-perturbed systems (Theorem \ref{T:im-pre-per}). Theorem \ref{T:im-pre} and Theorem \ref{T:im-pre-per} will play a crucial role in our approach for the nonexistence of observable chaos and its $C^1$-robustness in Section \ref{S:tran-chaos}.
For $C^1$-smooth strongly monotone mapping $F_0$, we have the following sharpened dynamics alternative:
\begin{lem}\label{P:im-da}
{\rm (Sharpened $C^1$-dynamics alternative).} Assume that \textnormal{(H1)-(H2)} hold. Assume also $F_{0}$ is pointwise dissipative. Then there is an integer $m>0$ such that, for any $x\in X$, either
\textnormal{(a)} $\omega(x,F_{0})$ is a linearly stable cycle of minimal period at most $m$; or,
\textnormal{(b)} there is a constant $\delta>$ 0 such that, for any $y \in X$ satisfying $y < x$ or $y > x$,
$$\mathop{\limsup}\limits_{n \to +\infty}\|F_{0}^{n}x-F_{0}^{n}y\|\geq\delta.$$
\end{lem}
\begin{proof}
See Wang and Yao \cite[Theorem A]{WY21-2}.
\end{proof}
\begin{rmk}\label{R:bdd-per-F0}
In fact, all the stable periods of $F_{0}$ are bounded above by $m$ (see \cite[Corollary 2.5]{WY21-2}).
\end{rmk}
\begin{thm}\label{T:im-pre}
{\rm (Improved prevalent dynamics).} Assume that \textnormal{(H1)-(H2)} hold. Assume also $F_{0}$ is pointwise dissipative with an attractor $A$. Then there is an integer $m>0$ such that the set
$$Q_{0}:=\{x\in X:\omega(x,F_{0})\text{ is a linearly stable cycle of minimal period at most }m\}$$
is prevalent in $X$. In particular, if $X=\mathbb{R}^{d}$, then the set $Q_0$ is of full Lebesgue measure in $X$.
\end{thm}
\begin{rmk}\label{R:Wang et al.}
For $C^1$-smooth strongly monotone discrete-time dynamical systems, Wang et al. \cite[Theorem A]{WYZ20-2} obtained the ``convergence to linearly stable cycles'' is a prevalent asymptotic behavior in the measure theoretic sense. Here, Theorem \ref{T:im-pre} improves \cite[Theorem A]{WYZ20-2} by showing that the set of minimal periods of linearly stable cycles is bounded by $m$. In addition, we succeed in proving the measurability of the sets involved by virtue of the boundedness of stable periods. Moreover, Theorem \ref{T:im-pre} is also shown to be robust under the $C^{1}$-perturbation (see Theorem \ref{T:im-pre-per}).
\end{rmk}
In order to prove the improved prevalent dynamics in Theorem \ref{T:im-pre}, it is important to prove first the set $Q_0$ is Borel. For this purpose, we need the following lemma.
\begin{lem}\label{L:borel-lsequi}
Let $B\subset X$ be a Borel subset and $h:B\to B$ be a $C^1$-map. Then the set $C_{s}:=\{x\in B: \omega(x,h)=z\ \text{for some linearly stable fixed point $z$ of $h$ }\}$ is Borel.
\end{lem}
\begin{proof}
Enciso, Hirsch and Smith \cite[Appendix, Lemma 10]{EHS08} proved this lemma for $C^1$-semiflows. One can follow the exactly same arguments and obtain an analogous proof for $C^1$-maps. We omit it here.
\end{proof}
\vskip 2mm
\noindent
{\it Proof of Theorem \ref{T:im-pre}.}
Let the integer $m>0$ be in Lemma \ref{P:im-da}. Define
$$Q_{0}:=\{x\in X:\omega(x,F_{0})\text{ is a linearly stable cycle of minimal period at most }m\},$$
and
$$Q_{0}^{\prime}:=\{x\in X:\omega(x,F_{0}^{m!})\text{ is a linearly stable fixed point}\}.$$
We first show that $Q_{0}=Q_{0}^{\prime}$. It is clear that if $\omega(x,F_{0})$ is a linearly stable cycle, then $\omega(x,F_{0}^{m!})$ is a linearly stable cycle. This entails that $Q_{0}\subset Q_{0}^{\prime}$. Notice from Remark \ref{R:bdd-per-F0} that all the stable periods of $F_{0}$ are bounded above by $m$. Then, $Q_{0}^{\prime}\subset Q_{0}$. Thus, $Q_{0}=Q_{0}^{\prime}$.
It follows from Lemma \ref{L:borel-lsequi} that $Q_{0}^{\prime}$ is Borel. So, $Q_{0}$ is also Borel. By virtue of Lemma \ref{P:im-da}, we can repeat the exactly same arguments in Pol\'{a}\v{c}ik and Tere\v{s}\v{c}\'{a}k \cite[Section 5]{PT92} to obtain that $X\setminus Q_{0}\subset\mathcal{U}_{2}(F_{0})$. Proposition \ref{P:unsta-count} implies that, for every simply ordered, open arc $J^{\prime}$, $J^{\prime}\cap\mathcal{U}_{2}(F_{0})$ is at most countable. So, $J^{\prime}\setminus Q_{0}$ is also at most countable. Thus, Proposition \ref{P:Borel-suffi-condi} entails that $X\setminus Q_{0}$ is shy. Therefore, $Q_{0}$ is prevalent.
In particular, if $X=\mathbb{R}^{d}$, then $Q_{0}$ is of full Lebesgue measure in $X$, since the property (v) of shyness in Section \ref{S:Not-pre}. We have completed the proof.\hfill $\square$
\vskip 3mm
Hereafter in this section, we consider the $C^1$-robustness for the improved prevalent dynamics. The following lemma reveals that the sharpened dynamics alternative of $F_0$ (Lemma \ref{P:im-da}) is robust under the $C^1$-perturbations.
\begin{lem}\label{P:im-da-perturb}
{\rm ($C^1$-robustness for sharpened $C^1$-dynamics alternative).} Assume that \textnormal{(H1)-(H3)} hold and $F_0$ is pointwise dissipative with an attractor $A$. Let $B_1$ be an open ball containing $A$. If
$$\sup\{\Vert F_{\epsilon}x-F_0x\Vert +\Vert DF_{\epsilon}(x)-DF_0(x)\Vert :\epsilon\in J, x\in B_1\}<\epsilon^{\prime}$$ for some sufficiently small $\epsilon^\prime >0$, then there is a solid cone $C_1\subset{\rm Int}C$, an open bounded set $D_1$ ($B_{1}\supset D_1\supset A$) and integers $q,m_1>0$ such that, for each $|\epsilon|$ sufficiently small
{\rm (i)}. For each $n\geq q$, $F_{\epsilon}^{n}(D_1)\subset D_1$ and $F_{\epsilon}^{n}x\ll_1F_{\epsilon}^{n}y$ whenever $x<_1y$ (with $x, y\in D_1$).
{\rm (ii)}. For each $x\in D_1$, either
\quad\quad\textnormal{(a)} $\omega(x,F_{\epsilon}^{q})$ is a linearly stable cycle of minimal period at most $m_1$; or,
\quad\quad\textnormal{(b)} there is a constant $\delta>$ 0 such that, for any $y \in D_1$ satisfying $y <_1x$ or $y >_1 x$,$$\mathop{\limsup}\limits_{n \to +\infty}\|F_{\epsilon}^{nq}x-F_{\epsilon}^{nq}y\|\geq\delta.$$
\end{lem}
\begin{proof}
For the proof of item (i), we refer to Tere\v{s}\v{c}\'{a}k \cite[Theorem 5.1]{T94}. For the proof of item (ii), see \cite[Theorem 3.1]{WY21-2}.
\end{proof}
\begin{rmk}\label{R:bdd-per-Fep}
In fact, for any $|\epsilon|$ sufficiently small, all the stable periods of ${F_{\epsilon}^{q}|}_{\overline{D}_1}$ are bounded above by $m_1$ (see \cite[Proposition 2.4]{WY21-2}). In addition, for any open bounded subset $D_2$ (satisfying $D_1\supset\overline{D}_2\supset D_{2}\supset A$), one can choose $\ep^\prime$ smaller and let the integer $q>0$ larger (if necessary) in Lemma \ref{P:im-da-perturb} such that, both $D_1$ and $D_2$ satisfies items (i)-(ii) in Lemma \ref{P:im-da-perturb} (see \cite[Remark 2.3]{WY21-2} or \cite[Eq.(5.11) on p.19]{T94}). Moreover, one can also choose $D_2$ to be connected, since $A$ is connected. If $X=\mathbb{R}^d$, this leads $\overline{D}_2$ to be a compact domain.
\end{rmk}
Hereafter in this paper, we always reserve the open bounded subsets $D_1,D_2$ (with $D_1\supset\overline{D}_2\supset D_{2}\supset A$) and the integers $q,m_1>0$ as in Lemma \ref{P:im-da-perturb} and Remark \ref{R:bdd-per-Fep}.
The following theorem reveals that the improved prevalent dynamics of $F_0$ (Theorem \ref{T:im-pre}) is robust under the $C^1$-perturbation.
\begin{thm}\label{T:im-pre-per}
{\rm ($C^1$-robustness for improved prevalent dynamics).} Let all hypotheses in Lemma \ref{P:im-da-perturb} hold. Then there exists an open bounded set $D_1\supset A$ and an integer $m>0$ such that, for any $|\epsilon|$ sufficiently small, the set
$$Q_{\epsilon}:=\{x\in D_1:\omega(x,F_{\epsilon})\text{ is a linearly stable cycle of minimal period at most }m\}$$
is prevalent in $D_1$. In particular, if $X=\mathbb{R}^{d}$, then the set $Q_{\epsilon}$ is of full Lebesgue measure in $D_1$.
\end{thm}
\begin{proof}
Let the open bounded subset $D_1$ and the integers $q,m_1>0$ be in Lemma \ref{P:im-da-perturb}. Clearly, it follows from (H3) that $F_\ep^q:D_1\to D_1$ is compact. Moreover, for each $|\epsilon|$ sufficiently small, Lemma \ref{P:im-da-perturb}(i) directly implies that $F_\ep^q:D_1\to D_1$ is {\it strongly monotone with respect to $C_{1}$}. By Lemma \ref{P:im-da-perturb}(ii), we can repeat the exactly same arguments in the proof of Theorem \ref{T:im-pre} (with $F_0$ replaced by $F_\ep^q$ there) to obtain that, the set
\begin{equation}\label{E:Fep-m-pre}
\tilde{Q}_{\epsilon}:=\{x\in D_{1}:\omega(x,F_{\epsilon}^{q})\text{ is a linearly stable cycle of minimal period at most }m_1\}
\end{equation}
is prevalent in $D_1$, for any $|\epsilon|$ sufficiently small.
Now, we define
$$Q_{\epsilon}:=\{x\in D_{1}:\omega(x,F_{\epsilon})\text{ is a linearly stable cycle of minimal period at most }m\},$$
where $m=m_1q$. On one hand, it is clear that if $\omega(x,F_{\epsilon})$ is a linearly stable cycle, then $\omega(x,F_{\epsilon}^{q})$ is a linearly stable cycle.
Then, the bound $m_1$ of stable periods of ${F_{\epsilon}^{q}|}_{\overline{D}_1}$ in Remark \ref{R:bdd-per-Fep} entails that, $Q_{\epsilon}\subset\tilde{Q}_{\epsilon}$. On the other hand, it follows from Lemma \ref{L:cycle-fixed pt} that $\tilde{Q}_{\epsilon}\subset Q_{\epsilon}$. Then, we have proved $Q_{\epsilon}=\tilde{Q}_{\epsilon}$. Thus, $\tilde{Q}_{\epsilon}$ is also prevalent in $D_{1}$, for any $|\epsilon|$ sufficiently small.
In particular, if $X=\mathbb{R}^{d}$, then $Q_{\ep}$ is of full Lebesgue measure in $X$, since the property (v) of shyness in Section \ref{S:Not-pre}.
This completes the proof.\hfill $\square$
\end{proof}
\begin{rmk}
We specially point out that, one can apply our theoretical result in this section (Theorem \ref{T:im-pre} and Theorem \ref{T:im-pre-per}) to obtain that, the improved prevalence of convergence to periodic solutions with a uniform bound of minimal periods, for time-periodic parabolic equations and their perturbed systems (see e.g. the equations in \cite{PH93,PT93,WY21-2,WYZ20-2}).
\end{rmk}
\section{Proof of the Main Results}\label{S:tran-chaos}
We focus on in this section and prove Theorem A-C. In order to prove these results, we need the following formula for the spectral radius of an operator.
\begin{lem}\label{L:spe-radi-norm}
Let $T\in\mathscr{L}(X)$. Then $r_{\sigma}(T)=\inf |T|$, where the infimum is taken over all norms $|\cdot|$ on $X$ equivalent to $\|\cdot\|$.
\end{lem}
\begin{proof}
See Holmes \cite[Theorem on p.164]{Holmes}. For finite-dimensional case, see also Horn and Johnson \cite[Lemma 5.6.10]{Johnson}.
\end{proof}
Now, we are in position to prove Theorem A-C.
\vskip 2mm
\noindent
{\it Proof of Theorem A.}
Let the integer $m>0$ be obtained in Theorem \ref{T:im-pre}. Take $T:=F_{0}^{m!}$. We first {\it assert} that: \emph{If $x\in X$ with $\omega(x,T)=z$ for some linearly stable fixed point $z$ of $T$, then $\lambda_{1}(x,F_{0})\leq0$}.
In fact, one has $r_{\sigma}(DT(z))\leq1$, since $z$ is a linearly stable fixed point of $T$. Then by Lemma \ref{L:spe-radi-norm}, for any $\tilde{\ep}>0$, there exists a norm $|\cdot|$ equivalent to $\|\cdot\|$ on $X$ such that
$$|DT(z)|<r_{\sigma}(DT(z))+\tilde{\ep}.$$
Hence, $|DT(z)|<1+\tilde{\ep}$. Recall that $T$ is $C^1$. Then there exists a $\delta>0$ such that,
$$|DT(y)|<1+2\tilde{\ep}, \text{ for any } y\in X \text{ with } |y-z|<\delta.$$
It follows from $\omega(x,T)=z$ that, there exists an integer $N>0$ such that, $|T^nx-z|<\delta$ for any $n\geq N$. Thus, the chain rule shows that
\begin{alignat*}{2}
\lambda_{1}(x,T)&=\limsup_{n \to +\infty}\frac{log|DT^{n}(x)|}{n}\leq\limsup_{n \to +\infty}\frac{log|DT^{N}(x)|+log|DT^{n-N}(T^Nx)|}{n}\\
&\leq\limsup_{n \to +\infty}\frac{log|DT^{N}(x)|+(n-N)log(1+2\tilde{\ep})}{n}\leq\log(1+2\tilde{\ep}).
\end{alignat*}
Since the arbitrary of $\tilde{\ep}$, we have $\lambda_{1}(x,T)\leq0$.
For any integer $n\geq1$, write $n=k_{n}m!+l_{n}$, where $k_{n}\geq0$ and $l_{n}\in\{0,1,2,\cdots,m!-1\}$. Then
\begin{eqnarray*}\label{E3.2}
\lambda_{1}(x,F_{0})&=&{\limsup_{n \to +\infty}\frac{log|DF_{0}^{n}(x)|}{n}}
={\limsup_{n \to +\infty}\frac{log|DF_{0}^{k_{n}m!+l_{n}}(x)|}{k_{n}m!+l_{n}}}\\
&\leq& {\limsup_{n \to +\infty}\frac{log(M|DF_{0}^{k_{n}m!}(x)|)}{k_{n}m!+l_{n}}}=\frac{1}{m!}{\limsup_{k \to +\infty}\frac{log|DT^{k}(x)|}{k}}\\
&=&\frac{1}{m!}\lambda_{1}(x,T)\leq0,
\end{eqnarray*}
where $M:=\max\{|DF_{0}^{l}(u)|:\ 0\leq l\leq m!-1,\ u\in\overline{O(x,F_{0})}\}$. Thus, we have proved the assertion.
Define
$$S_{0}:=\{x\in X:\ \lambda_{1}(x,F_{0})\leq 0\}.$$
Recall that $F_{0}$ is $C^1$. Then the function $x\mapsto \frac{log\|DF_0^n(x)\|}{n}$ is Borel-measurable, for any $n\geq1$. Hence, $x\mapsto \lambda_1(x,F_0)$ is Borel-measurable. Thus, $S_{0}$ is Borel.
Now, we define
$$Q_{0}^{\prime}:=\{x\in X:\omega(x,T)\text{ is a linearly stable fixed point}\}.$$
The assertion entails that $Q_{0}^{\prime}\subset S_{0}$. It follows from the proof of Theorem \ref{T:im-pre} that $Q_{0}^{\prime}$ is prevalent in $X$. Then $S_{0}$ is also prevalent in $X$, since the property (i) of shyness in Section \ref{S:Not-pre}.
In particular, if $X=\mathbb{R}^{d}$, $S_{0}$ is of full Lebesgue measure in $X$, since the property (v) of shyness in Section \ref{S:Not-pre}.
It follows that $F_{0}$ cannot have observable chaos. Thus, we have proved Theorem A.\hfill $\square$
\vskip 3mm
\noindent
{\it Proof of Theorem B.}
Let the open bounded subsets $D_1,D_2$ (with $D_1\supset\overline{D}_2\supset D_{2}\supset A$) and the integer $q>0$ be in Lemma \ref{P:im-da-perturb} and Remark \ref{R:bdd-per-Fep}. Define the closed bounded set $M:=\overline{D}_2$. We note that $F_{\ep}^{q}(M)\subset M$, since $F_{\ep}^{q}(D_2)\subset D_2$. Define
$$S_{\ep,D_1}:=\{x\in D_1:\ \lambda_{1}(x,F_{\ep}^{q})\leq 0\}.$$
By virtue of \eqref{E:Fep-m-pre}, one can follow the exactly same proof of Theorem A to obtain that, $S_{\ep,D_1}$ is prevalent in $D_1$. That is to say, $D_1\setminus S_{\ep,D_1}$ is shy. Now, we define
$$S_{\ep}:=\{x\in M:\ \lambda_{1}(x,F_{\epsilon})\leq 0\}.$$
Clearly,
$$S_{\ep}=\{x\in M:\ \lambda_{1}(x,F_{\ep}^q)\leq 0\}.$$
Then, $M\setminus S_{\ep}\subset D_1\setminus S_{\ep,D_1}$. Moreover, the same reason of $S_{0}$ is Borel in the proof of Theorem A also entails that, $S_{\ep}$ is Borel. So $M\setminus S_{\ep}$ is shy, since the property (i) of shyness in Section \ref{S:Not-pre}. Thus, $S_{\ep}$ is prevalent in $M$.
In particular, if $X=\mathbb{R}^{d}$, then $M$ is a compact domain in $\mathbb{R}^{d}$. Thus $S_{0}$ is of full Lebesgue measure in $M$, since the property (v) of shyness in Section \ref{S:Not-pre}. It follows that $F_{\ep}^{q}|_M$ cannot have observable chaos. The proof is completed.
\vskip 3mm
\noindent{\it Proof of Theorem C.}
Define the sets
$$C_{s}:=\{x\in X:\ \omega(x,\phi)\ \textnormal{is a linearly stable equilibrium}\},$$
and
$$S_{\phi}:=\{x\in X:\ \lambda_{1}(x,\phi)\leq 0\}.$$
Enciso, Hirsch and Smith \cite[Theorem 1]{EHS08} entails that, $C_{s}$ is prevalent in $X$. We are going to prove $C_{s}\subset S_{\phi}$. Suppose that $x\in C_{s}$ and $\om(x,\phi)=e$. Let $\tilde{t}>0$. Then $r_{\sigma}(D\phi_{\tilde{t}}(e))\leq1$. Thus we can repeat the exactly same arguments in the proof of Theorem A (with $F_{0}$ and $T$ replaced by $\phi_{\tilde{t}}$ there) to obtain that, $\lambda_{1}(x,\phi_{\tilde{t}})\leq0$. Thus, $\lambda_{1}(x,\phi)\leq0$ since the chain rule and the compactness of $\overline{O(x,\phi)}$. Hence, $x\in S_{\phi}$. That is to say, $C_{s}\subset S_{\phi}$.
Recall that $\phi$ is a $C^1$-semiflow. Then $x\mapsto \lambda_1(x,\phi)$ is Borel-measurable. Thus, $S_{\phi}$ is Borel. Therefore, $S_{\phi}$ is also prevalent in $X$, since the property (i) of shyness in Section \ref{S:Not-pre}.
In particular, if $X=\mathbb{R}^{d}$, $S_{\phi}$ is of full Lebesgue measure in $X$, since the property (v) of shyness in Section \ref{S:Not-pre}.
It follows that $\phi$ cannot have observable chaos. Thus, we have proved Theorem C.\hfill $\square$
\begin{rmk}
A large number of existing works have shown that monotone dynamical systems may have complicated dynamics, including horseshoes (see e.g., \cite{DP98,S76,S88,S95,S97,WJ01,WX10} and references therein). As a consequence of Theorem A-C, those chaotic behavior in monotone dynamical systems are all transient chaos.
\end{rmk}
| 173,508
|
TITLE: A coordinate free proof of the identity $Tr(A \otimes B)= Tr(A) \otimes Tr(B)$
QUESTION [5 upvotes]: I was going over some of the definitions of trace and was trying to find a way to prove one of the facts stated about the trace listed in the referenced Wikipedia article.
Let $M, N$ be finitely generated projective $R$-modules over a commutative ring $R$ with identity.
In particular I was wondering if there was a slick proof using the coordinate free language of the trace for the following fact.
How do we show that for any $f \in End_R(M), v \in End_R(N)$ then $ Tr( f \otimes_R g) = Tr(f) \otimes_R Tr(g)$?
REPLY [2 votes]: Warning: There will be some cheating below, in the sense that the phrase "the obvious map from ... to ..." will often be used. Thank you in advance for letting me know if I should be more explicit.
References will be made to H. Cartan and S. Eilenberg's classic book Homological algebra, legally available at Internet Archive.
LEMMA. Let $A$ be a ring (that is, an associative ring with $1$), let $\mathcal A$ be the category of $A$-modules, let $\mathcal B$ be the category opposite to $\mathcal A$, let $F$ and $G$ be additive functors from $\mathcal B^p\times\mathcal A^q$ to the category of abelian groups, and let $\varphi$ be a functorial morphism from $F$ to $G$. Assume that
$$
\varphi(A,\dots,A):F(A,\dots,A)\to G(A,\dots,A)
$$
is an isomorphism, and that $P_1,\dots,P_{p+q}$ are finitely generated projective $A$-modules. Then
$$
\varphi(P_1,\dots,P_{p+q}):F(P_1,\dots,P_{p+q})\to G(P_1,\dots,P_{p+q})
$$
is an isomorphism.
The statement results immediately from the following two facts:
Additive functors are compatible with finite direct sums in the sense of Proposition II.1.1 of Cartan-Eilenberg.
Let $f_1,\dots,f_n$ be $A$-linear maps. Then $\bigoplus f_i$ is bijective if and only if each $f_i$ is.
Assume that $A$ is commutative. The letters $U,V,X,Y$ will refer to "variable" $A$-modules. The letters $P,Q$ will refer to "variable" projective $A$-modules. Write respectively
$$
\otimes,\quad\quad\text{Hom},\quad\quad\text{End},\quad\quad X^*,\quad\quad\text{map}
$$
for
$$
\otimes_A,\quad\text{Hom}_A,\quad\text{End}_A,\quad \text{Hom}_A(X,A),\quad\text{$A$-linear map}.
$$
Associativity and commutativity of the tensor product will be tacitly used.
Recall that the trace of an endomorphism $f$ of $P$ is defined as follows. On applying the Lemma to the obvious functorial morphism
$$
U^*\otimes X\to\text{Hom}(U,X),
$$
we get a canonical isomorphism
$$
P^*\otimes P\ \overset{\sim}{\to}\ \text{End}(P).
$$
Then the trace of $f$ is defined by transporting to $\text{End}(P)$ the obvious map from $P^*\otimes P$ to $A$.
On applying the Lemma to the obvious functorial morphism
$$
\text{Hom}(U,X)\otimes\text{Hom}(V,Y)\to\text{Hom}(U\otimes V,X\otimes Y),
$$
we get canonical isomorphisms
$$
\text{End}(P)\otimes\text{End}(Q)\ \overset{\sim}{\to}\ \text{End}(P\otimes Q),\quad
P^*\otimes Q^*\ \overset{\sim}{\to}\ (P\otimes Q)^*.
$$
Let $f$ be in $\text{End}(P)$ and $g$ be in $\text{End}(Q)$. To check
$$\text{trace}(f\otimes g)=\text{trace}(f)\ \text{trace}(g),$$
it suffices to verify that the compositions of canonical isomorphisms
$$
P^*\otimes P\otimes Q^*\otimes Q\ \overset{\sim}{\to}\
\text{End}(P)\otimes\text{End}(Q)\ \overset{\sim}{\to}\
\text{End}(P\otimes Q)
$$
and
$$
P^*\otimes P\otimes Q^*\otimes Q\ \overset{\sim}{\to}\
(P\otimes Q)^*\otimes P\otimes Q\ \overset{\sim}{\to}\
\text{End}(P\otimes Q)
$$
coincide. But this is clear.
| 172,500
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Do you know someone who went the extra mile during Harrogate’s spectacular 2014?
The district’s tourism industry enjoyed a bumper year, as thousands of people arrived from across the globe to enjoy the Tour de France.
Hotels and businesses enjoyed a record-breaking year, and now you have the chance to nominate those on the front line for an exclusive
award.
Now in its seventh year, Harrogate’s Hospitality and Tourism Awards aim to celebrate and reward outstanding individuals and teams who truly represent Harrogate as one of the country’s leading destinations for both business and tourism.
The awards are organised by David Ritson and Simon Cotton of Destination Harrogate.
Mr Cotton said: “2014 has to be one of the most successful years Harrogate has ever had in the tourism sector off the back of the Tour de
France.
“In the hotel sector I know my business had a record year and I know that was celebrated by many other people in the tourism sector.
“We only did that with having fantastic people on the front line giving exceptional customer service to all these visitors day in, day out and we want to say thank you to them.
“Make sure you put them forward for a Harrogate Hospitality and Tourism Award.”.
How to enter
Nominate online at or post an entry form at the Tourist Information Centre.
You will also find voting forms in hotels, bars, restaurants and the library.
Closing date for entries is March 31, 2015. After this date a panel of judges will create a shortlist from each category who will be invited to attend a prestigious gala dinner at the Royal Hall on June 1, where winners will be announced.
Get involved on Twitter: Use #HHTA15
| 105,060
|
\begin{document}
\arraycolsep=1pt
\title{\bf\Large Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces
and Their Applications
\footnotetext{\hspace{-0.35cm} 2010 {\it
Mathematics Subject Classification}. Primary 42B35;
Secondary 42B30, 42B25, 42B20, 30L99.
\endgraf {\it Key words and phrases.}
anisotropic Euclidean space, (mixed-norm) Hardy space, Calder\'on-Zygmund decomposition,
discrete Calder\'{o}n reproducing
formula, grand maximal function, atom,
Littlewood-Paley function, Calder\'{o}n-Zygmund operator.
\endgraf This project is supported by the National
Natural Science Foundation of China
(Grant Nos.~11761131002, 11571039, 11726621 and 11471042).}}
\author{Long Huang, Jun Liu, Dachun Yang
and Wen Yuan\footnote{Corresponding author / January 22, 2018.}}
\date{}
\maketitle
\vspace{-0.8cm}
\begin{center}
\begin{minipage}{13cm}
{\small {\bf Abstract}\quad
Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$,
$\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$
and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$
be the anisotropic mixed-norm Hardy space associated with $\vec{a}$
defined via the non-tangential grand maximal function. In this article,
via first establishing a
Calder\'{o}n-Zygmund decomposition and a discrete Calder\'{o}n reproducing
formula, the authors then characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$,
respectively, by means of atoms,
the Lusin area function, the Littlewood-Paley $g$-function or
$g_{\lambda}^\ast$-function.
The obtained Littlewood-Paley $g$-function characterization of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ coincidentally confirms a conjecture proposed
by Hart et al. [Trans. Amer. Math. Soc. (2017), DOI: 10.1090/tran/7312].
Applying the aforementioned Calder\'{o}n-Zygmund decomposition as well as
the atomic characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$,
the authors establish a finite atomic characterization of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which further induces
a criterion on the boundedness of sublinear operators
from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space.
Then, applying this criterion, the authors obtain the boundedness of anisotropic
Calder\'{o}n-Zygmund operators from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ to
itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. The obtained
atomic characterizations of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$
and boundedness of anisotropic Calder\'{o}n-Zygmund
operators on these Hardy-type spaces positively answer
two questions mentioned by Cleanthous et al. in [J. Geom. Anal. 27 (2017), 2758-2787].
All these results are new even for the isotropic mixed-norm Hardy spaces on $\mathbb{R}^n$.
}
\end{minipage}
\end{center}
\vspace{0.215cm}
\section{Introduction\label{s1}}
The real-variable theory of Hardy spaces on the Euclidean space $\rn$ certainly plays an
important role in analysis, including harmonic analysis, partial differential equations
and geometrical analysis, and has been systematically studied and
developed so far; see, for example, \cite{fs72,lg14b,lu,s93,sw60}. It is well known that
Hardy spaces are good substitutes of Lebesgue spaces $L^p(\rn)$, with $p\in(0,1]$,
particularly, in the study on the boundedness of maximal functions
and Calder\'{o}n-Zygmund operators.
Notice that, as a generalization of the classical Lebesgue space $L^p(\rn)$,
the mixed-norm Lebesgue space $\lv$, in which the constant exponent
$p$ is replaced by an exponent vector $\vp\in [1,\fz]^n$, was studied by Benedek and
Panzone \cite{bp61} in 1961, which can be traced back to H\"{o}rmander \cite{h60}.
Later, in 1970, Lizorkin \cite{l70} further studied both the theory of
multipliers of Fourier integrals and estimates of convolutions
in the mixed-norm Lebesgue spaces.
Moreover, very recently, there appears a renewed increasing interest in
the theory of mixed-norm function spaces, including mixed-norm Lebesgue spaces,
mixed-norm Hardy spaces, mixed-norm Besov spaces and mixed-norm Triebel-Lizorkin spaces;
see, for example, \cite{cgn17,cgn17-2,gn16,htw17,jms13,jms14,jms15}.
For more developments of mixed-norm function spaces, we refer the reader to
\cite{cs,cgn17bs,f87,gjn17,js07,js08}.
On the other hand, due to the celebrated work \cite{c77,ct75,ct77} of Calder\'{o}n and
Torchinsky on parabolic Hardy spaces, there has been an increasing interest
in extending classical function spaces from Euclidean spaces to
some more general underlying spaces; see, for example,
\cite{mb03,ds16,fs82,hmy06,hmy08,s13,s16,s17,st87,t06,t15,yyh13}.
In particular, Bownik \cite{mb03} studied the anisotropic Hardy space $H_A^p(\rn)$ with
$A$ being a general expansive matrix on $\rn$ and $p\in(0,\fz)$,
which was a generalization of parabolic Hardy spaces introduced in \cite{c77}.
Later on, Bownik et al. \cite{blyz08} further extended the anisotropic Hardy space on $\rn$
to the weighted setting. For more progresses about the theory of anisotropic function spaces on $\rn$,
we refer the reader to \cite{blyz10,fhly17,lby14,lbyz10,lfy15,lwyy17,lyy16,lyy16LP,lyy17hl}
for anisotropic Hardy-type spaces and to \cite{mb05,mb07,bh06,lbyy12,lbyy14,lyy17}
for anisotropic Besov and Triebel-Lizorkin spaces.
Very recently, Cleanthous et al. \cite{cgn17} introduced the anisotropic mixed-norm Hardy space $\vh$
with $\va\in [1,\fz)^n$ and $\vp\in (0,\fz)^n$ via the non-tangential grand maximal function and
established its radial or non-tangential maximal function characterizations; moreover, they
mentioned several natural questions to be studied, which include the atomic characterizations
of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ and the boundedness of anisotropic Calder\'{o}n-Zygmund
operators on these Hardy-type spaces.
For more progresses about this theory, we refer the reader to
\cite{bn08,cgn17,f00,fjs00,jms13,jms14,jms15,yam86a,yam86b}.
Notice that, when $\vp:=(p,\ldots,p)\in(0,\fz)^n$, the anisotropic mixed-norm Hardy space $\vh$
becomes the anisotropic Hardy space $H_{\va}^p(\rn)$. Here,
we should point out that, in this case, $H_{\va}^p(\rn)$
and the anisotropic Hardy space $H_A^p(\rn)$ (see \cite{mb03})
coincide with equivalent quasi-norms, where $A$ is as in \eqref{4e5} below
(see Proposition \ref{2r4'} below).
In addition, Hart et al. \cite{htw17} introduced the mixed-norm Hardy space
$H^{p,q}(\mathbb{R}^{n+1})$ with $p,\,q\in(0,\fz)$ via the Littlewood-Paley $g$-function and
showed that $H^{p,q}(\mathbb{R}^{n+1})$, when $p,\,q\in(1,\fz)$, coincides,
in the sense of equivalent quasi-norms,
with $H_{\va}^{\vp}(\mathbb{R}^{n+1})$ from \cite{cgn17} when
$$\va:=(\overbrace{1,\ldots, 1}^{n+1\ \mathrm{times}})\quad {\rm and}\quad
\vp:=(\overbrace{p,\ldots,p}^{n\ \mathrm{times}},q)\in(1,\fz)^{n+1},$$
which is defined via the non-tangential grand maximal function
(see \cite[Definition 3.3]{cgn17} or Definition 2.11 below); moreover,
Hart et al. in \cite[p.\,9]{htw17} stated that ``We do not know if such mixed Hardy spaces coincide with the
$H^{p,q}(\mathbb{R}^{n+1})$ above for other values of $p$ and $q$, but it is likely",
in which such mixed Hardy spaces mean the Hardy-type spaces $\vh$.
In addition, recall that the classical isotropic singular integral operator
was first introduced by Calder\'{o}n and Zygmund \cite{cz52}, in which they
established the boundedness of these operators on $L^p(\rn)$ for any $p\in(1,\fz)$.
Later, Fern\'{a}ndez \cite{f87} investigated the corresponding boundedness
of some classical isotropic singular integral operators on the mixed-norm Lebesgue space
$\lv$ with $\vp\in (1,\fz)^n$ (see also Stefanov and Torres \cite{st04}).
For more developments of the boundedness of the classical isotropic singular
integral operators, we refer the reader to Torres \cite{t91}.
On the other hand, in 1966, Besov et al. \cite{bil66} and, independently,
Fabes and Rivi\`{e}re \cite{f66} introduced a class of anisotropic singular
integral operators and obtained the $L^p(\rn)$ boundedness of these operators
for any $p\in(1,\fz)$. Moreover, the boundedness of these anisotropic singular
integral operators from \cite{f66} on generalized Morrey spaces was studied by
Guliyev and Mustafayev \cite{gm11} in 2011, which extends the corresponding
results obtained by Besov et al. \cite{bil66} as well as Fabes and Rivi\`{e}re
\cite{f66}. However, the boundedness of the anisotropic singular integral operators
on the mixed-norm Lebesgue space $\lv$ with $\vp\in (1,\fz)^n$ and
from the anisotropic mixed-norm Hardy space $\vh$ (even from the isotropic mixed-norm Hardy space)
to itself or to $\lv$ is still unknown so far, where $\va\in [1,\fz)^n$ and $\vp\in (0,1]^n$.
Let
$$\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n,\ \vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$$
and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$
be the anisotropic mixed-norm Hardy space associated with $\vec{a}$
introduced by Cleanthous et al. in \cite{cgn17}, via the non-tangential grand maximal function.
In this article, we confirm the aforementioned conjecture proposed by Hart et al. \cite{htw17}
via completing the real-variable theory
of $\vh$ initially studied by Cleanthous et al. in \cite{cgn17}. To be precise, via first establishing a
Calder\'{o}n-Zygmund decomposition and a discrete Calder\'{o}n reproducing
formula, we then characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$,
respectively, by means of atoms,
the Lusin area function, the Littlewood-Paley $g$-function or
$g_{\lambda}^\ast$-function.
The obtained Littlewood-Paley $g$-function characterization of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ coincidentally confirms the aforementioned conjecture of
Hart et al. Applying the aforementioned Calder\'{o}n-Zygmund decomposition as well as
the atomic characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$,
we establish a finite atomic characterization of
$H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which further induces
a criterion on the boundedness of sublinear operators
from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space.
Then, applying this criterion, we obtain the boundedness of anisotropic
convolutional $\delta$-type and non-convolutional $\bz$-order
Calder\'{o}n-Zygmund operators from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ to
itself [or to $\lv$] with
$\delta\in(0,1]$, $\beta\in(0,\fz)\setminus\mathbb{N}$, $\vec{p}\in (0,1]^n$
and $\widetilde{p}_-\in(\frac{\nu}{\nu+\delta},1]$
or $\widetilde{p}_-\in(\frac{\nu}{\nu+\beta},\frac{\nu}{\nu+\lfloor\beta\rfloor a_-}]$,
where $\nu:=a_1+\cdots+a_n$, $\widetilde{p}_-:=\min\{p_1,\ldots,p_n\}$,
$a_-:=\min\{a_1,\ldots,a_n\}$ and $\lfloor\beta\rfloor$ denotes the largest integer
not greater than $\beta$. We should point out that the obtained
atomic characterizations of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$
and boundedness of anisotropic Calder\'{o}n-Zygmund
operators on these Hardy-type spaces positively answer
two questions mentioned by Cleanthous et al. in \cite[p.\,2760]{cgn17}. All these results are
new even for the isotropic mixed-norm Hardy spaces on $\mathbb{R}^n$.
This article is organized as follows.
In Section \ref{s2}, we first present some notation and notions used in this article,
including the anisotropic homogeneous quasi-norm, the anisotropic bracket, the mixed-norm Lebesgue space
and their basic properties. Then we recall the definition of anisotropic mixed-norm Hardy
spaces $\vh$ via the non-tangential grand maximal function from \cite{cgn17}.
The aim of Section \ref{s3} is to establish the atomic characterizations of $\vh$.
Recall that, in the proof of the atomic decomposition for the classical isotropic Hardy space $H^p(\rn)$,
we need to use the Calder\'on-Zygmund decomposition to decompose any element of $H^p(\rn)$ into
a sum of atoms (see, for example, \cite{s93}). Thus, in this section, we first establish a Calder\'{o}n-Zygmund decomposition
in anisotropic $\rn$ (see Lemma \ref{3l4} below) by borrowing some ideas from
the proof of Stein \cite[p.\,101, Proposition]{s93}; the obtained Calder\'{o}n-Zygmund
decomposition actually extends Stein \cite[p.\,101, Proposition]{s93}
and Grafakos \cite[Theorem 5.3.1]{lg14} as well as Sawano et al. \cite[Lemma 2.23]{shyy17} to the present setting.
Then, applying this Calder\'{o}n-Zygmund decomposition, we show the density of the subset
$L^{\vp/\wz{p}_-}(\rn)\cap\vh$ in $\vh$ for any $\vp\in(0,\fz)^n$ with $\wz{p}_-$ as
in \eqref{3e1} below (see Lemma \ref{3l7} below). By this density and the anisotropic
Calder\'{o}n-Zygmund decomposition associated with non-tangential grand maximal
functions as well as an argument similar to that used in
the proof of Nakai and Sawano \cite[Theorem 4.5]{ns12}, we then prove that $\vh$ is continuously
embedded into $\vahfz$ and hence also into $\vah$ due to the fact that each
$(\vp,\infty,s)$-atom is also a $(\vp,r,s)$-atom for any $r\in(1,\fz)$,
where $\vah$ denotes the anisotropic mixed-norm atomic Hardy space (see Definition \ref{3d2} below).
Conversely, via borrowing some ideas from the proofs of
Sawano \cite[Theorem 1]{sawa13} and Zhuo at al.
\cite[Proposition 2.11]{zsy16}, we first show that some estimates related to $\lv$ norms
for some series of functions can be reduced into dealing with the $L^r(\rn)$ norms of
the corresponding functions with $r\in (\max\{1,p_+\},\fz]$ and $p_+$ as in \eqref{2e10}
below (see Lemma \ref{3l6} below), which plays a key role in the proof of the atomic
characterizations of $\vh$ (see Theorem \ref{3t1} below)
and is also of independent interest. Indeed, using this key
lemma, the anisotropic Fefferman-Stein vector-valued inequality of the Hardy-Littlewood
maximal operator $\HL$ on $\lv$ (see Lemma \ref{3l2} below) and an argument similar to that used in
the proof of Sawano et al. \cite[Theorem 3.6]{shyy17}, we prove that
$\vah\st\vh$ and the inclusion is continuous, which then completes the proof
of the atomic characterizations of $\vh$. We point out that the obtained
atomic characterizations of $\vh$ gives a positive answer to a question
mentioned by Cleanthous et al. in \cite[p.\,2760]{cgn17}.
In Section \ref{s4}, as an application of the atomic characterization of
$\vh$ obtained in Theorem \ref{3t1}, we establish characterizations of
$\vh$ via Littlewood-Paley functions, including the Lusin
area function, the Littlewood-Paley $g$-function or $g^*_{\lambda}$-function.
Indeed, via borrowing some ideas from the proof of
Bownik et al. \cite[Lemma 2.12]{blyz10}, we establish a discrete Calder\'{o}n reproducing
formula (see Lemma \ref{4l5} below) associated to the anisotropic homogeneous quasi-norm on
$\rn$ for distributions vanishing weakly at infinity, which was introduced by
Folland and Stein \cite{fs82} on homogeneous groups.
Applying this discrete Calder\'{o}n reproducing formula
and an argument similar to that used in the proof
of Theorem \ref{3t1}, we first establish the Lusin area function
characterization of $\vh$ (see Theorem \ref{4t1} below).
Then, using this characterization and an approach initiated by Ullrich \cite{u12}
and further developed by Liang et al. \cite{lsuyy} and Liu et al. \cite{lyy17},
together with the anisotropic Fefferman-Stein
vector-valued inequality of the Hardy-Littlewood maximal operator
$\HL$ on $\lv$ (see Lemma \ref{3l2} below), we establish the Littlewood-Paley
$g$-function and $g_{\lambda}^\ast$-function characterizations of $\vh$
(see, respectively, Theorems \ref{4t2} and \ref{4t3} below).
Indeed, by the aforementioned approach from Ullrich \cite{u12}, via a key lemma
(see Lemma \ref{4l6} below) and an auxiliary function $g_{t,\ast}(f)$
(see \eqref{4e20} below), we show that the $\lv$ quasi-norm of the Lusin area
function can be controlled by that of the Littlewood-Paley $g$-function.
Moreover, the Littlewood-Paley $g$-function characterization of $\vh$ obtained in
Theorem \ref{4t2} below confirms the conjecture proposed by
Hart et al. in \cite[p.\,9]{htw17}.
Section \ref{s5} is devoted to establishing a finite atomic characterization of $\vh$.
In what follows, we use $C_c^\fz(\rn)$ to denote the set of all infinitely differentiable
functions with compact supports.
For any triplet $(\vp,r,s)$ as in Theorem \ref{3t1} below, we first show that $\vh\cap L^q(\rn)$,
with $q\in[1,\fz]$, and $\vh\cap C_c^\fz(\rn)$ are both dense in $\vh$ (see Lemma \ref{5l6} below),
and we then establish the finite atomic characterizations of $\vh$ (see Theorem \ref{5t1} below).
To be precise, via borrowing some ideas from the proofs of
\cite[Theorem 5.7]{lyy16} and \cite[Theorem 2.14]{lwyy17hlLP}, we prove that, for any given finite linear
combination of $(\vp,r,s)$-atoms with $r\in(\max\{p_+,1\},\fz)$ (or continuous $(\vp,\fz,s)$-atoms),
its quasi-norm in $\vh$ can be achieved via all its finite combinations of atoms of the same type.
This extends Meda et al.
\cite[Theorem 3.1 and Remark 3.3]{msv08} and Grafakos et al. \cite[Theorem 5.6]{gly08}
to the present setting of anisotropic mixed-norm Hardy spaces.
In Section \ref{s6}, applying the finite atomic
characterizations for $\vh$ obtained in Section \ref{s5},
we establish a criterion on the boundedness of
sublinear operators from $\vh$ into a quasi-Banach space
(see Theorem \ref{6t1} below), which is of independent interest;
moreover, using this criterion, we further show that,
if $T$ is a sublinear operator and maps all $(\vp,r,s)$-atoms
with $r\in(1,\fz)$ (or all continuous $(\vp,\fz,s)$-atoms) into uniformly bounded
elements of some $\gamma$-quasi-Banach space $\mathcal{B}_{\gamma}$ with $\gamma\in (0,1]$,
then $T$ has a unique bounded $\mathcal{B}_{\gamma}$-sublinear extension from $\vh$ into $\mathcal{B}_{\gamma}$
(see Corollary \ref{6c1} below).
This extends the corresponding results of Meda et al. \cite[Corollary 3.4]{msv08}
and Grafakos et al. \cite[Theorem 5.9]{gly08} as well as Ky \cite[Theorem 3.5]{ky14}
(see also \cite[Theorem 1.6.9]{ylk17})
to the present setting. Finally, via borrowing some ideas from the proofs
of Yan et al. \cite[Theorems 7.4 and 7.6]{yyyz16} and the criterion established
in Theorem \ref{6t1} and Corollary \ref{6c1} below, we also obtain the boundedness
of anisotropic convolutional $\delta$-type and anisotropic non-convolutional
$\bz$-order Calder\'{o}n-Zygmund operators
from $\vh$ to itself (see Theorems \ref{6t2} and \ref{6t4} below) or to $\lv$
(see Theorems \ref{6t3} and \ref{6t5} below),
where $\delta\in(0,1]$, $\bz\in(0,\fz)\setminus\nn$, $\vp\in (0,1]^n$ and $\widetilde{p}_-\in(\frac{\nu}{\nu+\delta},1]$
or $\widetilde{p}_-\in(\frac{\nu}{\nu+\bz},\frac{\nu}{\nu+\lfloor\bz\rfloor a_-}]$
with $\widetilde{p}_-$ as in \eqref{3e1} and $a_-$ as in \eqref{2e9} below.
We point out that Theorem \ref{6t2} extends the corresponding results of
Fefferman and Stein \cite[Theorem 12]{fs72} and that
Theorems \ref{6t4} and \ref{6t5} extend the corresponding results of
Stefanov and Torres \cite[Theorem 1]{st04} as well as Yan et al.
\cite[Theorem 7.6]{yyyz16} to the present setting. We also point out that
the obtained boundedness of these anisotropic Calder\'{o}n-Zygmund operators
on $\vh$ positively answers a question
mentioned by Cleanthous et al. in \cite[p.\,2760]{cgn17}.
Finally, we make some conventions on notation.
We always let
$\mathbb{N}:=\{1,2,\ldots\}$,
$\mathbb{Z}_+:=\{0\}\cup\mathbb{N}$
and $\vec0_n$ be the \emph{origin} of $\rn$.
For any multi-index
$\bz:=(\bz_1,\ldots,\bz_n)\in(\mathbb{Z}_+)^n=:\mathbb{Z}_+^n$,
let
$|\bz|:=\bz_1+\cdots+\bz_n$ and
$\pa^{\bz}:=(\f{\pa}{\pa x_1})^{\bz_1} \cdots (\f{\pa}{\pa x_n})^{\bz_n}.$
We denote by $C$ a \emph{positive constant}
which is independent of the main parameters,
but may vary from line to line. We also use $C_{(\az,\bz,\ldots)}$ to denote a positive constant
depending on the indicated parameters $\az,\,\bz,\ldots$.
The notation $f\ls g$ means $f\le Cg$ and, if $f\ls g\ls f$,
then we write $f\sim g$. For any $q\in[1,\fz]$, we denote by $q'$
its \emph{conjugate index}, namely, $1/q+1/q'=1$.
Moreover, if $\vec{q}:=(q_1,\ldots,q_n)\in[1,\fz]^n$, we denote by
$\vec{q}':=(q_1',\ldots,q_n')$ its \emph{conjugate index}, namely,
for any $i\in\{1,\ldots,n\}$, $1/q_i+1/q_i'=1$. In addition,
for any set $E\subset\rn$, we denote by $E^\complement$ the
set $\rn\setminus E$, by $\chi_E$ its \emph{characteristic function},
by $|E|$ its \emph{n-dimensional Lebesgue measure}
and by $\sharp E$ its \emph{cardinality}.
For any $s\in\mathbb{R}$, we denote by $\lfloor s\rfloor$
the \emph{largest integer not greater than $s$}. In what follows, $C^{\fz}(\rn)$
denotes the set of all \emph{infinite differentiable functions} on $\rn$
and $C_c^{\fz}(\rn)$ the set of all $C^{\fz}(\rn)$ functions with compact supports.
\section{Preliminaries \label{s2}}
In this section, we present the definition of the anisotropic mixed-norm Hardy space
via the non-tangential grand maximal function from \cite{cgn17}. To this end,
we first recall the notion of anisotropic homogeneous quasi-norms
and then state some of their basic conclusions to be used in this article.
We begin with recalling the definition of anisotropic homogeneous quasi-norms
from \cite{bil66, f66} (see also \cite{sw78}) as follows.
For any $b:=(b_1,\ldots,b_n)$, $x:=(x_1,\ldots,x_n)\in \rn$ and $t\in[0,\fz)$,
let $t^b x:=(t^{b_1}x_1,\ldots,t^{b_n}x_n)$.
\begin{definition}\label{2d1}
Let $\va:=(a_1,\ldots,a_n)\in [1,\fz)^n$.
The \emph{anisotropic homogeneous quasi-norm} $|\cdot|_{\va}$,
associated with $\va$, is a non-negative measurable function on $\rn$
defined by setting $|\vec0_n|_{\va}:=0$ and, for any $x\in \rn\setminus\{\vec0_n\}$,
$|x|_{\va}:=t_0$, where $t_0$ is the unique positive number such that $|t_0^{-\va}x|=1$,
namely,
$$\f{x_1^2}{t_0^{2a_1}}+\cdots+\f{x_n^2}{t_0^{2a_n}}=1.$$
\end{definition}
We also need the following notion of the anisotropic bracket
and the homogeneous dimension from \cite{sw78}, which plays an important role
in the study on anisotropic function spaces.
\begin{definition}\label{2d2}
Let $\va:=(a_1,\ldots,a_n)\in [1,\fz)^n$. The \emph{anisotropic bracket}, associated with $\va$,
is defined by setting, for any $x\in \rn$, $$\lg x\rg_{\va}:=|(1,x)|_{(1,\va)}.$$
Furthermore, the \emph{homogeneous dimension} $\nu$ is defined as
$$\nu:=|\vec{a}|:=a_1+\cdots +a_n.$$
\end{definition}
\begin{remark}\label{2r1}
\begin{enumerate}
\item[{\rm(i)}] It is easy to see that, for any $x\in \rn$, $|x|_{\va} < \lg x\rg_{\va}$.
\item[{\rm(ii)}] By $\lg \vec0_n\rg_{\va} = 1$ and the fact that,
for any $x\in \rn\setminus\{\vec0_n\}$, $\lg x\rg_{\va} > 1$, we find that,
for any $x\in \rn$, $\lg x\rg_{\va} \geq 1$.
\item[{\rm(iii)}] By \cite[(2.7)]{cgn17}, we know that $\lg \cdot \rg_{\va} $ belongs to $C^{\fz}(\rn)$ and,
for any $s\in \rr$ and multi-index $\bz \in \mathbb{Z}_+^n$, there exists a positive
constant $C_{(\va,s,\bz)}$, depending on $\va$, s and $\bz$, such that, for any $x\in \rn$,
\begin{align*}
\lf|\pa^{\bz}\lg x\rg_{\va}^s\r| \le C_{(\va,s,\bz)}\lg x\rg_{\va}^{s-\va\cdot\bz},
\end{align*}
here and hereafter, for any $\az:=(\az_1,\ldots,\az_n)$,
$\bz:=(\bz_1,\ldots,\bz_n) \in \rn$, $\az\cdot\bz:=\sum_{i=1}^{n}\az_i \bz_i$.
\end{enumerate}
\end{remark}
To compare the anisotropic homogeneous quasi-norm with the Euclidean norm,
we need the following Lemma \ref{2l1}, which is easy to be proved by using
Definition \ref{2d1}, the details being omitted.
\begin{lemma}\label{2l1}
Let $\va\in [1,\fz)^n$ and $x\in \rn$. Then
\begin{enumerate}
\item[{\rm(i)}] $|x|_{\va} > 1$ if and only if $|x| > 1$;
\item[{\rm(ii)}] $|x|_{\va} < 1$ if and only if $|x| < 1.$
\end{enumerate}
\end{lemma}
For any $\va:=(a_1,\ldots,a_n)\in [1,\fz)^n$, let
\begin{align}\label{2e9}
a_-:=\min\{a_1,\ldots,a_n\}\hspace{0.35cm}
{\rm and}\hspace{0.35cm} a_+:=\max\{a_1,\ldots,a_n\}.
\end{align}
Now let us recall some basic properties of $|\cdot|_{\va}$
and $\lg \cdot\rg_{\va}$;
see \cite{cgn17, js07, js08, sw78} for more details.
\begin{lemma}\label{2l2}
Let $\va:=(a_1,\ldots,a_n) \in [1,\fz)^n$, $t\in[0,\fz)$ and $a_-$ and $a_+$ be as
in \eqref{2e9}. Then, for any $x,\ y\in \rn$,
\begin{enumerate}
\item[{\rm(i)}] $\lf|t^{\va}x\r|_{\va}= t|x|_{\va}$;
\item[{\rm(ii)}] $|x+y|_{\va}\le |x|_{\va}+|y|_{\va}$;
\item[{\rm(iii)}] $\max\{|x_1|^{1/a_1},\ldots,|x_n|^{1/a_n}\}
\le |x|_{\va}\le \sumn |x_i|^{1/a_i}$;
\item[{\rm(iv)}] when $|x| \geq 1$, $|x|^{1/a_+} \le |x|_{\va} \le|x|^{1/a_-}$;
\item[{\rm(v)}] when $|x| < 1$, $|x|^{1/a_-} \le |x|_{\va} \le|x|^{1/a_+}$;
\item[{\rm(vi)}] $(\f{1}{2})^{a_-}(1+|x|_{\va})^{a_-} \le 1+|x| \le 2(1+|x|_{\va})^{a_+}$;
\item[{\rm(vii)}] $ \lg x\rg_{\va} \le 1+|x|_{\va}\le 2\lg x\rg_{\va}$;
\item[{\rm(viii)}] $\lg x+y\rg_{\va} \le 4\lg x\rg_{\va} \lg y\rg_{\va}$;
\item[{\rm(ix)}] for any measurable function $f$ on $\rn$,
$$\int_{\rn}f(x)\,dx=\int_0^{\fz}\int_{S^{n-1}}f(\rho^{\va}\xi)\rho^{\nu-1}\,d\sigma(\rho)\,d\rho,$$
where $S^{n-1}$ denotes the $n-1$ dimension unit sphere of $\rn$
and $\sigma(\rho)$ the spherical measure.
\end{enumerate}
\end{lemma}
\begin{remark}\label{2r2}
\begin{enumerate}
\item[{\rm(i)}] By Lemma \ref{2l2}(i), we easily know that the anisotropic
quasi-homogeneous norm $|\cdot|_{\va}$ is a norm if and only if
$\va=(\overbrace{1,\ldots,1}^{n\ \mathrm{times}})$ and, in this case,
the homogeneous quasi-norm $|\cdot|_{\va}$ becomes the Euclidean norm $|\cdot|$.
\item[{\rm(ii)}] It is easy to see that $\va\in [1,\fz)^n$ guarantees Lemma \ref{2l2}(ii).
\end{enumerate}
\end{remark}
For any $\va\in [1,\fz)^n$, $r\in (0,\fz)$ and $x\in \rn$,
we define the \emph{anisotropic ball} $B_{\va}(x,r)$, with center $x$ and radius $r$, by
setting $B_{\va}(x,r):=\{y\in \rn:\,|y-x|_{\va} < r\}$.
Then $B_{\va}(x,r)= x+r^{\va}B_{\va}(\vec0_n,1)$ and
$|B_{\va}(x,r)|=\nu_n r^{\nu}$, where $\nu_n:=|B_{\va}(\vec0_n,1)|$ (see \cite[(2.12)]{cgn17}).
Moreover, from Lemma \ref{2l1}(ii), we deduce that $B_0:=B_{\va}(\vec0_n,1)=B(\vec0_n,1)$,
where $B(\vec0_n,1)$ denotes the unit ball of $\rn$, namely, $B(\vec0_n,1):=\{y\in \rn:\,|y|<1\}$.
In what follows, we always let $\mathfrak{B}$ be the set of all anisotropic balls, namely,
\begin{align}\label{2e2}
\mathfrak{B}:=\lf\{B_{\va}(x,r):\ x\in\rn,\ r\in(0,\fz)\r\}.
\end{align}
For any $B\in\mathfrak{B}$ centered at $x\in\rn$ with radius $r\in (0,\fz)$ and $\delta\in(0,\fz)$,
let
\begin{align}\label{2e2'}
B^{(\delta)}:=B^{(\delta)}_{\va}(x,r):=B_{\va}(x,\delta r).
\end{align}
In addition, for any $x\in \rn$ and $r\in (0,\fz)$, the \emph{anisotropic cube}
$Q_{\va}(x,r)$ is defined by setting $Q_{\va}(x,r):=x + r^{\va}(-1,1)^n,$
whose Lebesgue measure $|Q_{\va}(x,r)|$ equals $2^n r^\nu$.
Denote by $\mathfrak{Q}$ the set of all anisotropic cubes, namely,
\begin{align}\label{2e3}
\mathfrak{Q}:=\lf\{Q_{\va}(x,r):\ x\in\rn,\ r\in(0,\fz)\r\}.
\end{align}
Now let us recall the definition of mixed-norm Lebesgue spaces from \cite{bp61}.
\begin{definition}\label{2d3}
Let $\vp:=(p_1,\ldots,p_n)\in (0,\fz]^n$. The \emph{mixed-norm Lebesgue space} $\lv$ is
defined to be the set of all measurable functions $f$ such that their quasi-norms
$$\|f\|_{\lv}:=\left\{\int_{\rr}\cdots\left[\int_{\rr}\left\{\int_{\rr}|f(x_1,\ldots,x_n)|^{p_1}
\,dx_1\right\}^{\f{p_2}{p_1}}\,dx_2\right]^{\f{p_3}{p_2}}\cdots\, dx_n\right\}^{\f{1}{p_n}}<\fz$$
with the usual modifications made when $p_i=\fz$, $i\in \{1,\ldots,n\}$.
\end{definition}
\begin{remark}\label{2r3}
\begin{enumerate}
\item[{\rm(i)}] Obviously, when $\vp=(p,\ldots,p)\in (0,\fz]^n$, $\lv$ coincides with
the classical Lebesgue space $L^p(\rn)$.
\item[{\rm(ii)}] For any $\vp\in(0,\fz]^n$, $(\lv,\|\cdot\|_{\lv})$
is a quasi-Banach space and, for any $\vp \in [1,\fz]^n$,
$(\lv,\|\cdot\|_{\lv})$ becomes a Banach space; see \cite[p.\,304, Theorem 1]{bp61}.
\item[{\rm(iii)}] Let $\vp\in (0,\fz]^n$. Then, for any $s\in (0,\fz)$ and $f\in\lv$,
\begin{align}\label{2e8}\lf\||f|^s\r\|_{\lv}=\|f\|_{L^{s \vp}(\rn)}^s.\end{align}
In addition, for any $\lz\in{\mathbb C}$, $\theta\in [0,\min(1,p_1,\ldots,p_n)]$
and $f,\ g\in\lv$, $\|\lz f\|_{\lv}=|\lz|\|f\|_{\lv}$ and
\begin{align*}
\|f+g\|_{\lv}^{\theta}\le \|f\|_{\lv}^{\theta}
+\|g\|_{\lv}^{\theta}
\end{align*}
(see \cite[p.\,188]{js07}).
\item[{\rm(iv)}] Let $\vp\in [1,\fz]^n$. Then, for any $f\in\lv$ and $g\in L^{\vp'}(\rn)$,
it is easy to see that
\begin{align*}
\int_{\rn}|f(x)g(x)|\,dx \le \|f\|_{\lv}\|g\|_{\lvv},
\end{align*}
where $\vp'$ denotes the conjugate index of $\vp$,
namely, for any $i\in \{1,\ldots,n\}$, $1/p_i+1/p_i'=1$.
\end{enumerate}
\end{remark}
For any $\vp:=(p_1,\ldots,p_n)\in (0,\fz)^n$, we always let
\begin{align}\label{2e10}
p_-:=\min\{1,p_1,\ldots,p_n\}\hspace{0.35cm}
{\rm and}\hspace{0.35cm} p_+:=\max\{p_1,\ldots,p_n\}.
\end{align}
Recall that a \emph{Schwartz function} is a $C^\infty(\rn)$
function $\varphi$ satisfying,
for any $N\in\zz_+$ and multi-index $\az\in\zz_+^n$,
$$\|\varphi\|_{N,\alpha}:=
\sup_{x\in\rn}\lf\{(1+|x|)^N
|\partial^\alpha\varphi(x)|\r\}<\infty.$$
Denote by
$\cs(\rn)$ the set of all Schwartz functions, equipped
with the topology determined by
$\{\|\cdot\|_{N,\alpha}\}_{N\in\zz_+,\az\in\zz_+^n}$,
and $\cs'(\rn)$ the \emph{dual space} of $\cs(\rn)$, equipped
with the weak-$\ast$ topology.
For any $N\in\mathbb{Z}_+$, let
$$\cs_N(\rn):=\lf\{\varphi\in\cs(\rn):\
\|\varphi\|_{\cs_N(\rn)}:=
\sup_{x\in\rn}\lf[\lg x\rg_{\va}^N\sup_{|\az|\le N}
|\partial^\alpha\varphi(x)|\r]\le 1\r\}.$$
In what follows, for any $\varphi \in \cs(\rn)$ and $t\in (0,\fz)$,
let $\varphi_t(\cdot):=t^{-\nu}\varphi(t^{-\va}\cdot)$.
\begin{definition}\label{2d4}
Let $\varphi\in\cs(\rn)$ and $f\in\cs'(\rn)$. The
\emph{non-tangential maximal function} $M_\varphi(f)$,
with respect to $\varphi$, is defined by setting, for any $x\in\rn$,
$$
M_\varphi(f)(x):= \sup_{y\in B_{\va}(x,t),
t\in (0,\fz)}|f\ast\varphi_t(y)|.
$$
Moreover, for any given $N\in\mathbb{N}$, the
\emph{non-tangential grand maximal function} $M_N(f)$ of
$f\in\cs'(\rn)$ is defined by setting, for any $x\in\rn$,
\begin{equation*}
M_N(f)(x):=\sup_{\varphi\in\cs_N(\rn)}
M_\varphi(f)(x).
\end{equation*}
\end{definition}
\begin{remark}\label{2r5}
Obviously, for any given $N\in \nn$ and any $f\in \cs'(\rn)$, $\varphi\in\cs(\rn)$,
$t\in (0,\fz)$ and $x\in \rn$,
$$|f\ast\varphi_t(x)|\le \|\varphi\|_{\cs_N(\rn)}M_N(f)(x).$$
\end{remark}
We now recall the notion of anisotropic mixed-norm Hardy spaces as follows,
which is just \cite[Definition 3.3]{cgn17}.
\begin{definition}\label{2d5}
Let $\va:=(a_1,\ldots,a_n)\in[1,\fz)^n$, $\vp\in(0,\fz)^n$,
$N_{\vp}:=\lfloor(\nu\frac{a_+}{a_-}
(\f{1}{p_-}+1)+\nu+2a_+\rfloor+1$ and
\begin{align}\label{2e11}
N\in\mathbb{N}\cap \lf[N_{\vp},\fz\r),
\end{align}
where $a_-,\,a_+$ are as in \eqref{2e9} and $p_-$ is as in \eqref{2e10}.
The \emph{anisotropic mixed-norm Hardy space} $\vh$ is defined by setting
\begin{equation*}
\vh:=\lf\{f\in\cs'(\rn):\ M_N(f)\in\lv\r\}
\end{equation*}
and, for any $f\in\vh$, let
$\|f\|_{\vh}:=\| M_N(f)\|_{\lv}$.
\end{definition}
\begin{remark}\label{2r4}
The quasi-norm of $\vh$ in Definition \ref{2d5} depends on $N$, however,
by Theorem \ref{3t1} below, we know that the space $\vh$ is independent
of the choice of $N$ as long as $N$ is as in \eqref{2e11}.
In addition, if $\va:=(\overbrace{1,\ldots,1}^{n\ \rm times})$
and $\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$, where $p\in(0,\fz)$,
then $\vh$ coincides with the classical isotropic Hardy space $H^p(\rn)$ of Fefferman and Stein \cite{fs72}.
\end{remark}
\section{Atomic characterizations of $\vh$\label{s3}}
In this section, we establish the atomic characterizations of $\vh$.
We begin with introducing the definition of anisotropic mixed-norm
$(\vp,r,s)$-atoms. In what follows, for any $r\in(0,\fz]$,
we use $L^r(\rn)$ to denote the space of all measurable functions $f$ such that
$$\|f\|_{L^r(\rn)}:=\lf\{\int_{\rn}|f(x)|^r\,dx\r\}^{1/r}<\fz$$
with the usual modification made when $r=\fz$.
\begin{definition}\label{3d1}
Let $\va\in[1,\fz)^n$, $\vp:=(p_1,\ldots,p_n)\in(0,\fz)^n$, $r\in (1,\fz]$ and
\begin{align}\label{3e1}
s\in\lf[\lf\lfloor\f{\nu}{a_-}\lf(\f{1}{\widetilde{p}_-}-1\r) \r\rfloor,\fz\r)\cap\zz_+,
\end{align}
where $a_-$ is as in \eqref{2e9} and $\widetilde{p}_-:=\min\{p_1,\ldots,p_n\}$.
An \emph{anisotropic mixed-norm $(\vp,r,s)$-atom} $a$ is
a measurable function on $\rn$ satisfying
\begin{enumerate}
\item[{\rm (i)}] $\supp a \st B$, where
$B\in\mathfrak{B}$ with $\mathfrak{B}$ as in \eqref{2e2};
\item[{\rm (ii)}] $\|a\|_{L^r(\rn)}\le \frac{|B|^{1/r}}{\|\chi_B\|_{\lv}}$;
\item[{\rm (iii)}] $\int_{\mathbb R^n}a(x)x^\az\,dx=0$ for any $\az\in\zz_+^n$
with $|\az|\le s$.
\end{enumerate}
\end{definition}
Throughout this article, we always call an anisotropic mixed-norm
$(\vp,r,s)$-atom simply by a $(\vp,r,s)$-atom.
Now, using $(\vp,r,s)$-atoms, we introduce the anisotropic
mixed-norm atomic Hardy space $\vah$ as follows.
\begin{definition}\label{3d2}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$, $r\in (1,\fz]$
and $s$ be as in \eqref{3e1}. The \emph{anisotropic mixed-norm
atomic Hardy space} $\vah$ is defined to be the
set of all $f\in\cs'(\rn)$ satisfying that there exist
$\{\lz_i\}_{i\in\nn}\st\mathbb{C}$
and a sequence of $(\vp,r,s)$-atoms, $\{a_i\}_{i\in\nn}$,
supported, respectively, on
$\{B_i\}_{i\in\nn}\st\mathfrak{B}$ such that
\begin{align*}
f=\sum_{i\in\nn}\lz_ia_i
\quad\mathrm{in}\quad\cs'(\rn).
\end{align*}
Moreover, for any $f\in\vah$, let
\begin{align*}
\|f\|_{\vah}:=
{\inf}\lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{p_-}\r\}^{1/{p_-}}\r\|_{\lv},
\end{align*}
where the infimum is taken over all decompositions of $f$ as above.
\end{definition}
Let $L_{\rm loc}^1(\rn)$ denote the collection of
all locally integrable functions on $\rn$.
\begin{definition}\label{3d3}
The \emph{Hardy-Littlewood maximal operator} $M_{{\rm HL}}(f)$ of $f\in L_{\rm loc}^1(\rn)$ is defined by setting,
for any $x\in\rn$,
\begin{align}\label{3e2}
M_{{\rm HL}}(f)(x):=\sup_{x\in Q\in\mathfrak{Q}}
\frac1{|Q|}\int_Q|f(y)|\,dy,
\end{align}
where $\mathfrak{Q}$ is as in \eqref{2e3}.
\end{definition}
\begin{remark}\label{3r1}
For any $f\in L_{\rm loc}^1(\rn)$ and $x\in \rn$, let
\begin{align*}
M(f)(x):=\sup_{I_n\in \mathbb{I}_{x_n}}\lf\{\f{1}{|I_n|}\int_{I_n}
\cdots\sup_{I_2\in \mathbb{I}_{x_2}}\lf[\f{1}{|I_2|}\int_{I_2}
\sup_{I_1\in \mathbb{I}_{x_1}}\lf\{\f{1}{|I_1|}\int_{I_1}
|f(y_1,y_2\ldots,y_n)|\,dy_1\r\}\,dy_2\r]\cdots \,dy_n\r\},
\end{align*}
where, for any $k\in \{1,\ldots,n\}$, ${\mathbb I}_{x_k}$ denotes the set of all intervals in $\rr_{x_k}$ containing
$x_k$. Then, it is easy to see that, for any $x\in \rn$, $$ M_{{\rm HL}}(f)(x)\le M(f)(x).$$
\end{remark}
To establish the atomic characterizations of $\vh$, we need several technical lemmas.
We begin with the following boundedness of the Hardy-Littlewood maximal operator
$M_{\rm HL}$ on $\lv$ with $\vp\in(1,\fz]^n$.
\begin{lemma}\label{3l1}
Let $\vp\in (1,\fz]^n$. Then there exists a positive
constant C, depending on $\vp$, such that, for any $f\in \lv$,
$$\|M_{{\rm HL}}(f)\|_{\lv}\le C\|f\|_{\lv},$$
where $M_{{\rm HL}}$ is as in \eqref{3e2}.
\end{lemma}
\begin{proof}
We first assume that $\vp:=(p_1,\ldots,p_n)\in(1,\fz)^n$. In this case, for any given
$m_1,~m_2\in \mathbb{Z}_+$ with $m_1+m_2=n$ and any $f\in\lv$, $s\in \rr^{m_1}$ and $t\in\rr^{m_2}$,
let $$f^*(s,t):=\sup_{r\in (0,\fz)}\f{1}{|B(s,r)|}
\int_{B(s,r)}|f(y,t)|\,dy.$$
In addition, for any given $\vec{p}_{m_2}:=(p_1,\ldots,p_{m_2})\in(1,\fz)^{m_2}$ and any $s\in \rr^{m_1}$, let
\begin{align*}
T_{L^{\vec{p}_{m_2}}(\rr^{m_2})}(f)(s)&:=\|f(s,\cdot)
\|_{L^{\vec{p}_{m_2}}(\rr^{m_2})}\\
&:=\left\{\int_{\rr}\cdots\left[\int_{\rr}\left\{\int_{\rr}
|f(s,t_1\ldots,t_{m_2})|^{p_1}\,dt_1\right\}^{\f{p_2}{p_1}}
\,dt_2\right]^{\f{p_3}{p_2}}\cdots \,dt_{m_2}\right\}^{\f{1}{p_{m_2}}}.
\end{align*}
Then it holds true that, for any $q\in (1,\fz)$,
\begin{align}\label{3e4}
\int_{\rr^{m_1}}\lf[T_{L^{\vec{p}_{m_2}}(\rr^{m_2})}(f^*)(s)\r]^q \,dx
\ls \int_{\rr^{m_1}}\lf[T_{L^{\vec{p}_{m_2}}(\rr^{m_2})}(f)(s)\r]^q \,dx
\end{align}
(see \cite[p.\,421]{b75}).
For any $k\in\{1,\ldots,n\}$ and $x:=(x_1,\cdots,x_n)\in \rn$, let
$$M_k(f)(x):=\sup_{I\in \mathbb{I}_{x_k}}\f{1}{|I|}\int_I
|f(x_1,\ldots,y_k,\ldots,x_n)|\,dy_k,$$
where $\mathbb{I}_{x_k}$ is as in Remark \ref{3r1}.
Then, for any $x\in\rn$, we have
\begin{align*}
M(f)(x)=M_n\lf(\cdots\lf(M_1(f)\r)\cdots\r)(x).
\end{align*}
By this, Remark \ref{3r1} and \eqref{3e4} with
$m_1:=1,~m_2:=n-1$, $s:=x_n$, $t:=(x_1,\ldots,x_{n-1})$
and $q:=p_n$, we conclude that
\begin{align}\label{3e5}
\|M_{{\rm HL}}(f)\|_{\lv}&\le\|M_n\lf(\cdots\lf(M_1(f)\r)\cdots\r)\|_{\lv}\\
&\ls\lf\{\int_{\rr}\lf[T_{L^{\vec{p}_{n-1}}(\rr^{n-1})}
\lf(\lf[M_{n-1}\lf(\cdots\lf(M_1(f)\r)\cdots\r)\r]^*\r)(x_n)\r]^{p_n}\, dx_n\r\}^{\f{1}{p_n}}\noz\\
&\ls \lf\{\int_{\rr}\lf[T_{L^{\vec{p}_{n-1}}(\rr^{n-1})}
(M_{n-1}\lf(\cdots\lf(M_1(f)\r)\cdots\r))(x_n)\r]^{p_n} \,dx_n\r\}^{\f{1}{p_n}}\noz\\
&\sim \|M_{n-1}\lf(\cdots\lf(M_1(f)\r)\cdots\r)\|_{\lv}\noz.
\end{align}
Repeating the above estimate $n-1$ times, we easily find that Lemma \ref{3l1} holds true
in the case when $\vp\in (1,\fz)^n$.
If $p_{i_0}=\fz$ for some $i_0\in \{1,\ldots,n\}$ and, for any $i\in \{1,\ldots,n\}$ with $i\neq i_0$,
$p_i\in (1,\fz)$, then, by an argument similar to that used in the estimation of \eqref{3e5} and the boundedness of $M_{\rm HL}$ on
$L^{\fz}(\rr)$ (see \cite[p.\,14]{mb03}), we easily conclude that Lemma \ref{3l1} also
holds true in this case. This finishes the proof of Lemma \ref{3l1}.
\end{proof}
By \cite[(2.24)]{gn16} and Remark \ref{3r1},
we immediately obtain the following Fefferman-Stein vector-valued inequality of
$M_{\rm HL}$ on $\lv$, the details being omitted.
\begin{lemma}\label{3l2}
Let $\vp\in (1,\fz)^n$ and $u\in(1,\fz]$.
Then there exists a positive
constant $C$ such that, for any
sequence $\{f_k\}_{k\in\nn}\st L^1_{\rm loc}(\rn)$,
$$\lf\|\lf\{\sum_{k\in\nn}
\lf[\HL(f_k)\r]^u\r\}^{1/u}\r\|_{\lv}
\le C\lf\|\lf(\sum_{k\in\nn}|f_k|^u\r)^{1/u}\r\|_{\lv}$$
with the usual modification made when $u=\fz$,
where $\HL$ denotes the Hardy-Littlewood maximal operator as in \eqref{3e2}.
\end{lemma}
\begin{remark}\label{3r2}
By Lemma \ref{3l2} and an argument similar to that used in the proof of \cite[Remark 2.5]{yyyz16},
we easily conclude that, for any given $\vp\in(0,\fz)^n$ and $r\in(0,p_-)$ with $p_-$
as in \eqref{2e10}, there exists a positive constant $C$ such that, for any $\bz\in[1,\fz)$
and sequence $\{B_i\}_{i\in\nn}\st\mathfrak{B}$,
$$\lf\|\sum_{i\in\nn}\chi_{B_i^{(\bz)}}\r\|_{\lv}\le C \bz^{\f{\nu}{r}}
\lf\|\sum_{i\in\nn}\chi_{B_i}\r\|_{\lv},$$
where $B_i^{(\bz)}$ is as in \eqref{2e2'}.
\end{remark}
\begin{definition}\label{3d4}
Let $\phi\in\cs(\rn)$ satisfying $\int_{\rn}\phi(x)\,dx\neq0$ and $f\in\cs'(\rn)$. The
\emph{radial maximal function} $M_\phi^0(f)$
of $f$, with respect to $\phi$, is defined by setting, for any $x\in\rn$,
\begin{equation*}
M_\phi^0(f)(x):= \sup_{t\in (0,\fz)}
|f\ast\phi_t(x)|.
\end{equation*}
\end{definition}
The following Lemma \ref{3l8} is from \cite[Theorem 3.4]{cgn17}.
\begin{lemma}\label{3l8}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$
and $N$ be as in \eqref{2e11}.
Then, for any given $\phi\in\cs(\rn)$
with $\int_{\rn}\phi(x)\,dx\neq0$ and any $f\in\cs'(\rn)$,
the following statements are equivalent:
\begin{enumerate}
\item[{\rm(i)}] $f\in\vh;$
\item[{\rm(ii)}] $M_\phi^0(f)\in\lv.$
\end{enumerate}
Moreover, there exists a positive constant $C$ such that, for any $f\in\vh$,
\begin{align*}
\|f\|_{\vh}\le C\lf\|M_\phi^0(f)
\r\|_{\lv}\le C\|f\|_{\vh}.
\end{align*}
\end{lemma}
Observe that $(\rn,\,|\cdot|_{\va},\,dx)$ is an RD-space (see \cite{hmy08,zy11}). From this
and \cite[Lemma 4.6]{gly08} (see also \cite[lemma 4.5]{zsy16}), we deduce the following lemma,
the details being omitted.
\begin{lemma}\label{3l3}
Let $\Omega\st \rn$ be an open subset with $|\Omega|<\infty$ and, for any $x\in \rn$, let
$$d(x,\boz):=\inf\{|x-y|_{\vec{a}}:\ y\notin \boz\}.$$
Then there exists a sequence $\{x_k\}_{k\in\nn}\st \boz$ such that,
for any $A\in [1,\fz)$ and
$$r_k:=d(x_k,\boz)/(2A)\hspace{0.2cm} with\hspace{0.2cm} k\in \nn,$$
it holds true that
\begin{enumerate}
\item[\rm(i)] $\boz=\bigcup_{k\in\nn} B_{\va}(x_k,r_k)$;
\item[\rm(ii)] $\{B_{\va}(x_k,r_k/4)\}_{k\in\nn}$
are pairwise disjoint;
\item[\rm(iii)] for any given $k\in \nn$,
$B_{\va}(x_k,Ar_k)\st \boz$;
\item[\rm(iv)] $Ar_k<d(x,\boz)<3Ar_k$ whenever
$k\in \nn$ and $x\in B_{\va}(x_k,Ar_k)$;
\item[\rm(v)] for any given $k\in \nn$,
there exists a $y_k\notin \boz$ such that
$|x_k-y_k|_{\vec{a}}<3Ar_k$;
\item[\rm(vi)] there exists a positive constant $R$, independent of $\Omega$,
such that, for any $k\in\nn$, $$\sharp \lf\{j\in\nn:\ \lf[B_{\va}(x_k,r_k)
\bigcap B_{\va}(x_j,Ar_j)\r]\neq\emptyset\r\}\le R.$$
\end{enumerate}
\end{lemma}
Let $\Phi$ be some fixed $C^\fz(\rn)$ function satisfying $\supp \Phi\st B(\vec{0}_n,1)$
and $\int_{\rn} \Phi(x)dx \neq 0$. For any $f\in\cs'(\rn)$ and $x\in \rn$, we always let
\begin{equation}\label{3e16}
M_0(f)(x):=M_\Phi^0(f)(x),
\end{equation}
where $M_\Phi^0(f)$ is as in Definition \ref{3d4} with $\phi$ replaced by $\Phi$.
In what follows, for any given $s\in \mathbb{Z}_+$,
the \emph{symbol $\cp_s(\rn)$} denotes the linear space of all polynomials
on $\rn$ with degree not greater than $s$.
The following Calder\'on-Zygmund decomposition extends the corresponding results of
Stein \cite[p.\,101, Proposition]{s93} and Grafakos \cite[Theorem 5.3.1]{lg14}
as well as Sawano et al. \cite[Lemma 2.23]{shyy17} to the present setting.
\begin{lemma}\label{3l4}
Let $\va \in [1,\fz)^n$, $\vp\in(0,\fz)^n$,
$s\in \mathbb{Z}_+$ and
$N$ be as in \eqref{2e11}.
For any $\sa\in (0,\fz)$ and $f\in \vh$,
let $$\CO := \{x\in \rn:\ M_N(f)(x)>\sa\},$$
where $M_N$ is as in Definition \ref{2d4}. Then
the following statements hold true:
\begin{enumerate}
\item[\rm(i)] There exists a sequence
$\{B_k^*\}_{k\in\nn}\subset \mathfrak{B}$
with $\mathfrak{B}$ as in \eqref{2e2},
which has finite intersection property,
such that $$\CO = \bigcup_{k\in \nn}\Qkk.$$
\item[\rm(ii)] There exist two distributions $g$ and $b$
such that $f=g+b$ in $\cs'(\rn)$.
\item[\rm(iii)] For the distribution $g$ as in (ii) and any $x\in \rn$,
\begin{equation}\label{3e6}
M_0(g)(x)\ls M_N(f)(x)\chi_{\CO^{\com}}(x)+\sum_{k\in\nn}
\f{\sa r_k^{\nu+(s+1)a_-}}{(r_k+|x-x_k|_{\va})^{\nu+(s+1)a_-}},
\end{equation}
where $a_-$ and $M_0$ are as in \eqref{2e9}, respectively, \eqref{3e16}, and the implicit positive
constant is independent of $f$ and $g$.
Moreover, for any $k\in \nn$, $x_k$ denotes
the center of $B_k^*$ and there exists a constant
$A^*\in (1,\fz)$, independent of $k$, such that $A^*-1$ is small enough
and $A^* r_k$ equals the radius of $B_k^*$.
\item[\rm(iv)] If $f\in L_{\loc}^1(\rn)$, then the distribution $g$ as in (ii)
belongs to $L^{\fz}(\rn)$
and $\|g\|_{L^{\fz}(\rn)} \ls \sa$ with the implicit positive
constant independent of $f$ and $g$.
\item[\rm(v)] If $s$ is as in \eqref{3e1} and $b$ as in (ii), then $b=\sum_{k\in\nn}b_k$ in $\cs'(\rn)$,
where, for each $k\in \nn,~b_k:=(f-c_k)\eta_k$, $\{\eta_k\}_{k\in \nn}$
is a partition of unity with respect to $\{B_k^*\}_{k\in \nn}$,
namely, for any $k\in\nn$, $\eta_k\in C_c^{\fz}(\rn)$,
$\supp \eta_k\st B_k^*$, $0\le\eta_k \le 1$
and $$\chi_{\CO}=\sum_{k\in \nn}\eta_k,$$ and
$c_k\in \cp_s(\rn)$ is a polynomial such that, for any $q\in\cp_s(\rn)$,
$$\langle f-c_k,q\eta_k\rangle=0.$$
Moreover, for any $k\in \nn$ and $x\in \rn$,
\begin{equation}\label{3e7}
M_0(b_k)(x)\ls M_N(f)(x)\chi_{B_k^*}(x)+\f{\sa r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}}\chi_{({B_k^*})^{\com}}(x),
\end{equation}
where $a_-$ and $M_0$ are as in \eqref{2e9}, respectively, \eqref{3e16}, and the implicit
positive constant is independent of $f$ and $k$.
\end{enumerate}
\end{lemma}
To show Lemma \ref{3l4}, we need an auxiliary inequality as follows,
which is a slight modification of \cite[p.\,100, (21)]{s93},
the details being omitted.
\begin{lemma}\label{3l5}
Let $\va\in [1,\fz)^n$ and $N\in \nn$. Assume that $\varphi$ is a function supported on
$B\in\mathfrak{B}$ with $\mathfrak{B}$ as in \eqref{2e2} and satisfies that, for each multi-index
$\bz\in \mathbb{Z}_+^n $ with $|\bz|\le N$, $|\pa^{\bz}\varphi|\le C_{(N)}r^{-\nu-\va \cdot \bz}$,
where $r$ is the radius of $B$ and $C_{(N)}$ a positive constant independent of $B$,
but depending on $N$. Then there exists a positive constant $C_{(N)}$, depending on $N$, such that, for any
$f\in \cs'(\rn)$ and $x\in B$,
$$\lf|\langle f,\varphi \rangle\r|\le C_{(N)} M_N (f)(x),$$
where $M_N$ is as in Definition \ref{2d4}.
\end{lemma}
Now we prove Lemma \ref{3l4}.
\begin{proof}[Proof of Lemma \ref{3l4}]
We prove this lemma by five steps.
\emph{Step 1.} In this step, we show (i). To this end, applying Lemma \ref{3l3} to
$\CO$ with $\CO$ as in Lemma \ref{3l4}, we obtain a collection of cubes, $\{B_k\}_{k\in \nn} \subset\mathfrak{B}$ with $\mathfrak{B}$ as in \eqref{2e2},
which has finite intersection property, such that $\CO=\bigcup_{k\in \nn}B_k$. Next, fix two numbers
$\de$ and $A^*$ satisfying $1<\de<A^*<\fz$. For any $k\in\nn$, let $\Qk:=B_k^{(\de)}$
and $\Qkk:=B_k^{(A^*)}$, then $$B_k\subset \Qk \subset \Qkk,$$
where $B_k^{(\de)}$ and $B_k^{(A^*)}$ are as in \eqref{2e2'} with $\dz$ replaced, respectively, by
$\de$ and $A^*$.
Moreover, by choosing $A^*$ sufficiently close to $1$ such that $\CO=\bigcup_{k\in \nn}\Qkk$
and $\{\Qkk\}_{k\in \nn}$ also has finite intersection property which is guaranteed by Lemma \ref{3l3}(vi), we know that (i) holds true.
\emph{Step 2.} In this step, we prove (iv). Let $\zeta\in C^{\fz}(\rn)$ satisfy that
$\supp\zeta\subset B_{\va}(\vec0_n,\de)$, $0\le\zeta\le1$ and $\zeta\equiv1$ on $B(\vec0_n,1)$.
In addition, for any $k\in\nn$ and $x\in\rn$, let $\zeta_k(x):=\zeta(r_k^{-\va}(x-x_k))$,
where $x_k$ denotes the center of $B_k$ and $r_k$ its radius, and
\begin{align}\label{3e8}
\eta_k(x):=
\frac {\zeta_k(x)}{\Sigma_{k\in\nn}\zeta_k(x)}.
\end{align}
Then it is easy to see that $\{\eta_k\}_{k\in\mathbb{N}}$ forms a smooth partition of unity
of $\CO$, namely, for any $k\in \nn$, $\eta_k\in C_c^{\fz}(\rn)$, $\supp\eta_k\subset\Qk$, $0\le \eta_k\le 1$
and $$\chi_{\CO}=\sum_{k\in \nn}\eta_k.$$
Notice that, for any multi-index $\bz\in \mathbb{Z}_+^n $, $k\in \nn$ and $x\in \rn$,
\begin{equation}\label{3e18}
\lf|\pa^{\beta}\eta_k(x)\r|\ls r_k^{-\beta\cdot\va}
\end{equation}
and $\int_{\rn}\eta_k(x)\,dx\thicksim |\Qk|\thicksim r_k^{\nu}\in (0,\fz)$.
Due to this, for any $k\in\nn$ and $x\in \rn$, let
$$\widetilde{\eta}_k(x):=\f{\eta_k}{\int_{\rn}\eta_k(y)\,dy}.$$
On the other hand, for any $k\in \nn$, let $\widetilde{b}_k:=(f-\widetilde{c}_k)\eta_k$,
where $\widetilde{c}_k$ is a constant determined by the requirement
that $\int_{\rn}\widetilde{b}_k(x)\,dx=0$, namely, $\widetilde{c}_k=\langle f,\widetilde{\eta}_k\rangle$.
We then conclude that, for any $k\in \nn$,
\begin{align}\label{3e9}
\lf|\widetilde{c}_k\r|\ls \sigma.
\end{align}
Indeed, for any $k\in \nn$, by Lemma \ref{3l3}(iv), we find that there exists some
$\widetilde{x}\in B_k^{(3A)}\cap\CO^{\com}$ with $A\in[1,\fz)$ as in Lemma \ref{3l3}.
Then \eqref{3e9} follows from Lemma \ref{3l5} with $\varphi:=\widetilde{\eta}_k$
and $B:=B_k^{(3A)}$. Similarly, from the fact that $\Qkk\st\CO$
and Lemma \ref{3l5}, we deduce that, for any $N,\,k\in \nn$ and $x\in \Qkk$,
\begin{align}\label{3e10}
|\widetilde{c}_k|\ls M_N(f)(x).
\end{align}
Now, for any $x\in \rn$, let $$g(x):=f(x)\chi_{\CO^{\com}}(x)+\sum_{k\in \nn}\widetilde{c}_k\eta_k (x).$$
If $x\in \CO^{\com}$, then $g(x)=f(x)$ and $M_N(f)(x)\le \sigma$, which imply that $|g(x)|\ls \sigma$;
if $x\in \CO$, then, by \eqref{3e9} and the finite intersection property of $\{\widetilde{B}_{k}\}_{k\in\nn}$,
we know that $|g(x)|\ls \sigma$. Thus, $g\in L^{\fz}(\rn)$
and $\|g\|_{L^{\fz}(\rn)} \ls \sa$, which completes the proof of (iv).
\emph{Step 3.} In this step, we show (v).
For any $k\in \nn$, let $b_k:=(f-c_k)\eta_k$ with $\eta_k$ as in \eqref{3e8}, where $c_k\in\cp_s(\rn)$
such that, for any $q\in\cp_s(\rn)$,$$\langle f-c_k,q\eta_k\rangle=0.$$
The existence of $c_k$ can be verified by an argument similar to that used in \cite[p.\,104]{s93}.
Now we establish the following two estimates, namely,
for any $N,\,k\in \nn$ and $x\in \Qkk$,
\begin{equation}\label{3e11}
M_0(b_k)(x)\ls M_N(f)(x)
\end{equation}
and, for any $k\in \nn$ and $x\in {(B_k^*)}^{\com}$,
\begin{equation}\label{3e12}
M_0(b_k)(x)\ls \f{\sa r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}}.
\end{equation}
To this end, we first claim that,
for any $k\in \nn$ and $x\in \rn$,
\begin{align}\label{3e13}
|c_k(x)\eta_k(x)|\ls \sigma
\end{align}
and, for any $N,\,k\in \nn$ and $x\in \Qkk$,
\begin{align}\label{3e14}
|c_k(x)\eta_k(x)|\ls M_N(f)(x).
\end{align}
Indeed, for any $k\in \nn$
and $q\in \cp_s(\rn)$, if we define
\begin{align*}
\|q\|_{\mathcal{H}}:=
\lf[\frac1{\int_\rn\eta_{k}(x)\,dx}\int_\rn
|q(x)|^2\eta_{k}(x)\,dx\r]^{1/2},
\end{align*}
then, by an argument similar to that used in the proof of \cite[p.\,104, (28)]{s93}, we find that,
for any multi-index $\bz\in\mathbb{Z}_+^n$, $k\in\nn$ and $q\in\cp_s(\rn)$,
\begin{align}\label{3e15}
\sup_{x\in \Qkk}\lf|\pa^{\bz} q(x)\r|\ls r_k^{-\bz\cdot\va}\|q\|_\mathcal{H},
\end{align}
where $\mathcal{H}$ denotes the Hilbert space $L^2(\Qkk,\widetilde{\eta}_kdx)$.
By \eqref{3e15} and an argument similar to that used in the estimations of \eqref{3e9} and \eqref{3e10},
we easily know that \eqref{3e13} and \eqref{3e14} hold true.
Let $k\in \nn$, $x\in \Qkk$, $t\in (0,\fz)$ and $\varphi(\cdot):=\eta_k(\cdot)\Phi_t(x-\cdot)$,
where $\Phi$ is as in \eqref{3e16}.
Then $(f\eta_k)*\Phi_t(x)=\langle f,\varphi\rangle$. When $t\in (0,r_k]$, from Lemma \ref{3l5} with $B:=B_{\va}(x,t)$
and the fact that, for any $y\in \rn$, $|\pa^\beta\varphi(y)|\ls t^{-\nu-\beta\cdot\va}$, it follows that
$\lf|(f\eta_k)*\Phi_t(x)\r|=|\langle f,\varphi\rangle|\ls_{(N)} M_N(f)(x)$ for any $N\in\nn$,
here and hereafter, the symbol $\ls_{(N)}$ means that the implicit positive constant may depend on $N$. When $t\in (r_k,\fz)$,
notice that, for any $y\in \rn$,
$|\pa^\beta\varphi(y)|\ls r_k^{-\nu-\beta\cdot\va}$. Then, by Lemma \ref{3l5} with $B:=B_{\va}(x,cr_k)$,
where $c$ is a positive constant large enough such that $\Qkk\st B_{\va}(x,cr_k)$,
we conclude that $\lf|(f\eta_k)*\Phi_t(x)\r|=|\langle f,\varphi\rangle|\ls_{(N)} M_N(f)(x)$ for any $N\in\nn$. Thus,
for any $N,\,k\in \nn$ and $x\in \Qkk$,
$$M_0(f\eta_k)(x)=\sup_{t\in (0,\fz)}
|f\eta_k\ast\Phi_t(x)|\ls_{(N)} M_N(f)(x).$$
On the other hand, by \eqref{3e16} and \eqref{3e14}, we find that, for any $N,\,k\in \nn$ and $x\in \Qkk$,
$M_0(c_k\eta_k)(x)\ls_{(N)} M_N(f)(x)$. Therefore, for any $N,\,k\in \nn$ and $x\in \Qkk$,
$$M_0(b_k)(x)\le M_0(f\eta_k)(x)+M_0(c_k \eta_k)(x)\ls_{(N)} M_N(f)(x),$$
which implies that \eqref{3e11} holds true.
Let $\Phi$ be as in \eqref{3e16}, $k\in\nn$ and $x\in(\Qkk)^{\com}$.
Then, by the fact that, for any $q\in \cp_s(\rn)$, $\langle b_k,q \rangle=0$, we know that, for any $k\in\nn$ and $t\in(0,\fz)$,
\begin{align}\label{3e3}
b_k*\Phi_t(x)&=b_k*\Phi_t(x)-\langle b_k,q_0 \rangle\\
&=\lf[f\eta_k*\Phi_t(x)-\langle f\eta_k,q_0 \rangle\r] -\lf[c_k\eta_k*\Phi_t(x)
-\langle c_k\eta_k,q_0 \rangle\r]=:\textrm{I}-\textrm{II},\noz
\end{align}
where $q_0$ is the Taylor expansion of $\Phi_t$ at the point $x-x_k$ of order $s$ and $x_k$
denotes the center of $B_k^*$.
For any $x\in(\Qkk)^{\com}$, let $\widetilde{\varphi}(\cdot):=\eta_k(\cdot)[\Phi_t(x-\cdot)-q_0(\cdot)]$.
Then, from the Taylor remainder theorem,
we deduce that, for any $k\in\nn$, $t\in(0,\fz)$ and $y\in \rn$,
\begin{align}\label{3e29}
\lf|\widetilde{\varphi}(y)\r|&=\lf|t^{-\nu}\eta_k(y)\lf[\Phi\lf(\dfrac{x-y}{t^{\va}}\r)-\sum_{|\az|\le s}
\dfrac{\pa^\az\Phi(\dfrac{x-x_k}{t^{\va}})}
{\az!}\lf(\dfrac{x_k-y}{t^{\va}}\r)^\az\r]\r|\\
&\ls \lf|t^{-\nu}\eta_k(y)
\sum_{|\az|=s+1}\pa^\az\Phi
\lf(\dfrac{\xi}{t^{\va}}\r)
\lf(\dfrac{x_k-y}{t^{\va}}\r)^\az\r|,\noz
\end{align}
where $\xi:=x-x_k+\theta(x_k-y)$ for some $\theta\in [0,1]$.
Recall that, for any $k\in\nn$, $\supp\eta_k\subset\Qk$ and $\Qk\subsetneqq\Qkk$.
Thus, for any $k\in \nn$ and $y\in \rn$, if $x\in(\Qkk)^{\com}$ and $\eta_k(y)\neq 0$,
then, it holds true that $|x-y|_{\va}\sim|x-x_k|_{\va}\gtrsim r_k$, which, combined with Lemma \ref{2l2}(ii), implies that
$|\xi|_{\va}\geq|x-x_k|_{\va}-|x_k-y|_{\va}\gtrsim |x-x_k|_{\va}$.
By this, the fact $|\xi/t^{\va}|_{\va}<1$ [which is deduced from $\widetilde{\varphi}(\cdot)\neq 0$
and $\supp \Phi\st B(\vec{0}_n,1)$], and Lemma \ref{2l2}(i), we conclude that $t\gs |x-x_k|_{\va}$.
Thus, we claim that, for any multi-index $\bz\in \mathbb{Z}_+^n $, $k\in \nn$ and $y\in \rn$,
\begin{equation}\label{3e17}
\lf|\pa^\bz \widetilde{\varphi}(y)\r| \ls \f{r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}} r_k^{-\nu-\va\cdot\bz}.
\end{equation}
Indeed, by \eqref{3e29}, for any multi-index $\bz\in \mathbb{Z}_+^n $, $k\in \nn$ and $y\in \rn$,
\begin{align}\label{3e30}
\lf|\pa^\bz \widetilde{\varphi}(y)\r| \ls& \lf|t^{-\nu}\pa^\bz
\eta_k(y)\sum_{|\az|=s+1}\pa^\az\Phi
\lf(\dfrac{\xi}{t^{\va}}\r)
\lf(\dfrac{x_k-y}{t^{\va}}\r)^\az\r|
+\lf|t^{-\nu}\eta_k(y)
\sum_{|\az|=s+1}\pa^{\az+\bz}\Phi
\lf(\dfrac{\xi}{t^{\va}}\r)
\lf(\dfrac{x_k-y}{t^{\va}}\r)^\az\r|\\
&+\lf|t^{-\nu}\eta_k(y)
\sum_{|\az|=s+1}\pa^\az\Phi
\lf(\dfrac{\xi}{t^{\va}}\r)
\pa^\bz\lf(\dfrac{x_k-y}{t^{\va}}\r)^\az\r|
=:\textrm{I}_1+\textrm{I}_2+\textrm{I}_3.\noz
\end{align}
Then, by \eqref{3e18}, $\Phi\in \cs(\rn)$, the fact that $|\xi|_{\va}\gtrsim |x-x_k|_{\va}$,
and (i), (v) and (vi) of Lemma \ref{2l2}, we find that, for any $K\in \mathbb{Z}_+$,
\begin{align}\label{3e31}
\textrm{I}_1 &\ls t^{-\nu}r_k^{-\va\cdot\bz}
\dfrac1{(1+|\frac{\xi}{t^{\va}}|)^K}
\lf|\dfrac{x_k-y}{t^{\va}}\r|^{s+1}
\ls t^{-\nu}r_k^{-\va\cdot\bz}\lf(\frac{t}
{|\xi|_{\va}}\r)^{K a_- } \lf(\frac{r_k}{t}\r)^{(s+1)a_-}\\
&\ls t^{-\nu}r_k^{-\va\cdot\bz}\lf(\dfrac{t}
{|x-x_k|_{\va}}\r)^{ K a_-} \lf(\frac{r_k}{t}\r)^{(s+1)a_-}.\noz
\end{align}
Let $K:=s+1$. From \eqref{3e31} and the fact that $t\gs |x-x_k|_{\va}$, we further deduce that
\begin{align}\label{3e32}
\textrm{I}_1
\ls \f{r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}} r_k^{-\nu-\va\cdot\bz}.
\end{align}
Similarly, we conclude that, for any $i\in\{2,3\}$,
\begin{align*}
\textrm{I}_i
\ls \f{r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}} r_k^{-\nu-\va\cdot\bz}.
\end{align*}
This, together with \eqref{3e30} and \eqref{3e32}, implies that \eqref{3e17} holds true. For any $k\in \nn$,
from Lemma \ref{3l3}(iv), it is easy to see that $B_k^{(3A)}\cap\CO^{\com}\neq\emptyset$.
Thus, by Lemma \ref{3l5} with $B:=B_k^{(3A)}$ and \eqref{3e17}, we know that, for any $x\in(\Qkk)^{\com}$,
\begin{equation}\label{3e19}
\lf|\textrm{I}\r|=\lf|\langle f,\widetilde{\varphi}\rangle\r|\ls \f{\sa r_k^{\nu+(s+1)a_-}}
{|x-x_k|_{\va}^{\nu+(s+1)a_-}}.
\end{equation}
On the other hand, from \eqref{3e13}, \eqref{3e17} with $\bz=\vec0_n$ and the fact that $0\le \eta_k\le 1$, it follows that, for any $x\in(\Qkk)^{\com}$,
\begin{align*}
|\textrm{II}|\le\int_{\rn}\lf|c_k(y)\eta_k(y)\r|\lf|\lf[\Phi_t(x-y)-q_0(y)\r]\r|\,dy
\ls \f{\sa r_k^{\nu+(s+1)a_-}}{|x-x_k|_{\va}^{\nu+(s+1)a_-}}\int_{\Qk}r_k^{-\nu}\,dy\sim
\f{\sa r_k^{\nu+(s+1)a_-}}{|x-x_k|_{\va}^{\nu+(s+1)a_-}},
\end{align*}
which, combined with \eqref{3e19}, \eqref{3e3}, \eqref{3e16} and Definition \ref{3d4}, further implies that \eqref{3e12} holds true.
By this and \eqref{3e11}, we find that \eqref{3e7} holds true.
Next we show the series $\{\sum_{k=1}^r b_k\}_{r\in\nn}$ converges in $\cs'(\rn)$.
Indeed, by Lemma \ref{3l8}, \eqref{3e7} and \eqref{2e8}, we conclude that,
for any given $N$ as in \eqref{2e11} and any $k\in \nn$,
\begin{equation*}\begin{aligned}
\lf\|b_k\r\|_{\vh} &\sim \lf\|M_0(b_k)\r\|_{\lv}\ls \lf\|M_N(f)\chi_{B_k^*}\r\|_{\lv}+
\sa\lf\|\f{ r_k^{\nu+(s+1)a_-}}
{|\cdot-x_k|_{\va}^{\nu+(s+1)a_-}}
\chi_{(B_k^*)^{\com}}\r\|_{\lv}\\
&\ls \lf\|M_N(f)\chi_{B_k^*}\r\|_{\lv}+\sa\lf\|
M_{\rm HL}(\chi_{\Qkk})\r\|_{L^{\frac{\nu+(s+1)a_-}
{\nu}\vp}(\rn)}^{\frac{\nu+(s+1)a_-}{\nu}}.
\end{aligned}\end{equation*}
Notice that $\widetilde{p}_->\frac{\nu}{\nu+(s+1)a_-}$ with $\widetilde{p}_-$
as in \eqref{3e1}. Then, by Lemma \ref{3l1} and \eqref{2e8}, we find that,
for any given $N$ as in \eqref{2e11} and any $k\in \nn$,
\begin{equation*}
\lf\|b_k\r\|_{\vh} \ls \lf\|M_N(f)\chi_{B_k^*}\r\|_{\lv}+\sa\lf\|
\chi_{\Qkk}\r\|_{\lv} \ls \lf\|M_N(f)\chi_{B_k^*}\r\|_{\lv},
\end{equation*}
which, combined with \cite[p.\,304, Theorem 2]{bp61}, implies that there exists some $g\in L^{\vp'}(\rn)$
with $\|g\|_{L^{\vp'}(\rn)}\le 1$ such that, for any $m,\,p\in \nn$,
\begin{align*}
\lf\|\sum_{k=m}^{m+p}b_k\r\|_{\vh}
\ls& \sum_{k=m}^{m+p}\lf\|b_k\r\|_{\vh}
\ls\sum_{k=m}^{m+p}\lf\|M_N(f)\chi_{B_k^*}\r\|_{\lv}\\
\ls&\int_{\rn}\lf|\sum_{k=m}^{m+p}M_N(f)(x)\chi_{B_k^*}(x)g(x)\r|\,dx
\ls \lf\|\sum_{k=m}^{m+p}M_N(f)\chi_{B_k^*}\r\|_{\vh}.
\end{align*}
Thus,
\begin{equation}\label{3e20}
\lf\|\sum_{k\in\nn}b_k\r\|_{\vh} \ls
\lf\|\sum_{k\in\nn}M_N(f)\chi_{B_k^*}\r\|_{\lv}
\ls \lf\|M_N(f)\chi_{\CO}\r\|_{\lv}\ls \|f\|_{\vh}<\fz,
\end{equation}
which implies that the series $\{\sum_{k=1}^r b_k\}_{r\in\nn}$ converges in $\vh$
and hence converges in $\cs'(\rn)$.
This finishes the proof of (v).
\emph{Step 4.} In this step, we prove (ii). Let $b:=\sum_{k\in\nn}b_k$ in $\cs'(\rn)$,
which is well defined by Step 3. From this, it further follows that the distribution
\begin{align}
\label{3e33}g:=f-b:=f-\sum_{k\in\nn}b_k
\end{align}
is well defined. Thus, $f=g+b$ in $\cs'(\rn)$,
which completes the proof of (ii).
\emph{Step 5.} In this step, we show (iii).
If $x\in \CO^{\com}$, then, from (v) and the facts that $M_0(f)\ls M_N(f)$ and
$|x-x_k|_{\va}\gs r_k$, we deduce that, for any given $N$ as in \eqref{2e11},
\begin{align*}
M_0(g)(x)\le M_0(f)(x)+\sum_{k\in \nn}M_0(b_k)(x)\ls M_N(f)(x)+\sum_{k\in\nn}
\f{\sa r_k^{\nu+(s+1)a_-}}{(r_k+|x-x_k|_{\va})^{\nu+(s+1)a_-}},
\end{align*}
which implies that \eqref{3e6} holds true when $x\in \CO^{\com}$.
If $x\in \CO$, then there exists some $m\in \nn$ such that $x\in B_m$. Let
$$E_1:=\{k\in\nn:\ \Qkk\cap B_m^*\neq\emptyset\}\quad{\rm and}\quad
E_2:=\{k\in\nn:\ \Qkk\cap B_m^*=\emptyset\}.$$
By \eqref{3e33}, we rewrite $$g=\lf(f-\sum_{k\in E_1}b_k\r)-\sum_{k\in E_2}b_k.$$
If $k\in E_2$, then $x\in B_m\cap(\Qkk)^{\com}$
and $|x-x_k|_{\va}\gs r_k$. By this, \eqref{3e12} and the fact that $\{\sum_{k=1}^r b_k\}_{r\in\nn}$ converges in $\cs'(\rn)$,
we conclude that, for any $x\in B_m$,
\begin{align}\label{3e34}
M_0\lf(\sum_{k\in E_2}b_k\r)(x)\ls\sum_{k\in E_2}
\f{\sa r_k^{\nu+(s+1)a_-}}{(r_k+|x-x_k|_{\va})^{\nu+(s+1)a_-}}.
\end{align}
Notice that
\begin{align}\label{3e35}
f-\sum_{k\in E_1}b_k=
\lf(f-\sum_{k\in E_1}f \eta_k\r)-\sum_{k\in E_1}c_k\eta_k
=:L-\sum_{k\in E_1}c_k\eta_k.
\end{align}
For the second term, by the finite intersection property of $\{\Qkk\}_{k\in \nn}$, we know that there exists
a positive constant $R$, independent of $m$, such that $\sharp E_1\le R$. Then, by \eqref{3e13} and the fact that
$x\in B_m$, we have
$$M_0\lf(\sum_{k\in E_1}c_k\eta_k\r)(x)\le
\sum_{k\in E_1}M_0\lf(c_k\eta_k\r)(x)\ls\sa\sim
\f{\sa r_m^{\nu+(s+1)a_-}}{(r_m+|x-x_m|_{\va})^{\nu+(s+1)a_-}}.$$
Finally, we estimate $M_0(L)(x)$ by
considering $L*\Phi_t(x)$ with $\Phi$ as in \eqref{3e16} and $t\in (0,\fz)$.
If $t\in (0,c_0 {\rm dist}(B_m,\CO^\com)]$,
where $${\rm dist}(B_m,\CO^\com):=\inf\lf\{|x-y|_{\va}:\ x\in B_m,~y\in \CO^\com\r\}$$
and $c_0\in (0,\fz)$ is small enough such that, for any $x\in B_m$ and
$t\in (0,c_0 {\rm dist}(B_m,\CO^\com)]$, $B_{\va}(x,t)\subset \widetilde{B}_m$, then,
for any $x\in B_m$, from the fact that $1-\sum_{k\in E_1}\eta_k(x)=1-\chi_{\CO}(x)=0$
and $\supp\Phi_t\st B_{\va}(0,t)$, we deduce that
\begin{align}\label{3e36}
L*\Phi_t(x)=\lf[f\lf(1-\sum_{k\in E_1}\eta_k\r)\r]*\Phi_t(x)=0.
\end{align}
On the other hand, if $t\in (c_0 {\rm dist}(B_m,\CO^\com),\fz)$, then, for any $x\in B_m$,
let $$\psi(\cdot):=\lf[1-\sum_{k\in E_1}\eta_k(\cdot)\r]\Phi_t(x-\cdot).$$
Obviously, $L*\Phi_t(x)=\langle f,\psi \rangle$.
By this and an argument similar to that used in the estimation of \eqref{3e11},
we find that, for any $t\in (c_0 {\rm dist}(B_m,\CO^\com),\fz)$ and $x\in B_m$,
$$\lf|L*\Phi_t(x)\r|=\lf|\langle f,\psi \rangle\r|\ls\sa\sim
\f{\sa r_m^{\nu+(s+1)a_-}}{(r_m+|x-x_m|_{\va})^{\nu+(s+1)a_-}}.$$
This, together with \eqref{3e35}, \eqref{3e36} and \eqref{3e34}, implies that \eqref{3e6} holds true, which
completes the proof of (iii) and hence of Lemma \ref{3l4}.
\end{proof}
From Lemma \ref{3l4} and its proof, we deduce the following result on the density,
which plays a key role in the proof of Theorem \ref{3t1} below.
\begin{lemma}\label{3l7}
Let $\va\in [1,\fz)^n$, $\vp\in(0,\fz)^n$ with $\widetilde{p}_-$ as
in \eqref{3e1} and $N$ be as in \eqref{2e11}.
Then $\vh\cap L^{\vp/\widetilde{p}_-}(\rn)$ is dense in $\vh$.
\end{lemma}
\begin{proof}
Let all the notation be the same as those used in the proof of Lemma \ref{3l4}.
For any $f\in\vh$, by (ii) and (v) of Lemma \ref{3l4}, we know that
$$f=g+b=g+\sum_{k\in \nn} b_k\quad\mathrm{in}\quad\cs'(\rn),$$
where $g,\,b$ and $\{b_k\}_{k\in\nn}$ are as in Lemma \ref{3l4}.
By \eqref{3e20}, we have
\begin{equation*}
\lf\|b\r\|_{\vh}=
\lf\|\sum_{k\in \nn}b_k\r\|_{\vh}
\ls \lf\|M_N(f)\chi_{\CO}\r\|_{\lv}\to 0
\quad{\rm as}\quad \sigma\to\fz,
\end{equation*}
where $\CO$ is as in Lemma \ref{3l4}.
Thus, for any $\varepsilon\in (0,\fz)$,
there exists some $\sa\in (0,\fz)$ such that
$$\|f-g\|_{\vh}=\|b\|_{\vh}<\varepsilon,$$
which, combined with the fact that $f\in\vh$, implies that $g\in\vh$. Therefore, to
complete the proof of Lemma \ref{3l7}, it suffices to show that $g\in L^{\vp/\widetilde{p}_-}(\rn)$.
To this end, by \cite[Theorem 6.1]{cgn17}, Definition \ref{2d5}, Lemma \ref{3l8} and \eqref{3e16},
we only need to prove that $M_0(g)\in L^{\vp/\widetilde{p}_-}(\rn)$. Indeed, by
\eqref{3e6}, \eqref{2e8}, Lemma \ref{3l2}, the finite intersection property of $\{B_k^*\}_{k\in\nn}$ and Lemma \ref{3l4}(i),
we conclude that
\begin{equation*}\begin{aligned}
\lf\|M_0(g)\r\|_{L^{\vp/{\widetilde{p}_-}}(\rn)}&\ls
\lf\|M_N(f)\chi_{\CO^{\com}}\r\|_{L^{\vp/{\widetilde{p}_-}}(\rn)}+
\sa\lf\|\sum_{k\in\nn}\f{ r_k^{\nu+(s+1)a_-}}
{(r_k+|\cdot-x_k|_{\va})^{\nu+(s+1)a_-}}\r\|_{L^{\vp/{\widetilde{p}_-}}(\rn)}\\
&\ls \sa^{1-{\widetilde{p}_-}}\lf\|M_N(f)\chi_{\CO^{\com}}\r\|_{\lv}^{\widetilde{p}_-}+
\sa\lf\|\sum_{k\in\nn}\lf[M_{\rm HL}(\chi_{\Qkk})
\r]^{\f{\nu+(s+1)a_-}{\nu}}\r\|_{L^{\vp/{\widetilde{p}_-}}(\rn)}\\
&\ls \sa^{1-{\widetilde{p}_-}}\lf\|M_N(f)\r\|_{\lv}^{\widetilde{p}_-}+
\sa\lf\|\lf\{\sum_{k\in\nn}\lf[M_{\rm HL}(\chi_{\Qkk})
\r]^{\f{\nu+(s+1)a_-}{\nu}}\r\}^{\f{\nu}{\nu+(s+1)a_-}}\r\|_{L^{{
\f{\nu+(s+1)a_-}{\nu}}\f{\vp}{{\widetilde{p}_-}}}(\rn)}^{\f{\nu+(s+1)a_-}{\nu}}\\
&\ls \sa^{1-{\widetilde{p}_-}}\lf\|M_N(f)\r\|_{\lv}^{\widetilde{p}_-}+
\sa\lf\|\sum_{k\in\nn}\chi_{\Qkk}\r\|_{L^{\vp/{\widetilde{p}_-}}(\rn)}\\
&\ls \sa^{1-{\widetilde{p}_-}}\lf\|M_N(f)\r\|_{\lv}^{\widetilde{p}_-}+
\lf\|M_N(f)\chi_{\CO}\r\|_{\lv}^{\widetilde{p}_-}<\fz.
\end{aligned}\end{equation*}
This implies that $M_0(g)\in L^{\vp/\widetilde{p}_-}(\rn)$ and hence
finishes the proof of Lemma \ref{3l7}.
\end{proof}
Via borrowing some ideas from the proofs of \cite[Proposition 2.11]{zsy16} and \cite[Theorem 1.1]{sawa13},
we obtain the following lemma, which is of independent interest.
\begin{lemma}\label{3l6}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$, $\bz\in(0,\fz)$ and $r\in[1,\fz]\cap(p_+,\fz]$
with $p_+$ as in \eqref{2e10}. Assume that
$\{\lz_i\}_{i\in\nn}\st\mathbb{C}$, $\{B_i\}_{i\in\nn}\st\mathfrak{B}$
and $\{m_i\}_{i\in\nn}\st L^r(\rn)$ satisfy, for any $i\in\nn$,
$\supp m_i\st B_i^{(\bz)}$ with $B_i^{(\bz)}$ as in \eqref{2e2'},
\begin{align}\label{3e37}
\|m_i\|_{L^r(\rn)}
\le\frac{|B_i|^{1/r}}{\|\chi_{B_i}\|_{\lv}}
\end{align}
and
$$\lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{{p_-}}\r\}^{1/{p_-}}\r\|_{\lv}<\fz.$$
Then
\begin{align}\label{3e38}
\lf\|\lf[\sum_{i\in\nn}\lf|\lz_im_i\r|^{{p_-}}\r]
^{1/{p_-}}\r\|_{\lv}
\le C\lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{{p_-}}\r\}^{1/{p_-}}\r\|_{\lv},
\end{align}
where $p_-$ is as in \eqref{2e10} and $C$ a positive constant independent
of $\lz_i$, $B_i$ and $m_i$.
\end{lemma}
\begin{proof}
By \cite[p.\,304, Theorem 2]{bp61}, we find that there exists some $g\in L^{(\vp/{p_-})'}(\rn)$
with norm not greater than 1 such that
$$\lf\|\lf[\sum_{i\in\nn}\lf|\lz_im_i\r|^{{p_-}}\r]
^{1/{p_-}}\r\|_{\lv}^{p_-}=\lf\|\sum_{i\in\nn}\lf|\lz_im_i\r|^{{p_-}}
\r\|_{L^{\vp/{p_-}}(\rn)}\ls \int_{\rn}\sum_{i\in\nn}
\lf|\lz_im_i(x)\r|^{{p_-}}\lf|g(x)\r|\,dx.$$
Moreover, from the H\"{o}lder inequality and \eqref{3e37}, we deduce that,
for any $r\in[1,\fz]$,
\begin{equation*}\begin{aligned}
\int_{\rn}\sum_{i\in\nn}\lf|\lz_im_i(x)\r|^{{p_-}}\lf|g(x)\r|\,dx
&\le \sum_{i\in \nn}|\lz_i|^{p_-}\|m_i\|_{L^r (\rn)}^{p_-}
\lf\|g\r\|_{L^{(r/{p_-})'}(B_i^{(\bz)})}\\
&\le \sum_{i\in \nn}\frac{|\lz_i|^{p_-}|B_i|^{{p_-}/r}}
{\|\chi_{B_i}\|_{\lv}^{p_-}}\lf\|g\r\|_{L^{(r/{p_-})'}(B_i^{(\bz)})}\\
&\ls \sum_{i\in \nn}\frac{|\lz_i|^{p_-}|B_i^{(\bz)}|}
{\|\chi_{B_i}\|_{\lv}^{p_-}}\inf_{z\in B_i^{(\bz)}}
\lf[M_{\rm HL}\lf(|g|^{(r/{p_-})'}\r)(z)\r]^{1/{(r/{p_-})'}}\\
&\ls \int_{\rn} \sum_{i\in \nn}\frac{|\lz_i|^{p_-}
\chi_{B_i^{(\bz)}}(x)}{\|\chi_{B_i}\|_{\lv}^{p_-}}
\lf[M_{\rm HL}\lf(|g|^{(r/{p_-})'}\r)(x)\r]^{1/{(r/{p_-})'}}\,dx,
\end{aligned}\end{equation*}
which, together with the H\"{o}lder inequality [see Remark \ref{2r3}(iv)], Remark \ref{3r2}, Lemma \ref{3l1} and the fact that $r\in (p_+,\fz]$,
implies that
\begin{equation*}\begin{aligned}
\int_{\rn}\sum_{i\in\nn}\lf|\lz_im_i(x)\r|^{{p_-}}\lf|g(x)\r|\,dx
&\ls \lf\|\sum_{i\in \nn}\frac{|\lz_i|^{p_-}
\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}^{p_-}}
\r\|_{L^{\vp/{p_-}}(\rn)}
\lf\|\lf[M_{\rm HL}\lf(|g|^{(r/{p_-})'}\r)\r]^{1/{(r/{p_-})'}}
\r\|_{L^{(\vp/{p_-})'}(\rn)}\\
&\ls \lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}
\|_{\lv}}\r]^{{p_-}}\r\}^{1/{p_-}}
\r\|_{L^{\vp}(\rn)}^{p_-}
\lf\|g\r\|_{L^{(\vp/{p_-})'}(\rn)}.
\end{aligned}\end{equation*}
From this and the fact that $\lf\|g\r\|_{L^{(\vp/{p_-})'}(\rn)}\le 1$, it follows that \eqref{3e38} holds true.
This finishes the proof of Lemma \ref{3l6}.
\end{proof}
Now we state the main result of this section as follows.
\begin{theorem}\label{3t1}
Let $\va\in [1,\fz)^n,\,\vp\in (0,\fz)^n,\,r\in(\max\{p_+,1\},\fz]$
with $p_+$ as in \eqref{2e10}, $N$ be as in \eqref{2e11} and $s$
as in \eqref{3e1}.
Then $\vh=\vah$ with equivalent quasi-norms.
\end{theorem}
\begin{remark}
By Proposition \ref{2r4'} below, we know that,
when $\va:=(a_1,\ldots, a_n)\in [1,\fz)^n$
and $\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$,
where $p\in(0,1]$, Theorem \ref{3t1}
is just \cite[p.\,39, Theorem 6.5]{mb03}
with
\begin{align}\label{4e5}
A:=\left(
\begin{array}{cccc}
2^{a_1} & 0 & \cdots & 0\\
0 & 2^{a_2} & \cdots & 0\\
\vdots & \vdots& &\vdots \\
0 & 0 & \cdots & 2^{a_n} \\
\end{array}
\right).
\end{align}
\end{remark}
Now we proof Theorem \ref{3t1}.
\begin{proof}[Proof of Theorem \ref{3t1}]
First, we show that
\begin{align}\label{3e21}
\vah\subset\vh.
\end{align}
To this end, for any $f\in\vah$, by Definition \ref{3d2}, we know that
there exist $\{\lz_i\}_{i\in\nn}\st\mathbb{C}$
and a sequence of $(\vp,r,s)$-atoms, $\{a_i\}_{i\in\nn}$,
supported, respectively, on
$\{B_i\}_{i\in\nn}\st\mathfrak{B}$ such that
\begin{align*}
f=\sum_{i\in\nn}\lz_ia_i
\quad\mathrm{in}\quad\cs'(\rn)
\end{align*}
and
\begin{align*}
\|f\|_{\vah}\sim
\lf\|\lf\{\sum_{i\in\nn}\lf[\frac{|\lz_i|
\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]
^{p_-}\r\}^{p_-}\r\|_{\lv}.
\end{align*}
Let $M_0$ be as in \eqref{3e16}. Then, by Lemma \ref{3l8}, we have
$$\|f\|_{\vh}\sim\lf\|M_0(f)\r\|_{\lv}.$$
Thus, to prove \eqref{3e21}, it suffices
to show that
\begin{align}\label{3e22}
\lf\|M_0\lf(\sum_{i\in\nn}\lz_ia_i\r)\r\|_{\lv}\ls \lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}
\|_{\lv}}\r]^{p_-}\r\}^{p_-}\r\|_{\lv}.
\end{align}
For this purpose, first, for any $i\in\nn$ and $x\in B_i^{(2)}$,
where $B_i^{(2)}$ is as in \eqref{2e2'} with $\dz=2$, it is easy to see that
\begin{align}\label{3e23}
M_0(a_i)(x)\ls M_{\rm HL}(a_i)(x).
\end{align}
On the other hand, for any $i\in\nn$, by the vanishing
moment condition of $a_i$, we conclude that, for any $t\in(0,\fz)$ and $x\in (B_i^{(2)})^{\com}$,
\begin{align}\label{3e39}
\lf|a_i*\Phi_t(x)\r|&\le\int_{B_i}\lf|a_i(y)
\Phi_t(x-y)\r|\,dy\\
&=t^{-\nu}\int_{B_i}\lf|a_i(y)\r|
\lf|\Phi\lf(\dfrac{x-y}{t^{\va}}\r)-\sum_{|\az|\le s}
\dfrac{\pa^\az\Phi(\dfrac{x-x_i}{t^{\va}})}
{\az!}\lf(\dfrac{x_i-y}{t^{\va}}\r)^\az\r|\,dy\noz\\
&\ls t^{-\nu}\int_{B_i}\lf|a_i(y)\r|
\lf|\sum_{|\az|=s+1}\pa^\az\Phi
\lf(\dfrac{\xi}{t^{\va}}\r)
\lf(\dfrac{x_i-y}{t^{\va}}\r)^\az\r|\,dy,\noz
\end{align}
where $\Phi$ is as in \eqref{3e16}, for any $i\in \nn$, $x_i$ and $r_i$ denote the
center, respectively, the radius of $B_i$ and
$\xi:=x-x_i+\theta(x_i-y)$ for some $\theta\in [0,1]$.
For any $i\in\nn$, $x\in (B_i^{(2)})^{\com}$ and $y\in B_i$, we easily find that $|x-y|_{\va}\sim|x-x_i|_{\va}$ and $|\xi|_{\va}\geq
|x-x_i|_{\va}-|x_i-y|_{\va}\gtrsim |x-x_i|_{\va}$.
From this, \eqref{3e39}, the fact that $\Phi\in \cs(\rn)$, (i),
(v) and (vi) of Lemma \ref{2l2}, the H\"{o}lder inequality and Definition \ref{3d1}(ii),
we deduce that, for any $i\in\nn$, $t\in(0,\fz)$, $K\in\mathbb{Z}_+$ and $x\in (B_i^{(2)})^{\com}$,
\begin{align}\label{3e40}
\lf|a_i*\Phi_t(x)\r|&\ls t^{-\nu}\int_{B_i}\lf|a_i(y)\r|
\dfrac1{(1+|\xi/t^{\va}|)^K}
\lf|\dfrac{x_i-y}{t^{\va}}\r|^{s+1}\,dy\\
&\ls t^{-\nu}\lf(\frac{t}{|\xi|_{\va}}\r)^{K a_-}
\lf(\dfrac{r_i}{t}\r)^{(s+1)a_-}\int_{B_i}\lf|a_i(y)\r|\,dy\noz\\
&\ls t^{-\nu}\lf(\frac{t}{|x-x_i|_{\va}}\r)^{K a_-}
\lf(\dfrac{r_i}{t}\r)^{(s+1)a_-}\|a_i\|_{L^r(\rn)}\lf|B_i\r|^{1/{r'}}\noz\\
&\ls t^{-\nu}\lf(\frac{t}{|x-x_i|_{\va}}\r)^{K a_-}
\lf(\dfrac{r_i}{t}\r)^{(s+1)a_-}\frac{|B_i|}{\|\chi_{B_i}\|_{\lv}}.\noz
\end{align}
Without loss of generality, we may assume that, for any $i\in\nn$
and $x\in (B_i^{(2)})^{\com}$, $a_i*\Phi_t(x)\neq0$.
By this and the fact that $\supp\Phi\st B(\vec0_n,1)$,
we know that $t\gs|x-x_i|_{\va}$. From this and \eqref{3e40} with
$K:=s+1$, it follows that, for any $i\in\nn$,
$t\in(0,\fz)$ and $x\in (B_i^{(2)})^{\com}$,
\begin{align*}
\lf|a_i*\Phi_t(x)\r|\ls \frac1{\|\chi_{B_i}\|_{\lv}}
\lf(\f{r_i}{|x-x_i|_{\va}}\r)^{\nu+(s+1)a_-},
\end{align*}
which implies that, for any $i\in \nn$ and $x\in (B_i^{(2)})^{\com}$,
\begin{align*}
M_0(a_i)(x)\ls \frac1{\|\chi_{B_i}\|_{\lv}}
\lf(\f{r_i}{|x-x_i|_{\va}}\r)^{\nu+(s+1)a_-}\ls
\frac1{\|\chi_{B_i}\|_{\lv}}\lf[M_{\rm HL}\lf(\chi_{B_i}\r)(x)\r]^
{\frac{\nu+(s+1)a_-}{\nu}}.
\end{align*}
By this and \eqref{3e23}, we conclude that, for any $i\in \nn$ and $x\in \rn$,
\begin{align}\label{3e24}
M_0(a_i)(x)\ls M_{\rm HL}(a_i)(x)\chi_{B_i^{(2)}}(x)+
\frac1{\|\chi_{B_i}\|_{\lv}}\lf[M_{\rm HL}\lf(\chi_{B_i}\r)(x)\r]^
{\frac{\nu+(s+1)a_-}{\nu}}.
\end{align}
Notice that $r\in (\max\{p_+,1\},\fz]$. Then, by \eqref{3e23} and Lemma \ref{3l1},
we find that
$$\lf\|M_0(a_i)\chi_{B_i^{(2)}}\r\|_{L^r(\rn)}
\ls\lf\|M_{\rm HL}(a_i)\chi_{B_i^{(2)}}\r\|_{L^r(\rn)}
\ls\frac{|B_i|^{1/r}}{\|\chi_{B_i}\|_{\lv}},$$
which, combined with Lemma \ref{3l6}, further implies that
$$\lf\|\lf\{\sum_{i\in\nn}\lf[\lf|\lz_i\r|M_0(a_i)
\chi_{B_i^{(2)}}\r]^{{p_-}}\r\}^{1/{p_-}}\r\|_{\lv}
\ls \lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{{p_-}}\r\}^{1/{p_-}}\r\|_{\lv}.$$
From this, Remark \ref{2r3}(ii),
\eqref{3e24}, \eqref{2e8} and Lemma \ref{3l2},
we deduce that
\begin{align*}
\lf\|M_0\lf(\sum_{i\in\nn}\lz_ia_i\r)\r\|_{\lv}&\ls
\lf\|\sum_{i\in\nn}|\lz_i| M_0(a_i)\r\|_{\lv}\\
&\ls \lf\|\sum_{i\in\nn}|\lz_i| M_{\rm HL}(a_i)\chi_{B_i^{(2)}}\r\|_{\lv}\noz\\
&\hs+
\lf\|\sum_{i\in\nn}\frac{|\lz_i|}{\|\chi_{B_i}\|_{\lv}}\lf[M_{\rm HL}\lf(\chi_{B_i}\r)\r]^
{\frac{\nu+(s+1)a_-}{\nu}}\r\|_{\lv}\\
&\ls \lf\|\lf\{\sum_{i\in\nn}\lf[\lf|\lz_i\r|M_{\rm HL}(a_i)
\chi_{B_i^{(2)}}\r]^{{p_-}}\r\}^{1/{p_-}}\r\|_{\lv}\noz\\
&\hs+
\lf\|\sum_{i\in\nn}\lf\{\frac{|\lz_i|}{\|\chi_{B_i}\|_{\lv}}\lf[M_{\rm HL}\lf(\chi_{B_i}\r)\r]^
{\frac{\nu+(s+1)a_-}{\nu}}\r\}^{\frac{\nu}{\nu+(s+1)a_-}}\r\|_{L^
{\frac{\nu+(s+1)a_-}{\nu}\vp}(\rn)}^
{\frac{\nu+(s+1)a_-}{\nu}}\\
&\ls\lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}
\|_{\lv}}\r]^{p_-}\r\}^{1/p_-}\r\|_{\lv},
\end{align*}
which implies that \eqref{3e22} holds true. This proves \eqref{3e21}.
We now prove that $\vh\st\vah$. To this end, it suffices
to show that
\begin{align}\label{3e25}
\vh\st H_{\va}^{\vp,\fz,s}(\rn),
\end{align}
due to the fact that each $(\vp,\infty,s)$-atom is also a $(\vp,r,s)$-atom
and hence $H_{\va}^{\vp,\fz,s}(\rn)\st\vah$.
Next we prove \eqref{3e25} by two steps.
\emph{Step 1.} In this step, we show that,
for any $f\in\vh\cap L^{\vp/{p_-}}(\rn)$,
\begin{align}\label{3e26}
\|f\|_{H_{\va}^{\vp,\fz,s}(\rn)}\ls\|f\|_{\vh}
\end{align}
holds true.
For any $j \in \mathbb{Z}$, $N$ as in \eqref{2e11}
and $f\in\vh\cap L^{\vp/{\widetilde{p}_-}}(\rn)$, let
$\CO_j:=\{x\in\rn:\ M_N(f)(x)>2^j\}$.
Then, by Lemma \ref{3l4} with $\sigma=2^j$,
we know that there exist two distributions $g_j$ and $b_j$ such that
$$f=g_j+b_j
\quad {\rm in}\quad \cs'(\rn),
$$
and $b_j=\sum_{k\in\nn}b_{j,k}$ in $\cs'(\rn)$, where, for any $j\in \zz$ and
$k\in\nn$, $b_{j,k}:=(f-c_{j,k})\eta_{j,k}$, supported on $B_{j,k}\in\mathfrak{B}$,
$c_{j,k}\in \cp_s(\rn)$ and $\eta_{j,k}$ is constructed
as in \eqref{3e8} with $B_k$ replaced by $B_{j,k}$.
Moreover, by an estimation similar to that of \eqref{3e20}, we conclude that
\begin{align}\label{3e41}
\lf\|f-g_j\r\|_{\vh}=\lf\|b_j\r\|_{\vh}\ls
\lf\|M_N(f)\chi_{\CO_j}\r\|_{\lv}\to 0
\end{align}
as $j\to \fz$.
In addition, from the fact that $f\in L^{\vp/{\widetilde{p}_-}}(\rn)$ and
the H\"{o}lder inequality [see Remark \ref{2r3}(iv)],
it follows that $f\in L_{\rm loc}^1(\rn)$. By this and
Lemma \ref{3l4}(iv), we find that $\|g_j\|_{L^{\fz}(\rn)}\ls 2^j$.
This, together with \eqref{3e41}, further implies that
\begin{align}\label{3e42}
f=\sum_{j\in \mathbb{Z}}\lf(g_{j+1}-g_{j}\r)
\quad\mathrm{in}\quad\cs'(\rn).
\end{align}
On the other hand, for any $j\in \zz,\,k\in\nn$
and $q\in\cp_s(\rn)$, let
\begin{align}\label{3e28}
\|q\|_{j,k}:=
\lf[\frac1{\int_\rn\eta_{j,k}(x)\,dx}\int_\rn
|q(x)|^2\eta_{j,k}(x)\,dx\r]^{1/2}
\end{align}
and, for any $i$, $k\in\nn$ and $j\in\zz$,
$c_{j+1,k,i}$ be the orthogonal projection of
$(f-c_{j+1,i})\eta_{j,k}$ on $\cp_{s}(\rn)$
with respect to the norm in \eqref{3e28}.
Then, by \eqref{3e42} and an argument similar to that used
in \cite[pp.\,108-109]{s93}, we know that
$$f=\sum_{j\in \mathbb{Z}}\lf(g_{j+1}-g_{j}\r)=
\sum_{j\in \mathbb{Z}}\sum_{k\in\nn}\lf[b_{j,k}-\sum_{i\in \nn}\lf(b_{j+1,i}\eta_{j,k}-c_{j+1,k,i}\eta_{j+1,i}\r)\r]
=:\sum_{j\in \mathbb{Z}}\sum_{k\in\nn}
A_{j,k}\quad\mathrm{in}\quad\cs'(\rn),$$
and, for any $j\in\zz$ and $k\in\nn$, $A_{j,k}$ is supported on $B_{j,k}$ and
satisfies $\|A_{j,k}\|_{L^{\fz}(\rn)}\le C_02^j$ with $C_0$ being some positive constant,
independent of $j$ and $k$, and, for any $q\in \cp_s(\rn)$,
$$\int_{\rn}A_{j,k}(x)q(x)\,dx=0. $$
For any $j\in\zz$ and $k\in\nn$, let
\begin{align}\label{3e27}
{\kappa_{j,k}}:=C_0 2^j\lf\|B_{j,k}\r\|_{\lv}\quad {\rm and}\quad
a_{j,k}:=\f{A_{j,k}}{\kappa_{j,k}}.
\end{align}
Then it is easy to see that each $a_{j,k}$ is
a $(\vp,\fz,s)$-atom, namely,
\begin{align*}
\supp a_{j,k}\subset B_{j,k},\quad
\lf\|a_{j,k}\r\|_{L^{\infty}(\rn)}\le\frac1{\|B_{j,k}\|_{\lv}}
\end{align*}
and, for any $q\in\cp_{s}(\rn)$,
\begin{align*}
\int_\rn a_{j,k}(x)q(x)\,dx=0.
\end{align*}
Moreover, we have
$$f=\sum_{j\in \mathbb{Z}}\sum_{k\in\nn}
\kappa_{j,k}a_{j,k}\quad\mathrm{in}\quad\cs'(\rn).$$
In addition, from Definition \ref{3d2}, \eqref{3e27}, the fact that $\bigcup_{k\in\nn}B_{j,k}=\CO_j$,
the finite intersection property of $\{B_{j,k}\}_{k\in\nn}$
and the definition of $\CO_j$, we further deduce that
\begin{align*}
\|f\|_{H_{\va}^{\vp,\fz,s}(\rn)}&\ls
\lf\|\lf\{\sum_{j\in \mathbb{Z}}\sum_{k\in\nn}
\lf[\frac{\kappa_{j,k}\chi_{B_{j,k}}}{\|\chi_{B_{j,k}}
\|_{\lv}}\r]^{p_-}\r\}^{1/p_-}\r\|_{\lv}\\
&\sim \lf\|\lf[\sum_{j\in \mathbb{Z}}\sum_{k\in\nn}
\lf(2^j\chi_{B_{j,k}}\r)^{p_-}\r]^{1/p_-}\r\|_{\lv}
\ls \lf\|\lf[\sum_{j\in \mathbb{Z}}
\lf(2^j\chi_{\CO_{j}}\r)^{p_-}\r]^{1/p_-}\r\|_{\lv}\\
&\sim \lf\|\lf[\sum_{j\in \mathbb{Z}}
\lf(2^j\chi_{\CO_{j}\backslash
{\CO_{j+1}}}\r)^{p_-}\r]^{1/p_-}\r\|_{\lv}
\sim \lf\|\lf\{\sum_{j\in \mathbb{Z}}
\lf[M_N(f)\chi_{\CO_{j}\backslash
{\CO_{j+1}}}\r]^{p_-}\r\}^{1/p_-}\r\|_{\lv}\\
&\ls \lf\|M_N(f)\r\|_{\lv}\sim \|f\|_{\vh}.
\end{align*}
This implies that \eqref{3e26} holds true.
\emph{Step 2.} In this step, we prove that \eqref{3e26} also holds true
for any $f\in\vh$.
To this end, let $f\in\vh$. Then, by Lemma \ref{3l7},
we find that there exists a sequence
$\{f_i\}_{i\in\nn}\st\vh\cap L^{\vp/{\widetilde{p}_-}}(\rn)$
such that $f=\sum_{i\in\nn}f_i$ in $\vh$ and, for any $i\in\nn$,
$$\lf\|f_i\r\|_{\vh}\le2^{2-i}\|f\|_{\vh}.$$
Notice that, for any $i\in\nn$, by the conclusion obtained in Step 1,
we conclude that $f_i$ has an atomic decomposition, namely,
$$f_i=\sum_{j\in\zz}\sum_{k\in\nn}\kappa_{j,k,i}a_{j,k,i}
\hspace{0.4cm} {\rm in}\hspace{0.3cm} \cs'(\rn),$$
where $\{\kappa_{j,k,i}\}_{j\in\zz,\,k\in\nn}$
and $\{a_{j,k,i}\}_{j\in\zz,\,k\in\nn}$
are constructed as in \eqref{3e27}. Thus,
$\{a_{j,k,i}\}_{j\in\zz,\,k\in\nn}$ are $(\vp,\fz,r)$-atoms and
hence we have
$$f=\sum_{i\in\nn}\sum_{j\in\zz}\sum_{k\in\nn}\kappa_{j,k,i}a_{j,k,i}
\hspace{0.4cm} {\rm in}\hspace{0.3cm} \cs'(\rn)$$
and
$$\|f\|_{H_{\va}^{\vp,\fz,r}(\rn)}
\le\lf[\sum_{i\in\nn}\lf\|f_i\r\|_{\vh}^{p_-}\r]^{1/{p_-}}
\ls\|f\|_{\vh},$$
which implies that \eqref{3e26} holds true for any $f\in\vh$ and
hence completes the proof of Theorem \ref{3t1}.
\end{proof}
\section{Littlewood-Paley function characterizations of $\vh$\label{s4}}
In this section, as an application of the atomic
characterizations of $\vh$ obtained in Theorem \ref{3t1},
we establish the Littlewood-Paley function
characterizations of $\vh$.
Let $\va\in [1,\fz)^n$. Assume that $\phi\in\cs(\rn)$ is a radial function such that,
for any multi-index $\alpha\in\zz_+^n$ with $|\alpha|\le s$,
where $s$ is as in \eqref{3e1},
\begin{align}\label{4e1}
\int_{\rn}\phi(x)x^\alpha\,dx=0
\end{align}
and, for any
$\xi\in\rn\backslash\{\vec{0}_n\}$,
\begin{align}\label{4e2}
\sum_{k\in\zz}\lf|\widehat{\phi}\lf(2^{k\va}\xi\r)\r|^2=1,
\end{align}
here and hereafter, $\widehat{\phi}$ denotes the \emph{Fourier transform} of $\phi$,
namely, for any $\xi\in\rn$,
\begin{align*}
\widehat \phi(\xi) := \int_{\rn} \phi(x) e^{-2\pi\imath x \cdot \xi} \, dx,
\end{align*}
where $\imath:=\sqrt{-1}$.
Then, for any $\lambda\in(0,\fz)$ and
$f\in\cs'(\rn)$, the \emph{anisotropic Lusin area function} $S(f)$,
the \emph{Littlewood-Paley} $g$-\emph{function} $g(f)$ and
the \emph{Littlewood-Paley} $g_\lambda^\ast$-\emph{function} $g_\lambda^\ast(f)$
are defined, respectively, by setting, for any $x\in\rn$,
\begin{align*}
S(f)(x):=\lf[\sum_{k\in\mathbb{Z}}2^{-k\nu}\int_{B_{\va}(x,2^k)}
\lf|f\ast\phi_{k}(y)\r|^2\,dy\r]^{1/2},
\end{align*}
\begin{align}\label{4e25}
g(f)(x):=\lf[\sum_{k\in\mathbb{Z}}
\lf|f\ast\phi_{k}(x)\r|^2\r]^{1/2}
\end{align}
and
\begin{align*}
g_\lambda^\ast(f)(x):=
\lf\{\sum_{k\in\mathbb{Z}}2^{-k\nu}\int_{\rn}
\lf[\frac{2^{k}}{2^{k}+|x-y|_{\va}}\r]^{\lambda\nu}
\lf|f\ast\phi_{k}(y)\r|^2\,dy\r\}^{1/2},
\end{align*}
where, for any $k\in \mathbb{Z}$, $\phi_{k}(\cdot)
:=2^{-k\nu}\phi(2^{-k\va}\cdot)$.
Recall that $f\in\cs'(\rn)$ is said to
\emph{vanish weakly at infinity} if, for any $\phi\in\cs(\rn)$,
$f\ast\phi_{k}\to0$ in $\cs'(\rn)$ as $k\to \fz$.
In what follows, we always let $\cs'_0(\rn)$ be the set of all $f\in\cs'(\rn)$
vanishing weakly at infinity.
Then the main results of this section are the following
succeeding three theorems.
\begin{theorem}\label{4t1}
Let $\va\in [1,\fz)^n$, $\vp\in(0,\fz)^n$ and
$N$ be as in \eqref{2e11}.
Then $f\in\vh$ if and only if
$f\in\cs'_0(\rn)$ and $S(f)\in\lv$. Moreover,
there exists a positive constant $C$ such that,
for any $f\in\vh$,
$$C^{-1}\|S(f)\|_{\lv}\le\|f\|_{\vh}\le C\|S(f)\|_{\lv}.$$
\end{theorem}
\begin{theorem}\label{4t2}
Let $\va,\,\vp$ and $N$ be as in Theorem \ref{4t1}.
Then $f\in\vh$ if and only if
$f\in\cs'_0(\rn)$ and $g(f)\in\lv$. Moreover,
there exists a positive constant $C$ such that,
for any $f\in\vh$,
$$C^{-1}\|g(f)\|_{\lv}\le\|f\|_{\vh}\le C\|g(f)\|_{\lv}.$$
\end{theorem}
\begin{theorem}\label{4t3}
Let $\va,\,\vp$ and $N$ be as in Theorem \ref{4t1} and
$\lambda\in(1+\frac{2}{{\min\{\widetilde{p}_-,2\}}}, \fz)$,
where $\widetilde{p}_-:=\min\{p_1,\ldots,p_n\}$.
Then $f\in\vh$ if and only if $f\in\cs'_0(\rn)$ and
$g_\lambda^{\ast}(f)\in\lv$. Moreover,
there exists a positive constant $C$ such that,
for any $f\in\vh$,
$$C^{-1}\lf\|g_\lz^\ast(f)\r\|_{\lv}\le\|f\|_{\vh}
\le C\lf\|g_\lz^\ast(f)\r\|_{\lv}.$$
\end{theorem}
\begin{remark}\label{4r2}
\begin{enumerate}
\item[{\rm (i)}]
We should point out that Theorem \ref{4t2} gives a positive answer to the conjecture
proposed by Hart et al. in \cite[p.\,9]{htw17}, namely, the mixed-norm Hardy space
$H^{p,q}(\mathbb{R}^{n+1})$, with $p,\,q\in(0,\fz)$, introduced by Hart et al. \cite{htw17} via the
Littlewood-Paley $g$-function coincides, in the sense of equivalent quasi-norms, with $H^{\vp}_{\va}(\mathbb{R}^{n+1})$,
where $\va:=(\overbrace{1,\ldots, 1}^{n+1\ \mathrm{times}})$ and
$\vp:=(\overbrace{p,\ldots,p}^{n\ \mathrm{times}},q)$.
\item[{\rm (ii)}]
We should also point out that
the range of $\lz$ in Theorem \ref{4t3} does not coincide with the best known one,
namely, $\lz\in(2/p,\fz)$, of the $g_\lz^\ast$-function characterization
of the classical Hardy space $H^p(\rn)$ and it is still unclear whether or not the
$g_\lz^\ast$-function, when
$\lz\in(\frac{2}{\min\{\widetilde{p}_-,2\}},1+\frac{2}{\min\{\widetilde{p}_-,2\}}]$,
can characterize $\vh$, because the method used in the proof of
Theorem \ref{4t3} does not work in this case.
\end{enumerate}
\end{remark}
The following proposition establishes the relation between $\vh$ and $H_A^p(\rn)$,
where $H_A^p(\rn)$ is the anisotropic Hardy space introduced by Bownik in
\cite[p.\,17, Definition 3.11]{mb03}.
\begin{proposition}\label{2r4'}
Let $\va:=(a_1,\ldots,a_n)\in[1,\fz)^n$ and
$\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$, where $p\in(0,\fz)$.
Then $\vh$ and the anisotropic Hardy space $\vAh$
coincide with equivalent quasi-norms, where
$A$ is as in \eqref{4e5}.
\end{proposition}
\begin{proof}
Let $A$ be as in \eqref{4e5}. Then, by \cite[Remark 2.5(i)]{lwyy17}
and \cite[Theorem 6.2]{lwyy17} with $p(\cdot):=p\in(0,\fz)$, we conclude that
$f\in\vAh$ if and only if $f\in\cs'_0(\rn)$ and
$$\lf[\sum_{k\in\mathbb{Z}}
\lf||\det A|^{-k}f\ast\phi(A^{-k}\cdot)\r|^2\r]^{1/2}
=\lf[\sum_{k\in\mathbb{Z}}
\lf|2^{-k\nu}f\ast\phi(2^{-k\va}\cdot)\r|^2\r]^{1/2}
=g(f)\in L^p(\rn),$$
where $\phi$ is as in \eqref{4e25}.
This, combined with Theorem \ref{4t2} and the obvious fact that,
when $\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$ with $p\in(0,\fz)$,
$L^{\vp}(\rn)=L^p(\rn)$, further implies that, in this case, $f\in\vAh$
if and only if $f\in\vh$. Thus,
when $\va:=(a_1,\ldots,a_2)\in[1,\fz)^n$ and
$\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$, where $p\in(0,\fz)$,
$\vh=\vAh$ with equivalent quasi-norms, where
$A$ is as in \eqref{4e5}. This finishes the proof of
Proposition \ref{2r4'}.
\end{proof}
\begin{remark}
Recall that, via the Lusin-area function,
the Littlewood-Paley $g$-function or $g_\lambda^\ast$-function,
Li et al. in \cite[Theorems 2.8, 3.1 and 3.9]{lfy15} characterized the anisotropic Musielak-Orlicz Hardy
space $H_A^\varphi(\rn)$ with $\varphi:\ \rn\times[0,\fz)\to[0,\fz)$
being an anisotropic growth function (see \cite[Definition 2.3]{lfy15}).
As was mentioned in \cite[p.\,285]{lfy15}, if, for any given $p\in(0,1]$
and any $x\in\rn$ and $t\in(0,\fz)$,
\begin{align}\label{4e3}
\varphi(x,t):=t^p,
\end{align}
then $H_A^\varphi(\rn)=\vAh$.
From this and Proposition \ref{2r4'}, we deduce that,
when $\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$,
where $p\in(0,1]$, Theorems \ref{4t1}, \ref{4t2} and \ref{4t3}
are just \cite[Theorems 2.8, 3.1 and 3.9]{lfy15},
respectively, with $A$ as in \eqref{4e5} and $\varphi$ as in \eqref{4e3}.
\end{remark}
To prove Theorem \ref{4t1},
we need several technical lemmas. First, it is easy
to see that the following conclusion holds
true, the details being omitted.
\begin{lemma}\label{4l7}
Let $\va:=(a_1,\ldots,a_n)\in [1,\fz)^n$, $\vp:=(p_1,\ldots,p_n)\in(0,\fz)^n$, $r\in (0,\fz)$,
$x\in \rn$ and $Q_{\va}(x,r)\in \mathfrak{Q}$
with $\mathfrak{Q}$ as in \eqref{2e3}.
Then $\|\chi_{Q_{\va}(x,r)}\|_{\lv}=\prod_{i=1}^{n}2^{1/p_i}r^{a_i/p_i}$.
\end{lemma}
Moreover, using Lemma \ref{4l7} and borrowing some ideas from the proof of
\cite[Lemma 6.5]{yyyz16}, we obtain the following conclusion.
\begin{lemma}\label{4l1}
Let $\va\in [1,\fz)^n$, $\vp\in(0,\fz)^n$ and
$N$ be as in \eqref{2e11}.
Then $\vh\subset\cs'_0(\rn)$.
\end{lemma}
\begin{proof}
Let $f\in \vh$. Then, by Remark \ref{2r5}, we know that,
for any $\phi\in \cs(\rn)$, $k\in \mathbb{Z}$,
$x\in \rn$ and $y\in Q_{\va}(x,2^k)$,
$|f\ast\phi_k(x)|\ls M_N(f)(y)$ with $N$ as in \eqref{2e11}.
Thus, there exists a positive constant $C$ such that
\begin{align}\label{4e4}
Q_{\va}(x,2^k)\subset \{y\in \rn:\, M_N(f)(y)\geq C|f\ast\phi_k(x)|\}.
\end{align}
On the other hand, from Lemma \ref{4l7}, it follows that, for any $k\in \mathbb{Z}_+$,
$\|\chi_{Q_{\va}(x,2^k)}\|_{\lv}\gs 2^{k\nu/p_+}$
with $p_+$ as in \eqref{2e10}. By this and \eqref{4e4},
we conclude that, for any $x\in \rn$,
\begin{align*}
\lf|f\ast\phi_k(x)\r|&=\lf|Q_{\va}(x,2^k)\r|^{-1/{p_+}}
\lf|Q_{\va}(x,2^k)\r|^{1/{p_+}}\lf|f\ast\phi_k(x)\r|\\
&\ls 2^{-k\nu/p_+}\lf\|\chi_{Q_{\va}(x,2^k)}\r\|_{\lv}|f\ast\phi_k(x)|
\ls 2^{-k\nu/p_+}\lf\|M_N(f)\r\|_{\lv}\to 0
\end{align*}
as $k\to \fz$. This implies $f\in \cs'_0(\rn)$ and hence finishes the proof of Lemma \ref{4l1}.
\end{proof}
The following lemma is a special case of \cite[Lemma 2.3]{blyz10},
which is a variant of \cite[Theorem 11]{mc90}. Indeed, let $(a_1,\ldots,a_n)\in[1,\fz)^n$. Then, applying
\cite[Lemma 2.3]{blyz10} with $A$ as in \eqref{4e5},
we immediately obtain the following conclusions, the details being omitted.
\begin{lemma}\label{4l2}
Let $\va\in [1,\fz)^n$. Then there exists a set
$$\mathcal{Q}:=\lf\{Q_\alpha^k\subset\rn:\ k\in\mathbb{Z},
\,\alpha\in E_k\r\}$$
of open subsets, where $E_k$ is some index set, such that
\begin{enumerate}
\item[{\rm (i)}] for each $k\in\zz$,
$\lf|\rn\setminus\bigcup_{\alpha}Q_\alpha^k\r|=0$
and, when $\alpha\neq\beta$,
$Q_\alpha^k\cap Q_\beta^k=\emptyset$;
\item[{\rm(ii)}] for any $\alpha,\,\beta,\,k,\,\ell$ with $\ell\geq k$,
either $Q_\alpha^k\cap Q_\beta^\ell=\emptyset$ or
$Q_\alpha^\ell\subset Q_\beta^k$;
\item[{\rm(iii)}] for each $(\ell,\beta)$ and each $k<\ell$,
there exists a unique $\alpha$ such that
$Q_\beta^\ell\subset Q_\alpha^k$;
\item[{\rm(iv)}] there exist some $w\in\zz\setminus\zz_+$
and $u\in\nn$ such that, for any $Q_\alpha^k$
with $k\in\mathbb{Z}$ and $\alpha\in E_k$,
there exists $x_{Q_\alpha^k}\in Q_\alpha^k$
such that, for any $x\in Q_\alpha^k$,
$$x_{Q_\alpha^k}+2^{(wk-u)\va}B_0
\subset Q_\alpha^k\subset x+2^{(wk+u)\va}B_0,$$
where $B_0$ denotes the unit ball of $\rn$.
\end{enumerate}
\end{lemma}
In what follows, we call
$\mathcal{Q}:=
\{Q_\alpha^k\}_{k\in\mathbb{Z},\,\alpha\in E_k}$
from Lemma \ref{4l2} \emph{dyadic cubes} and
$k$ the \emph{level}, denoted by $\ell(Q_\alpha^k)$,
of the dyadic cube $Q_\alpha^k$
for any $k\in\mathbb{Z}$ and $\alpha\in E_k$.
\begin{remark}\label{4r1}
In the definition of $(\vp,r,s)$-atoms (see Definition \ref{3d1}),
if we replace anisotropic balls $\mathfrak{B}$ by
dyadic cubes, then, from Lemma \ref{4l2}, we deduce that
the corresponding variable anisotropic atomic Hardy space
coincides with the original one (see Definition \ref{3d2})
in the sense of equivalent quasi-norms.
\end{remark}
Now we establish the following Calder\'{o}n reproducing formula.
\begin{lemma}\label{4l3}
Let $\va \in [1,\fz)^n$ and $\varphi\in \cs(\rn)$ satisfy that
$\supp \widehat{\varphi}$ is compact and bounded away from the origin
and, for any $\xi\in\rn\setminus\{\vec{0}_n\}$,
\begin{align}\label{4e6}
\sum_{k\in\mathbb{Z}}
\widehat{\varphi}\lf(2^{k\va}\xi\r)=1.
\end{align}
Then, for any $f\in L^2(\rn)$,
$f=\sum_{k\in\mathbb{Z}}f\ast\varphi_k$ in $L^2(\rn)$.
The same holds true in $\cs'(\rn)$ for any $f\in \cs'_0(\rn)$.
\end{lemma}
To show Lemma \ref{4l3}, we need the following Lemma \ref{4l4},
which is just a variant of \cite[Lemma 3.8]{mb03},
the details being omitted.
\begin{lemma}\label{4l4}
Let $\varphi\in \cs(\rn)$ and $\int_{\rn}\varphi(x)\,dx=1$.
Then, for any $f\in \cs(\rn)$, $f\ast\varphi_k\to f$ in $\cs(\rn)$
as $k\to -\fz$. The same holds true in $\cs'(\rn)$ for any $f\in \cs'(\rn)$.
\end{lemma}
Now we prove Lemma \ref{4l3}.
\begin{proof}[Proof of Lemma \ref{4l3}]
We show this lemma by two steps.
\emph{Step 1.}
Assume $f\in L^2(\rn)$. For any $\xi\in \rn$, let $F(\xi):=\sum_{k\in\mathbb{Z}}
|\widehat{\varphi}(2^{k\va}\xi)|$.
Obviously, for any $\xi\in \rn$, $F(\xi)=F(2^{\va}\xi)$, which implies that, to show
$F\in L^{\fz}(\rn)$, it suffices to consider the values of
$F$ on $2^{\va}B_0\setminus B_0$, where $B_0$ denotes the unit ball of $\rn$.
Since $\widehat{\varphi}\in \cs(\rn)$ and $\supp \widehat{\varphi}$ is bounded away from $\vec0_n$,
it follows that, for any
$\xi\in \rn\setminus B_0$, $|\widehat{\varphi}(\xi)|\ls\frac1{1+|\xi|}$ and,
for any $\xi\in 2^{\va}B_0$, $|\widehat{\varphi}(\xi)|\ls|\xi|$.
Then, by (i), (v) and (vi) of Lemma \ref{2l2}, we find that, for any
$\xi \in 2^{\va}B_0\setminus B_0$,
\begin{align*}
F(\xi)&=\sum_{k\geq 0}\lf|\widehat{\varphi}\lf(2^{k\va}\xi\r)\r|+
\sum_{k< 0}\lf|\widehat{\varphi}\lf(2^{k\va}\xi\r)\r|
\ls \sum_{k\geq 0}\frac1{1+|2^{k\va}\xi|}+
\sum_{k< 0}\lf|2^{k\va}\xi\r|\\
&\ls\sum_{k\geq 0}\frac1{(2^k|\xi|_{\va})^{a_-}}+
\sum_{k< 0}\lf(2^k|\xi|_{\va}\r)^{a_-}
\ls\sum_{k\geq 0}\frac1{2^{ka_-}}+\sum_{k< 0}2^{(k+1)a_-}\ls 1,
\end{align*}
which implies $F\in L^{\fz}(\rn)$. By this, the Lebesgue
dominated convergence theorem and \eqref{4e6}, we conclude that, for any
$f\in L^2(\rn)$ and $\xi\in \rn$,
$$\widehat{f}(\xi)=\sum_{k\in \zz}\widehat{\varphi}\lf(2^{k\va}\xi\r)
\widehat{f}(\xi)\quad\mathrm{in}\quad L^2(\rn)$$
and hence $f=\sum_{k\in\mathbb{Z}}f\ast\varphi_k$ in $L^2(\rn)$.
\emph{Step 2.} Assume $f\in \cs'_0(\rn)$.
Let $\phi:=\sum_{k=0}^{\fz}\varphi_k$. Since $\varphi\in \cs(\rn)$
and $\varphi_{k}(\cdot):=2^{-k\nu}\varphi(2^{-k\va}\cdot)$,
it then follows that $\phi$ is well defined pointwise on $\rn$.
We now claim that \begin{align}\label{4e24}
\phi\in \cs(\rn)\quad \mathrm{and} \quad\int_{\rn}\phi(x)\,dx=1.
\end{align}
Assume that this claim holds true for the moment.
Then, by this and Lemma \ref{4l4}, we know that $f\ast\phi_{-N}\to f$
in $\cs'(\rn)$ as $N\to \fz$. On the other hand,
for any $f\in \cs'_0(\rn)$, it is easy to see
that $f\ast\phi_{N}\to 0$ in $\cs'(\rn)$ as $N\to \fz$.
Therefore, for any $f\in \cs'_0(\rn)$, as $N\to\fz$,
$$f\ast\phi_{-N}-f\ast\phi_{N}\to f\quad\mathrm{in}
\quad \cs'(\rn).$$
Moreover, since, for any $j\in \zz$,
$\phi_j=\sum_{k=0}^{\fz}(\varphi_k)_j=\sum_{k=j}^{\fz}\varphi_k$,
it follows that $\sum_{k=-N}^{N}\varphi_k=\phi_{-N}-\phi_{N+1}$. Therefore,
$$\lim_{N\to \fz}\sum_{k=-N}^{N}f\ast\varphi_k=\lim_{N\to \fz}f\ast
\lf(\sum_{k=-N}^{N}\varphi_k\r)=\lim_{N\to \fz}\lf(f\ast\phi_{-N}-f\ast\phi_{N+1}\r)=f$$
in $\cs'(\rn)$, which implies that, for any $f\in \cs'_0(\rn)$,
$f=\sum_{k\in\zz}f\ast\varphi_k$ holds true in $\cs'(\rn)$.
Let us now prove the above claim \eqref{4e24}. To this end, for any $\xi\in \rn$, let $G(\xi):=\sum_{k=0}^{\fz}
\widehat{\varphi}(2^{k\va}\xi)$.
Then, to show \eqref{4e24}, it suffices to prove that
$G\in \cs(\rn)$, $\phi=\mathcal{F}^{-1}G$ and
$\int_{\rn}\phi(x)\,dx=1$, where $\mathcal{F}^{-1}$ denotes
the inverse Fourier transform, namely, for any $\xi\in\rn$,
$\mathcal{F}^{-1}G(\xi):=\widehat{G}(-\xi)$.
Indeed, since $\supp \widehat{\varphi}$ is compact, we may assume that
$\supp \widehat{\varphi}\st 2^{k_0\va}B_0$ for some $k_0\in \zz$.
Then, for any $k\in\zz_+$, it is easy to see that
$\supp \widehat{\varphi}(2^{k\va}\cdot)\st 2^{(k_0-k)\va}B_0\st 2^{k_0\va}B_0$,
which implies that $\supp G\st 2^{k_0\va}B_0$. To prove $G\in C^{\fz}(\rn)$,
for any multi-index $\az\in \zz_+^n$ and $\xi\in \rn$, let
$$F_{\az}(\xi):=\sum_{k\in \zz}\lf|\pa^{\az}\lf[\widehat{\varphi}\lf(2^{k\va}\xi\r)\r]\r|.$$
We first show $F_{\az}\in L^{\fz}(\rn)$. Notice that, for any $\xi\in \rn$,
$$F_{\az}(2^{\va}\xi)=\sum_{k\in \zz}\lf|\pa^{\az}\lf[\widehat{\varphi}\lf(2^{(k+1)\va}\xi\r)\r]\r|
=\sum_{k\in \zz}\lf|\pa^{\az}\lf[\widehat{\varphi}\lf(2^{k\va}\xi\r)\r]\r|=F_{\az}(\xi),$$
which implies that, to show $F_{\az}\in L^{\fz}(\rn)$, we only need to consider the value
of $F_{\az}$ on $2^{\va}B_0\setminus B_0$.
From the fact that $\widehat{\varphi}\in \cs(\rn)$, (i) and (vi) of Lemma \ref{2l2}, we deduce that,
for any $\xi\in 2^{\va}B_0\setminus B_0$,
$$\lf|\pa^{\az}\lf[\widehat{\varphi}\lf(2^{k\va}\xi\r)\r]\r|\ls \frac1{1+|2^{k\va}\xi|}
\ls \frac1{(2^k|\xi|_{\va})^{a_-}}\ls 2^{-k a_-}$$
when $k\in\nn$, and $|\pa^{\az}\widehat{\varphi}(2^{k\va}\xi)|\ls 2^{k|\az|a_-}$
when $k\in \zz\setminus\nn$. By this, we further conclude that, for any $\xi\in 2^{\va}B_0\setminus B_0$,
$$F_{\az}(\xi)\ls \sum_{k\in\nn}{2^{-ka_-}}+\sum_{k\in \zz\setminus\nn}2^{k|\az|a_-}\ls 1.$$
Thus, $F_{\az}\in L^{\fz}(\rn)$. This implies that, for any $\xi\in \rn$,
$\pa^{\az} G(\xi)=\sum_{k=0}^{\fz}\pa^{\az}[\widehat{\varphi}(2^{k\va}\xi)]$.
Therefore, $G\in C^{\fz}(\rn)$. From this and $\supp G\st 2^{k_0\va}B_0$,
it follows that $G\in \cs(\rn)$.
Moreover, by the facts that
$\supp\widehat{\varphi}(2^{k\va}\cdot)\st 2^{(k_0-k)\va}B_0$ and $\supp (\sum_{k=0}^{\fz}
|\widehat{\varphi}(2^{k\va}\cdot)|)\st 2^{k_0\va}B_0$,
the H\"{o}lder inequality and the Minkowski inequality, we find that
\begin{align*}
\int_{\rn} \sum_{k=0}^{\fz}\lf|\widehat{\varphi}\lf(2^{k\va}\xi\r)\r|\,d\xi
&\le \lf|2^{k_0\va}B_0\r|^{1/2}\lf\{\int_{\rn} \lf[\sum_{k=0}^{\fz}
\lf|\widehat{\varphi}\lf(2^{k\va}\xi\r)\r|\r]^2\,d\xi\r\}^{1/2}
\ls 2^{\nu k_0/2} \sum_{k=0}^{\fz}\lf[\int_{\rn}
\lf|\widehat{\varphi}\lf(2^{k\va}\xi\r)\r|^2\,d\xi\r]^{1/2}\\
&\ls 2^{\nu k_0/2} \sum_{k=0}^{\fz}\lf[\int_{\rn}
\chi_{2^{(k_0-k)\va}B_0}(\xi)\,d\xi\r]^{1/2}
\ls 2^{\nu k_0} \sum_{k=0}^{\fz}2^{-k\nu/2}\ls 1.
\end{align*}
Then, by the Fubini theorem, we obtain $\mathcal{F}^{-1}G=\sum_{k=0}^{\fz}
\mathcal{F}^{-1}[\widehat{\varphi}(2^{k\va}\cdot)]=\phi$ and hence $\phi\in \cs(\rn)$.
Let $e_1:=(1,0,\ldots,0)\in \rn$. Since $\widehat{\varphi}\in \cs(\rn)$, from \eqref{4e6},
we deduce that
$$\int_{\rn}\phi(x)\,dx=\widehat{\phi}(\vec0_n)=\lim_{j\to -\fz}\widehat{\phi}\lf(2^{j\va}e_1\r)=
\lim_{j\to -\fz}\sum_{k=0}^{\fz}\widehat{\varphi}\lf(2^{(j+k)\va} e_1\r)=\sum_{k\in\mathbb{Z}}
\widehat{\varphi}\lf(2^{k\va}e_1\r)=1,$$
which completes the proof of \eqref{4e24} and hence of Lemma \ref{4l3}.
\end{proof}
Using Lemma \ref{4l3}, we obtain the following Calder\'{o}n reproducing formula.
\begin{lemma}\label{4l5}
Let $\va\in [1,\fz)^n$ and $s\in\mathbb{Z_+}$.
Then there exist $\varphi,\,\psi\in\cs(\rn)$ satisfying
\begin{enumerate}
\item[{\rm(i)}] $\supp\varphi\subset B_0,
\,\int_{\rn}x^\gamma\varphi(x)\,dx=0$ for any
$\gamma\in\zz_+^n$ with $|\gamma|\le s,
\,\widehat{\varphi}(\xi)\geq C$
for any $\xi\in\{x\in\rn:\ m\le|x|_{\va}\le t\}$,
where $0<m<t<1$ and $C\in(0,\fz)$ are constants;
\item[{\rm(ii)}] $\supp \widehat{\psi}$
is compact and bounded away from the origin;
\item[{\rm(iii)}] for any $\xi\in\rn\setminus\{\vec{0}_n\}$,
$\sum_{k\in\mathbb{Z}}
\widehat{\psi}(2^{k\va}\xi)\widehat{\varphi}(2^{k\va}\xi)=1$.
\end{enumerate}
Moreover, for any $f\in L^2(\rn),\,f=
\sum_{k\in\mathbb{Z}}f\ast\psi_k\ast\varphi_k$ in $L^2(\rn)$.
The same holds true in $\cs'(\rn)$ for any $f\in \cs'_0(\rn)$.
\end{lemma}
We point out that the existences of such $\varphi$ and $\psi$
in Lemma \ref{4l5} can be verified by an argument similar to that used in the proof of
\cite[Theorem 5.8]{bh06}. Then the conclusions of Lemma \ref{4l5}
follow immediately from Lemma \ref{4l3} via replacing $\varphi$ by $\varphi\ast\psi$.
Now we prove Theorem \ref{4t1}.
\begin{proof}[Proof of Theorem \ref{4t1}]
We first show the sufficiency of this theorem. For this purpose,
let $f\in\cs'_0(\rn)$ and $S(f)\in\lv$. Then we need
to prove that $f\in\vh$ and
\begin{align}\label{4e7}
\|f\|_{\vh}\ls\|S(f)\|_{\lv}.
\end{align}
To this end, for any $k\in\mathbb{Z}$, let
$\Omega_k:=\{x\in\rn:\ S(f)(x)>2^k\}$ and
$$\mathcal{Q}_k:=\lf\{Q\in\mathcal{Q}:
\ |Q\cap\Omega_k|>\frac{|Q|}2\ \ {\rm and}\
\ |Q\cap\Omega_{k+1}|\le\frac{|Q|}2\r\}.$$
It is easy to see that, for any $Q\in\mathcal{Q}$,
there exists a unique $k\in\mathbb{Z}$
such that $Q\in\mathcal{Q}_k$.
For any given $k\in\zz$, denote by $\{Q_i^k\}_i$ the collection of all \emph{maximal dyadic cubes}
in $\mathcal{Q}_k$,
namely, there exists no $Q\in\mathcal{Q}_k$
such that $Q_i^k\subsetneqq Q$ for any $i$.
For any $Q\in\mathcal{Q}$, let
$$\widehat{Q}:=\lf\{(y,t)\in\rn\times\mathbb{R}:\
y\in Q\ \ {\rm and}\
\ t\sim w\ell(Q)+u\r\},$$
here and hereafter, $t\sim w\ell(Q)+u$ always means
\begin{align}\label{4e8}
w\ell(Q)+u+1\le t<w[\ell(Q)-1]+u+1,
\end{align}
where $w$ and $u$ are as in Lemma \ref{4l2}(iv) and $\ell(Q)$ denotes the level of $Q$.
Clearly, $\{\widehat{Q}\}_{Q\in\mathcal{Q}}$ are mutually disjoint and
\begin{align}\label{4e9}
\rn\times\mathbb{R}=\bigcup_{k\in\mathbb{Z}}\bigcup_i B_{k,\,i},
\end{align}
where, for any $k\in\mathbb{Z}$ and $i$,
$B_{k,\,i}:=\bigcup_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}\widehat{Q}$.
Then, by Lemma \ref{4l2}(ii), we easily know that $\{B_{k,i}\}_{k\in\zz,\,i}$
are mutually disjoint.
Let $\psi$ and $\varphi$ be as in Lemma \ref{4l5}. Then
$\varphi$ has the vanishing moments up to order $s$ as in \eqref{3e1}. By
Lemma \ref{4l5}, the properties of the tempered distributions
(see \cite[Theorem 2.3.20]{lg14} or \cite[Theorem 3.13]{sw71}) and \eqref{4e9},
we find that, for any $f\in\cs'_0(\rn)$ with
$S(f)\in \lv$ and $x\in\rn$,
\begin{align*}
f(x)
&=\sum_{k\in\mathbb{Z}}f\ast\psi_k\ast\varphi_k(x)
=\int_{\rn\times\mathbb{R}}
f\ast\psi_t(y)\varphi_t(x-y)\,dy\,dm(t)\\
&=\sum_{k\in\mathbb{Z}}\sum_i\int_{B_{k,\,i}}
f\ast\psi_t(y)\varphi_t(x-y)\,dy\,dm(t)
=:\sum_{k\in\mathbb{Z}}\sum_i h_i^k(x)
\end{align*}
in $\cs'(\rn)$, where, for any $k\in\mathbb{Z},\,i$ and $x\in\rn$,
\begin{align}\label{4e10}
h_i^k(x)
:=&\int_{B_{k,\,i}}f\ast\psi_t(y)\varphi_t(x-y)\,dy\,dm(t)\\
=&\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\int_{\widehat{Q}}f\ast\psi_t(y)\varphi_t(x-y)\,dy\,dm(t)
=:\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}e_{Q}(x)\noz
\end{align}
with convergence in $\cs'(\rn)$, and $m(t)$ denotes
the \emph{counting measure} on $\mathbb{R}$, namely,
for any set $E\st \rr$, $m(E):=\sharp E$ if $E$ has
only finite elements, or else $m(E):=\fz$.
Using \cite[(3.23)]{lyy16LP}
with the dilation $A$ as in \eqref{4e5}, we conclude that,
for any $x\in\rn$,
\begin{align}\label{4e11}
\lf[S\lf(\sum_{Q\in\mathcal{R}}e_Q\r)(x)\r]^2\ls
\sum_{Q\in\mathcal{R}}\lf[M_{{\rm HL}}(c_Q\chi_Q)(x)\r]^2,
\end{align}
where $\mathcal{R}\st\mathcal{Q}$ is an arbitrary set of dyadic cubes, $e_Q$ is as in \eqref{4e10}
and, for any $Q\in\mathcal{R}$,
$$c_Q:=\lf[\int_{\widehat{Q}}|\psi_t\ast f(y)|^2
\,dy\frac{dm(t)}{2^{\nu t}}\r]^{1/2}.$$
Next we show that, for any $k\in\zz$ and $i$,
$h_i^k$ is a $(\vp,r,s)$-atom multiplied by a harmless constant. This is completed
by Steps 1 through 3 below.
\emph{Step 1.}
For any $x\in\supp h_i^k$, by \eqref{4e10}, $h_i^k(x)\neq0$
implies that there exists $Q\subset Q_i^k$ and
$Q\in\mathcal{Q}_k$ such that $e_{Q}(x)\neq0$.
Then there exists $(y,t)\in\widehat{Q}$ such that
$2^{-t\va}(x-y)\in B_0$, where $B_0$ denotes the unit ball of $\rn$.
By this, Lemma \ref{4l2}(iv),
\eqref{4e8} and Lemma \ref{2l2}(ii), we have
$$x\in y+2^{t\va}B_0\subset x_Q+2^{(w\ell(Q)+u)\va}B_0
+2^{(w[\ell(Q)-1]+u+1)\va}B_0\subset x_Q+2^{(w[\ell(Q)-1]+u+2)\va}B_0.$$
Thus,
$$\supp e_Q\subset x_Q+2^{(w[\ell(Q)-1]+u+2)\va}B_0.$$
From this, the fact that
$h_i^k=\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}e_{Q}$,
(ii) and (iv) of Lemma \ref{4l2} and Lemma \ref{2l2}(ii),
we further deduce that
\begin{align}\label{4e12}
\supp h_i^k
&\subset\bigcup_{Q\subset Q_i^k,\,Q\in\mathcal
{Q}_k}x_Q+2^{(w[\ell(Q)-1]+u+2)\va}B_0\\
&\subset x_{Q_i^k}+2^{w[\ell(Q_i^k)+u]\va}B_0+
2^{(w[\ell(Q_i^k)-1]+u+2)\va}B_0
\subset x_{Q_i^k}+2^{(w[\ell(Q_i^k)-1]+u+2)\va}B_0=:B_i^k.\noz
\end{align}
\emph{Step 2.}
For any $Q\in\mathcal{Q}_k$ and
$x\in Q$, by Lemma \ref{4l2}(iv), we find that
$$M_{{\rm HL}}\lf(\chi_{\Omega_k}\r)(x)\ge\frac1{2^{[w\ell(Q)+u]\nu}}
\int_{x_Q+2^{[w\ell(Q)+u]\va}B_0}\chi_{\Omega_k}(z)\,dz>2^{-2u\nu}
\frac{|\Omega_k\cap Q|}{|Q|}> 2^{-2u\nu-1},$$
which implies that
\begin{align}\label{4e13}
\bigcup_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}Q
\subset\widehat{\Omega}_k:=\lf\{x\in\rn:\
M_{{\rm HL}}\lf(\chi_{\Omega_k}\r)(x)> 2^{-2u\nu-1}\r\}.
\end{align}
In addition, for any $Q\in\mathcal{Q}_k$ and $x\in Q$,
by Lemma \ref{4l2}(iv) and
$Q\subset\widehat{\Omega}_k$, we know that
$$M_{{\rm HL}}\lf(\chi_{Q\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}\r)(x)
\geq\frac1{|Q|}\int_{Q}\chi_{\widehat{\Omega}_k
\setminus\Omega_{k+1}}(z)\,dz
\gs\frac{|Q|-|Q|/2}{|Q|}\gs\frac{\chi_Q(x)}2.$$
From this, \cite[Theorem 3.2]{blyz10} with the
dilation $A$ as in \eqref{4e5}, \eqref{4e11}, Lemma \ref{3l2}
and an argument similar to that used in
the proof of \cite[(3.26)]{lyy16LP},
it follows that, for any $r\in(1,\fz)$,
\begin{align}\label{4e14}
\lf\|\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
e_{Q}\r\|_{L^r(\rn)}
\ls\lf\|\lf[\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\lf(c_Q\r)^2\chi_{Q\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}\r]^{1/2}\r\|_{L^r(\rn)}.
\end{align}
On the other hand, for any $Q\in\mathcal{Q}_k,\,x\in Q$
and $(y,t)\in\widehat{Q}$, by Lemma
\ref{4l2}(iv), Lemma \ref{2l2}(ii) and \eqref{4e8},
we easily know that
$$x-y\in 2^{[w\ell(Q)+u]\va}B_0+2^{[w\ell(Q)+u]\va}B_0
\subset 2^{[w\ell(Q)+u+1]\va}B_0\subset 2^{t\va}B_0.$$
By this and the disjointness of
$\{\widehat{Q}\}_{Q\subset Q_i^k}$,
we find that
\begin{align}\label{4e15}
\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\lf(c_Q\r)^2\chi_{Q\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}(x)
&=\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\int_{\widehat{Q}}
|\psi_t\ast f(y)|^2\,dy\frac{dm(t)}{2^{\nu t}}
\chi_{Q\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}(x)\\
&\ls\lf[S(f)(x)\r]^2\chi_{Q_i^k\cap(\widehat
{\Omega}_k\setminus\Omega_{k+1})}(x).\noz
\end{align}
From the definition of $\widehat{\Omega}_k$
(see \eqref{4e13}), it follows that, for any $r\in(1,\fz)$,
\begin{align*}
\lf|\widehat{\Omega}_k\r|\le2^{(2u\nu+1)r}\int_{\rn}
\lf[M_{{\rm HL}}\lf(\chi_{\Omega_k}\r)(x)\r]^r\,dx
\ls|\Omega_k|,
\end{align*}
which, combined with \eqref{4e15}, implies that
\begin{align}\label{4e16}
&\lf\|\lf\{\sum_{Q
\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\lf(c_Q\r)^2\chi_{Q\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}\r\}^
{\frac12}\r\|^r_{L^r(\rn)}\\
&\hs\le\int_{\rn}
\lf[\chi_{Q_i^k\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}(x)
\int_{\bigcup_{Q\subset Q_i^k,\,
Q\in\mathcal{Q}_k}\widehat{Q}}
|\psi_t\ast f(y)|^2\,dy\frac{dm(t)}
{2^{\nu t}}\r]^{r/2}\,dx\noz\\
&\hs\ls 2^{kr}\lf|\widehat{\Omega}_k\r|
\ls2^{kr}\lf|\Omega_k\r|<\fz.\noz
\end{align}
For any $N\in\mathbb{N}$, let
$\mathcal{Q}_{k,\,N}:=
\{Q\in\mathcal{Q}_k:\ |\ell(Q)|>N\}$.
Then, replacing
$\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}e_{Q}$
by $\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_{k,N}}e_Q$
in \eqref{4e14}, we obtain
\begin{align*}
\lf\|\sum_{Q\subset Q_i^k,\,Q\in
\mathcal{Q}_{k,\,N}}e_{Q}\r\|_{L^r(\rn)}^r
&\ls\lf\|\lf[\sum_{Q\subset Q_i^k,\,Q\in
\mathcal{Q}_{k,\,N}}\lf(c_Q\r)^2\chi_{Q\cap
(\widehat{\Omega}_k\setminus\Omega_{k+1})}
\r]^{1/2}\r\|_{L^r(\rn)}^r\noz\\
&\ls\int_{\rn}\chi_{Q_i^k\cap
(\widehat{\Omega}_k\setminus\Omega_{k+1})}(x)
\lf[\int_{\bigcup_{Q\subset Q_i^k,\,Q\in
\mathcal{Q}_{k,\,N}}\widehat{Q}}|\psi_t\ast f(y)|^2\,dy
\frac{dm(t)}{2^{\nu t}}\r]^{r/2}\,dx.\noz
\end{align*}
From this, \eqref{4e16} and the
Lebesgue dominated convergence theorem, we deduce that
$$\lf\|\sum_{Q\subset Q_i^k,\,Q\in
\mathcal{Q}_{k,\,N}}e_{Q}\r\|_{L^r(\rn)}\rightarrow0$$
as $N\rightarrow\fz$, and hence
$h_i^k=\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}e_{Q}$
in $L^r(\rn)$. This, together with \eqref{4e14},
\eqref{4e15}, the definition of $B_i^k$ (see \eqref{4e12})
and Lemma \ref{4l2}(iv), implies that
\begin{equation}\label{4e17}
\lf\|h_i^k\r\|_{L^r(\rn)}\ls\lf\{\int_{\rn}\lf[S(f)(x)\r]^r
\chi_{Q_i^k\cap(\widehat{\Omega}_k
\setminus\Omega_{k+1})}(x)\,dx\r\}^{1/r}\ls2^k\lf|Q_i^k\r|^{1/r}
\le C_1 2^k\lf|B_i^k\r|^{1/r},
\end{equation}
where $C_1$ is a positive constant
independent of $f$, $k$ and $i$.
\emph{Step 3.}
Recall that $\varphi$ has the vanishing moments up to
$s\ge\lfloor\nu/a_-(1/\widetilde{p}_--1)\rfloor$
and so does $e_Q$. For any
$k\in\mathbb{Z},\,i$, $\gamma\in\zz_+^n$ with
$|\gamma|\le s$ and $x\in\rn$,
let $g(x):=x^\gamma\chi_{B_i^k}(x)$.
Clearly, $g\in L^{r'}(\rn)$ with $r\in(1,\fz)$.
Thus, by \eqref{4e12} and the facts that
$(L^{r'}(\rn))^\ast=L^r(\rn)$ and
$$\supp e_Q\subset x_Q+
2^{(w[\ell(Q)-1]+u+2)\va}B_0,$$
we conclude that
\begin{equation*}
\int_{\rn}h_i^k(x)x^\gamma\,dx
=\langle h_i^k,g\rangle
=\sum_{Q\subset Q_i^k,\,Q\in
\mathcal{Q}_k}\langle e_{Q},g\rangle
=\sum_{Q\subset Q_i^k,\,Q\in\mathcal{Q}_k}
\int_{\rn}e_{Q}(x)x^\gamma\,dx=0,
\end{equation*}
namely, $h_i^k$ has the vanishing moments up to $s$,
which, combined with \eqref{4e12} and \eqref{4e17},
implies that $h_i^k$ is a harmless constant multiple of a $(\vp,r,s)$-atom
supported on $B_i^k$.
For any $k\in\mathbb{Z}$ and $i$,
let $\lambda_i^k:=C_1 2^k\|\chi_{B_i^k}\|_{\lv}$ and
$a_i^k:=(\lambda_i^k)^{-1}h_i^k$, where
$C_1$ is as in \eqref{4e17}.
Then
$$f=\sum_{k\in\mathbb{Z}}\sum_i h_i^k=
\sum_{k\in\mathbb{Z}}\sum_i \lambda_i^k a_i^k\qquad {\rm in}\quad \cs'(\rn).$$
Moreover, it is easy to see that, for any $k\in\zz$ and $i$, $a_i^k$ is a $(\vp,r,s)$-atom.
For any $k\in\mathbb{Z}$ and $i$, by the fact that
$|\Qik\cap\Omega_k|\ge\frac{|\Qik|}{2}$
and Lemma \ref{4l7}, we find that
$$\lf\|\chi_{\Qik}\r\|_{\lv}\ls
\lf\|\chi_{\Qik\cap\Omega_k}\r\|_{\lv}.$$
From this, Theorem \ref{3t1},
the mutual disjointness of $\{Q_i^k\}_{k\in\mathbb{Z},\,i}$ and
Lemma \ref{4l2}(iv), we further deduce that
\begin{align*}
\|f\|_{\vh}
&\ls\lf\|\lf\{\sum_{k\in\zz}\sum_{i}
\lf[\frac{\lz_i^k\chi_{\Bik}}{\|\chi_{\Bik}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}\\
&\sim\lf\|\lf[\sum_{k\in\zz}\sum_{i}\lf(2^{k}
\chi_{\Bik}\r)^{{p_-}}\r]^{{1/p_-}}\r\|_{\lv}
\sim\lf\|\lf[\sum_{k\in\zz}\sum_{i}\lf(2^{k}
\chi_{\Qik}\r)^{p_-}\r]^{{1/p_-}}\r\|_{\lv}\\
&\sim\lf\|\sum_{k\in\zz}\sum_{i}
\lf(2^{k}\chi_{\Qik}\r)^{p_-}\r\|
_{L^{\vp/p_-}(\rn)}^{{1/p_-}}
\ls\lf\|\sum_{k\in\zz}\sum_{i}
\lf(2^{k}\chi_{\Qik\cap\Omega_k}\r)^{p_-}\r\|
_{L^{\vp/p_-}(\rn)}^{{1/p_-}}\\
&\ls\lf\|\lf[\sum_{k\in\zz}\lf(2^k\chi_{\Omega_k}\r)^
{p_-}\r]^{1/{p_-}}\r\|_{\lv}
\sim\lf\|\lf[\sum_{k\in\zz}
\lf(2^k\chi_{\Omega_k\setminus\Omega_{k+1}}\r)^
{p_-}\r]^{1/{p_-}}\r\|_{\lv}\\
&\ls\lf\|S(f)\lf[\sum_{k\in\zz}
\chi_{\Omega_k\setminus\Omega_{k+1}}
\r]^{1/{p_-}}\r\|_{\lv}
\sim\lf\|S(f)\r\|_{\lv},
\end{align*}
which implies that $f\in\vh$ and \eqref{4e7} holds true.
This finishes the proof of the sufficiency of Theorem \ref{4t1}.
Next we show the necessity of this theorem.
To this end, let $f\in\vh$. Then, by Lemma \ref{4l1}, we know that $f\in\cs'_0(\rn)$.
On the other hand, by Theorem \ref{3t1}, we conclude that
there exist $\{\lz_i\}_{i\in\nn}\st\mathbb{C}$
and a sequence of $(\vp,r,s)$-atoms, $\{a_i\}_{i\in\nn}$,
supported, respectively, on
$\{B_i\}_{i\in\nn}\st\mathfrak{B}$ such that
\begin{align*}
f=\sum_{i\in\nn}\lz_ia_i
\quad\mathrm{in}\quad\cs'(\rn)
\end{align*}
and
\begin{align*}
\|f\|_{\vh}\sim
\lf\|\lf\{\sum_{i\in\nn}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}.
\end{align*}
Let $w$ and $u$ be as in Lemma \ref{4l2}(iv).
Then, by an argument similar to that used in the proof of
\cite[(5.10)]{lyy17hl}, we find that,
for any $i\in \nn$ and $x\in (B_i^{(2^{u-w+2})})^\com$
with $B_i^{(2^{u-w+2})}$ as in \eqref{2e2'},
$$S(a_i)(x)\ls \lf\|\chi_{B_i}\r\|_{\lv}^{-1}
\lf[\HL(\chi_{B_i})(x)\r]^{\frac{\nu+(s+1)a_-}{\nu}},$$
where $M_{\rm HL}$ denotes the Hardy-Littlewood maximal operator as in \eqref{3e2}.
From this, we further deduce that, for any $x\in \rn$,
\begin{align}\label{4e19}
S(f)(x)&\le\sum_{i\in\nn}|\lz_i|S(a_i)(x)\chi_{B_i^{(2^{u-w+2})}}(x)
+\sum_{i\in\nn}|\lz_i|S(a_i)(x)\chi_{(B_i^{(2^{u-w+2})})^\com}(x)\\
&\ls\lf\{\sum_{i\in\nn}\lf[|\lz_i|S(a_i)(x)\chi_{B_i^{(2^{u-w+2})}}(x)\r]
^{p_-}\r\}^{1/p_-}\noz\\
&\qquad+\sum_{i\in\nn}\frac{|\lz_i|}{\|\chi_{B_i}\|_{\lv}}
\lf[\HL(\chi_{B_i})(x)\r]^{\frac{\nu+(s+1)a_-}{\nu}}.\noz
\end{align}
By \cite[Theorem 3.2]{blyz10} with the dilation $A$ as in \eqref{4e5},
we know that, for any $r\in (1,\fz)$ and $i\in \nn$,
$$\lf\|S(a_i)\r\|_{L^r(\rn)}\ls\lf\|a_i\r\|_{L^r(\rn)}.$$
Then, by \eqref{4e19} and an argument similar to that
used in the proof of Theorem \ref{3t1}, we further conclude that
$$\|S(f)\|_{\lv}\ls\|f\|_{\vh},$$
which completes the proof of the necessity
and hence of Theorem \ref{4t1}.
\end{proof}
In what follows, for any $x\in\rn$, let
\begin{equation*}
\rho_{\va}(x):=\sum_{j\in\mathbb{Z}}
2^{\nu j}\chi_{2^{(j+1)\va}B_0\setminus 2^{j\va}B_0}(x)\hspace{0.25cm}
{\rm when}\ x\neq\vec0_n,\hspace{0.35cm} {\rm or\ else}
\hspace{0.25cm}\rho_{\va}(\vec0_n):=0.
\end{equation*}
Recall that, for any given $\va\in [1,\fz)^n$,
$\phi\in\cs(\rn)$, $t\in(0,\fz)$,
$k\in\zz$ and any $f\in\cs'(\rn)$,
the\emph{ anisotropic Peetre maximal function}
$(\phi_{k}^*f)_t$ is defined by setting,
for any $x\in\rn$,
\begin{align*}
\lf(\phi_{k}^*f\r)_t(x)
:=\esup_{y\in\rn}\frac{|(\phi_{-k}\ast f)(x+y)|}
{[1+2^{\nu k}\rho_{\va}(y)]^t}
\end{align*}
and the \emph{$g$-function associated with $(\phi_{k}^*f)_t$}
is defined by setting, for any $x\in\rn$,
\begin{align}\label{4e20}
g_{t,\ast}(f)(x)
:=\lf\{\sum_{k\in\zz}\lf[\lf(\phi_{k}^*f\r)_t(x)\r]^2\r\}^{1/2},
\end{align}
where, for any $k\in \mathbb{Z}$, $\phi_{k}(\cdot)
:=2^{-k\nu}\phi(2^{-k\va}\cdot)$.
The following estimate is just a variant of
\cite[(3.13)]{lyy17}, which originates from
\cite[(2.66)]{u12} and the argument
used in the proof of \cite[Theorem 2.8]{u12},
the details being omitted.
\begin{lemma}\label{4l6}
Let $\phi\in\cs(\rn)$ be a radial function satisfying
\eqref{4e1} and \eqref{4e2}.
Then, for any given $N_0\in\nn$ and $r\in(0,\fz)$, there exists a positive
constant $C_{(N_0,r)}$, which may depends on $N_0$ and $r$, such that,
for any $t\in(0,N_0)$, $\ell\in\zz$, $f\in\cs'(\rn)$ and $x\in\rn$,
it holds true that
\begin{align*}
\lf[\lf(\phi^*_\ell f\r)_t(x)\r]^r
\le C_{(N_0,r)}\sum_{k\in\zz_+}2^{-\nu kN_0r}2^{\nu(k+\ell)}
\int_\rn\frac{|(\phi_{-k-\ell}\ast f)(y)|^r}
{[1+2^{\nu\ell}\rho_{\va}(x-y)]^{tr}}\,dy.
\end{align*}
\end{lemma}
We now prove Theorem \ref{4t2}.
\begin{proof}[Proof of Theorem \ref{4t2}]
First, let $f\in\vh$. Then Lemma \ref{4l1} implies that
$f\in\cs'_0(\rn)$. In addition, repeating the proof
of the necessity of Theorem \ref{4t1} with some slight
modifications, we easily conclude that $g(f)\in \lv$
and $\lf\|g(f)\r\|_{\lv}\ls\|f\|_{\vh}$.
Thus, by Theorem \ref{4t1}, we know that, to prove Theorem \ref{4t2},
it suffices to show that, for any $f\in\cs'_0(\rn)$
with $g(f)\in\lv$,
\begin{align}\label{4e21}
\|S(f)\|_{\lv}\ls\|g(f)\|_{\lv}
\end{align}
holds true.
Indeed, from the fact that, for any $f\in\cs'_0(\rn)$, $t\in(0,\fz)$ and almost every $x\in\rn$,
$S(f)(x)\ls g_{t,\ast}(f)(x)$, it follows that, to show \eqref{4e21}, we only need to prove that
\begin{align}\label{4e22}
\lf\|g_{t,*}(f)\r\|_{\lv}\ls\lf\|g(f)\r\|_{\lv}
\end{align}
holds true for any $f\in\cs'_0(\rn)$ and some $t\in(\frac1{\min\{\widetilde{p}_-,2\}},\fz)$.
Now we show \eqref{4e22}. To this end, assume that $\phi\in\cs(\rn)$
is a radial function and satisfies \eqref{4e1} and \eqref{4e2}.
Obviously, $t\in(\frac1{\min\{\widetilde{p}_-,2\}},\fz)$ implies that there exists
$r\in\lf(0,{\min\{\widetilde{p}_-,2\}}\r)$ such that $t\in(\frac1{r},\fz)$.
Fix $N_0\in(\frac1r,\fz)$.
By this, Lemma \ref{4l6} and the Minkowski inequality,
we know that, for any $x\in\rn$,
\begin{align*}
g_{t,*}(f)(x)
&=\lf\{\sum_{k\in\zz}\lf[\lf(\phi_k^*f\r)_t(x)\r]^2\r\}^{1/2}\\
&\ls\lf[\sum_{k\in\zz}\lf\{\sum_{j\in\zz_+}2^{-\nu jN_0r}2^{\nu(j+k)}
\int_\rn\frac{|(\phi_{-j-k}\ast f)(y)|^r}
{[1+2^{\nu k}\rho_{\va}(x-y)]^{tr}}\,dy\r\}^{2/r}\r]^{1/2}\\
&\ls\lf\{\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)}\lf[\sum_{k\in\zz}2^{\frac{2k\nu}r}
\lf\{\int_\rn\frac{|(\phi_{-j-k}\ast f)(y)|^r}
{[1+2^{\nu k}\rho_{\va}(x-y)]^{tr}}\,dy\r\}^{2/r}\r]^{r/2}\r\}^{1/r},
\end{align*}
which, combined with \eqref{2e8}, implies that
\begin{align*}
&\lf\|g_{t,*}(f)\r\|_{\lv}^{rp_-}\\
&\hs\ls\lf\|\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)}
\lf[\sum_{k\in\zz}2^{\frac{2k\nu}r}
\lf\{\int_\rn\frac{|(\phi_{-j-k}\ast f)(y)|^r}
{[1+2^{\nu k}\rho_{\va}(\cdot-y)]^{tr}}\,dy\r\}^{2/r}\r]^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-}\\
&\hs\ls\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)p_-}
\lf\|\lf[\sum_{k\in\zz}2^{\frac{2k\nu}r}
\lf\{\int_\rn\frac{|(\phi_{-j-k}\ast f)(y)|^r}
{[1+2^{\nu k}\rho_{\va}(\cdot-y)]^{tr}}\,dy\r\}^{2/r}\r]^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-}\\
&\hs\ls\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)p_-}
\lf\|\lf\{\sum_{k\in\zz}2^{\frac{2k\nu}r}
\lf[\sum_{i\in\nn}2^{-\nu itr}\int_{\rho_{\va}(\cdot-y)\sim 2^{\nu(i-k)}}
\lf|(\phi_{-j-k}\ast f)(y)\r|^r\,dy\r]^{2/r}\r\}^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-},
\end{align*}
where $\rho_{\va}(\cdot-y)\sim 2^{\nu(i-k)}$ means that
$\{x\in\rn:\ \rho_{\va}(x-y)<2^{-\nu k}\}$ when $i=0$,
or $\{x\in\rn:\ 2^{\nu(i-k-1)}\le\rho_{\va}(x-y)<2^{\nu(i-k)}\}$ when $i\in\nn$.
Then, from the Minkowski inequality and Lemma \ref{3l2},
we further deduce that
\begin{align*}
&\lf\|g_{t,*}(f)\r\|_{\lv}^{rp_-}\\
&\hs\ls\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)p_-}
\lf\|\sum_{i\in\nn}2^{-\nu itr}\lf\{\sum_{k\in\zz}2^{\frac{2k\nu}r}
\lf[\int_{\rho_{\va}(\cdot-y)\sim 2^{\nu(i-k)}}
\lf|(\phi_{-j-k}\ast f)(y)\r|^r\,dy\r]^{2/r}\r\}^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-}\\
&\hs\ls\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)p_-}
\lf\|\sum_{i\in\nn}2^{\nu i(1-tr)}\lf\{\sum_{k\in\zz}
\lf[\HL\lf(\lf|\phi_{-j-k}\ast f\r|^r\r)\r]^{2/r}\r\}^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-}\\
&\hs\ls\sum_{j\in\zz_+}2^{-j\nu(N_0r-1)p_-}
\sum_{i\in\nn}2^{\nu ip_-(1-tr)}\lf\|\lf(\sum_{k\in\zz}
\lf|\phi_{-j-k}\ast f\r|^2\r)^{r/2}\r\|
_{L^{\frac{\vp}{r}}(\rn)}^{p_-}
\ls\|g(f)\|_{\lv}^{rp_-}.
\end{align*}
This proves \eqref{4e22} and hence
finishes the proof of Theorem \ref{4t2}.
\end{proof}
Applying Theorems \ref{4t1} and \ref{4t2}, we now
prove Theorem \ref{4t3}.
\begin{proof}[Proof of Theorem \ref{4t3}]
By Theorem \ref{4t1} and the fact that, for any $f\in\cs'(\rn)$ and $x\in\rn$,
$S(f)(x)\le g_\lz^{\ast}(f)(x)$,
we find that
the sufficiency of this theorem is obvious. Thus,
to prove this theorem, it suffices to show the necessity.
To this end, let $f\in\vh$ and $\varphi$ be as in the proof
of Theorem \ref{4t2}. Then, by Lemma \ref{4l1},
we find that $f\in\cs'_0(\rn)$. By the fact that
$\lambda\in(1+\frac{2}{{\min\{\widetilde{p}_-,2\}}}, \fz)$,
we conclude that there exists some
$t\in(\frac1{{\min\{\widetilde{p}_-,2\}}},\fz)$ such that $\lz\in(1+2t,\fz)$
and, for any $x\in\rn$,
\begin{align*}
g_\lambda^\ast(f)(x)&=
\lf\{\sum_{k\in\mathbb{Z}}2^{-k\nu}\int_{\rn}
\lf[\frac{2^{k}}{2^{k}+|x-y|_{\va}}\r]^{\lambda\nu}
\lf|f\ast\varphi_{k}(y)\r|^2\,dy\r\}^{1/2}\\
&\ls\lf\{\sum_{k\in\mathbb{Z}}2^{-k\nu}
\lf[\lf(\varphi_{-k}^*f(x)\r)_t\r]^2\int_{\rn}
\lf[1+\frac{\rho_{\va}(x-y)}{2^{\nu k}}\r]^{2t-\lz}\,dy\r\}^{1/2}\\
&\ls\lf\{\sum_{k\in\mathbb{Z}}
\lf[\lf(\varphi_{-k}^*f(x)\r)_t\r]^2\r\}^{1/2}
\sim g_{t,*}(f)(x).
\end{align*}
This, together with \eqref{4e22} and Theorem \ref{4t2},
further implies that $g_\lambda^\ast(f)\in\lv$ and
$$\lf\|g_\lambda^\ast(f)\r\|_{\lv}
\ls\|f\|_{\lv},$$
which completes the proof Theorem \ref{4t3}.
\end{proof}
\section{Finite atomic characterizations of $\vh$\label{s5}}
In this section, we establish the finite atomic characterizations of $\vh$.
We begin with introducing the following notion
of anisotropic mixed-norm finite atomic Hardy spaces $\vfah$.
\begin{definition}\label{5d1}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$, $r\in (1,\fz]$
and $s$ be as in \eqref{3e1}. The \emph{anisotropic mixed-norm
finite atomic Hardy space} $\vfah$ is defined to be the set of all
$f\in\cs'(\rn)$ satisfying that there exist $I\in\nn$,
$\{\lz_i\}_{i\in[1,I]\cap\nn}\st\mathbb{C}$ and
a finite sequence of $(\vp,r,s)$-atoms,
$\{a_i\}_{i\in[1,I]\cap\nn}$, supported, respectively, on
$\{B_i\}_{i\in[1,I]\cap\nn}\st\mathfrak{B}$
such that
\begin{align*}
f=\sum_{i=1}^I\lambda_ia_i
\quad\mathrm{in}\quad\cs'(\rn).
\end{align*}
Moreover, for any $f\in\vfah$, let
\begin{align*}
\|f\|_{\vfah}:=
{\inf}\lf\|\lf\{\sum_{i=1}^{I}
\lf[\frac{|\lz_i|\chi_{B_i}}{\|\chi_{B_i}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv},
\end{align*}
where $p_-$ is as in \eqref{2e10} and the
infimum is taken over all decompositions of $f$ as above.
\end{definition}
By \cite[p.\,13, Theorem 3.6]{mb03} with $A$ as in \eqref{4e5},
we immediately obtain the following conclusion, the details being omitted.
\begin{lemma}\label{5l5} For any given $N\in\nn$, let $M_N$ be as in Definition
\ref{2d4}.
\begin{enumerate}
\item[\rm(i)]
Let $p\in(1,\fz]$. Then, for any given $N\in \nn$, there exists a positive constant $C_{(p,N)}$,
depending on $p$ and $N$, such that, for any $f\in L^p(\rn)$,
\begin{align*}
\lf\|M_N(f)\r\|_{L^p(\rn)}
\le C_{(p,N)}\|f\|_{L^p(\rn)}.
\end{align*}
\item[\rm(ii)] For any given $N\in \nn$, there exists a positive constant $C_{(N)}$,
depending on $N$, such that, for any $\lambda\in (0,\fz)$ and $f\in L^1(\rn)$,
\begin{align*}
\lf|\lf\{x\in\rn:\ M_N(f)(x)>\lz\r\}\r|\le C_{(N)}\frac{\|f\|_{L^1(\rn)}}\lambda.
\end{align*}
\end{enumerate}
\end{lemma}
Obviously, by Theorem \ref{3t1}, we easily know that,
for any $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$,
$s\in\zz_+$ as in \eqref{3e1} and
$r\in(\max\{p_+,1\},\fz]$ with $p_+$ as in \eqref{2e10},
the set $H_{\va,{\rm fin}}^{\vp,r,s}(\rn)$ is dense in $\vh$ with
respect to the quasi-norm $\|\cdot\|_{\vh}$.
From this, we deduce the following density of $\vh$.
\begin{lemma}\label{5l6}
If $\va\in [1,\fz)^n$ and $\vp\in (0,\fz)^n$, then,
\begin{enumerate}
\item[{\rm (i)}] for any $q\in[1,\fz]$, $\vh\cap L^q(\rn)$
is dense in $\vh$;
\item[{\rm (ii)}] $\vh\cap C_c^\fz(\rn)$
is dense in $\vh$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove (i).
For any $\vp\in(0,\fz)^n$, by the density
of the set $H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)$
in $\vh$ and $H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)\subset L^q(\rn)$ for any $q\in[1,\fz]$,
we easily know that
$\vh\cap L^q(\rn)$ is dense in $\vh$. This finishes the proof of (i).
Next we show (ii).
To this end, we first prove that, for any $\varphi\in\cs(\rn)$ with
$\int_{\rn}\varphi(x)\,dx\neq0$ and $f\in \vh$, as $k\to-\fz$,
\begin{align}\label{5e22}
f\ast\varphi_k\rightarrow f \hspace{0.5cm} {\rm in}\quad \vh.
\end{align}
To show this, we first assume that $f\in \vh\cap L^2(\rn)$.
In this case, to prove \eqref{5e22}, it suffices to show that, for almost every $x\in\rn$,
as $k\to-\fz$,
\begin{align}\label{5e23}
M_N\lf(f\ast\varphi_k-f\r)(x)\rightarrow0,
\end{align}
where $N:=N_{\vp}+2$. Indeed, by the fact that, for any $k\in \nn$,
$f\ast\varphi_k-f\in L^2(\rn)$ and Lemma \ref{5l5}(i), we find that, for any $k\in\mathbb{Z}$,
$M_N(f\ast\varphi_k-f)\in L^2(\rn)$. By this,
\cite[p.\,39, Lemma 6.6]{mb03} with $A$ as in \eqref{4e5}, \eqref{5e23}
and the Lebesgue dominated convergence theorem, we know that, for any $f\in \vh\cap L^2(\rn)$,
\eqref{5e22} holds true.
Now we show \eqref{5e23}.
Notice that, if $h$ is continuous and has compact support,
then $h$ is uniformly continuous on $\rn$. Therefore, for any $\delta\in(0,\fz)$,
there exists $\eta\in(0,\fz)$ such that, for any $y\in\rn$
satisfying $|y|_{\va}<\eta$ and $x\in\rn$,
$$|h(x-y)-h(x)|<\frac\delta{2\|\varphi\|_{L^1(\rn)}}.$$
Without loss of generality,
we may assume that $\int_{\rn}\varphi(x)\,dx=1$. Then, for any $k\in\zz$,
$\int_{\rn}\varphi_k(x)\,dx=1$. This further implies that, for any $k\in\zz$ and $x\in\rn$,
\begin{align}\label{5e24}
|h\ast\varphi_k(x)-h(x)|
&\le\int_{|y|_{\va}<\eta}|h(x-y)-h(x)||\varphi_k(y)|\,dy
+\int_{|y|_{\va}\ge\eta}\cdots\\
&<\frac\delta2+2\|h\|_{L^\fz(\rn)}
\int_{|y|_{\va}\ge 2^{-k}\eta}|\varphi(y)|\,dy.\noz
\end{align}
On the other hand, by the integrability of $\varphi$, we know that
there exists $\widetilde{k}\in\zz$ such that,
for any $k\in(-\fz,\widetilde{k}]\cap\zz$,
$$2\|h\|_{L^\fz(\rn)}
\int_{|y|_{\va}\ge 2^{-k}\eta}|\varphi(y)|\,dy<\frac\delta2,$$
which, combined with \eqref{5e24}, implies that
$\lim_{k\to-\fz}|h\ast\varphi_k(x)-h(x)|=0$ holds true uniformly for any $x\in\rn$.
Thus, $\|h\ast\varphi_k-h\|_{L^\fz(\rn)}\to0$ as $k\to-\fz$. From this and Lemma \ref{5l5}(i), we deduce that
\begin{align}\label{5e25}
\lf\|M_N\lf(h\ast\varphi_k-h\r)\r\|_{L^\fz(\rn)}\ls
\|h\ast\varphi_k-h\|_{L^\fz(\rn)}\to0\hspace{0.3cm} {\rm as}\ k\to-\fz.
\end{align}
For any given $\epsilon\in(0,\fz)$, there exists a continuous function $h$
with compact support such that
$$\|f-h\|^2_{L^2(\rn)}<\epsilon.$$
Then \eqref{5e25} and \cite[p.\,39, Lemma 6.6]{mb03} with $A$ as
in \eqref{4e5} imply that there exists a positive constant $C_2$ such that,
for any $x\in\rn$,
\begin{align*}
&\limsup_{k\to-\fz}M_N\lf(f\ast\varphi_k-f\r)(x)\\
&\hs\le\sup_{k\in\zz}M_N\lf((f-h)\ast\varphi_k\r)(x)
+\limsup_{k\to-\fz}M_N\lf(h\ast\varphi_k-h\r)(x)
+M_N(h-f)(x)\le C_2M_{N_{\vp}}(h-f)(x).
\end{align*}
By this and Lemma \ref{5l5}(ii),
we conclude that there exists a
positive constant $C_3$ such that, for any $\lz\in(0,\fz)$,
\begin{align*}
&\lf|\lf\{x\in\rn:\ \limsup_{k\to-\fz}
M_N\lf(f\ast\varphi_k-f\r)(x)>\lz\r\}\r|\\
&\hs\le\lf|\lf\{x\in\rn:\
M_{N_{\vp}}(h-f)(x)>\frac\lz{C_2}\r\}\r|
\le C_3\frac{\|f-h\|^2_{L^2(\rn)}}{\lz^2}\le C_3\frac{\epsilon}{\lz^2}.
\end{align*}
This implies that, for any
$f\in \vh\cap L^2(\rn)$, \eqref{5e23} holds true.
When $f\in \vh$, by an argument similar to that used in the proof of
\cite[Lemma 5.2(ii)]{lyy16}, we easily find that \eqref{5e22} also holds true.
Notice that, if $f\in H_{\va,{\rm fin}}^{\vp,r,s}(\rn)$
and $\varphi\in C_c^\fz(\rn)$
with $\int_{\rn}\varphi(x)\,dx\neq0$, then, for any $k\in\zz$,
$$f\ast\varphi_k\in C_c^\fz(\rn)\cap \vh$$
and, by \eqref{5e22},
\begin{align*}
f\ast\varphi_k\rightarrow f \quad\mathrm{in}\quad \vh
\quad\mathrm{as}\quad k\to-\fz.
\end{align*}
From this and the density of the set $H_{\va,{\rm fin}}^{\vp,r,s}(\rn)$ in
$\vh$, it follows that
$C_c^\fz(\rn)\cap \vh$ is dense in $\vh$. This finishes the proof
of (ii) and hence of Lemma \ref{5l6}.
\end{proof}
The following Lemmas \ref{5l4} and \ref{5l1} are from Theorem \ref{3t1}
and its proof, which are of independent interest and are needed
in the proof of Theorem \ref{5t1} below.
\begin{lemma}\label{5l4}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$, $r\in(\max\{p_+,1\},\fz]$
with $p_+$ as in \eqref{2e10}, $s$ be as in \eqref{3e1}
and $N$ as in \eqref{2e11}. Then there exists a positive
constant $C$ such that, for any $(\vp,r,s)$-atom $a$,
$$\lf\|M_N(a)\r\|_{\lv}\le C.$$
\end{lemma}
\begin{lemma}\label{5l1}
Let $\va\in[1,\fz)^n$, $\vp\in(0,\fz)^n$, $r\in (1,\fz]$
and $s$ be as in \eqref{3e1}. Then,
for any $f\in \vh\cap L^{r}(\rn)$, there exist
$\{\lz_{j,k}\}_{j\in\zz,\,k\in\nn}\subset\mathbb{C}$,
$\{B_{j,k}\}_{j\in \zz,k\in\nn}\st\mathfrak{B}$ and
$(\vp,\fz,s)$-atoms $\{a_{j,k}\}_{j\in\zz,\,k\in\nn}$ such that
\begin{align*}
f=\sum_{j\in\zz}\sum_{k\in\nn}
\lz_{j,k}a_{j,k}\quad{\rm in}\quad \cs'(\rn),
\end{align*}
\begin{align}\label{5e1}
\supp a_{j,k}\subset B_{j,k}\hspace{0.2cm}
for\ any\ j\in\zz\ and\ k\in\nn,\hspace{0.2cm}
\CO_j=\bigcup_{k\in\mathbb{N}}B_{j,k}\hspace{0.2cm}
for\ any\ j\in\zz,
\end{align}
here $\CO_j:=\{x\in\rn:\ M_N(f)(x)>2^j\}$
with $N$ as in \eqref{2e11},
\begin{align}\label{5e2}
B_{j,k}^{(1/4)}\bigcap B_{j,m}^{(1/4)}
&=\emptyset\hspace{0.3cm} for\ any\ j\in\zz\ and\ k,\,m\in\nn\ with\ k\neq m,\\
&\qquad and\ B_{j,k}^{(1/4)}\ and B_{j,m}^{(1/4)}\ as\ in\ \eqref{2e2'}\ with\ \dz=1/4,\noz
\end{align}
and
\begin{align}\label{5e3}
\sharp\lf\{m\in\mathbb{N}:\
B_{j,k}\cap B_{j,m}\neq\emptyset\r\}\le
R\hspace{0.2cm} for\ any\hspace{0.2cm} k\in\mathbb{N}
\end{align}
with $R$ being a
positive constant independent of $j$ and $f$.
Moreover, there exists a positive constant $C$, independent of $f$, such that,
for any $j\in\zz$, $k\in\nn$ and almost every $x\in\rn$,
\begin{align}\label{5e4}
\lf|\lz_{j,k}a_{j,k}(x)\r|\le C2^j
\end{align}
and
\begin{align}\label{5e5}
\lf\|\lf\{\sum_{j\in\zz}\sum_{k\in\nn}
\lf[\frac{|\lz_{j,k}|\chi_{B_{j,k}}}
{\|\chi_{B_{j,k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
\le C\|f\|_{\vh},
\end{align}
where $p_-$ is as in \eqref{2e10}.
\end{lemma}
\begin{remark}\label{5r1}
For any $j\in\zz$, $k\in\nn$ and $s$ as in \eqref{3e1},
let $\eta_{j,k}$ be the same as in
the proof of Theorem \ref{3t1}. Then, for any
$f\in \vh\cap L^{r}(\rn)$ with $r\in(1,\fz]$,
by an argument similar to that used in the
proof of Theorem \ref{3t1}, together with Lemma \ref{3l4},
we know that there exists a unique polynomial $c_{j,k}\in\cp_s(\rn)$
such that, for any $q\in\cp_s(\rn)$,
\begin{align}\label{5e6}
\lf\langle f,q\eta_{j,k}\r\rangle=
\lf\langle c_{j,k},q\eta_{j,k}\r\rangle=
\int_\rn c_{j,k}(x)q(x)\eta_{j,k}(x)\,dx.
\end{align}
In addition,
for any $i$, $k\in\nn$ and $j\in\zz$, let
$c_{j+1,k,i}$ be the orthogonal projection of
$(f-c_{j+1,i})\eta_{j,k}$ on $\cp_{s}(\rn)$
with respect to the norm defined by \eqref{3e28}, namely,
$c_{j+1,k,i}$ is the unique element of $\cp_s(\rn)$
such that, for any $q\in\cp_s(\rn)$,
\begin{align}\label{5e7}
\int_\rn \lf[f(x)-c_{j+1,i}(x)\r]\eta_{j,k}(x)q(x)
\eta_{j+1,i}(x)\,dx=\int_\rn c_{j+1,k,i}(x)q(x)
\eta_{j+1,i}(x)\,dx
\end{align}
and, for any $k\in\mathbb{N}$ and $j\in\mathbb{Z}$,
\begin{align}\label{5e8}
\lz_{j,k}a_{j,k}=(f-c_{j,k})\eta_{j,k}-
\sum_{i\in\mathbb{N}}\lf[(f-c_{j+1,i})\eta_{j,k}
-c_{j+1,k,i}\r]\eta_{j+1,i}.
\end{align}
\end{remark}
From \eqref{3e13}, \eqref{3e14} and their proofs,
we deduce the following Lemmas \ref{5l2}
and \ref{5l3} (see also the proofs of \cite[p.\,104, ($23'$) and p.\,108, (35)]{s93}), the details being omitted.
\begin{lemma}\label{5l2}
Let $\va\in[1,\fz)^n$ and $\vp\in(0,\fz)^n$. Then
there exists a positive constant $C$ such that,
for any $j\in\zz$, $k\in\nn$ and $f\in\vh$,
$$\sup_{y\in\rn}\lf|c_{j,k}(y)\eta_{j,k}(y)\r|\le C\sup_{y\in U_j^k}
M_N(f)(y)\le C2^j,$$
where $N\in \nn$, $M_N$ is as in Definition \ref{2d4} and, for any $j\in\zz$ and $k\in\nn$,
$\eta_{j,k}$ is as in the proof of Theorem \ref{3t1}, $c_{j,k}$ as in Remark \ref{5r1},
$\CO_j$ and $B_{j,k}$ as in the proof of Theorem \ref{3t1}, $B_{j,k}^{(2)}$ as in \eqref{2e2'}
with $\dz=2$,
and $U_j^k:=B_{j,k}^{(2)}\cap (\CO_j)^\com$.
\end{lemma}
\begin{lemma}\label{5l3}
Let $\va$, $\vp$, $f$ and $M_N$ be as in Lemma \ref{5l2}. Then there exists a positive constant $C$,
independent of $f$, such that, for any $j\in\zz$ and $i,\,k\in\nn$,
$$\sup_{y\in\rn}\lf|c_{j+1,k,i}(y)\eta_{j+1,i}(y)\r|\le C\sup_
{y\in\widetilde{U}_j^k}M_N(f)(y)\le C2^{j+1},$$
where, for any $j\in\zz$ and $i,\,k\in\nn$,
$\eta_{j+1,i}$ is as in the proof of Theorem \ref{3t1}, $c_{j+1,k,i}$ as in Remark \ref{5r1},
$\CO_{j+1}$ and $B_{j+1,k}$ as in the proof of Theorem \ref{3t1} with $j$ replaced by $j+1$, $B_{j+1,k}^{(2)}$ as in \eqref{2e2'} with $\dz=2$,
and $\widetilde{U}_j^k:=B_{j+1,k}^{(2)}\cap(\CO_{j+1})^\com$.
\end{lemma}
In what follows, denote by $C(\rn)$
the \emph{set of all continuous functions}.
Then we have the following finite atomic characterizations
of $\vh$, which extends \cite[Theorem 3.1 and Remark 3.3]{msv08} and
\cite[Theorem 5.6]{gly08} to the present
setting of anisotropic mixed-norm Hardy spaces.
\begin{theorem}\label{5t1}
Let $\va\in [1,\fz)^n,\,\vp\in(0,\fz)^n$,
$r\in(\max\{p_+,1\},\fz]$ with $p_+$ as in
\eqref{2e10} and $s$ be as in \eqref{3e1}.
\begin{enumerate}
\item[{\rm (i)}]
If $r\in(\max\{p_+,1\},\fz)$, then $\|\cdot\|_{\vfah}$
and $\|\cdot\|_{\vh}$ are equivalent quasi-norms on $\vfah$;
\item[{\rm (ii)}]
$\|\cdot\|_{\vfahfz}$
and $\|\cdot\|_{\vh}$ are equivalent quasi-norms on
$\vfahfz\cap C(\rn)$.
\end{enumerate}
\end{theorem}
\begin{remark}
Recall that
Bownik et al. in \cite[Theorem 6.2]{blyz08}
established the finite atomic characterizations of the weighted
anisotropic Hardy space $H_w^p(\rn;A)$ with $w$
being a Muckenhoupt weight (see \cite[Definition 2.5]{blyz08}).
As was mentioned in \cite[p.\,3077]{blyz08}, if $w:\equiv1$,
then $H_w^p(\rn;A)=\vAh$.
By this and Proposition \ref{2r4'}, we know that,
when $\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})$,
where $p\in(0,1]$, Theorem \ref{5t1} is just \cite[Theorem 6.2]{blyz08}
with the weight $w:\equiv1$ and $A$ as in \eqref{4e5}.
\end{remark}
Now we proof Theorem \ref{5t1}.
\begin{proof}[Proof of Theorem \ref{5t1}]
Let $\va\in [1,\fz)^n,\,\vp\in(0,\fz)^n$,
$r\in(\max\{p_+,1\},\fz]$ with $p_+$ as in
\eqref{2e10} and $s$ be as in \eqref{3e1}.
Then, by Theorem \ref{3t1}, we find that
$\vfah\subset \vh$ and, for any $f\in \vfah$,
$\|f\|_{\vh}\ls\|f\|_{\vfah}$.
Therefore, to prove Theorem \ref{5t1}, it suffices to show that,
for any $f\in \vfah$ when $r\in(\max\{p_+,1\},\fz)$
and, for any $f\in [\vfahfz\cap C(\rn)]$ when $r=\fz$,
$$\|f\|_{\vfah}\ls\|f\|_{\vh}.$$
We prove this by the following three steps.
\emph{Step 1.}
Let $r\in(\max\{p_+,1\},\fz]$. Without loss of generality,
we may assume that $f\in \vfah$ and $\|f\|_{\vh}=1$. Clearly, there
exists some $j_0\in\zz$ such that $\supp f\subset 2^{j_0\va}B_{0}$,
due to the fact that $f$ has compact support, where $B_{0}$ denotes the unit ball of $\rn$.
In the remainder of this section, we always let
$N:=N_{\vp}$ with $N_{\vp}$ as in Definition \ref{2d5} and, for any $j\in\zz$, let
$$\CO_j:=\lf\{x\in\rn:\ M_N(f)(x)>2^j\r\}.$$
Notice that
$f\in \vh\cap L^{\widetilde{r}}(\rn)$, where
$\widetilde{r}:=r$ when $r\in(\max\{p_+,1\},\fz)$ and
$\widetilde{r}:=2$ when $r=\fz$. Then, by Lemma \ref{5l1},
we conclude that there exist $\{\lz_{j,k}\}_{j\in\zz,\,k\in\nn}
\subset\mathbb{C}$ and a sequence of
$(\vp,\fz,s)$-atoms, $\{a_{j,k}\}_{j\in\zz,\,k\in\nn}$, such that
\begin{align}\label{5e9}
f=\sum_{j\in\zz}\sum_{k\in\nn}\lz_{j,k}a_{j,k}\quad{\rm in}\quad \cs'(\rn),
\end{align}
and \eqref{5e1} through \eqref{5e5} also hold true.
By this and an argument similar to that used in the proof of Step 2
of the proof of \cite[Theorem 5.7]{lyy16},
we know that there exists a positive constant $C_4$ such that, for any
$x\in (2^{ (j_0+4)\va}B_0)^\com$,
\begin{align}\label{5e10}
M_N(f)(x)\le C_4\lf\|\chi_{2^{j_0\va}B_{0}}\r\|_{\lv}^{-1}.
\end{align}
Let
\begin{align}\label{5e11}
\widetilde{j}:=
\sup\lf\{j\in\zz:\ 2^k<C_4\lf\|\chi_{2^{j_0\va}B_{0}}\r\|_{\lv}^{-1}\r\}
\end{align}
with $C_4$ as in \eqref{5e10}. Then, from \eqref{5e10}, we deduce that,
for any $j\in(\widetilde{j},\fz]\cap\zz$,
\begin{align}\label{5e12}
\CO_j\subset 2^{ (j_0+4)\va}B_0.
\end{align}
Using $\wz{j}$ as in \eqref{5e11}, we rewrite \eqref{5e9} as
\begin{align}\label{5e13}
f=\sum_{j=-\fz}^{\widetilde{j}}\sum_{k\in\nn}\lz_{j,k}a_{j,k}+
\sum_{j=\widetilde{j}+1}^\fz\sum_{k\in\nn}\lz_{j,k}a_{j,k}=:h+\ell
\quad{\rm in}\quad \cs'(\rn).
\end{align}
In the remainder of this step, we devote to proving that $h$ is a
$(\vp,\fz,s)$-atom multiplied by a harmless constant independent of $f$. For this purpose,
from \eqref{5e12}, it is easy to see that
$\supp \ell\subset\cup_{j=\widetilde{j}+1}^\fz\CO_j
\subset 2^{ (j_0+4)\va}B_0$. By this, the fact that
$\supp f\subset 2^{ (j_0+4)\va}B_0$ and \eqref{5e13}, we know
that $\supp h\subset 2^{ (j_0+4)\va}B_0$.
On the other hand, by the H\"{o}lder inequality, we find that,
for any $r\in(\max\{p_+, 1\}, \fz]$ and $r_1\in(\max\{p_+, 1\},r)$,
$$\int_{\rn}|f(x)|^{r_1}\,dx
\le\lf|2^{j_0\va}B_{0}\r|^{1-\frac{r_1}r}\|f\|_{L^r(\rn)}^{r_1}<\fz.$$
This, together with the facts that
$\supp f\subset 2^{j_0\va}B_{0}$ and that $f$
has vanishing moments up to order $s$, further implies that
$f$ is a harmless constant multiple of a $(1,r_1,s)$-atom.
By this and Lemma \ref{5l4},
we know that
$M_N(f)\in L^1(\rn)$. Therefore, by \eqref{5e3}, \eqref{5e1}, \eqref{5e12} and
\eqref{5e4}, we conclude that
$$\int_{\rn}\sum_{j=\widetilde{j}+1}^\fz\sum_{k\in\nn}
\lf|\lz_{j,k}a_{j,k}(x)x^\alpha\r|\,dx
\ls\sum_{j\in\zz}2^j|\CO_j|\ls\lf\|M_N(f)\r\|_{L^1(\rn)}<\fz.$$
From this and the vanishing moments of $a_{j,k}$, we deduce that $\ell$
has vanishing moments up to $s$ and hence so does $h$ by \eqref{5e13}.
Moreover, from \eqref{5e3}, \eqref{5e4} and \eqref{5e11},
it follows that, for any $x\in\rn$,
$$|h(x)|\ls\sum_{j=-\fz}^{\widetilde{j}}2^j
\ls\lf\|\chi_{2^{j_0\va}B_{0}}\r\|_{\lv}^{-1}.$$
Thus, there exists a positive constant $C_5$, independent of $f$, such that
$h/C_5$ is a $(\vp,\fz,s)$-atom and also a
$(\vp,r,s)$-atom for any $\vp\in (0,\fz)^n$, $r\in(\max\{p_+,1\},\fz]$ and $s$
as in \eqref{3e1}.
\emph{Step 2.}
This step is aimed to prove (i). To this end,
for any $J\in(\widetilde{j},\fz)\cap\zz$ and
$j\in[\widetilde{j}+1,J]\cap\zz$ with $\wz{j}$
as in \eqref{5e11}, let
\begin{align*}
I_{(J,j)}:=\lf\{k\in\nn:\ |k|+|j|\le J\r\}
\hspace{0.3cm} {\rm and}\hspace{0.3cm} \ell_{(J)}:=\sum_{j=\widetilde{j}+1}^{J}
\sum_{k\in I_{(J,j)}}\lz_{j,k}a_{j,k}.
\end{align*}
For any $r\in(\max\{p_+,1\},\fz)$,
we first show that $\ell\in L^r(\rn)$.
Indeed, for any $x\in\rn$,
since $\rn=\bigcup_{i\in\zz}(\CO_i\setminus\CO_{i+1})$,
it follows that there exists an $i_0\in\zz$ such that
$x\in(\CO_{i_0}\setminus\CO_{i_0+1})$. Notice that,
for any $j\in(i_0,\fz)\cap\zz$,
$\supp a_{j,k}\subset B_{j,k}\subset \CO_j\subset\CO_{i_0+1}$.
Then \eqref{5e3} and \eqref{5e4} imply that,
for any $x\in(\CO_{i_0}\setminus\CO_{i_0+1})$,
$$\lf|\ell(x)\r|\le\sum_{j=\widetilde{j}+1}^\fz\sum_{k\in\nn}|\lz_{j,k}a_{j,k}(x)|
\ls\sum_{j\le i_0}2^j\ls2^{i_0}\ls M_N(f)(x).$$
Since $f\in L^r(\rn)$, from Lemma \ref{5l5}(i), it follows that
$M_N(f)\in L^r(\rn)$. Therefore, by the Lebesgue
dominated convergence theorem, we find that $\ell_{(J)}$
converges to $\ell$ in $L^r(\rn)$ as $J\to\fz$.
This implies that, for any given
$\epsilon\in(0,1)$, there exists
a $J\in[\widetilde{j}+1,\fz)\cap\zz$ large enough,
depending on $\epsilon$, such that
$[\ell-\ell_{(J)}]/\epsilon$ is a $(\vp,r,s)$-atom and hence
$f=h+\ell_{(J)}+[\ell-\ell_{(J)}]$
is a finite linear combination of $(\vp,r,s)$-atoms.
By this, Step 1 and \eqref{5e5}, we conclude that
$$\|f\|_{\vfah}
\ls C_5+
\lf\|\lf\{\sum_{j=\wz{j}+1}^{J}\sum_{k\in I_{(J,j)}}
\lf[\frac{|\lz_{j,k}|\chi_{B_{j,k}}}
{\|\chi_{B_{j,k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
+\epsilon\ls1,$$
which completes the proof of (i).
\emph{Step 3.}
In this step, we prove (ii). For this purpose,
let $f\in \vfahfz\cap C(\rn)$.
Then, by \eqref{5e8}, we find that, for any $j\in\zz$ and $k\in\nn$,
$a_{j,k}$ is continuous. Moreover, by the fact that
there exists a positive constant $C_{(n,N)}$,
depending only on $n$ and $N$, such that, for any $x\in\rn$,
\begin{align}\label{5e16}
M_N(f)(x)\le C_{(n,N)}\|f\|_{L^\fz(\rn)},
\end{align}
we easily know that, for any $j\in\zz$ satisfying
$2^j\ge C_{(n,N)}\|f\|_{L^\fz(\rn)}$, the level set $\CO_j$ is empty.
Let
$$\widehat{j}:=\sup\lf\{j\in\zz:\ 2^j< C_{(n,N)}\|f\|_{L^\fz(\rn)}\r\}.$$
Then the index $j$ in the sum defining
$\ell$ runs only over $j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$.
Let $\epsilon\in(0,\fz)$. Then the fact that $f$ is uniformly continuous implies that there exists
a $\delta\in(0,\fz)$ such that
$|f(x)-f(y)|<\epsilon$ whenever $|x-y|_{\va}<\delta$.
Furthermore, for this $\epsilon$, let
$$\ell_1^\epsilon:=\sum_{j=\widetilde{j}+1}^{\widehat{j}}
\sum_{k\in E_1^{(j,\delta)}}\lz_{j,k}a_{j,k}\hspace{0.4cm} {\rm and}\hspace{0.4cm}
\ell_2^\epsilon:=\sum_{j=\widetilde{j}+1}^{\widehat{j}}
\sum_{k\in E_2^{(j,\delta)}}\lz_{j,k}a_{j,k},$$
where, for any $j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$,
$$E_1^{(j,\delta)}:=\lf\{k\in\nn:\ r_{j,k}\ge\delta\r\}\quad
\mathrm{and}\quad E_2^{(j,\delta)}:=\lf\{k\in\nn:\ r_{j,k}<\delta\r\}$$
with $x_{j,k}$ and
$r_{j,k}$ being the center and the radius of $B_{j,k}$, respectively.
Next we give a finite decomposition of $f$.
Clearly, by \eqref{5e2} and \eqref{5e12}, we know that,
for any fixed $j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$,
$E_1^{(j,\delta)}$ is a finite set and hence $\ell_1^\epsilon$
is a finite linear combination of continuous $(\vp,\fz,s)$-atoms.
Then, by \eqref{5e5}, we have
\begin{align}\label{5e14}
\lf\|\lf\{\sum_{j=\widetilde{j}+1}^{\widehat{j}}\sum_{k\in E_1^{(j,\delta)}}
\lf[\frac{|\lz_{j,k}|\chi_{B_{j,k}}}{\|\chi_{B_{j,k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
\ls\|f\|_{\vh}.
\end{align}
In addition, for any $j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$,
$k\in\nn$ satisfying $r_{j,k}<\delta$ and $x\in B_{j,k}$,
we have $|f(x)-f(x_{j,k})|<\epsilon$. By \eqref{5e6} and the fact that
$\supp\eta_{j,k}\subset B_{j,k}$,
we conclude that, for any $q\in\cp_s(\rn)$,
$$\frac 1{\int_\rn\eta_{j,k}(x)\,dx}\int_\rn
\lf[\widetilde{f}(x)-\widetilde{c}_{j,k}(x)\r]q(x)\eta_{j,k}(x)\,dx=0,$$
where, for any $x\in\rn$,
$$\widetilde{f}(x):=\lf[f(x)-f(x_{j,k})\r]
\chi_{B_{j,k}}(x)
\hspace{0.3cm} {\rm and}\hspace{0.3cm}\widetilde{c}_{j,k}(x):=
c_{j,k}(x)-f(x_{j,k}).$$
Since \eqref{5e16} and the fact that, for any $x\in\rn$, $|\widetilde{f}(x)|<\epsilon$
imply that, for any $x\in\rn$, $M_N(\widetilde{f})(x)\ls\epsilon$,
from Lemma \ref{5l2}, it follows that
\begin{align}\label{5e17}
\sup_{y\in\rn}\lf|\widetilde{c}_{j,k}(y)\eta_{j,k}(y)\r|
\ls\sup_{y\in\rn}M_N\lf(\widetilde{f}\r)(y)\ls\epsilon.
\end{align}
Similarly to Remark \ref{5r1},
for any $j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$,
$k\in E_2^{(j,\delta)}$ and $i\in\nn$, let
$\widetilde{c}_{j+1,k,i}$
be the orthogonal projection of
$(\widetilde{f}-\widetilde{c}_{j+1,i})\eta_{j,k}$ on
$\cp_{s}(\rn)$ with
respect to the norm in \eqref{3e28}.
Then, for any $q\in\cp_{s}(\rn)$,
\begin{align}\label{5e18}
\int_\rn \lf[\widetilde{f}(x)-\widetilde{c}_{j+1,i}(x)\r]\eta_{j,k}(x)q(x)
\eta_{j+1,i}(x)\,dx=\int_\rn \widetilde{c}_{j+1,k,i}(x)q(x)
\eta_{j+1,i}(x)\,dx.
\end{align}
By the fact that $\supp\eta_{j,k}\subset B_{j,k}$, we have
$[\widetilde{f}-\widetilde{c}_{j+1,i}]\eta_{j,k}
=[f-c_{j+1,i}]\eta_{j,k}$.
From this, \eqref{5e7} and \eqref{5e18}, we deduce that
$\widetilde{c}_{j+1,k,i}=c_{j+1,k,i}$. Then, by Lemma \ref{5l3},
we know that
\begin{align}\label{5e19}
\sup_{y\in\rn}\lf|\widetilde{c}_{j+1,k,i}(y)\eta_{j+1,i}(y)\r|
\ls\sup_{y\in\rn}M_N(\widetilde{f})(y)\ls\epsilon.
\end{align}
Moreover, by \eqref{5e8} and
$\sum_{i\in\nn}\eta_{j+1,i}=\chi_{\CO_{j+1}}$, we conclude that
\begin{align*}
\lz_{j,k}a_{j,k}&=(f-c_{j,k})\eta_{j,k}-
\sum_{i\in\mathbb{N}}\lf[(f-c_{j+1,i})
\eta_{j,k}-c_{j+1,k,i}\r]\eta_{j+1,i}\\
&=\eta_{j,k}\widetilde{f}\chi_{\CO_{j+1}^
\com}-\widetilde{c}_{j,k}\eta_{j,k}+\eta_{j,k}\sum
_{i\in\nn}\widetilde{c}_{j+1,i}\eta_{j+1,i}+\sum_
{i\in\mathbb{N}}\widetilde{c}_{j+1,k,i}\eta_{j+1,i},
\end{align*}
which, combined with \eqref{5e17}, \eqref{5e19}
and \eqref{5e3}, further implies that, for any
$j\in\{\widetilde{j}+1,\ldots,\widehat{j}\}$,
$k\in E_2^{(j,\delta)}$ and $x\in B_{j,k}$,
$|\lz_{j,k}a_{j,k}(x)|\ls\epsilon$.
Then, by \eqref{5e1} and \eqref{5e3}, we easily know that
there exists a positive constant $C_6$, independent of $f$, such that,
for any $x\in \rn$
\begin{align}\label{5e15}
\lf|\ell_2^\epsilon(x)\r|\le C_6\sum_{j=\widetilde{j}+1}^{\widehat{j}}
\epsilon=C_6\lf[\widehat{j}-\widetilde{j}\r]\epsilon.
\end{align}
Therefore, the arbitrariness of $\epsilon \in (0, \fz)$ implies that we split $\ell$ into a continuous
part and a part which is pointwisely uniformly arbitrarily small, namely,
$\ell=\ell_1^\epsilon+\ell_2^\epsilon$. Thus,
$\ell$ is continuous and, by Step 1, $h=f-\ell$ is a $C_5$ multiple of a continuous
$(\vp,\fz,s)$-atom.
Notice that $\ell$ and $\ell_1^\epsilon$ are both continuous and have vanishing
moments up to order $s$ and hence so does $\ell_2^\epsilon$. This, combined with
the fact that $\supp\ell_2^\epsilon\subset 2^{ (j_0+4)\va}B_0$ and
\eqref{5e15}, further implies that we can choose
$\epsilon$ small enough such that $\ell_2^\epsilon$ is an arbitrarily
small multiple of a continuous $(\vp,\fz,s)$-atom. Indeed,
$\ell_2^\epsilon=\lz^{(\epsilon)} a^{(\epsilon)}$, where
$$\lz^{(\epsilon)}:=C_6\lf[\widehat{j}-\widetilde{j}\r]
\epsilon\lf\|\chi_{2^{ (j_0+4)\va}B_0}\r\|_{\lv}^{-1}$$
and $a^{(\epsilon)}$ is a continuous $(\vp,\fz,s)$-atom. In this case,
$f=h+\ell_1^\epsilon+\ell_2^\epsilon$ is just a finite atomic
decomposition of $f$. Then, by \eqref{5e14} and the fact that $h/C_5$ is
a $(\vp,\fz,s)$-atom, we find that
$$\|f\|_{H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)}\lesssim
\|h\|_{H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)}
+\lf\|\ell_1^\epsilon\r\|_{H_{\va,{\rm fin}}^{\va,\fz,s}(\rn)}
+\lf\|\ell_2^\epsilon\r\|_{H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)}\ls1,$$
which completes the proof of (ii) and hence of Theorem \ref{5t1}.
\end{proof}
\section{Some applications\label{s6}}
As applications, in this section, we first establish a criterion on the
boundedness of sublinear operators from $\vh$ into a quasi-Banach
space. Applying this criterion, we further obtain
the boundedness of anisotropic convolutional $\delta$-type and
non-convolutional $\bz$-order Calder\'on-Zygmund operators
from $\vh$ to itself [or to $\lv$].
Recall that a complete vector space is called a \emph{quasi-Banach space} $\mathcal{B}$ if its quasi-norm $\|\cdot\|_{\mathcal{B}}$ satisfies
\begin{enumerate}
\item[{\rm (i)}] $\|f\|_{\mathcal{B}}=0\Longleftrightarrow f$ is the zero element of $\mathcal{B}$;
\item[{\rm (ii)}] there exists a positive constant $H\in[1,\fz)$ such that, for any
$f,\,g\in\mathcal{B}$,
$$\|f+g\|_{\mathcal{B}}\le H(\|f\|_{\mathcal{B}}+\|g\|_{\mathcal{B}}).$$
\end{enumerate}
Clearly, when $H=1$, a quasi-Banach space $\mathcal{B}$ is just a Banach space.
Moreover, for any given $\gamma\in(0,1]$, a
\emph{$\gamma$-quasi-Banach space} ${\mathcal{B}_{\gamma}}$ is a quasi-Banach space equipped
with a quasi-norm $\|\cdot\|_{\mathcal{B}_{\gamma}}$ satisfying that there exists a constant $C\in[1,\fz)$
such that, for any $t\in \nn$ and $\{f_i\}_{i=1}^{t}\st\mathcal{B}_{\gamma}$, $\|\sum_{i=1}^t f_i\|_{\mathcal{B}_{\gamma}}^{\gamma}\le
C \sum_{i=1}^t \|f_i\|_{\mathcal{B}_{\gamma}}^{\gamma}$ holds true
(see \cite{zy08, zy09, ky14, ylk17}).
Let $\mathcal{B}_{\gamma}$ be a $\gamma$-quasi-Banach space with
$\gamma\in(0,1]$ and $\mathcal{Y}$ a linear space. An operator
$T$ from $\mathcal{Y}$ to $\mathcal{B}_{\gamma}$ is said to be
$\mathcal{B}_{\gamma}$-\emph{sublinear} if
there exists a positive constant $C$ such that, for any $t\in \nn$,
$\{\mu_{i}\}_{i=1}^t\st \mathbb{C}$ and $\{f_{i}\}_{i=1}^t\st\mathcal{Y}$,
$$\lf\|T\lf(\sum_{i=1}^t \mu_i f_i\r)\r\|_{\mathcal{B}_{\gamma}}^{\gamma}\le
C\sum_{i=1}^t |\mu_i|^{\gamma}\lf\|T(f_i)\r\|_{\mathcal{B}_{\gamma}}^{\gamma}$$
and, for any $f,\,g\in \mathcal{Y}$,
$\|T(f)-T(g)\|_{\mathcal{B}_{\gamma}}\le C\|T(f-g)\|_{\mathcal{B}_{\gamma}}$
(see \cite{zy08, zy09, ky14, ylk17}).
Clearly, for any
$\gamma\in (0,1]$, the linearity of $T$ implies its $\mathcal{B}_{\gamma}$-sublinearity.
As an application of the finite atomic characterizations of $\vh$ obtained
in Section \ref{s5} (see Theorem \ref{5t1}),
we establish the following criterion on the boundedness of sublinear
operators from $\vh$ into a quasi-Banach
space $\mathcal{B}_{\gamma}$.
\begin{theorem}\label{6t1}
Assume that $\va\in [1,\fz)^n$, $\vp\in (0,\fz)^n$, $r\in(\max\{p_+,1\},\fz]$
with $p_+$ as in \eqref{2e10}, $\gamma\in (0,1]$, $s$ is
as in \eqref{3e1} and $\mathcal{B}_{\gamma}$ a $\gamma$-quasi-Banach space.
If either of the following two statements holds true:
\begin{enumerate}
\item[{\rm (i)}] $r\in(\max\{p_+,1\},\fz)$ and
$T:\ \vfah\to\mathcal{B}_{\gamma}$
is a $\mathcal{B}_{\gamma}$-sublinear operator satisfying that
there exists a positive constant $C_7$ such that,
for any $f\in \vfah$,
\begin{align}\label{6e1}
\|T(f)\|_{\mathcal{B}_{\gamma}}\le C_7\|f\|_{\vfah};
\end{align}
\item[{\rm (ii)}]
$T:\ \vfahfz\cap C(\rn)\to\mathcal{B}_{\gamma}$
is a $\mathcal{B}_{\gamma}$-sublinear operator satisfying that
there exists a positive constant $C_8$ such that,
for any $f\in \vfahfz\cap C(\rn)$,
$$\|T(f)\|_{\mathcal{B}_{\gamma}}\le C_8\|f\|_{\vfahfz},$$
\end{enumerate}
then $T$ uniquely extends to a bounded $\mathcal{B}_{\gamma}$-sublinear operator from $\vh$
into $\mathcal{B}_{\gamma}$. Moreover, there exists a positive constant $C_9$ such that,
for any $f\in \vh$,
$$\|T(f)\|_{\mathcal{B}_{\gamma}}\le C_9\|f\|_{\vh}.$$
\end{theorem}
The following conclusion is an immediate corollary of Theorem \ref{6t1}, which
extends the corresponding results of Meda et al. \cite[Corollary 3.4]{msv08}
and Grafakos et al. \cite[Theorem 5.9]{gly08} as well as Ky \cite[Theorem 3.5]{ky14}
(see also \cite[Theorem 1.6.9]{ylk17})
to the present setting, the details being omitted.
\begin{corollary}\label{6c1}
Let $\va$, $\vp$, $r$, $\gamma$, $s$ and $\mathcal{B}_{\gamma}$ be as
in Theorem \ref{6t1}. If either of the following two statements holds true:
\begin{enumerate}
\item[{\rm (i)}] $r\in(\max\{p_+,1\},\fz)$ and $T$ is a $\mathcal{B}_{\gamma}$-sublinear
operator from $\vfah$ to $\mathcal{B}_{\gamma}$ satisfying
$$\sup\lf\{\|T(a)\|_{\mathcal{B}_{\gamma}}:\
a\ is\ any\ (\vp,r,s){\text-}atom\r\}<\fz;$$
\item[{\rm(ii)}] $T$ is a $\mathcal{B}_{\gamma}$-sublinear
operator defined on all continuous $(\vp,\fz,s)$-atoms satisfying
$$\sup\lf\{\|T(a)\|_{\mathcal{B}_{\gamma}}:\
a\ is\ any\ continuous\ (\vp,\fz,s){\text-}atom\r\}<\fz,$$
\end{enumerate}
then $T$ has a unique bounded $\mathcal{B}_{\gamma}$-sublinear
extension $\widetilde{T}$ from $\vh$ to $\mathcal{B}_{\gamma}$.
\end{corollary}
We now prove Theorem \ref{6t1}.
\begin{proof}[Proof of Theorem \ref{6t1}]
To show (i), let $r\in(\max\{p_+,1\},\fz)$ and $f\in\vh$.
Then, by the density of
$\vfah$ in $\vh$, we know that there exists a Cauchy sequence
$\{f_k\}_{k\in\nn}\subset \vfah$ such that
$$\lim_{k\to\fz}\lf\|f_k-f\r\|_{\vh}=0.$$
By this, \eqref{6e1} and Theorem \ref{5t1}(i),
we conclude that, as $k$, $\ell\to\fz$,
\begin{align*}
\lf\|T(f_k)-T(f_{\ell})\r\|_{\mathcal{B}_{\gamma}}
\ls\lf\|T(f_k-f_{\ell})\r\|_{\mathcal{B}_{\gamma}}\ls
\lf\|f_k-f_{\ell}\r\|_{\vfah}\sim\lf\|f_k-f_{\ell}\r\|_{\vh}\to0,
\end{align*}
which implies that $\{T(f_k)\}_{k\in\nn}$ is a Cauchy sequence in $\mathcal{B}_{\gamma}$.
Therefore, by the completeness of $\mathcal{B}_{\gamma}$, we find that there exists
some $h\in\mathcal{B}_{\gamma}$ such that $h=\lim_{k\to\fz}T(f_k)$
in $\mathcal{B}_{\gamma}$.
Then let $T(f):=h$. From this, \eqref{6e1} and Theorem \ref{5t1}(i) again, we
further deduce that
\begin{align*}
\|T(f)\|_{\mathcal{B}_{\gamma}}^{\gamma}&\ls\limsup_{k\to\fz}\lf[\lf\|T(f)-T(f_k)\r\|_{\mathcal{B}_{\gamma}}^{\gamma}
+\lf\|T(f_k)\r\|_{\mathcal{B}_{\gamma}}^{\gamma}\r]\ls\limsup_{k\to\fz}\lf\|T(f_k)\r\|_{\mathcal{B}_{\gamma}}^{\gamma}\\
&\ls\limsup_{k\to\fz}\lf\|f_k\r\|_{\vfah}^{\gamma}
\sim\lim_{k\to\fz}\lf\|f_k\r\|_{\vh}^{\gamma}\sim\|f\|_{\vh}^{\gamma},
\end{align*}
which completes the proof of (i).
We now prove (ii).
First, by the proof of \cite[Theorem 6.13(ii)]{lyy16} with some slight modifications,
we easily know that
$H_{\va,{\rm fin}}^{\vp,\fz,s}(\rn)\cap C(\rn)$ is dense in $\vh$.
Then, from this and an argument similar to that used in the proof of (i),
we conclude that (ii) holds true.
This finishes the proof of (ii) and hence of Theorem \ref{6t1}.
\end{proof}
Let $\va\in [1,\fz)^n$. For any $\delta\in (0,1]$,
an \emph{anisotropic convolutional
$\delta$-type Calder\'{o}n-Zygmund operator} $T$ from \cite{bil66,f66}
is a linear operator, which is bounded on $L^2(\rn)$ with kernel
$k\in \cs'(\rn)$ coinciding with a locally integrable
function on $\rn\setminus\{\vec{0}_n\}$ and satisfying that
there exists a positive constant $C$ such that,
for any $x,\,y\in \rn$ with $|x|_{\va}>2|y|_{\va}$,
$$|k(x-y)-k(x)|\le C\frac{|y|_{\va}^{\delta}}{|x|_{\va}^{\nu+\delta}}$$
and, for any $f\in L^2(\rn)$, $T(f)(x):={\rm p.\,v.}\,k\ast f(x)$.
Via borrowing some ideas from the proof of Yan et al. \cite[Theorem 7.4]{yyyz16} and
the criterion established in Theorem \ref{6t1} and Corollary \ref{6c1},
we obtain the boundedness of anisotropic convolutional $\delta$-type Calder\'{o}n-Zygmund operators
from $\vh$ to itself (see Theorem \ref{6t2} below) or to $\lv$ (see Theorem \ref{6t3} below),
which extends the corresponding results of Fefferman and Stein \cite[Theorem 12]{fs72}
as well as Yan et al. \cite[Theorem 7.4]{yyyz16} to the present setting.
\begin{theorem}\label{6t2}
Let $\va\in [1,\fz)^n$, $\vp\in (0,1]^n$, $\delta\in(0,1]$
and $\widetilde{p}_-\in(\frac\nu{\nu+\delta},1]$ with $\widetilde{p}_-$ as in \eqref{3e1}.
Let $T$ be an anisotropic convolutional $\dz$-type Calder\'on-Zygmund operator.
Then there exists a positive constant $C$
such that, for any $f\in \vh$,
$$\|T(f)\|_{\vh}\le C\|f\|_{\vh}.$$
\end{theorem}
\begin{theorem}\label{6t3}
Let $\va\in [1,\fz)^n$, $\vp\in (0,1]^n$, $\delta\in(0,1]$
and $\widetilde{p}_-\in(\frac\nu{\nu+\delta},1]$ with $\widetilde{p}_-$ as in \eqref{3e1}.
Let $T$ be an anisotropic convolutional $\dz$-type Calder\'on-Zygmund operator,
then there exists a positive constant $C$
such that, for any $f\in \vh$,
$$\|T(f)\|_{\lv}\le C\|f\|_{\vh}.$$
\end{theorem}
\begin{remark}\label{6r3}
We point out that the boundedness of the anisotropic convolutional
$\dz$-type Calder\'on-Zygmund operator on the mixed-norm Lebesgue space
$\lv$ with $\vp\in(1,\fz)^n$ is still unknown so far.
\end{remark}
Now we prove Theorem \ref{6t2}.
\begin{proof}[Proof of Theorem \ref{6t2}]
Let $f\in H^{\vp,2,s}_{\va,\rm {fin}}(\rn)$ with $s$ as in \eqref{3e1}.
Then, without loss of generality,
we may assume that $\|f\|_{\vh}=1$. Notice that
$f\in \vh\cap L^2(\rn)$,
by an argument similar to that used in the proof of Theorem \ref{3t1},
we find that there exist a sequence of $(\vp,2,s)$-atoms,
$\{a_{k}\}_{k\in\mathbb{N}}$, supported, respectively, on
$\{B_k\}_{k\in\nn}:=\{B_{\va}(x_k,r_k)\}_{k\in\nn}\st \mathfrak{B}$ and
$\{\lz_{k}\}_{k\in\mathbb{N}}\subset\mathbb{C}$
such that
\begin{align}\label{6e2}
f=\sum_{k\in\mathbb{N}}\lambda_{k}a_{k}\quad{\rm in}\quad L^2(\rn)
\end{align}
and
\begin{align*}
\lf\|\lf\{\sum_{k\in\nn}
\lf[\frac{|\lz_{k}|\chi_{B_{k}}}
{\|\chi_{B_{k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
\lesssim\|f\|_{\vh}\ls 1
\end{align*}
with $p_-$ as in \eqref{2e10}. From the boundedness of $T$ on $L^2(\rn)$ and \eqref{6e2},
we deduce that, for any $f\in H^{\vp,2,s}_{\va,\rm {fin}}(\rn)$,
\begin{align}\label{6e4}
T(f)=\sum_{k\in\mathbb{N}}\lambda_{k}T(a_{k})\quad{\rm in}\quad L^2(\rn).
\end{align}
Thus, by Theorem \ref{6t1}(i) and Lemma \ref{3l8},
to prove Theorem \ref{6t2}, we only need to show that, for any $f\in H^{\vp,2,s}_{\va,\rm {fin}}(\rn)$,
\begin{align}\label{6e3}
\|T(f)\|_{\vh}\sim\|M_0(T(f))\|_{\lv}\ls 1,
\end{align}
where $M_0$ is as in \eqref{3e16}.
To this end,
from \eqref{6e4}, it is easy to see that
$$
\lf\|M_0(T(f))\r\|_{\lv}\ls \lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))
\chi_{B_k^{(2)}}\r\|_{\lv}+\lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))
\chi_{(B_k^{(2)})^\com}\r\|_{\lv}=:\textrm{I}+\textrm{II},
$$
where $B_k^{(2)}$ is as in \eqref{2e2'} with $\dz=2$.
For $\textrm{I}$,
by Lemma \ref{3l1} and the fact that $T$ is bounded on $L^2(\rn)$,
we conclude that, for any $k\in \nn$,
$$
\lf\|M_0\lf(T(a_k)\r)\chi_{B_k^{(2)}}\r\|_{L^2(\rn)}\ls \lf\|M_{\rm {HL}}
(T(a_k))\chi_{B_k^{(2)}}\r\|_{L^2(\rn)}\ls\lf\|T(a_k)\r\|_{L^2(\rn)}
\ls \lf\|a_k\r\|_{L^2(\rn)}\ls \frac{|B_k|^{1/2}}{\|\chi_{B_k}\|_{\lv}},
$$
where $M_{\rm {HL}}$ denotes the Hardy-Littlewood maximal operator
as in \eqref{3e2}.
This, combined with Lemma \ref{3l6}, implies that
\begin{align}\label{6e16}
\textrm{I}&\le \lf\|\lf\{\sum_{k\in\mathbb{N}}\lf[|\lambda_{k}|M_0(T(a_{k}))
\chi_{B_k^{(2)}}\r]^{p_-}\r\}^{1/p_-}\r\|_{\lv}
\ls\lf\|\lf\{\sum_{k\in\nn}
\lf[\frac{|\lz_{k}|\chi_{B_{k}}}
{\|\chi_{B_{k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
\ls 1.
\end{align}
Next, we deal with \textrm{II}. To this end, for any $t\in (0,\fz)$,
let $k^{(t)}:=k*\Phi_t$, where $k$ is the kernel of $T$ and $\Phi$ as in \eqref{3e16}.
Then, we claim that $k^{(t)}$ satisfies the same conditions as $k$.
Indeed, since, for any $t \in (0,\fz)$ and $f\in L^2(\rn)$, $k^{(t)}*f=k*\Phi_t*f$,
we have
\begin{align*}
\lf\|k^{(t)}*f\r\|_{L^2(\rn)}&=\|k*\Phi_t*f\|_{L^2(\rn)}
=\|k*(\Phi_t*f)\|_{L^2(\rn)}\\
&\ls \|\Phi_t*f\|_{L^2(\rn)}\ls\|f\|_{L^2(\rn)}.
\end{align*}
On the other hand, by an argument similar to that
used in the proof of \cite[p.\,117, Lemma]{s93},
we conclude that, for any $x,\,y\in \rn$ with
$|x|_{\va}>2|y|_{\va}$,
$$\lf|k^{(t)}(x-y)-k^{(t)}(x)\r|\le C\frac{|y|_{\va}^{\delta}}{|x|_{\va}^{\nu+\delta}},$$
where $C$ is a positive constant independent of $t,\,x$ and $y$.
Therefore, the above claim holds true.
Now, by the vanishing moment condition of $a_k$ and the H\"{o}lder
inequality, we know that, for any $x\in(B_k^{(2)})^{\com}$,
\begin{align*}
M_0(T(a_k))(x)&=\sup_{t\in (0,\fz)}\lf|\Phi_t*(k*a_k)(x)\r|
=\sup_{t\in (0,\fz)}\lf|k^{(t)}*a_k(x)\r|\\
&\le \sup_{t\in (0,\fz)}\int_{B_k}\lf|k^{(t)}(x-y)-k^{(t)}(x-x_k)\r||a_k(y)|\,dy\\
&\ls\int_{B_k}\frac{|y-x_k|_{\va}^{\delta}}{|x-x_k|_{\va}^{\nu+\delta}}|a_k(y)|\,dy
\ls \frac{r_k^{\delta}}{|x-x_k|_{\va}^{\nu+\delta}}\|a_k\|_{L^2(\rn)}|B_k|^{1/2}\\
&\ls \frac{r_k^{\nu+\delta}}{|x-x_k|_{\va}^{\nu+\delta}}\frac1{\|\chi_{B_k}\|_{\lv}}
\ls \lf[M_{\rm HL}\lf(\chi_{B_k}\r)(x)\r]^{\f{\nu+\delta}{\nu}}\frac1{\|\chi_{B_k}\|_{\lv}},
\end{align*}
which implies that
\begin{align}\label{6e8}
M_0(T(a_k))(x)\chi_{(B_k^{(2)})^{\com}}(x)\ls
\lf[M_{\rm HL}\lf(\chi_{B_k}\r)(x)\r]^{\f{\nu+\delta}{\nu}}\frac1{\|\chi_{B_k}\|_{\lv}}.
\end{align}
Therefore, by \eqref{2e8}, the fact that $\widetilde{p}_-\in(\frac\nu{\nu+\delta},1]$ and Lemma \ref{3l2},
we find that
\begin{align*}
\textrm{II}&\ls \lf\|\sum_{k\in\mathbb{N}}\frac{|\lambda_{k}|}{\|\chi_{B_k}\|_{\lv}}
\lf[M_{\rm HL}\lf(\chi_{B_k}\r)\r]^{\f{\nu+\delta}{\nu}}\r\|_{\lv}
\sim \lf\|\lf\{\sum_{k\in\mathbb{N}}\frac{|\lambda_{k}|}{\|\chi_{B_k}\|_{\lv}}
\lf[M_{\rm HL}\lf(\chi_{B_k}\r)\r]^{\f{\nu+\delta}{\nu}}\r\}^{\frac{\nu}{\nu+\delta}}
\r\|_{L^{\f{\nu+\delta}{\nu}\vp}(\rn)}^{\f{\nu+\delta}{\nu}}\\
&\ls \lf\|\lf\{\sum_{k\in\mathbb{N}}\frac{|\lambda_{k}|\chi_{B_k}}{\|\chi_{B_k}\|_{\lv}}
\r\}^{\frac{\nu}{\nu+\delta}}
\r\|_{L^{\f{\nu+\delta}{\nu}\vp}(\rn)}^{\f{\nu+\delta}{\nu}}\noz
\ls \lf\|\lf\{\sum_{k\in\nn}
\lf[\frac{|\lz_{k}|\chi_{B_{k}}}
{\|\chi_{B_{k}}\|_{\lv}}\r]^
{p_-}\r\}^{1/p_-}\r\|_{\lv}
\ls 1.
\end{align*}
Finally, combining the above estimates of \textrm{I} and \textrm{II},
we obtain \eqref{6e3}.
This finishes the proof of Theorem \ref{6t2}.
\end{proof}
Now we prove Theorem \ref{6t3}.
\begin{proof}[Proof of Theorem \ref{6t3}]
Let $\vp\in (0,1]^n$ and $s$ be as in \eqref{3e1}. By Corollary \ref{6c1}(i),
to prove this theorem, we know that it suffices to show that, for any $(\vp,2,s)$-atom $a$,
\begin{align}\label{6e5}
\lf\|T(a)\r\|_{\lv}\ls 1.
\end{align}
Now we show \eqref{6e5}. Let $\supp a\st B\in \mathfrak{B}$.
From the fact that $T$ is bounded on $L^2(\rn)$ and $a\in L^2(\rn)$,
we deduce that
$$\lf\|T(a)\chi_{B^{(2)}}\r\|_{L^2(\rn)}\ls \|a\|_{L^2(\rn)}
\ls \frac{|B|^{1/2}}{\|\chi_{B}\|_{\lv}},$$
where $B^{(2)}$ is as in \eqref{2e2'} with $\dz=2$, which, together with Lemma \ref{3l6}, implies that
\begin{align}\label{6e6}
\lf\|T(a)\chi_{B^{(2)}}\r\|_{\lv}\ls 1.
\end{align}
On the other hand, when $x\in (B^{(2)})^{\com}$,
by an argument similar to that used in the estimation
of \eqref{6e8}, we conclude that
$$\lf|T(a)(x)\r|\ls \lf[M_{\rm HL}(\chi_{B})(x)\r]
^{\f{\nu+\delta}{\nu}}\frac1{\|\chi_{B}\|_{\lv}}.$$
Therefore, by \eqref{2e8}, the fact that $\widetilde{p}_-\in(\frac\nu{\nu+\delta},1]$ and Lemma \ref{3l1},
we know that
\begin{align*}
\lf\|T(a)\chi_{(B^{(2)})^{\com}}\r\|_{\lv}&\ls \lf\|\lf[M_{\rm HL}(\chi_{B})\r]
^{\f{\nu+\delta}{\nu}}\frac1{\|\chi_{B}\|_{\lv}}\r\|_{\lv}\ls 1,
\end{align*}
which, combined with \eqref{6e6}, further implies \eqref{6e5} holds true and hence
completes the proof of Theorem \ref{6t3}.
\end{proof}
We now introduce a class of anisotropic $\beta$-order Calder\'{o}n-Zygmund operators
as follows.
\begin{definition}
Let $\va\in[1,\fz)^n$. For any given $\beta\in (0,\fz)\setminus\nn$, a linear operator
$T$ is called an \emph{anisotropic $\beta$-order Calder\'{o}n-Zygmund operator} if $T$ is bounded
on $L^2(\rn)$ and its kernel
$$\mathcal{K}:\ (\rn\times\rn)\setminus\{(x,x):\ x\in\rn\}\to \mathbb{C}$$
satisfies that there exists a positive constant $C$ such that, for any $\az\in\zz_+^n$ with
$|\alpha|\le \lfloor\bz\rfloor$ and $x,\,y,\,z\in \rn$,
\begin{align}\label{6e7}
|\pa^{\az}_x\mathcal{K}(x,y)-\pa^{\az}_x\mathcal{K}(x,z)|\le C\frac{|y-z|_{\va}^{\bz-\lfloor\bz\rfloor}}
{|x-y|_{\va}^{\nu+\bz}}
\quad{\rm when}\quad |x-y|_{\va}>2|y-z|_{\va}
\end{align}
and, for any $f\in L^2(\rn)$ with compact support and $x\notin \supp f$,
$$T(f)(x)=\int_{\supp f}\mathcal{K}(x,y)f(y)\,dy.$$
\end{definition}
For any $l\in\nn$, an operator $T$ is said to have the \emph{vanishing moment
condition up to order $l$} if, for any $a\in L^2(\rn)$ with compact support and
satisfying that, for any $\gamma\in\zz^+_n$ with $|\gamma|\le l$, $\int_{\rn}x^{\gamma}a(x)\,dx=0$,
it holds true that $\int_{\rn}x^{\gamma}T(a)(x)\,dx=0$.
Then we have the following boundedness of anisotropic $\beta$-order Calder\'{o}n-Zygmund operators
$T$ from $\vh$ to itself (see Theorem \ref{6t4} below) or to $\lv$ (see Theorem \ref{6t5} below),
which extends the corresponding results of Stefanov and Torres \cite[Theorem 1]{st04}
as well as Yan et al. \cite[Theorem 7.6]{yyyz16} to the present setting.
\begin{theorem}\label{6t4}
Let $\va\in[1,\fz)^n,\,\vp\in(0,1]^n$, $\bz\in(0,\fz)\setminus\nn$, $\widetilde{p}_-\in(\frac{\nu}{\nu+\bz},\frac{\nu}{\nu+\lfloor\bz\rfloor a_-}]$
with $\widetilde{p}_-$ as in $\eqref{3e1}$ and $a_-$ as in \eqref{2e9} and $T$ be an anisotropic $\beta$-order Calder\'{o}n-Zygmund operator having the
vanishing moment conditions up to order $\lfloor\bz\rfloor$. Then there exists a positive constant $C$
such that, for any $f\in \vh$,
$$\|T(f)\|_{\vh}\le C\|f\|_{\vh}.$$
\end{theorem}
\begin{theorem}\label{6t5}
Let $\va\in[1,\fz)^n,\,\vp\in(0,1]^n$, $\bz\in(0,\fz)\setminus\nn$, $\widetilde{p}_-\in(\frac{\nu}{\nu+\bz},\frac{\nu}{\nu+\lfloor\bz\rfloor a_-}]$
with $\widetilde{p}_-$ as in $\eqref{3e1}$ and $a_-$ as in \eqref{2e9} and $T$ be an anisotropic $\beta$-order Calder\'{o}n-Zygmund operator. Then there exists a positive constant $C$
such that, for any $f\in \vh$,
$$\|T(f)\|_{\lv}\le C\|f\|_{\vh}.$$
\end{theorem}
\begin{remark}\label{6r2}
\begin{enumerate}
\item[(i)]
When $\bz:=\delta\in(0,1)$, then $\az=(\overbrace{0,\ldots,0}^{n\ \mathrm{times}})$ and the operator $T$ in
Theorem \ref{6t4} (or Theorem \ref{6t5}) becomes an anisotropic non-convolutional
$\delta$-type Calder\'{o}n-Zygmund operator.
Thus, from Theorem \ref{6t4} (or Theorem \ref{6t5}), we deduce that, for any $\va\in[1,\fz)^n,\,\vp\in(0,1]^n,\,\delta\in(0,1]$ and
$\widetilde{p}_-\in(\frac{\nu}{\nu+\delta},1]$ with $\widetilde{p}_-$ as in \eqref{3e1},
the anisotropic non-convolutional $\delta$-type
Calder\'{o}n-Zygmund operator is bounded from $\vh$ to itself [or to $\lv$].
In addition, we point out that the boundedness of the anisotropic
$\bz$-order Calder\'{o}n-Zygmund operators on
the mixed-norm Lebesgue space $\lv$ with $\vp\in(1,\fz)^n$ is still unknown so far.
\item[(ii)] When $\va:=(\overbrace{1,\ldots,1}^{n\ \rm times})$ and $
\vp:=(\overbrace{p,\ldots,p}^{n\ \rm times})\in (0,\fz)^n$, $\vh$ and $\lv$
become the classical isotropic Hardy space $H^p(\rn)$ and Lebesgue space $L^p(\rn)$, respectively, and
$T$ becomes the classical $\dz$-type Calder\'on-Zygmund operator.
In this case, we know that, if $\delta\in(0,1]$ and
$p\in(\frac n{n+\delta},1]$, then Theorems \ref{6t2} and \ref{6t3}
and (i) of this remark imply the boundedness of
the classical $\dz$-type Calder\'on-Zygmund operator
from $H^p(\rn)$ to itself and
from $H^p(\rn)$ to $L^p(\rn)$ for any $\delta\in(0,1]$ and $p\in(\frac n{n+\delta},1]$,
which is a well-known result (see, for example, \cite{a86,lu,s93}).
\end{enumerate}
\end{remark}
Now we prove Theorem \ref{6t4}.
\begin{proof}[Proof of Theorem \ref{6t4}]
By an argument similar to that used in
the proof of Theorem \ref{6t2}, we know that, to show Theorem \ref{6t4}, we only need to prove that
\begin{align}\label{6e15}
\lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))\r\|_{\lv}\ls 1,
\end{align}
where $\{\lambda_k\}_{k\in\nn}$ and $\{a_k\}_{k\in\nn}$ are the same as in the
proof of Theorem \ref{6t2} and $M_0$ is as in \eqref{3e16}. For this purpose, first, it is easy to see that
\begin{align*}
\lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))\r\|_{\lv}
&\ls \lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))
\chi_{B_k^{(4)}}\r\|_{\lv}+\lf\|\sum_{k\in\mathbb{N}}|\lambda_{k}|M_0(T(a_{k}))
\chi_{(B_k^{(4)})^\com}\r\|_{\lv}\\
&=:\textrm{I}+\textrm{II},
\end{align*}
where, for any $k\in\nn$, $B_k:=B_{\va}(x_k,r_k)$
is the same as in the proof of Theorem \ref{6t2} and $B_k^{(4)}$ as in \eqref{2e2'} with $\dz=4$.
For \textrm{I}, by a proof similar to that of \eqref{6e16}, we conclude that $\textrm{I}\ls 1$.
Next, we deal with \textrm{II}. To this end, from the vanishing moment condition of $T$ and the fact
that $\lfloor\bz\rfloor\le \frac{\nu}{a_-}(\frac1{\widetilde{p}_-}-1)$, which implies $\lfloor\bz\rfloor\le s$,
it follows that, for any $k\in \nn,\,t\in (0,\fz)$ and $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e9}
\lf|\Phi_t*T(a_k)(x)\r|=&\frac1{t^{\nu}}\lf|\int_{\rn}\Phi\lf(\frac{x-y}{t^{\va}}\r)T(a_k)(y)\,dy\r|\\
\le &\frac1{t^{\nu}}\int_{\rn}\lf|\Phi\lf(\frac{x-y}{t^{\va}}\r)-\sum_{|\az|\le \lfloor\bz\rfloor}
\frac{\pa^{\az}\Phi(\frac{x-x_k}{t^{\va}})}{\az!}\lf(\frac{y-x_k}{t^{\va}}\r)^{\az}\r|\lf|T(a_k)(y)\r|\,dy\noz\\
=&\frac1{t^{\nu}}\lf(\int_{|y-x_k|_{\va}<2r_k}+\int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}+
\int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}\r)\noz\\
&\times \lf|\Phi\lf(\frac{x-y}{t^{\va}}\r)-\sum_{|\az|\le \lfloor\bz\rfloor}
\frac{\pa^{\az}\Phi(\frac{x-x_k}{t^{\va}})}{\az!}\lf(\frac{y-x_k}{t^{\va}}\r)^{\az}\r|\lf|T(a_k)(y)\r|\,dy\noz\\
=&:\textrm{II}_1+\textrm{II}_2+\textrm{II}_3,\noz
\end{align}
where $\Phi$ is as in \eqref{3e16}.
For $\textrm{II}_1$, by the Taylor remainder theorem and (vi), (iv) and (v)
of Lemma \ref{2l2}, we conclude that, for any $k\in\nn$, $N\in\nn$,
$t\in(0,\fz)$, $x\in (B_k^{(4)})^\com$ and $y\in\rn$ with $|y-x_k|_{\va}<2r_k$, there exists $\theta_1(y)\in B_k^{(2)}$ such that
\begin{align}\label{6e10}
\textrm{II}_1
\le&\frac1{t^{\nu}}\int_{|y-x_k|_{\va}<2r_k}\lf|\sum_{|\az|= \lfloor\bz\rfloor+1}
\pa^{\az}\Phi\lf(\frac{x-\theta_1(y)}{t^{\va}}\r)\r|
\lf|\frac{y-x_k}{t^{\va}}\r|^{\lfloor\bz\rfloor+1}\lf|T(a_k)(y)\r|\,dy\\
\ls& \frac1{t^{\nu}}\int_{|y-x_k|_{\va}<2r_k}\frac1{(1+|\frac{x-\theta_1(y)}{t^{\va}}|)^N}\lf|\frac{y-x_k}
{t^{\va}}\r|^{\lfloor\bz\rfloor+1}\lf|T(a_k)(y)\r|\,dy\noz\\
\ls& \frac1{t^{\nu}}\int_{|y-x_k|_{\va}<2r_k}\lf(\frac{t}{|x-x_k|_{\va}}\r)^{Na_-}\noz\\
&\times\max\lf\{\lf(\frac{|y-x_k|_{\va}}{t}\r)^{(\lfloor\bz\rfloor+1)a_-},\,
\lf(\frac{|y-x_k|_{\va}}{t}\r)^{(\lfloor\bz\rfloor+1)a_+}\r\}\lf|T(a_k)(y)\r|\,dy.\noz
\end{align}
When $t\le |x-x_k|_{\va}$, let
\begin{align*}
N:=\left\{
\begin{array}{cl}
\vspace{0.25cm}
&\lf\lfloor\dfrac{\nu+(\lfloor\bz\rfloor+1)a_-}{a_-}\r\rfloor+1
\hspace{0.5cm} {\rm when}\hspace{0.5cm} |y-x_k|_{\va}<t,\\
&\lf\lfloor\dfrac{\nu+(\lfloor\bz\rfloor+1)a_+}{a_-}\r\rfloor+1
\hspace{0.5cm}{\rm when}\hspace{0.5cm} |y-x_k|_{\va}\ge t
\end{array}\r.
\end{align*}
in \eqref{6e10}.
Then, by this, the H\"{o}lder inequality and the fact that $T$ is bounded on $L^2(\rn)$, we know that,
for any $k\in \nn,\,t\in (0,\fz)$ and $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e11}
\textrm{II}_1\ls& \int_{|y-x_k|_{\va}<2r_k}\max\lf\{\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\,\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_+}}
{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\lf|T(a_k)(y)\r|\,dy\\
\ls&\max\lf\{\frac{r_k^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\, \frac{r_k^{(\lfloor\bz\rfloor+1)a_+}}{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}
\lf\|T(a_k)\r\|_{L^2(\rn)}\lf|B_k\r|^{1/2}\noz\\
\ls&\max\lf\{\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+
(\lfloor\bz\rfloor+1)a_-},\,\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+(\lfloor\bz\rfloor+1)a_+}\r\}
\frac1{\|\chi_{B_k}\|_{\lv}}.\noz
\end{align}
When $t>|x-x_k|_{\va}$, let $N:=\lfloor\frac{\nu+(\lfloor\bz\rfloor+1)a_-}{a_-}\rfloor$
in \eqref{6e10}. Then it is easy to see that \eqref{6e11} also holds true.
For $\textrm{II}_2$, by the Taylor remainder theorem, some arguments
similar to those used in the estimations of \eqref{6e10} and
\eqref{6e11}, the vanishing moment condition of $a_k$,
the fact that $\lfloor\bz\rfloor\le s$, \eqref{6e7}, the H\"{o}lder
inequality and Lemma \ref{2l2}(ix), we find
that, for any $z\in B_k$, there exists $\theta_2(z)\in B_k$ such that,
for any $t\in(0,\fz)$ and $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e12}
\textrm{II}_2
\ls& \int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}
\max\lf\{\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\,\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_+}}
{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\lf|T(a_k)(y)\r|\,dy\\
\ls& \int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}
\max\lf\{\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\, \frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_+}}{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\noz\\
&\times\lf[\int_{B_k}|a_k(z)|\lf|\mathcal{K}(y,z)-\sum_{|\az|< \lfloor\bz\rfloor}
\frac{\pa^{\az}_y\mathcal{K}(y,x_k)}{\az!}(z-x_k)^{\az}\r|\,dz\r]\,dy\noz\\
\sim& \int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}
\max\lf\{\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\, \frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_+}}{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\noz\\
&\times\int_{B_k}|a_k(z)|\lf|\sum_{|\az|= \lfloor\bz\rfloor}
\frac{\pa^{\az}_y\mathcal{K}(y,x_k)-\pa^{\az}_y\mathcal{K}(y,\theta_2(z))}{\az!}(z-x_k)^{\az}\r|\,dz\,dy\noz\\
\ls& \int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}\max
\lf\{\frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\, \frac{|y-x_k|_{\va}^{(\lfloor\bz\rfloor+1)a_+}}{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\noz\\
&\times\int_{B_k}|a_k(z)|\frac{r_k^{\bz}}{|y-x_k|_{\va}^{\nu+\bz}}\,dz\,dy\noz\\
\ls& r_k^{\bz}\lf\|a_k\r\|_{L^2(\rn)}\lf|B_k\r|^{1/2}\noz\\ &\times\int_{2r_k\le|y-x_k|_{\va}<\frac{|x-x_k|_{\va}}{2}}
\max\lf\{\frac{|y-x_k|_{\va}^{-\nu-\bz+(\lfloor\bz\rfloor+1)a_-}}{|x-x_k|_{\va}^{\nu+
(\lfloor\bz\rfloor+1)a_-}},\,\frac{|y-x_k|_{\va}^{-\nu-\bz+(\lfloor\bz\rfloor+1)a_+}}
{|x-x_k|_{\va}^{\nu+(\lfloor\bz\rfloor+1)a_+}}\r\}\,dy\noz\\
\ls&\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B_k}\|_{\lv}}.\noz
\end{align}
For $\textrm{II}_3$, from the Taylor remainder theorem, the vanishing moment condition of $a_k$,
the fact that $\lfloor\bz\rfloor\le s$, \eqref{6e7}, the H\"{o}lder inequality and Lemma \ref{2l2}(vi), we deduce
that, for any $z\in B_k$, there exists $\theta_3(z)\in B_k$ such that, for any $t\in(0,\fz)$ and $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e13}
\textrm{II}_3\le&\frac1{t^{\nu}}\int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}
\lf|\Phi\lf(\frac{x-y}{t^{\va}}\r)-\sum_{|\widetilde{\az}|\le \lfloor\bz\rfloor}
\frac{\pa^{\widetilde{\az}}\Phi(\frac{x-x_k}{t^{\va}})}{\widetilde{\az}!}\lf(\frac{y-x_k}{t^{\va}}\r)^{\widetilde{\az}}\r|\\
&\times\lf[\int_{B_k}|a_k(z)|\lf|\mathcal{K}(y,z)-\sum_{|\az|< \lfloor\bz\rfloor}
\frac{\pa^{\az}_y\mathcal{K}(y,x_k)}{\az!}(z-x_k)^{\az}\r|\,dz\r]\,dy\noz\\
\sim& \int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}\lf|\frac1{t^{\nu}}\lf[\Phi\lf(\frac{x-y}
{t^{\va}}\r)-\sum_{|\widetilde{\az}|\le \lfloor\bz\rfloor}
\frac{\pa^{\widetilde{\az}}\Phi(\frac{x-x_k}{t^{\va}})}{\widetilde{\az}!}
\lf(\frac{y-x_k}{t^{\va}}\r)^{\widetilde{\az}}\r]\r|\noz\\
&\times\int_{B_k}|a_k(z)|\lf|\sum_{|\az|= \lfloor\bz\rfloor}
\frac{\pa^{\az}_y\mathcal{K}(y,x_k)-\pa^{\az}_y\mathcal{K}(y,\theta_3(z))}{\az!}(z-x_k)^{\az}\r|\,dz\,dy\noz\\
\ls& \int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}\lf|\Phi_t(x-y)\r|\int_{B_k}
|a_k(z)|\frac{r_k^{\bz}}{|y-x_k|_{\va}^{\nu+\bz}}\,dz\,dy\noz\\
&+ \int_{|y-x_k|_{\widetilde{\va}}\geq\frac{|x-x_k|_{\va}}{2}}\lf|\frac1{t^{\nu}}\sum_{|\widetilde{\az}|\le \lfloor\bz\rfloor}
\frac{\pa^{\widetilde{\az}}\Phi(\frac{x-x_k}{t^{\va}})}{\widetilde{\az}!}
\lf(\frac{y-x_k}{t^{\va}}\r)^{\widetilde{\az}}\r|\int_{B_k}
|a_k(z)|\frac{r_k^{\bz}}{|y-x_k|_{\va}^{\nu+\bz}}\,dz\,dy\noz\\
\ls& \frac{r_k^{\bz}}{|x-x_k|_{\va}^{\nu+\bz}}\|a_k\|_{L^2(\rn)}|B_k|^{1/2}\int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}
\lf|\Phi_t(x-y)\r|\,dy\noz\\
&+r_k^{\bz}\|a_k\|_{L^2(\rn)}|B_k|^{1/2}\int_{|y-x_k|_{\va}\geq\frac{|x-x_k|_{\va}}{2}}
\frac1{t^{\nu}}\sum_{|\widetilde{\az}|\le \lfloor\bz\rfloor}\lf(\frac{t}{|x-x_k|_{\va}}\r)
^{Na_-}\lf|\frac{y-x_k}{t^{\va}}\r|^{|\widetilde{\az}|}\,dy\noz\\
=&:\textrm{II}_{3,1}+\textrm{II}_{3,2}.\noz
\end{align}
For $\textrm{II}_{3,1}$, from the size condition of $a_k$ and the fact that $\Phi$ is as in \eqref{3e16},
it follows that, for any $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e14'}
\textrm{II}_{3,1}\ls\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B_k}\|_{\lv}}.
\end{align}
In addition, by an argument similar to that used in the estimation of \eqref{6e11} and Lemma \ref{2l2}(ix),
we find that, for any $x\in (B_k^{(4)})^\com$,
$$\textrm{II}_{3,2}\ls\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B_k}\|_{\lv}},$$
which, combined with \eqref{6e13} and \eqref{6e14'}, further implies that, for any $x\in (B_k^{(4)})^\com$,
\begin{align}\label{6e14}
\textrm{II}_{3}\ls\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B_k}\|_{\lv}}.
\end{align}
Combining \eqref{6e9}, \eqref{6e11}, \eqref{6e12} and \eqref{6e14}, we conclude that, for any $x\in (B_k^{(4)})^\com$,
\begin{align*}
M_0(T(a_k))(x)
&=\sup_{t\in(0,\fz)}\lf|\Phi_t*T(a_k)(x)\r|
\ls\lf(\frac{r_k}{|x-x_k|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B_k}\|_{\lv}}\\
&\ls \lf[M_{\rm HL}\lf(\chi_{B_k}\r)(x)\r]^{\frac{\nu+\bz}{\nu}}\frac1{\|\chi_{B_k}\|_{\lv}},
\end{align*}
which implies that
$$M_0(T(a_k))(x)\chi_{(B_k^{(4)})^\com}(x)\ls \lf[M_{\rm HL}\lf(\chi_{B_k}\r)(x)\r]^{\frac{\nu+\bz}{\nu}}\frac1{\|\chi_{B_k}\|_{\lv}}.$$
Then, by the fact that $\widetilde{p}_-<\frac{\nu+\bz}{\nu}$ and an argument similar to that used in the proof of Theorem \ref{6t2}, we know that
\eqref{6e15} holds true. This finishes the proof of Theorem \ref{6t4}.
\end{proof}
Now we prove Theorem \ref{6t5}.
\begin{proof}[Proof of Theorem \ref{6t5}]
Let $\vp\in(0,1]^n$ and $s$ be as in \eqref{3e1}. By an argument similar to that
used in the proof of Theorem \ref{6t3}, we know that, to show Theorem \ref{6t5},
it suffices to prove that, for any $(\vp,2,s)$-atom $a$ and $x\in\rn$,
\begin{align}\label{6e1'}
\lf|T(a)(x)\chi_{B^{(2)}}(x)\r|\ls \lf[M_{\rm HL}(\chi_{B})(x)\r]
^{\f{\nu+\bz}{\nu}}\frac1{\|\chi_{B}\|_{\lv}},
\end{align}
where $B$ and $B^{(2)}$ are as in the proof of Theorem \ref{6t3}.
Indeed, let $x_0$ and $r$ denote the center and the radius of $B$, respectively.
From the Taylor remainder theorem, the vanishing moment condition of $a$,
the fact that $\lfloor\bz\rfloor\le \frac{\nu}{a_-}(\frac1{\widetilde{p}_-}-1)$,
which implies $\lfloor\bz\rfloor\le s$, and the H\"{o}lder inequality,
we deduce that, for any $z\in B$,
there exists $\theta(z)\in B$ such that, for any $x\in B^{(2)}$,
\begin{align*}
\lf|T(a)(x)\r|&\leq\int_B |a(z)|\lf|\mathcal{K}(x,z)\r|\,dz\\
&=\int_{B}|a(z)|\lf|\mathcal{K}(x,z)-\sum_{|\az|< \lfloor\bz\rfloor}
\frac{\pa^{\az}_x\mathcal{K}(x,x_0)}{\az!}(z-x_0)^{\az}\r|\,dz\\
&\sim \int_{B}|a(z)|\lf|\sum_{|\az|= \lfloor\bz\rfloor}
\frac{\pa^{\az}_x\mathcal{K}(x,x_0)-\pa^{\az}_x\mathcal{K}(x,\theta(z))}{\az!}(z-x_0)^{\az}\r|\,dz\\
&\ls \int_{B}|a(z)|\frac{r^{\bz}}{|x-x_0|_{\va}^{\nu+\bz}}\,dz
\ls \frac{r^{\bz}}{|x-x_0|_{\va}^{\nu+\bz}}\|a\|_{L^2(\rn)}|B|^{1/2}\\
&\ls \lf(\frac{r}{|x-x_0|_{\va}}\r)^{\nu+\bz}\frac1{\|\chi_{B}\|_{\lv}}
\ls \lf[M_{\rm HL}(\chi_{B})(x)\r]^{\f{\nu+\bz}{\nu}}\frac1{\|\chi_{B}\|_{\lv}},
\end{align*}
which implies that \eqref{6e1'} holds true and hence completes the proof of Theorem \ref{6t5}.
\end{proof}
| 73,773
|
So this is something that's been severely peeving me at my gym lately. (Sorry if the ladies feel left out, but I've never witnessed this with them.) I'm not talking about,"Damn! Missed that PR attempt...(sets BB down on safety rack)...better luck next week..."
I mean looking like a retarded camel giving birth on the last 4-5 reps of EVERY working set of your new and improved total body lat pulldowns or asking me to give you a spot and after I've literally lifted half the weight up for you for two straight reps you still have the balls to mutter, "one more!" even though you're only lifting up the right half of the loaded BB anymore.
I honestly don't understand, and can't say I've ever had this mentality since I started training. I understand that there's a thing called pride and men want to appear strong, but does the "perks" of hitting a number seriously outweigh looking like the biggest turd in the universe while doing it?
I think the newb with a buck 35 on his back squatting to proper depth and knowing how to recruit his hips properly looks more badass than any douche who can throw 3 plates on and...well...like I said...retarded camel. Hell, the newb with a broomstick on his back looks better!
Any insight?
Machismo.
I this a lot at my gym, where guys come down with ILS (Imaginary Lat Syndrome) the moment they walk in the door before ever touching a weight.
I have a video I'll link later ( or search yotube for "nardpuncher" and "how not to work out") of a guy I see that ONLY benches (from what I've seen) and ONLYdoes 225lbs with terrible form. I mean this guy doesn't warm up and hasn't gone above 225 the whole 2 years or so I've been seeing him at the gym.
My gym is typically okay about this, although not great. The big guys squat properly, the little guys don't. The mentality, I believe, isn't just male specific.
I walked into the gym once to notice a girl benching. That's cool, I remember thinking. We don't see a lot of females weight lifting, they're often fucking around on the bosu balls or doing some lameass core workout, if they're not on the treadmills.
Few even wander in to the weight training area, except when they're sending some kind of message to a guy friend, and even then often in packs. This girl was doing something that just isn't very popular among women here, (although I doubt this is big with women in other places- someone please prove me wrong).
She only has 85 pounds on the bar. That's cool though, she's small, and I hadn't seen her around after all the time I've spent at the gym, so it was likely she was new. She pulls the bar off the rack. Her arms are shaking. Her jaw is clenched. She begins lowering and...
She gets halfway down and then pushes back up.
I frown. Maybe it was too heavy. I know that guys who don't lift can start at that level, assuming they're unathletic. It would make sense for a girl who doesn't participate in physical activity to need to start less. Likely she would put the bar down and lower the wei-
She goes in for another rep. Then another. Then another. Only getting halfway down each time. I stopped watching after that.
I saw something similar when a girl in the squat rack next to me tried to match my weight, but only got halfway to parallel during her sets. She stopped after she saw me add more weight for my next set.
I didn't understand it myself. Maybe they're afraid of being humiliated? Maybe they want to seem stronger than they are? The thing is, when I see a weak guy struggle, it motivates me. That guy is having trouble, but he's here because he wants to grow and he's not taking no for an answer, no matter how humiliating the process is. The guy who goes in there and works at it with a low weight but does it properly deserves respect in my book. I don't laugh at them or mock them, I try harder.
The first time I tried benching, at 110 pounds I remember a fairly big guy coming over and asking if I needed a spotter. When I told him I didn't know how to bench, as nobody I ever trained with thought I could even lift the bar, he showed me how and made sure I worked on my form, rather than adding weight to the bar. It was at this point that I realized I wasn't going to be laughed at for being small, there were people willing to help.
Other people, I think, don't quite understand that. They're afraid of seeming small, and want the attention and satisfaction of being big without working for it. A bigger number means they're stronger and better. If they can shortcut getting there, while justifying it in their head, then they'll do it.
Well god dammit, one day I'm going to learn how to type a post that isn't as long as a small essay.
And for enhanced effect I google image searched "retarded camel giving birth." For some reason, this guy showed up about halfway down the page. Coincidentally, I think he's the dude I spotted last week!
When I was in middle school, I used to attempt dunks that I knew I was going to miss just to convince onlooker that I could probably dunk sometimes. Which doesn't really have anything to do with this thread, and I only mention so you all know that I could almost dunk in middle school, which is kinda impressive.
Nards, I watched some of those videos and the assisted assisted-pull-ups cracked me up.I know a guy that fits the description of always trying to lift heavier than he should. Tries to do 225 for 5x5 without warmups but every week drops a rep or two, last time we trained it went: 5, 3, 1, 1, 1/2 WTF man?! I'm stronger than this usually!
He's pretty short (5'5'') and has good pressing levers so he progressed to this point within a few months but has been stagnant for the last 5 years from what he tells me. When I suggested eating more I got the old "can't lose my abs" speech and once I convinced him to do some warm ups with something less than 225 things started to get better. At least in his case it's a little man thing and likes to appear strong to people passing his bench.
I've seen a couple of guys (one in particular does it all the time) load the bench up with something that looks like it would crush them, sit down, do some stretches, then arm swings, lay down then get back up like they just lifted the weight. Orignially I thought that maybe the weight just got in their head and had a lapse of confidence but now I think it's for show as it's been repeated.
Because of wanting to impress your avatar.
Most people that go to the gym have no idea of what they are doing.This explains everything.
I dont think its only males. I was going to the squat rack last month and a skinny girl was doing squats with 60kg. I put 60kg to warmup and she put 80kg and didnt even go halfway down and was shaking bad. So whats that all about =)
Ha! While I was reading your post i thought about a guy at the gym back in Winnipeg that would do that...load the bar to 315 or more, sit down, get up, stretch and just basically hang around the bench so that people think he actually moved the weight.
Girls are less likely to have learned how to bench by goofing around in some high school friend's garage, who's dad had a weight set.
I don't know why people don't at least go watch a few Youtube videos first. That's the only way I actually got up the nerve to head over to the benches and squat racks. I appreciate that you aren't making fun of her for only benching 85 pounds. I'm little and that hits kinda close to home!
I talked about this in another thread, why are women who "lift weights" curling 10 pound dumbbells for a whole bunch of reps every time they come to the gym?
So they don't wake up the next day accidentally looking like Professor X, silly.
Thats so pathetic...wow. I use a university gym so I see a ton of this kind of stuff, but I've never seen anybody go to the gym to pretend to lift weights!
I used to do that.Why?
I didn't know what I looked like. (Is proproception the right term?) I couldn't "feel" what a squat to depth was. I had to have strangers come up and tell me "Hey, you're not squatting to depth" before I figured it out.
I didn't have any idea what the exercise was supposed to be. "Bench press is self-explanatory, right?" Well, at some point you have to discover that the bar is supposed to go all the way down. (My dad had the same issue -- he'd been "benching" for years and it had NEVER OCCURRED TO HIM that the bar is supposed to touch your chest. He was all, "do I really have to?" when I told him, and I said "yep" and he had to cut his bench weight in half. Then when I wasn't around to nag him he went back to benching halfway down.)
If you are really skinny I dont think its a good idea to bench press all the way to touch the chest. At that point your elbow might be alot below your bench and having that angle isnt really good for shoulders in my opinion.
really?
Haha
If you're in the proper technique your elbows shouldn't go below the bench no matter how concave your chest is.
I use a ton of technique when I lift. Some would call it cheating.
Unless your lats are wider than mine, which is a small percentage of people at my gym, I couldn't give a fuck what anyone thinks of my pulldowns.
There is more than one little bastard in my gym that think they are hot shit with there little guy form, slamming 135 into the rack after their set of rows... I find that more annoying than people putting in effort to step outside their comfort zone.
In addition: I stopping paying attention to what people who aren't making progress were doing awhile ago, and it really helped my own.
(Outside of those few freaks that make you notice them, slamming bars into the rack, stand in front of you, etc etc etc)
| 360,112
|
TITLE: What does $K^{1/p}$ for a field $K$ mean?
QUESTION [3 upvotes]: In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows.
$k$ is a field of characteristic $p<\infty$, and $P$ its prime field. Suppose that $k$ arose from $P$ by adjoining finitely many (algebraic or transcendental) elements. Then she writes:
Because the prime field $P$ is perfect, it follows that $k^{1/p}$ is finite with respect to $k$.
I would have suspected that $k^{1/p}=\{x\in\bar k:x^p\in k\}$. This was called "Wurzelkörper" earlier in the paper. What is it, and why does the above follow from $P$ being perfect?
Thanks!
REPLY [8 votes]: $P$ being perfect means that exponentiation by $p$ is an isomorphism of $P$, so we can take $p$-th roots of any element in $P$ . You are right with the definition of $k^{1/p}$. To prove the statement suppose $k$ is generated by $x_1,\ldots,x_n$ over $P$. I claim that $k^{1/p}$ is (finitely) generated by $x_1^{1/p},\ldots, x_n^{1/p}$. Indeed let $y\in k$ be of the form
$$
y=a_1x_1+\ldots+a_nx_n,\quad a_i\in P.
$$
If $b_i\in P$ is the $p$-th root of $a_i$ then the $p$-th root of $y$ in $\overline{k}$ is
$$
y^{1/p}=b_1x_1^{1/p}+\ldots+b_nx_n^{1/p}
$$
Remark: $p$-th roots of elements are unique in characteristic $p$. If $a^p=b^p$ then $a^p-b^p=(a-b)^p=0$, so $a=b$.
| 160,515
|
There Goes The Bride
Comedy, Romance | 88 Mins | Released: 1980
Director: Terry Marcel
Starring: Tommy Smothers, Twiggy Lawson, Phil Silvers, Sylvia Sims, Jim Backus, Broderick Crawford, Martin Balsom
Our Rating: 5
Color
A nervous ad executive (Tom Smothers) creates havoc on his daughter’s wedding day and becomes obsessed with a dream girl (Twiggy) he keeps seeing everywhere but whom he can’t catch.
Movie Notes:
Review of There Goes the Bride
Its just a fantasy. It’s not the real thing.
mark.waltz from United States - 6 October 2015
Is she a ghost? A figment of his imagination? Its up to the viewer to decide in one of many supernatural comedies to come out on the wake of “Heaven Can Wait”. Twiggy isn’t the bride, but perhaps the spirit of the character that she played in “The Boyfriend”. Unlike her character of Polly and that musical based on the hit British musical, her character of Polly here is a pranksterish spirit, appearing to the father of the bride Tom Smothers on the day of his daughter’s wedding, not letting him out of her sight, and causing all sorts of havoc for him and his family.
There are a lot of guest stars, many of whom are simply window dressing for a rather mediocre comedy that is amusing, but not quite as charming as other similar films of the same genre. Smother’s former in-laws are an obnoxious group, his ex wife a total shrew, and mother-in-law Hermione Baddely a carbon copy of the old school nasty mother-in-law. Then, there is his whining daughter, perplexed by her parents bickering and her father’s seemingly insane notion that some beautiful woman is chasing him and happens to be one that nobody but him can see.
“Heaven Can Wait”, a remake of “Here Comes Mr. Jordan”, had been a smash hit several years before this had an obviously minimal release. That film had an excellent screenplay and extremely funny performances, but this film is only moderately amusing on spots. Among the guest stars that pop in and out are Phil Silvers, seemingly having no purpose there other than to add name value, Jim Backus, playing a lecherous businessman feeding line to his sexy younger secretary. When he offers her more champagne, you half expect her to respond delicious!, like his real wife Henny had in a famous comedy sketch with her husband years before. Martin Balsam plays an obnoxious relative who keeps on getting into fights with innocent passers-by, and at times I wanted to see his character totally thrown off the screen.
One very funny moment has the grandfather of the bride arriving at the wrong church, and presenting the apparent other grandparents with flowers, unaware that this Spanish couple are obviously not his granddaughters other grandparents.
Getting a bit convoluted and testy in spots, it is easy to see why this was probably shelved, and only had a few bookings. Other similar films that did better than this include “Kiss Him Goodbye, a 1982 comedy with a dead husband coming back to harass his newly remarried wife, the hysterical “All of Me” with Steve Martin and Lily Tomlin apparently sharing the same body, and the sweet “Chances Are” where Cybil Shepherd’s ex husband comes back in the form of the man played by Robert Downey who is about to marry her own daughter.
Twiggy’s character reminded me a great deal of the character that Glenn Close played in the comedy “Maxie” where an innocent church secretary is possessed by the spirit of a long dead flapper. This film has to thrive on the chemistry of its two stars, both names in the late 1960s, but barely remembered by most people when it came out.there are a few moments when they do play delightfully off of each other, but the circumstances surrounding them seem so absurd and the choreography of Struthers by himself being pulled away by the invisible Twiggy, just you not seeing as well acted as when Steve Martin get it with Lily Tomlin manipulating his every move in “All of Me”.
| 249,574
|
Assessor in Business and Administration, Essex
Closing date:
Contract type: Part Time
Contract term: Permanent
Suitable for NQTs: Yes
Chelmsford College is seeking to appoint an outstanding Work Based Tutor who has a high level of skill and expertise within business administration. In addition, experience in assessing Apprenticeship frameworks and work based learning provision is essential. The successful candidate must hold a professional qualification in their sector and an Assessors Award. A teaching qualification and IV award is desirable..
| 216,853
|
\begin{document}
\title{A Direct Proof of Schwichtenberg's \\ Bar Recursion Closure Theorem}
\author{Paulo Oliva and Silvia Steila}
\maketitle
\begin{abstract} In \cite{Schwichtenberg}, Schwichtenberg showed that the System $\T$ definable functionals are closed under a rule-like version Spector's bar recursion of lowest type levels $0$ and $1$. More precisely, if the functional $Y$ which controls the stopping condition of Spector's bar recursor is $\T$-definable, then the corresponding bar recursion of type levels $0$ and $1$ is already $\T$-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for $\alpha < \varepsilon_0$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system $\T$ input, what the corresponding system $\T$ output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into $\T$-definitions under the conditions of Schwichtenberg's theorem. Finally, with the explicit construction we can also easily state a sharper result: if $Y$ is in the fragment $\T_i$ then terms built from $\BR^{\NN, \sigma}$ for this particular $Y$ are definable in the fragment $\T_{i + \max \{ 1, \level{\sigma} \} + 2}$.
\end{abstract}
\section{Introduction}
In \cite{Goedel}, G\"{o}del interpreted intuitionistic arithmetic in a quantifier-free type theory with primitive recursion in all finite types, the so-called System $\T$. This interpretation became known as \vgt{Dialectica}, the name of the journal where it was published. The Dialectica interpretation of arithmetic was extended by Spector to classical analysis in the system \vgt{$\T +$ bar recursion} \cite{Spector(62)}.
The schema of Spector's bar recursion (for a pair of finite types $\tau, \sigma$) is defined as
\begin{equation} \label{spector-def}
\BR^{\tau, \sigma}(G, H, Y)(s) \stackrel{\sigma}{=}
\left\{
\begin{array}{ll}
G(s) & {\rm if} \; Y(\hat s) < |s| \\[2mm]
H(s)(\lambda x^\tau . \BR(G,H,Y)(s * x)) & {\rm otherwise}
\end{array}
\right.
\end{equation}
where $s \colon \tau^*$, $G \colon \tau^* \to \sigma$, $H \colon \tau^* \to (\tau \to \sigma) \to \sigma$ and $Y \colon (\NN \to \tau) \to \NN$. As usual $\hat s$ denotes the infinite extension of the finite sequence $s$ with $0$'s of appropriate type. For clarify of exposition we prefer to separate the arguments that stay fixed during the recursion, namely $G, H$ and $Y$, from the mutable argument $s$.
In \cite{Schwichtenberg}, Schwichtenberg proved that if $Y, G$ and $H$ are closed terms of system $\T$, and if $\tau$ is of type level $0$ or $1$, then the functional $\lambda s . \BR^{\tau, \sigma}(G, H, Y)(s)$ is already $\T$-definable.
Schwichtenberg's original proof is based on the notion of infinite terms as introduced by Tait \cite{Tait} and his argument requires the normalization theorem for infinite terms and the valuation functional provided in \cite{Schwichtenberg73}. Schwichtenberg proves that bar recursions of type levels $0$ and $1$ are reducible to $\alpha$-recursion for some $\alpha < \varepsilon_0$. Hence, using an interdefinability result from Tait \cite{Tait}, he concludes that they are also reducible to primitive recursions of higher types. Such detour makes it extremely difficult to work out the $\T$-definition of $\lambda G, H, s . \BR^{\tau, \sigma}(G, H, Y)(s)$ for a given concrete $\T$ definable $Y$, for instance, $Y(\alpha) = \Rec^\NN(0, \lambda k. \alpha)(\alpha(0))$, where $\alpha \colon \NN \to \NN$ and $k$ is a fresh variable.
In here we present a direct inductive proof of Schwichtenberg's result which provides an explicit method to eliminate bar recursion of type levels $0$ and $1$ when $Y$ is a concrete system $\T$ term. The focus of our result is syntactic: We describe an effective construction that given a term in $\T + \BR$, satisfying the above restrictions, will produce an equivalent term in system $\T$. We also strengthen Schwichtenberg's result by showing that when $Y$ is $\T$-definable and $\tau$ is of type level $0$ or $1$, then the functional $\lambda G, H, s . \BR^{\tau, \sigma}(G, H, Y)(s)$ is already $\T$-definable (uniformly in $G$ and $H$).
Our proof is composed of two main parts. In the first part (Section \ref{sec-general-br}) we define a variant of bar recursion which we call \emph{general bar recursion} -- a family of bar recursive functions parametrized by bar predicates. We show that when the bar predicate ``secures'' the functional $Y$, then $\BR$ for that $Y$ can be defined from the general bar recursion. In the second part (Section \ref{sec-main-result}) we present the main construction: Given a $\T$-definable $Y$, we can $\T$-define a general bar recursion for a bar predicate which secures $Y$. The construction of the term which corresponds to the given $\T + \BR$ term is syntactic, as its definition is by induction on the structure of the input term. The proof of equivalence is carried out in intuitionistic Heyting arithmetic in all finite types $\HAomega$. One can, however, also view the result model-theoretically, by looking at models of $\HAomega$. Our result establishes that restricted bar-recursive terms have a denotation which falls within the subset of $\T$-definable elements.
\subsection{Spector's bar recursion}
The finite types are defined inductively, where $\NN$ is the basic finite type, $\tau_0 \to \tau_1$ is the type of functions from $\tau_0$ to $\tau_1$, and $\tau_0^*$ is the type of finite sequences whose elements are of type $\tau_0$. Note that we have, for convenience, enriched the type system with the type of finite sequences. As usual, we often write $\tau_1^{\tau_0}$ for the type $\tau_0 \to \tau_1$.
System $\T$ \cite{Goedel,Spector(62)} consists of the simply typed $\lambda$-calculus with natural numbers ($0$ and $\Suc$) and the recursor $\Rec^\rho$, for each finite type $\rho$, together with the associated equations:
\begin{equation}
\Rec^{\rho}(a, f)(n) \stackrel{\rho}{=}
\left\{
\begin{array}{ll}
a & {\rm if} \; n = 0 \\[2mm]
f(m,\Rec^\rho(a, f)(m)) & {\rm if} \; n = \Suc(m)
\end{array}
\right.
\end{equation}
where $a \colon \rho$ and $f \colon \NN \to \rho \to \rho$. When translating bar recursive terms into system $\T$ terms we will also make use of a definitional extension of $\T$ with finite products $\tau \times \sigma$. When $s \colon \tau$ and $t \colon \sigma$ we write $\apc{s}{t}$ for the element of type $\tau \times \sigma$.
As usual, $\NN$ has type level $0$; the type level of $\rho \to \eta$ is the maximum between the type level of $\rho$ plus $1$ and the type level of $\eta$; the type level of $\tau \times \sigma$ is the maximum between the type level of $\tau$ and the type level of $\sigma$; the type level of $\tau^*$ is the type level of $\tau$. We write $\level{\tau}$ for the type level of $\tau$. The fragment of $\T$ where the recursor $\Rec^\rho$ is restricted to types $\rho$ with $\level{\rho} \leq i$ is denoted $\T_i$.
\begin{definition}[Spector's bar recursion] For each pair of types $\tau, \sigma$, let $\SpectorEq^{\tau, \sigma}$ be the universal formula
\begin{equation*} \label{spector-def-eq}
\SpectorEq^{\tau, \sigma}(\xi,G,H,Y) \; \eqdef \; \forall s^{\tau^*}
\left\{
\begin{array}{lcl}
Y(\hat s) < |s| & \to & \xi(G, H, Y)(s) \stackrel{\sigma}{=} G(s) \\[1mm]
& \wedge& \\[1mm]
Y(\hat s) \geq |s| & \to & \xi(G, H, Y)(s) \stackrel{\sigma}{=} H(s)(\lambda x^\tau . \xi(G,H,Y)(s * x))
\end{array}
\right\}
\end{equation*}
where
\[ \xi \colon (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to (\tau^\NN \to \NN) \to \tau^* \to \sigma \]
The extension of system $\T$ with Spector's bar recursion consists of adding to the language of $\T$ a family of constants $\BR^{\tau, \sigma}$, for each pair of finite types $\tau, \sigma$, together with the defining axioms $\forall G, H, Y \, \SpectorEq^{\tau,\sigma}(\BR^{\tau,\sigma},G,H,Y)$. We speak of Spector bar recursion of type level $i$ when $\tau$ has type level $i$.
\end{definition}
When we omit an argument of $\SpectorEq^{\tau, \sigma}(\xi,G,H,Y)$ we will assume it is universally quantified, e.g.
\[
\begin{array}{lcl}
\SpectorEq^{\tau,\sigma}(\xi, G) & \eqdef & \forall H, Y \, \SpectorEq^{\tau,\sigma}(\xi,G,H,Y) \\[2mm]
\SpectorEq^{\tau,\sigma}(\xi) & \eqdef & \forall G, H, Y \, \SpectorEq^{\tau,\sigma}(\xi,G,H,Y)
\end{array}
\]
We will also use \emph{named parameters} in order to fix a particular parameter of $\xi$, e.g. if $t$ is a term having the same type as $Y$ then $\SpectorEq^{\tau,\sigma}(\xi, Y=t)$ stands for the formula
\[
\forall G, H \, \forall s^{\tau^*}
\left\{
\begin{array}{lcl}
t(\hat s) < |s| & \to & \xi(G, H)(s) \stackrel{\sigma}{=} G(s) \\[1mm]
& \wedge& \\[1mm]
t(\hat s) \geq |s| & \to & \xi(G, H)(s) \stackrel{\sigma}{=} H(s)(\lambda x^\tau . \xi(G,H)(s * x))
\end{array}
\right\}
\]
where we replace $Y$ by $t$ and omit the argument $Y$ from $\xi$. Finally, when clear from the context we will omit the superscript types, and write simply $\SpectorEq$.
\begin{remark}[Related work] A previous analysis by Kreisel (see e.g. \cite{Spector(62)}), together with the reduction provided by Howard \cite{Howard1}, guarantees that system $\T$ is not closed under the bar recursion rule when $\tau$ has type level greater or equal to $2$. Diller \cite{Diller} presented a reduction of bar recursion to $\alpha$-recursion for some bounded ordinal $\alpha$, while Howard \cite{Howard2, Howard3} provided an ordinal analysis of the constant of bar recursion of type level $0$. Kreuzer \cite{Kreuzer12} refined Howard's ordinal analysis of bar recursion in terms of Grzegorczyk's hierarchy. In \cite{Kohlenbach99}, Kohlenbach generalised Schwichtenberg's result by showing that if $Y[\vec{x},\vec{f}] \colon \NN$ is a term with variables $\vec{x}$ of type level $0$ and $\vec{f}$ of type level $1$, then the bar recursive functional of type level $0$ provided by $Y$, is $\T$-definable. Kohlenbach's argument is based on the observation that in Schwichtenberg's result no restrictions are put on the type $\sigma$, hence it is possible to relativize Schwichtenberg's proof where $Y$ is allowed to contains parameters of type levels $0$ and $1$ in system $\T$. The same argument can be carried over to our construction below.
\end{remark}
\begin{notation} Throughout the paper we adopt the following conventions:
\begin{itemize}
\item We use $\tau, \sigma, \rho, \eta$ to denote finite types.
\item We write $a \colon \tau$ or $a^\tau$ to indicate that $a$ is a term of type $\tau$.
\item A tuple of variables $x_1, \ldots, x_n$ will be denoted by $\vec{x}$.
\item The term $0^\tau$ denotes the standard inductively defined zero object of type $\tau$.
\item Given a finite sequence $s \colon \tau^*$, $\hat{s} \colon \NN \to \tau$ denotes the extension of $s$ with infinitely many $0^\tau$.
\item For any finite sequence $s \colon \tau^*$ and any $x \colon \tau$, $s*x$ denotes appending $x$ to $s$.
\item For any finite sequences $s, s' \colon \tau^*$, $s*s'$ denotes their concatenation.
\item Given $s \colon \tau^*$ and an infinite sequence $\alpha \colon \tau^\NN$, we also write $s*\alpha$ to denote their concatenation.
\item For any infinite sequence $\alpha \colon \NN \to \tau$, $\bar{\alpha}n$ denotes the finite sequence $\ap{\alpha(0), \dots, \alpha(n-1)}$. We also use the same notation for finite sequences $s \colon \tau^*$ when $n \leq |s|$.
\end{itemize}
\end{notation}
\section{General Bar Recursion}
\label{sec-general-br}
Let us start by observing that if $Y \colon (\NN \to \NN) \to \NN$ is a constant function then bar recursion for such $Y$ is $\T$-definable, for any types $\tau, \sigma$.
\begin{lemma}[$\HAomega$] \label{constant-lemma} For each $\tau, \sigma$, let $i = \max\{1 + \level{\tau}, \level{\sigma}\}$, there is a closed term
\[ \Psi \colon \NN \to (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma \]
in $\T_i$ such that for all $k \colon \NN$ we have $\SpectorEq(\Psi(k), Y = \lambda \alpha . k)$.
\end{lemma}
\begin{proof} We define a term $\Psi$ and show it satisfies
\[
\Psi(k)(G, H)(s) \stackrel{\sigma}{=}
\left\{
\begin{array}{ll}
G(s) & {\rm if} \; |s| > k \\[2mm]
H(s)(\lambda x^\tau . \Psi(k)(G,H)(s * x)) & {\rm if} \; |s| \leq k
\end{array}
\right.
\]
for all $k, G, H$ and $s$. First define the functional
\[
\varphi \colon (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \NN \to \tau^* \to \sigma
\]
by primitive recursion as
\begin{equation} \label{varphi-def}
\varphi(G, H)(n) \eqdef
\left\{
\begin{array}{ll}
G & {\rm if} \; n = 0 \\[2mm]
\lambda s^{\tau^*} . H(s)(\lambda x^\tau . \varphi(G,H)(n-1)(s * x)) & {\rm if} \; n > 0.
\end{array}
\right.
\end{equation}
Then, using $\varphi$, define the functional $\Psi$ by cases as
\begin{equation} \label{Psi-def}
\Psi(k)(G, H)(s) \eqdef
\left\{
\begin{array}{ll}
G(s) & {\rm if} \; |s| > k \\[2mm]
\varphi(G, H)(k + 1 - |s|)(s) & {\rm if} \; |s| \leq k.
\end{array}
\right.
\end{equation}
Clearly the functional $\Psi$ is $\T$-definable, and only requires primitive recursion of type $\tau^* \to \sigma$, so it is in fact definable in $\T_i$ for $i = \max\{1 + \level{\tau}, \level{\sigma}\}$. It remains for us to prove that $\Psi(k)(G,H)$ satisfies the above mentioned equation. \\[1mm]
Let $k, G, H$ and $s$ be fixed.
If $|s| > k$ then
\[ \Psi(k)(G, H)(s) \stackrel{(\ref{Psi-def})}{=} G(s). \]
When $|s| \leq k$, we distinguish two cases. If $(\dagger) \, |s| = k$ then
\[
\begin{array}{lcl}
\Psi(k)(G,H)(s)
& \stackrel{(\ref{Psi-def})}{=} & \varphi(G,H)(k+1-|s|)(s) \\
& \stackrel{(\dagger)}{=} & \varphi(G,H)(1)(s) \\
& \stackrel{(\ref{varphi-def})}{=} & H(s)(\lambda x^\tau . \varphi(G,H)(0)(s * x)) \\
& \stackrel{(\ref{varphi-def})}{=} & H(s)(\lambda x^\tau . G(s * x)) \\
& \stackrel{(\ref{Psi-def})}{=} & H(s)(\lambda x^\tau . \Psi(k)(G,H)(s * x)).
\end{array}
\]
If $|s| < k$, we have
\[
\begin{array}{lcl}
\Psi(k)(G,H)(s)
& \stackrel{(\ref{Psi-def})}{=} & \varphi(G,H)(k+1-|s|)(s) \\
& \stackrel{(\ref{varphi-def})}{=} & H(s)(\lambda x. \varphi(G,H)(k+1-|s|-1)(s*x)) \\[1mm]
& = & H(s)(\lambda x. \varphi(G,H)(k+1-|s*x|)(s*x)) \\
& \stackrel{(\ref{Psi-def})}{=} & H(s)(\lambda x. \Psi(k)(G,H)(s*x)).
\end{array}
\]
\end{proof}
A predicate $S(s^{\tau^*})$ is called a \emph{bar} if it satisfies the following three conditions:
\begin{itemize}
\item[$(i)$] \emph{Decidable}: $\forall s^{\tau^*} (S(s) \vee \neg S(s))$
\item[$(ii)$] \emph{Bar}: $\forall \alpha^{\tau^\NN} \exists n^\NN S(\bar{\alpha} n)$
\item[$(iii)$] \emph{Monotone}: $\forall s^{\tau^*}, t^{\tau^*} (S(s) \to S(s * t))$
\end{itemize}
We now introduce a variant of Spector's bar recursion, which we call \emph{general bar recursion}. These are parametrized by a bar predicate $S(s^{\tau^*})$.
\begin{definition}[General bar recursion] For each pair of types $\tau, \sigma$, and a bar predicate $S(s^{\tau^*})$, let $\GeneralEq{S}^{\tau, \sigma}$ be the formula
\begin{equation} \label{secure-def}
\GeneralEq{S}^{\tau, \sigma}(\xi,G,H) \; \eqdef \; \forall s^{\tau^*}
\left\{
\begin{array}{lcl}
S(s) & \to & \xi(G, H)(s) \stackrel{\sigma}{=} G(s) \\[1mm]
& \wedge& \\[1mm]
\neg S(s) & \to & \xi(G, H)(s) \stackrel{\sigma}{=} H(s)(\lambda x^\tau . \xi(G,H)(s * x))
\end{array}
\right\}
\end{equation}
where $\xi \colon (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma$.
\end{definition}
When clear from the context we will omit the superscript types, writing simply $\GeneralEq{S}$ instead of $\GeneralEq{S}^{\tau,\sigma}$. And once again, we write $\GeneralEq{S}(\xi)$ as a shorthand for $\forall G, H \, \GeneralEq{S}(\xi,G,H)$.
\begin{definition} We say that a bar $S$ \emph{secures $Y \colon \tau^\NN \to \NN$} if for all $s^{\tau^*}$
\[ S(s) \quad \Rightarrow \quad \mbox{$\lambda \beta. Y(s*\beta)$ is constant.} \]
\end{definition}
\begin{theorem}[$\HAomega$] \label{secure-br-thm} Let $\tau, \sigma$ be fixed, and $i = \max\{1 + \level{\tau}, \level{\sigma}\}$. Let also $t \colon \tau^\NN \to \NN$ be a fixed closed term in $\T_i$. There is a $\T_i$-term $\Phi^t$ such that for any bar $S$ securing $t$
\[ \GeneralEq{S}(\Delta) \quad \Rightarrow \quad \SpectorEq(\Phi^t(\Delta), Y = t) \]
\end{theorem}
\begin{proof} Let $t$ be fixed and assume ($\dagger$) $S$ is a bar securing $t$. First, define the construction
\[ {\mathcal H}^t \colon (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to (\tau \to \sigma) \to \sigma \]
as
\begin{equation} \label{H-eq}
{\mathcal H}^t(G,H)(s)(f^{\tau \to \sigma}) \eqdef
\begin{cases}
G(s) & {\rm if} \; t(\hat s) < |s| \\[2mm]
H(s)(f) & {\rm otherwise},
\end{cases}
\end{equation}
and let $\Phi^t$ be the $\T_i$-definable term:
\begin{equation} \label{Phi-Y-eq}
\Phi^t(\Delta)(G, H)(s) \eqdef \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s)
\end{equation}
where $\Psi$ is the construction given in the proof of Lemma \ref{constant-lemma} and $\Delta$ has type $$(\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma.$$
Suppose $\Delta$ is such that $(\ddagger) \; \GeneralEq{S}(\Delta)$. We must show $\SpectorEq(\Phi^t(\Delta),Y = t)$.
First we must show that if $t(\hat s) < |s|$ then $\Phi^t(\Delta) = G(s)$. So we assume $t(\hat s) < |s|$ and consider two cases (using the decidability of the bar): \\[1mm]
If $S(s)$ then
\eqleft{
\begin{array}{lcl}
\Phi^t(\Delta)(G, H)(s)
& \stackrel{(\ref{Phi-Y-eq})}{=} & \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s) \\
& \stackrel{(\ddagger)}{=} & \Psi(t \hat{s})(G, H)(s) \\
& \stackrel{(\textup{L}\ref{constant-lemma})}{=} & G(s)
\end{array}
}
whereas if $\neg S(s)$, then
\eqleft{
\begin{array}{lcl}
\Phi^t(\Delta)(G, H)(s)
& \stackrel{(\ref{Phi-Y-eq})}{=} & \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s) \\
& \stackrel{(\ddagger)}{=} & {\mathcal H}^t(G, H)(s)(\lambda x . \Phi^t(\Delta)(G,H)(s*x)) \\
& \stackrel{(\ref{H-eq})}{=} & G(s).
\end{array}
}
Secondly, we must show that when $t(\hat s) \geq |s|$ then $$\Phi^t(\Delta)(G,H)(s) = H(s)(\lambda x . \Phi^t(\Delta)(G,H)(s * x)).$$
Again we assume $t(\hat s) \geq |s|$ and consider two cases: \\[1mm]
If $S(s)$ then, by our assumption ($\dagger$), $\lambda \beta. t(s*\beta)$ is constant, and in particular $(*)$ $t(\widehat{s*x}) = t(\hat{s})$. By monotonicity of the bar we also have $S(s * x)$. Hence
\eqleft{
\begin{array}{lcl}
\Phi^t(\Delta)(G,H)(s)
& \stackrel{(\ref{Phi-Y-eq})}{=} & \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s), {\mathcal H}^t(G, H))(s) \\
& \stackrel{(\ddagger)}{=} & \Psi(t(\hat{s}))(G, H)(s) \\
& \stackrel{\textup{L}\ref{constant-lemma}}{=} & H(s)(\lambda x . \Psi(t(\hat{s}))(G, H)(s * x)) \\
& \stackrel{(*)}{=} & H(s)(\lambda x . \Psi(t(\widehat{s*x}))(G, H)(s * x)) \\
& \stackrel{(\ddagger)}{=} & H(s)(\lambda x . \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s*x)) \\
& \stackrel{(\ref{Phi-Y-eq})}{=} & H(s)(\lambda x . \Phi^t(\Delta)(G,H)(s * x)).
\end{array}
}
Otherwise, if $\neg S(s)$ then
\eqleft{
\begin{array}{lcl}
\Phi^t(\Delta)(G,H)(s)
& \stackrel{(\ref{Phi-Y-eq})}{=} & \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s) \\
& \stackrel{(\ddagger)}{=} & {\mathcal H}^t(G,H)(s)(\lambda x . \Delta(\lambda s' . \Psi(t(\widehat{s'}))(G, H)(s'), {\mathcal H}^t(G, H))(s*x)) \\
& \stackrel{(\ref{Phi-Y-eq})}{=} & {\mathcal H}^t(G,H)(s)(\lambda x . \Phi^t(\Delta)(G,H)(s*x)) \\
& \stackrel{(\ref{H-eq})}{=} & H(s)(\lambda x . \Phi^t(\Delta)(G,H)(s*x)).
\end{array}
}
\end{proof}
\section{Main Result}
\label{sec-main-result}
We have just shown that Spector's bar recursion, when $Y$ is a fixed $\T$-term $t$, is $\T$-definable in the general bar recursion for any predicate $S$ securing $t$. We will now prove that for $\tau=\NN$ or $\tau= \NN \to \NN$ and for any fixed term $t[\alpha]$, there exists some predicate $S$ securing the closed term $\lambda \alpha . t[\alpha]$ such that there is a $\T$-definable functional which satisfies the general bar recursion equation $\GeneralEq{S}$. For the rest of the section, let $\tau$ and $\sigma$ be fixed.
\begin{definition} For each finite type $\eta$ we associate inductively a new finite type $\eta^\circ$ as:
\[
\begin{array}{lcl}
\NN^\circ & = & (\tau^\NN \to \NN) \times ((\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma) \\[2mm]
(\rho_0 \to \rho_1)^\circ & = & \rho_0^\circ \to \rho_1^\circ
\end{array}
\]
\end{definition}
Since terms $t$ of type $\NN^\circ$ in fact consist of a pair of functionals, we will use the terminology
\begin{itemize}
\item $\Val_t \colon \tau^\NN \to \NN$ for the first component of $t$, and
\item $\BRSec_t \colon (\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma$ for the second component,
\end{itemize}
so that $t = \apc{\Val_t}{\BRSec_t}$.
\begin{lemma} \label{type-level-lemma} $\level{\eta^\circ} = 2 + \max \{ 1 + \level{\tau}, \level{\sigma} \} + \level{\eta}$.
\end{lemma}
\begin{proof} By induction on the structure of $\eta$.
\begin{itemize}
\item $\eta = \NN$. First notice that the type level of $\NN^\circ$ is dictated by the component $\tau^* \to (\tau \to \sigma) \to \sigma$. Since $$\level{\tau^* \to (\tau \to \sigma) \to \sigma} = \max \{ 2 + \level{\tau}, 1 + \level{\sigma} \}$$ we have that
\[ \level{(\tau^* \to \sigma) \to (\tau^* \to (\tau \to \sigma) \to \sigma) \to \tau^* \to \sigma} = 2 + \max \{ 1 + \level{\tau}, \level{\sigma} \} \]
so $\level{\NN^\circ} = 2 + \max \{ 1 + \level{\tau}, \level{\sigma} \} + \level{\NN}$.
\item $\eta= \rho_0 \to \rho_1$. By definition
$\level{\eta^{\circ}} = \max\{ 1 + \level{\rho_0^{\circ}}, \level{\rho_1^{\circ}}\}$.
By induction hypothesis for $i=0,1$ we have
$$\level{\rho_i^{\circ}} = 2 + \max \{ 1+\level{\tau} , \level{\sigma} \} + \level{\rho_i}.$$
Therefore
\[
\begin{array}{lcl}
\level{\eta^{\circ}} & = & \max\{ 1 + \level{\rho_0^{\circ}}, \level{\rho_1^{\circ}}\} \\
& = & 2 + \max \{ 1+\level{\tau} , \level{\sigma} \} + \max\{1+ \level{\rho_0}, \level{\rho_1}\}\\
& = & 2 + \max \{ 1+\level{\tau} , \level{\sigma} \} + \level{\eta}.
\end{array}
\]
\end{itemize}
\end{proof}
\subsection{Translation (case $\tau = \NN$)}
For the rest of this sub-section we shall also assume that $\tau = \NN$, and that $\alpha$ is a special variable of type $\NN \to \NN$. In Section \ref{sec-case-fct} we describe which small changes need to be made to treat the case $\tau = \NN \to \NN$. Moreover, we assume $\sigma$ to be an arbitrary but fixed finite type.
Given a term $t \colon \NN$ with the special variable $\alpha$ as the only free variable, our goal is to define a term $t^\circ \colon \NN^\circ$ in such a way that $\Val_{t^\circ} = \lambda \alpha . t$, allowing us to evaluate $t$ for concrete values of $\alpha$, and $\BRSec_{t^\circ}$ will be such that $\GeneralEq{S}(\BRSec_{t^\circ})$, for some bar $S$ which secures $\lambda \alpha . t$. For a term $t$ of a higher-type we will define $t^\circ$ in such a way that this property is preserved at ground type.
\begin{definition} \label{circ-def} Let $\Psi(k)$ be the $\T$-term defined in the proof of Lemma \ref{constant-lemma} (defining bar recursion in the special case when $Y$ is the constant functional $\lambda \alpha . k$). Assume a given mapping of variables $x \colon \eta$ to variables $x^\circ \colon \eta^\circ$, and let $\alpha$ be a special variable of type $\NN \to \tau$, where in this section $\tau$ is assume to be $\NN$. For any term $t \colon \rho$ in system $\T$, define $t^\circ \colon \rho^\circ$ inductively as follows:
\[
\begin{array}{lcl}
0^\circ & \eqdef & \apc{\lambda \alpha. 0}{\lambda G,H.G} \\[2mm]
\Suc^\circ & \eqdef & \lambda x^{\NN^\circ} . \apc{\lambda \alpha . \Suc(\Val_x(\alpha))}{\BRSec_x}\\[2mm]
\alpha^\circ & \eqdef & \lambda x^{\NN^\circ} . \apc{\lambda \alpha . \alpha(\Val_x(\alpha))}{\lambda G, H. \BRSec_x(\lambda s' . \Psi(\Val_x(\widehat{s'}))(G,H)(s'),H)} \\[2mm]
(x^\eta)^\circ & \eqdef & x^\circ \\[2mm]
(\lambda x^\eta . t)^\circ & \eqdef &\lambda x^\circ . t^\circ \\[2mm]
(u v)^\circ & \eqdef & u^\circ v^\circ \\[2mm]
(\Rec^\eta)^\circ & \eqdef & \lambda a^{\eta^\circ}, F^{\NN^\circ \to \eta^\circ \to \eta^\circ}, x^{\NN^\circ}, v^{\rho^\circ} . \apc{\lambda \alpha . \Val_{r[\Val_x(\alpha)]}(\alpha)}{{\sf B}}
\end{array}
\]
where in the case of the $\Rec^{\eta}$ we assume $\eta = \rho \to \NN$, and $r[n]$ and ${\sf B}$ are built from $a, F, x$ and $v$ as
\begin{itemize}
\item $r[n] \eqdef \Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n)(v)$
\item ${\sf B}(G,H)(s) \eqdef \BRSec_x(\lambda s' . \BRSec_{r[\Val_x(\widehat{s'})]}(G,H)(s'),H)(s)$
\end{itemize}
using the abbreviation $k^\circ \eqdef \apc{\lambda \alpha. k}{\lambda G, H .G}$ in the definition of $r[n]$. If $\eta = \NN$ then we may omit $\rho$ and the variable $v^{\rho^\circ}$, and should define $r[n] \eqdef \Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n)$.
\end{definition}
Note that if $t:\NN$ has variables $\alpha$ and $x_1,\ldots,x_n$ free, then $t^\circ$ will only have $x_1^\circ, \ldots, x_n^\circ$ free.
\subsection{Verification}
We will now show that for any term $t[\alpha] \colon \NN$, the second component of $(t[\alpha])^\circ$, i.e. $\BRSec_{(t[\alpha])^\circ}$, is a term in system $\T$ which defines a general bar recursion for some bar predicate $S$ which secures $\lambda \alpha . t[\alpha]$.
\begin{theorem}[$\HAomega + \AC_0$] \label{main-theorem} Let $\tau = \NN$ and $t \colon \NN$ be a term of system $\T$ with only $\alpha^{\NN \to \tau}$ as free variable. Then there exists a bar $S$ which secures $\lambda \alpha . t$ such that $\GeneralEq{S}(\BRSec_{t^\circ})$.
\end{theorem}
\begin{proof}
Let $\sim_\rho \, \subseteq \, \rho^\circ \times (\NN^\NN \to \rho)$ be the logical relation between terms of system $\T$ defined as:
\[
\begin{array}{ccc}
f^{\NN^\circ} \sim_{\NN} g^{\NN^\NN \to \NN} & \eqdef & \Val_f = g \,\wedge\, \exists S (S \mbox{ is a bar securing } g \mbox{ and } \GeneralEq{S}(\BRSec_f)) \\[2mm]
f^{\rho_0^\circ \to \rho_1^\circ} \sim_{\rho_0 \to \rho_1} g^{\NN^\NN \to (\rho_0 \to \rho_1)} & \eqdef &
\forall x^{\rho_0^\circ} \forall y^{\NN^\NN \to \rho_0} (x \sim_{\rho_0} y \implies f(x) \sim_{\rho_1} \lambda \alpha . g(\alpha)(y \alpha))
\end{array}
\]
We prove that for any $t$ with free variables $\vec{x}$ and (possibly) $\alpha$, $(\lambda \vec{x}.t)^\circ \sim_\rho \lambda \alpha \lambda \vec{x} . t$ by structural induction over $t$, where $\rho$ is the type of $\lambda \vec{x}.t$.
\begin{itemize}
\item $t = 0$. We need to show that
\[
0^\circ = \apc{\lambda \alpha. 0}{\lambda G,H, s . G(s)} \sim_{\NN} \lambda \alpha. 0.
\]
Clearly we have $\Val_{0^\circ} = \lambda \alpha . 0$. Let $S(s) \eqdef {\sf true}$, which is a bar securing $\lambda \alpha . 0$. Then, indeed we also have $\GeneralEq{S}(\BRSec_{0^\circ})$ since
\[
\BRSec_{0^\circ}(G,H)(s) = G(s).
\]
\item $t= \Suc$. Let us show that $\Suc^\circ \sim_{\NN \to \NN} \lambda \alpha . \Suc$, i.e. for all $x \colon \NN^\circ$ and $g^{\NN^\NN \to \NN}$
\[ x \sim_\NN g \quad \implies \quad \Suc^\circ(x) \sim_\NN \lambda \alpha . \Suc(g \alpha) \]
The premise ensures that $\Val_x = g$ and $\GeneralEq{S_x}(\BRSec_x)$ for some bar $S_x$ securing $g$. Hence, assuming the premise, and unfolding the definition of $\Suc^\circ$, we need to show
\[ \apc{\lambda \alpha . \Suc(g \alpha)}{\BRSec_x} \sim_\NN \lambda \alpha . \Suc(g \alpha). \]
The only non-trivial part is to observe that if $S_x$ secures $g$ then it also secures $\lambda \alpha . \Suc(g \alpha)$.
\item $t = z^\rho$. When $t$ is simply a free-variable $z$ we must show that $(\lambda z . z)^\circ \sim_{\rho \to \rho} \lambda \alpha \lambda z . z$. But this follows directly from the definition of $\sim_{\rho \to \rho}$, noticing that $(\lambda z^\rho . z)^\circ \eqdef \lambda z^{\rho^\circ} . z$
\item $t = \alpha$. We need to show that $\alpha^\circ \sim_{\NN \to \NN} \lambda \alpha . \alpha$, i.e. for all $x^{\NN^\circ}$ and $g^{\NN^\NN \to \NN}$
\[ x \sim_\NN g \quad \implies \quad \alpha^\circ(x) \sim_\NN \lambda \alpha . \alpha(g \alpha). \]
Again, the premise $x \sim_\NN g$ implies that $\Val_x = g$ and $\GeneralEq{S_x}(\BRSec_x)$, for some bar $S_x$ securing $g$. Hence, fix $x$ and $g$ such that $x \sim_\NN g$. Unfolding the definition of $\alpha^\circ$, we show
\[ \apc{\lambda \alpha . \alpha (g \alpha)}{\lambda G,H, s . \BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G, H)(s'),H)(s)} \sim_\NN \lambda \alpha . \alpha(g \alpha) \]
where $\Psi(g(\widehat{s'}))(G,H)$ is the $\T$-definition of $\BR^{\tau, \sigma}(G,H,\lambda \alpha. g(\widehat{s'}))$ (cf. Lemma \ref{constant-lemma}).
The first conjunct of the definition of $\sim_\NN$ is trivially satisfied. Let $$S(s) \eqdef S_x(s) \wedge g \hat{s} < |s|.$$ Since $S_x(s)$ is a bar, and $S_x$ secures $g$, it follows that $S(s)$ is also a bar. Moreover, since $S_x$ secures $g$, it also follows that $S$ secures $\lambda \alpha . \alpha(g \alpha)$. Using the hypothesis $(\dagger) \, \GeneralEq{S_x}(\BRSec_x)$, we need to show
\[ \GeneralEq{S}(\lambda G, H. \BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G,H)(s'),H)). \]
Fix $G, H$ and $s$. Consider two cases: \\[1mm]
If $S(s)$ then $S_x(s)$ and $g \hat s < |s|$. In this case we trivially have
\[ \BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G,H)(s'),H)(s) \stackrel{(\dagger)}{=} \Psi(g(\hat{s}))(G, H)(s) = G(s) \]
If $\neg S(s)$ then either $\neg S_x(s)$ or $g \hat s \geq |s|$. We consider two cases: \\[1mm]
If $S_x(s)$ holds then $g \hat s \geq |s|$.
Moreover, $(\ddagger) \, g \hat{s} = g (\widehat{s * y})$ for any $y$, since $S_x$ secures $g$. By monotonicity of $S_x$ we also have $S_x(s*y)$ for any $y$.
Hence
\begin{align*}
\BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G, H)(s'),H)(s)
&\stackrel{(\dagger)}{=} \Psi(g(\hat{s}))(G, H)(s) \\[1mm]
&= H(s)(\lambda y . \Psi(g(\hat{s}))(G, H)(s * y)) \\
&\stackrel{(\ddagger)}{=} H(s)(\lambda y . \Psi(g(\widehat{s*y}))(G, H)(s * y)) \\
&\stackrel{(\dagger)}{=} H(s)(\lambda y . \BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G, H)(s'),H)(s*y))
\end{align*}
If $\neg S_x(s)$ then
\begin{align*}
\BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G, H)(s'),H)(s)
&\stackrel{(\dagger)}{=} H(s)(\lambda y . \BRSec_x(\lambda s' . \Psi(g(\widehat{s'}))(G, H)(s'),H)(s*y))
\end{align*}
\item $t = \lambda x^\rho. u$. Trivial by induction hypothesis.
\item $t = u^{\rho \to \tau} v^\rho$. For simplicity let us assume $u$ has free-variables $x_1^{\sigma_1}$ and $x_2^{\sigma_2}$ and $v$ has free-variables $x_2^{\sigma_2}$ and $x_3^{\sigma_3}$, which is enough to illustrate how the difference in the set of free-variables of $u$ and $v$ is handled. We must show that
$$(\lambda x_1,x_2,x_3 . u v)^\circ \sim_{(\sigma_1\times\sigma_2\times\sigma_3) \to \tau} \lambda \alpha \lambda x_1,x_2,x_3. u v.$$
By the definition of $(\cdot)^\circ$ this is
\[ \lambda x_1^\circ, x_2^\circ, x_3^\circ. u^\circ v^\circ \sim_{(\sigma_1\times\sigma_2\times\sigma_3) \to \tau} \lambda \alpha \lambda x_1,x_2,x_3 . u v \]
By induction hypothesis we have $\lambda x_1^\circ, x_2^\circ. u^\circ \sim_{(\sigma_1\times \sigma_2) \to (\rho \to \tau)} \lambda \alpha, x_1, x_2. u$, i.e. for all $x_1^\circ$,$\tilde{x}_1^{\NN^\NN \to \sigma_1}$, $x_2^{\circ}$,$\tilde{x}_2^{\NN^\NN \to \sigma_2}$ and $y^{\circ}, \tilde{y}^{\NN^\NN \to \rho}$
\[
x_1^\circ \sim_{\sigma_1} \tilde{x}_1 \wedge x_2^\circ \sim_{\sigma_2} \tilde{x}_2 \wedge y^\circ \sim_\rho \tilde{y}
\quad \implies \quad u^\circ y^\circ \sim_{\tau} \lambda \alpha . u[\tilde{x}_1 \alpha/x_1][\tilde{x}_2 \alpha/x_2](\tilde{y} \alpha),
\]
and $\lambda x_2^\circ,x_3^\circ. v^\circ \sim_{(\sigma_2 \times \sigma_3) \to \rho} \lambda \alpha, x_2,x_3. v$, i.e. for all $x_2^\circ$, $\tilde{x}_2^{\NN^\NN \to \sigma_2}$, $x_3^\circ$ and $\tilde{x}_2^{\NN^\NN \to \sigma_3}$
\[ x_2^\circ \sim_{\sigma_2} \tilde{x}_2 \wedge x_3^\circ \sim_{\sigma_3} \tilde{x}_3 \implies v^\circ \sim_{\rho} \lambda \alpha . v[\tilde{x}_2 \alpha/x_2][\tilde{x}_3 \alpha/x_3]. \]
Therefore given for any $j \in \bp{1,2,3}$ $x_j^\circ$ and $ \tilde{x}_j^{\NN^\NN \to \sigma_j}$ such that $x_j \sim_{\sigma_j} \tilde{x}_j$, we have
\[ v^\circ \sim_{\rho} \lambda \alpha . v[\tilde{x}_2 \alpha/x_2][\tilde{x}_3 \alpha/x_3] \]
which we can plug into the first induction hypothesis to obtain
\[
\begin{array}{lcl}
u^\circ v^\circ & \sim_\tau & \lambda \alpha . u[\tilde{x}_1 \alpha/x_1][\tilde{x}_2 \alpha/x_2](v[\tilde{x}_2 \alpha/x_2][\tilde{x_3} \alpha/x_3])) \\[2mm]
& = & (\lambda \alpha \lambda x_1,x_2,x_3. u v)(\alpha)(\tilde{x}_1 \alpha)(\tilde{x}_2 \alpha)(\tilde{x}_3 \alpha).
\end{array}
\]
\item $t = \Rec^{\eta}$. Without loss of generality we can assume that the recursor has type $\eta = \rho \to \NN$ for some type $\rho$. It is easy to check that $k^\circ \sim_\NN \lambda \alpha . k$, for any variable $k \colon \NN$, where $k^\circ$ is the abbreviation introduced at the end of Definition \ref{circ-def}. \\[1mm]
Assume $x \sim_{\NN} g$, $a \sim_\eta A$, $F \sim_{\NN \to \eta \to \eta} \psi$, $v \sim_\rho V$. We must show that
\[ (\Rec^\eta)^\circ(a,F)(x)(v) \sim_\NN \lambda \alpha. \Rec^\eta(A \alpha, \psi \alpha)(g \alpha)(V \alpha). \]
Or, unfolding the definition of $(\Rec^\eta)^\circ$, that
\[ \apc{\lambda \alpha . \Val_{r[\Val_x(\alpha)]}(\alpha)}{{\sf B}} \sim_\NN \lambda \alpha. \Rec^\eta(A \alpha, \psi \alpha)(g \alpha)(V \alpha) \]
where $r[n]$ and ${\sf B}$ are as in Definition \ref{circ-def}. Again we note that the premise $x \sim_\NN g$ implies that $\Val_{x} = g$ and $(\dagger) \, \GeneralEq{S_x}(\BRSec_{x})$, for a bar $S_x$ securing $g$. \\[1mm]
{\bf Claim 1}. For all $n^\NN$, $\Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n) \sim_\eta \lambda \alpha. \Rec^\eta(A \alpha, \psi \alpha)(n)$. \\[1mm]
{\bf Proof}. By induction on $n$. If $n=0$, since $a \sim_{\eta} A$, then
\begin{align*}
\Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(0) \eqdef a \sim_\eta A \eqdef \lambda \alpha. \Rec^\eta(A \alpha,\psi \alpha)(0).
\end{align*}
For $n>0$, by induction hypothesis we have,
\[ \Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n-1) \sim_\eta \lambda \alpha. \Rec^{\eta}(A \alpha, \psi \alpha)(n-1) \]
Since $F \sim_{\NN \to \eta \to \eta} \psi$ and $(n-1)^\circ \sim_{\NN} \lambda \alpha . n - 1$, we have that for all $b \sim_\eta B$
\[ F((n-1)^\circ,b) \sim_\eta \lambda \alpha . \psi(\alpha)(n - 1, B \alpha). \]
Hence:
\begin{align*}
\Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n)
&\eqdef F((n-1)^\circ,\Rec^{\eta^\circ}(a, \lambda k^\NN . F(k^\circ))(n-1)) \\
& \sim_\eta \lambda \alpha . \psi(\alpha)(n - 1, \Rec^\eta(A \alpha, \psi \alpha)(n-1))\\
&\eqdef \lambda \alpha. \Rec^\eta(A \alpha,\psi \alpha)(n).
\end{align*}
This concludes the proof of the first claim. \\[2mm]
{\bf Claim 2}. For all $n^\NN$
\[ r[n] \sim_\NN \lambda \alpha . \Rec^\eta(A \alpha, \psi \alpha)(n)(V \alpha). \]
{\bf Proof}. Immediate from Claim 1 and the assumption $v \sim_\rho V$. \\[1mm]
Claim 2 in particular implies (by the definition of $\sim_\NN$) that for all $n^\NN$
\[ \Val_{r[n]}(\alpha) = \Rec(A \alpha, \psi \alpha)(n)(V \alpha) \]
and $(\ddagger) \, \GeneralEq{S}(\BRSec_{r[n]})$, for some bar $S$ securing $\lambda \alpha . \Rec^\eta(A \alpha, \psi \alpha)(n)(V \alpha)$. By countable choice $\AC_0$ we have a sequence of bars $(S_n)_{n \in \NN}$. Taking $n = g \alpha$ we have
\begin{itemize}
\item[] $(i)~\Val_{r[g \alpha]}(\alpha) = \Rec^\eta(A \alpha, \psi \alpha)(g \alpha)(V \alpha)$
\end{itemize}
Let $S(s) \eqdef S_x(s) \wedge S_{g \hat s}(s)$. We also have that
\begin{itemize}
\item[] $(ii)~S$ is a bar securing $\lambda \alpha. \Rec^\eta(A \alpha, \psi \alpha)(g \alpha)(V \alpha)$
\end{itemize}
Indeed, since $S(s)$ implies that both $g$ and $\lambda \alpha . \Rec^\eta(A \alpha, \psi \alpha)(g \hat s)(V \alpha)$ are secure, which implies that $\lambda \alpha. \Rec^\eta(A \alpha, \psi \alpha)(g \alpha)(V \alpha)$ is secure. \\[1mm]
{\bf Claim 3}. $\GeneralEq{S}(\lambda G, H . \BRSec_x(\lambda s' . \BRSec_{r[g(\widehat{s'})]}(G,H)(s'),H)).$ \\[1mm]
{\bf Proof}. Fix $G, H$ and $s$. We consider two cases: \\[1mm]
If $S(s)$, then $S_x(s)$ and $S_{g \hat s}(s)$ also hold. Hence,
\[
\begin{array}{lcl}
\BRSec_x(\lambda s' . \BRSec_{r[g(\widehat{s'})]}(G,H)(s'),H)(s)
& \stackrel{(\dagger)}{=} & \BRSec_{r[g(\hat{s})]}(G,H)(s) \\
& \stackrel{(\ddagger)}{=} & G(s).
\end{array}
\]
If $\neg S(s)$, then either $\neg S_x(s)$ or $\neg S_{g \hat s}(s)$. We consider two cases: \\[1mm]
If $S_x(s)$ then $\neg S_{g \hat s}(s)$. Then, using that $S_x(s)$ implies both $(*) \, g \hat s = g(\widehat{s * w})$ and $(**) \, S_x(s*w)$,
\[
\begin{array}{lcl}
\BRSec_x(\lambda s' . \BRSec_{r[g(\widehat{s'})]}(G,H)(s'),H)(s)
& \stackrel{(\dagger)}{=} & \BRSec_{r[g(\hat{s})]}(G,H)(s)) \\
& \stackrel{(\ddagger)}{=} & H(s, \lambda w . \BRSec_{r[g(\hat{s})]}(G,H)(s*w)) \\
& \stackrel{(*)}{=} & H(s, \lambda w . \BRSec_{r[g(\widehat{s*w})]}(G,H)(s*w)) \\
& \stackrel{(\dagger, **)}{=} & H(s, \lambda w . \BRSec_x(\lambda s' . \BRSec_{r[g(\widehat{s'})]}(G,H)(s'),H)(s*w)).
\end{array}
\]
Finally, if $\neg S_x(s)$ then the result follows directly by $(\dagger)$.
\end{itemize}
\end{proof}
By combining Theorems \ref{secure-br-thm} and \ref{main-theorem} we obtain:
\begin{corollary}[$\HAomega$] \label{main-cor} Let $\tau = \NN$ and $t \colon \NN$ be a $\T$-term with only $\alpha \colon \tau^\NN$ as free variable. Then $\SpectorEq(\Phi^t(B_{t^\circ}), Y = t)$. Moreover, if $t \in \T_i$ then $\Phi^t(B_{t^\circ}) \in \T_j$, where $j = 2 + \max\{1, \level{\sigma}\} + i$.
\end{corollary}
\begin{proof} By Theorem \ref{main-theorem}, we have $\GeneralEq{S}(\BRSec_{t^\circ})$ for a bar predicate $S$ securing $\lambda \alpha . t[\alpha]$. By Theorem \ref{secure-br-thm} it then follows that $\SpectorEq(\Phi^t(\BRSec_{t^\circ}), Y = t)$. It remains to notice that if $t$ uses a recursor of type $\eta$ then
$t^\circ$ uses a recursor of type $\eta^\circ$. Hence, if $i = \level{\eta}$, by Lemma \ref{type-level-lemma} we have that $\level{\eta^\circ} = 2 + \max \{ 1 + \level{\tau}, \level{\sigma} \} + \level{\eta}$. Since $\level{\tau} = 0$ and $\level{\eta} = i$ we are done. Although we have used countable choice $\AC_0$ in the proof of Theorem \ref{main-theorem}, by modified realizability we can eliminate it here since this corollary is purely universal, so the verification proof that $\Phi^t(B_{t^\circ})$ is a $\T$-definition of bar recursion for $Y = t$ can actually be carried out within $\HAomega$.
\end{proof}
\begin{remark} Note that our construction is parametric in $G$ and $H$, in the sense that we do not require $G$ and $H$ to be $\T$-definable. But once we consider concrete $\T$ terms $Y, G$ and $H$, we get as a corollary Schwichtenberg's result that the functional $\lambda s. \BR(G,H,Y)(s)$ is also $\T$-definable. Unfortunately our construction might not give the ``optimal'' $\T$-definition of $\lambda s. \BR(G,H,Y)(s)$. Indeed, when $Y, G$ and $H$ are in $\T_0$ we obtain a definition of $\lambda s. \BR(G,H,Y)(s)$ in $\T_3$. Howard's analysis \cite{Howard2} suggests that in such cases a definition $\lambda s. \BR(G,H,Y)(s)$ already in $\T_1$ exists. This seems to be the price we need to pay for having a more general construction that works uniformly in $G$ and $H$.
\end{remark}
\begin{remark} Our original motivation for this work started with our bar-recursive bound \cite{BRBOUND} for the Termination Theorem by Podelski and Rybalchenko \cite{Podelski}. The Termination Theorem characterizes the termination of transition-based programs as a properties of well-founded relations. Its classical proof requires Ramsey's Theorem for pairs \cite{Ramsey}. By using Schwichtenberg's result, we proved that under certain hypotheses our bound is in system $\T$. By applying the main construction from this paper we can obtain explicit constructions of the bounds in system $\T$.
\end{remark}
\subsection{Illustrative Example}
Corollary \ref{main-cor} is a generalization of the result obtained by Schwichtenberg in \cite{Schwichtenberg}, but note that our construction is much more explicit, and one we can easily replace bar recursive definitions by their equivalent system $\T$ ones (under the conditions of Schwichtenberg's result).
Let us go back to the example alluded to in the introduction, i.e. Spector's bar recursion for $t[\alpha] = \Rec^\NN(0, \lambda k.\alpha)(\alpha(0))$, where $\lambda k . \alpha$ is ignoring the first argument $k$ so that $$t[\alpha] = \alpha(\alpha(\ldots(\alpha(0))\ldots)$$ with $\alpha(0)$ applications of $\alpha$. In order to work out the $\T$-definition of the bar recursive functional $\lambda G, H, s . \BR^{\NN, \NN}(G, H, \lambda \alpha . t[\alpha])(s)$ we first calculate $\BRSec_{t^\circ}$,
\[
\BRSec_{(t[\alpha])^\circ}(G,H)(s)
= \BRSec_{(\alpha(0))^\circ}(\lambda s'. \BRSec_{r[\Val_{(\alpha(0))^\circ} (\widehat{s'})]}(G,H)(s'),H)(s)
= \BRSec_{(\alpha(0))^\circ}(\lambda s'. \BRSec_{r[\widehat{s'}(0)]}(G,H)(s'),H)(s)
\]
where $r[n] = \Rec^{\NN^\circ}(0^\circ, \lambda k.\alpha^\circ)(n)$. By Corollary \ref{main-cor}, $\GeneralEq{S}(\BRSec_{(t[\alpha])^\circ})$ for some bar $S$ which secures $\lambda \alpha . t[\alpha]$. Hence, $\BR^{\NN,\NN}(G,H,\lambda \alpha. t[\alpha])$ can be $\T$-defined as
\eqleft{
\BR^{\NN,\NN}(G,H,\lambda \alpha. t[\alpha]) = \Phi^{\lambda \alpha . t[\alpha]}(\lambda G, H. \BRSec_{t^\circ}(G,H))(G, H)
}
with $\Phi^{\lambda \alpha . t[\alpha]}$ as in the proof of Theorem \ref{secure-br-thm}, i.e.
\eqleft{
\BR^{\NN,\NN}(G,H,\lambda \alpha. t[\alpha]) = \BRSec_{t^\circ}(\lambda s'.\Psi(t[\widehat{s'}])(G, H)(s'), \mathcal{H}^{\lambda \alpha . t[\alpha]}(G, H) )
}
where
\eqleft{
{\mathcal H}^Y(G,H)(s)(f^{\NN \to \NN}) \eqdef
\begin{cases}
G(s) & {\rm if} \; Y(\hat s) < |s| \\[2mm]
H(s)(f) & {\rm otherwise}.
\end{cases}
}
\subsection{The Case $\tau = \NN \to \NN$}
\label{sec-case-fct}
We now discuss how to extend the construction given in Definition \ref{circ-def}, and the proof of Theorem \ref{main-theorem}, so that Corollary \ref{main-cor} also holds when $\tau = \NN \to \NN$. In this case $\alpha$ has type $\NN \to (\NN \to \NN)$. First, in Definition \ref{circ-def}, when $\tau=\NN \to \NN$ we modify the definition of $\alpha^\circ$ as
\[ \alpha^\circ \eqdef \lambda x^{\NN^{\circ}} y^{\NN^{\circ}} . \apc{\Val}{\BRSec} \]
where
\begin{itemize}
\item $\Val(\alpha) \eqdef \alpha(\Val_x(\alpha))(\Val_y(\alpha))$,
\item $\BRSec(G,H)(s) \eqdef \BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{\Val_x(\widehat{s'}), \Val_y(\widehat{s'})})(G,H)(s'),H),H)(s)$.
\end{itemize}
We also need to modify the proof of Theorem \ref{main-theorem} in the place where the case $\alpha$ is treated. Let $x^\circ \sim g$ and $y^\circ \sim h$. This implies that $(\dagger)$ $\GeneralEq{S_x}(\BRSec_x)$ for a bar $S_x$ securing $g$, and $(\ddagger)$ $\GeneralEq{S_y}(\BRSec_y)$ for a bar $S_y$ securing $h$. Define the predicate:
\[S(s) \eqdef S_x(s) \wedge S_y(s) \wedge \max\bp{\Val_x(\hat{s}), \Val_y(\hat{s})} < |s|.\]
That $S(s)$ is a bar follows directly from the assumptions that $S_x$ and $S_y$ are bars. We show that
$\GeneralEq{S}(\BRSec)$.
Consider two cases:
\noindent If $S(s)$ holds, then $S_x(s) \wedge S_y(s) \wedge \max\bp{g(\hat s), h(\hat{s}) }< |s|$. In this case we trivially have
\[
\begin{array}{lcl}
\BRSec(G,H)(s)
& \stackrel{(\ddagger)}{=} & \BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H)(s) \\[1mm]
& \stackrel{(\dagger)}{=} & \Psi(\max\bp{g(\hat s), h(\hat s)})(G,H)(s) \\[1mm]
& = & G(s).
\end{array}
\]
If $\neg S(s)$ holds, we consider three cases:
\noindent If $\neg S_y(s)$ then
\begin{align*}
\BRSec(G,H)(s)
&\eqdef \BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s)\\
&\stackrel{(\ddagger)}{=} H(s)( \lambda z.\BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s*z))\\
&\eqdef H(s)(\lambda z . \BRSec(G,H)(s * z)).
\end{align*}
\noindent If $S_y(s)$ but $\neg S_x(s)$ then, by monotonicity we have also $S_y(s*z)$ for every $z$. Thus:
\begin{align*}
\BRSec(G)(s)
&\eqdef \BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s)\\
&\stackrel{(\ddagger)}{=} \BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H)(s)\\
&\stackrel{(\dagger)}{=} H(s)(\lambda z . \BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H)(s * z)) \\
&\stackrel{(\ddagger)}{=} H(s)( \lambda z.\BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s*z))\\
&\eqdef H(s)(\lambda z . \BRSec(G,H)(s * z)).
\end{align*}
\noindent If $S_y(s)$ and $S_x(s)$ and $\max\bp{g(\hat s), h(\hat s)}\geq |s|$. From $S_y(s)$ and $S_x(s)$ we have $(*)$ $g(\widehat{s*z})= g(\hat s) \wedge h(\widehat{s*z})= h(\hat s)$ for every $z$. Moreover, by monotonicity we have also $S_y(s*z)$ and $S_x(s* z)$.
\begin{align*}
\BRSec(G,H)(s)
&\eqdef \BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s)\\
&\stackrel{(\ddagger)}{=} \BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H)(s)\\
&\stackrel{(\dagger)}{=} \Psi(\max\bp{g(\hat s), h(\hat s)})(G, H)(s) \\
&\eqdef H(s)(\lambda z . \Psi(\max\bp{g(\hat{s}), h(\hat{s})})(G, H)(s * z)) \\
&\stackrel{(*)}{=} H(s)(\lambda z . \Psi(\max\bp{g(\widehat{s*z}), h(\widehat{s*z})})(G, H)(s * z)) \\
&\stackrel{(\dagger)}{=} H(s)( \lambda z.\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H)(s*z)) \\
&\stackrel{(\ddagger)}{=} H(s)( \lambda z.\BRSec_y(\BRSec_x(\lambda s'.\Psi(\max\bp{g(\widehat{s'}), h(\widehat{s'})})(G,H)(s'),H),H)(s*z)) \\
&\eqdef H(s)(\lambda z . \BRSec(G,H)(s * z)).
\end{align*}
\vspace{3mm}
\noindent \textbf{Acknowledgements.} The authors are grateful to Stefano Berardi, Ulrich Kohlenbach, Helmut Schwichtenberg and the reviewer for various useful comments and suggestions.
\bibliographystyle{asl}
\bibliography{biblio}
\end{document}
| 76,356
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Georgia Rule (2007) Stream Online Free - 123 Movies
Watch Georgia Rule 2007 Full Movie Online free in HD on 123 movies,Georgia Rule follows a rebellious, uncontrollable teenager who is hauled off by her dysfunctional mo...You are watching the movie Georgia Rule (2007) Stream Online Free - 123 Movies produced in United States of America belongs in Genre Comedy, Drama, Romance with rating of 5.7 / 5.38 roadcast at Ww1.123movie.ag. Movies was first released in United States of America, dated 2007-05-11. Original title of the movie is "Georgia Rule" and Original Language is "En". The Popularity of this movies is 5.38.
| 284,208
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TITLE: A Question on Probability - Hunter and Rabbit
QUESTION [2 upvotes]: Suppose there are m different hunters and n different rabbits. Each hunter selects a rabbit uniformly at random independently as a target. Suppose all the hunters shoot at their chosen targets at the same time and every hunter hits his target.
(i) Consider a particular Rabbit $1$, what is the probability that Rabbit $1$ survives?
(ii) Suppose $m=7$, $n=5$. What is the probability that no rabbit survives?
Attempt for (i):
Consider 1st hunter,
No. of rabbits he can choose is $n-1$, since Rabbit $1$ survives.
Consider 2nd hunter,
No. of rabbits he can choose is $n-1$, since Rabbit $1$ survives.
....
So, for $m$ hunters, number of ways they choose rabbits such that they won't choose Rabbit 1
$= (n-1)^m$
And number of ways ways they choose rabbits = $n^m$
$$P(\text{Rabbit 1 survives)} = \frac{ (n-1)^m }{n^m} = \left[ \frac{(n-1)}{n} \right]^m$$
REPLY [0 votes]: I found out how to arrive at the numbers I got from my simulation, so I'll try my hand at answering.
First, my simulation. As I think we've killed enough rabbits by now, I'll try to make the world a better place by giving icecream to children.
We've got 7 children: Alice, Bob, Carol, Dave, Eve, Frank, and Gabrielle.
The icecream parlor has only 5 flavours. You can come up with any flavours you like, but we'll just number them 1 through 5.
The kids get one scoop each. These kids like to share, and they'd like to try each flavour. So if they can make sure that they have picked each flavour at least once among the seven of them, they all can taste every flavour by sharing.
The question now becomes, what is the probability of the 7 children having picked all 5 flavours between them (if they don't know what the others picked, of course).
Now here's my simulation of that in SQL (Oracle 11g).
CREATE OR REPLACE TYPE nums AS TABLE OF NUMBER;
/
WITH
icecream AS (
SELECT LEVEL AS flavour
FROM dual
CONNECT BY LEVEL <= :v_nr_of_flavours
),
children AS (
SELECT
a.flavour AS alice,
b.flavour AS bob,
c.flavour AS carol,
d.flavour AS dave,
e.flavour AS eve,
f.flavour AS frank,
g.flavour AS gabrielle,
CARDINALITY(
nums(
a.flavour,
b.flavour,
c.flavour,
d.flavour,
e.flavour,
f.flavour,
g.flavour
)
MULTISET UNION DISTINCT
nums()
) AS nr_of_flavours_picked
FROM icecream g
CROSS JOIN icecream f
CROSS JOIN icecream e
CROSS JOIN icecream d
CROSS JOIN icecream c
CROSS JOIN icecream b
CROSS JOIN icecream a
)
SELECT
COUNT(*) AS nr_of_combinations,
nr_of_flavours_picked,
CASE
WHEN GROUPING(nr_of_flavours_picked) = 1
THEN NULL
ELSE DECODE(nr_of_flavours_picked, :v_nr_of_flavours, 1, 0)
END AS all_flavours_picked
FROM children
GROUP BY ROLLUP (nr_of_flavours_picked);
This gives us a value of 78125 total possibilities of which 16800 have all rabbits killed all flavours picked.
But where do those numbers come from?
The number of total possibilities is easy, that's ($5^7$). But the other number is a bit more involved.
As it turns out, there are two ways to have 7 children pick all 5 flavours. Either two flavours are picked twice, or one flavour is picked thrice.
That last case is the easiest, as that is just $7 \cdot 6 \cdot 5 \cdot 4$ (the first four kids pick a flavour that hasn't been picked yet, the last three kids pick the one flavour left). Of course there are ${5 \choose 1} = 5$ flavours that can be picked thrice, so we get $7 \cdot 6 \cdot 5 \cdot 4 \cdot 5$ possibilities.
The first case is a little bit harder, but not much. Here we have $7 \cdot 6 \cdot 5 \cdot 6$ (the first three kids pick a flavour that hasn't been picked yet, after which there are 6 ways to distribute the remaining two pairs of flavours among the remaining four kids). Here we have ${5 \choose 2} = 10$ ways of deciding which two flavours get picked twice, so the total number here is $7 \cdot 6 \cdot 5 \cdot 6 \cdot 10$.
The total of these two cases is $7 \cdot 6 \cdot 5 \cdot 4 \cdot {5 \choose 1} + 7 \cdot 6 \cdot 5 \cdot 6 \cdot {5 \choose 2} = 7 \cdot 6 \cdot 5 \cdot (4 \cdot 5 + 6 \cdot 10) = 210 \cdot 80 = 16800$ which is indeed the number we got from our simulation.
So the probability of the children having picked all flavours is $\frac{16800}{78125}$.
Here are the results for 7 children with different numbers of flavours.
$$
\begin{array}{rrr}
\begin{array}{c}\text{Nr. of flavours}\end{array} &
\begin{array}{c}\text{Nr. of combinations} \\ \text{with all flavours chosen}\end{array} &
\begin{array}{c}\text{Nr. of possible combinations}\end{array} \\
\hline
1 & 1 & 1 \\
2 & 126 & 128 \\
3 & 1806 & 2187 \\
4 & 8400 & 16384 \\
5 & 16800 & 78125 \\
6 & 15120 & 279936 \\
7 & 5040 & 823543 \\
\end{array}
$$
The general question remains to find a formula for different $m$ (hunters or children) and $n$ (rabbits or icecream flavours). I can explain all the numbers in the table above, but so far I haven't been able to formulate the general formula.
| 77,300
|
Sheriff Witt dedicates the Graffiti Van
On Tuesday, May 24th, 2005, members of Lexington's Project Cease-Fire unveiled The Graffiti Van,
Lexington's newest tool to combat graffiti
vandalism.
"Graffiti is an outward and visible sign to the community that we aren't winning the fight against crime
on our streets," said Ray Larson, Commonwealth Attorney. "This Graffiti Eradication Program sends a strong
message to these vandals, that graffiti or any other type of vandalism will not be tolerated and will be
prosecuted to the fullest extent of the law."
It is estimated that graffiti costs the country $12 billion a year to clean up," said Kathy Witt,
Fayette County Sheriff. "This and any other types of vandalism lowers property values and the general
quality of life in the affected part of the community, and ultimately raises taxes. Using inmate labor saves
tax dollars and provides a means for those who are incarcerated to make efficient use of their time while
paying their debt to the community."
Inmates help clean up after vandals
Project Cease-fire is a coalition of law enforcement and criminal justice agencies who meet regularly to
discuss gun-related crimes committed in Lexington: The U.S. Attorney's Office; the Commonwealth
Attorney's Office; the Fayette County Attorney's Office; the Office of the Fayette County Sheriff; the
Lexington Divison of Police; Alcohol, Tobacco, Firearms and Explosives; and the Department of Homeland
Security-Immigration Customs Enforcement.
| 318,075
|
Supported »
Clash Royale v1.2.0 For iPhone,iPad,iP: …Read More »
Love You To Bits for iPhone,iPad,iPod touch
“Love »
Implosion – Never Lose Hop v1.2.2 For iPhone,iPad,iPod – …Read More »
FINAL FANTASY Ⅸ v1.0.8 for iPhone,iPad,iPod touch
FINAL FANTASY Ⅸ IPA for iPhone,iPad,iPod touch ========== ●Special sale price for the FINAL FANTASY IX release! ●20% off from February 10 to February 21, 2016! ========== ————————————————– – As this application is very large, it will take some time to download. – This application takes up around 4GB of »
| 126,429
|
TITLE: Prove that if $x^TAx = 0 \Rightarrow Ax = 0$
QUESTION [0 upvotes]: A is $n \times n$ symmetric and positive semidefinite. So we can use $A^T = A$ and $x^TAx \geq 0$ for all x $\in \mathbb{R}^n$.
Proof by contradiction comes to mind, or $x^TAx = 0 \; \land \; Ax \neq 0$. This might seem like a given at first but I'm thinking what if there is some linear dependence between $x^T$ and the $Ax$ vector?
Edit: Ideally, I'm supposed to prove this using the rank of a matrix.
If I were to prove it by contradiction, $x^TAx = 0 \; \land \; Ax \neq 0$, I'd start off by saying that:
$x$ cannot be a zero vector in this case. The original statement $x^TAx = 0 \Rightarrow Ax = 0$ is trivial for a zero $x$ vector and needs no proof.
from $Ax \neq 0$ we have $rank(Ax) = 1$ since $Ax$ is also a non-zero vector.
problem: how to show that the $x^T$ vector and the $Ax$ vector cannot be orthogonal? (this would be the point of the contradiction proof)
problem 2: the only idea I have for a proof barely utilizes the rank of a matrix, which is my assignment.
REPLY [1 votes]: I think you have the best hint from Gerry Myerson in the link from @projectilemotion.
Symmetric $A \Rightarrow$ we can choose orthogonal eigenvectors $v$ for A. These form a basis for $\mathcal{R}^n$, so you can write $x=\sum b_jv_j$.
Positive semi-definite $A \Rightarrow$ its eigenvalues are non-negative. Express $x^TAx$ in terms of the eigenvector basis and use these two facts, you should be home.
| 11,027
|
Julien Absalon ruled cross country competition for the lion’s share of his 20 year career. Born in 1980 in Remiremont, France, near the German and Swiss borders, he literally grew up with the sport. Absalon was officially introduced to mountain biking by a family friend at 14 and began racing shortly after with limited success. […] […]
In the year 2000, a rumor began circulating around bike shops and mountain bike races. A guy in New York had figured out a way to replace traditional inner tubes with a liquid that could seal punctures almost instantly, improve traction and control, and even make tires roll faster. On top of all that, first Editor in Chief and part owner of Britain’s first mountain […]
[…]
| 27,288
|
\begin{document}
\maketitle
\begin{abstract}
We show that every structurally submodular separation system admits a canonical tree set which distinguishes its tangles.
\end{abstract}
\section{Introduction}
The concept of~\emph{tangles} has its origins in the graph minors project of Robertson and Seymour~\cite{GMX}, where tangles were introduced as a unifying framework with which to describe and study highly cohesive substructures in graphs. A central theorem in their work is a~\emph{tree-of-tangles theorem}, which roughly says that the tangles of a graph give rise to a tree-decomposition of that graph with each tangle in a different bag.
Since their inception the theory of tangles has seen a number of advancements: it has been discovered~\cites{AbstractSepSys,ProfilesNew} that the notion of tangles can be formulated more abstractly and does not require an underlying graph structure, making it applicable to a wider range of combinatorial structures. A~\emph{separation system} in this abstract set-up is an axiomatisation of the properties of separations of well-known structures such as graphs or matroids: it is a set, whose elements we call~\emph{separations}, that is equipped with a partial ordering in terms of which all the properties important for tangle theory, such as `nested', `orientation', or `consistency', can be expressed~\cite{AbstractSepSys}.
This higher level of abstraction, and of stripping away the superfluous information about the underlying graph, have facilitated a number of cleaner proofs and stronger results. A recent result~\cite{ProfilesNew} of Diestel, Hundertmark, and Lemanczyk extends the tree-of-tangles theorem of Robertson and Seymour~\cite{GMX} to tangles outside graph theory by finding a~\emph{tree set}, a set of pairwise nested separations, and in addition to this achieves a significant strengthening: the tree set found in~\cite{ProfilesNew} can be built~canonically. The latter means that the construction of the tree set can be carried out using only invariants of the given combinatorial structure. Having a canonical way of constructing the tree set is desirable, for instance, for reproducibility of results when implementing an algorithm for this construction: the canonicity guarantees that the algorithm will construct the same tree set regardless of how the separation system and tangles to be distinguished are presented to it as input.
Establishing canonical tree-of-tangles theorems has been a long-standing goal in tangle theory, since the original proof in~\cite{GMX} relied on a technique that is unable to produce canonical results. A first breakthrough towards this goal was achieved in~\cite{CDHH13CanonicalAlg}, which managed to establish such a canonical theorem for tangles in graphs. With a similar overall strategy Diestel, Hundertmark and Lemanczyk~\cite{ProfilesNew} could then extend this canonical result to arbitrary separation systems and the most general class of tangles, which are called~\emph{profiles}.
A central ingredient in~\cite{ProfilesNew} is an~\emph{order function} on the separations considered, similar to the order~$ \abs{A\cap B} $ of a separation~$ (A,B) $ of a graph that was already used in~\cite{GMX} and~\cite{CDHH13CanonicalAlg}. In this setting one then considers the separation system~$ \vS_k $ of all separations of order less than~$ k $ and studies its tangles. In analogy to the function~$ {(A,B)\mapsto\abs{A\cap B}} $ from graphs this order function is usually assumed to be submodular. This submodularity of the order function has a structural effect on the separation system~$ \vS_k $: for any two separations in~$ \vS_k $ at least one of their pairwise join and meet (which are separations given by opposite `corners' of these separations) again lies in~$ \vS_k $.
Later Diestel, Erde, and Weißauer~\cite{AbstractTangles} showed that the latter structural condition by itself is already sufficiently strong for proving tree-of-tangles theorems: tangle theory can be meaningfully studied without the hitherto usual assumption of a submodular order function, further widening its applicability. If a separation system has this structural property but not necessarily a submodular order function then it is~\emph{structurally submodular} or simply~\emph{submodular} if the context is clear. The tree-of-tangles theorem established in~\cite{AbstractTangles} then reads as follows:
\begin{THM}[\cite{AbstractTangles}*{Theorem 6}]\label{thm:Daniel}
Let~$ \vS $ be a structurally submodular separation system and~$ \cP $ a set of profiles of~$ S $. Then~$ \vS $ contains a tree set~$ N $ that distinguishes~$ \cP $.
\end{THM}
\noindent Here, a tree set~$ N $~\emph{distinguishes}~$ \cP $ if each pair of profiles in~$ \cP $ lies on different sides of some separation in~$ N $. For formal definitions see~\cref{sec:defs}.
This~\cref{thm:Daniel} is even more widely applicable than the result of~\cite{ProfilesNew}, but has one major downside: it does not yield canonicity, since the proof in~\cite{AbstractTangles} chooses certain separations arbitrarily.
In this note we present a new proof, which establishes the following canonical version of~\cref{thm:Daniel}:
\begin{THM}\label{thm:canonical_old}
Let~$ \vS $ be a structurally submodular separation system and~$ \cP $ a set of profiles of~$ S $. Then there is a nested set~$ N=N(\vS,\cP)\sub S $ which distinguishes~$ \cP $. This~$ N(\vS,\cP) $ can be chosen canonically: if~$ \phi\colon\vS\to\vS' $ is an isomorphism of separation systems and~$ \cP'\coloneqq\menge{\phi(P)\mid P\in\cP} $ then~$ \phi(N(\vS,\cP))=N(\vS',\cP') $.
\end{THM}
There are some technical subtleties in the formulation of~\cref{thm:canonical_old} due to the fact that neither the profile property nor structural submodularity need be preserved by isomorphisms of separation systems. To avoid these difficulties, we obtain~\cref{thm:canonical_old} by first establishing the following more general but somewhat more technical result, which slightly weakens the definitions of submodularity and profiles in order to make them compatible with such isomorphisms:
\begin{THM}\label{thm:canonical}
Let~$ \vS $ be a separation system and $\cP$ a collection of consistent orientations of $\vS$ such that $\vS$ is $\cP$-submodular. Then there is a nested set~$ N=N(\vS,\cP)\sub S $ which distinguishes~$ \cP $. This~$ N(\vS,\cP) $ can be chosen canonically: if~$ \phi\colon\vS\to\vS' $ is an isomorphism of separation systems and~$ \cP'\coloneqq\menge{\phi(P)\mid P\in\cP} $ then~$ \phi(N(\vS,\cP))=N(\vS',\cP') $.
\end{THM}
\cref{thm:canonical_old} is then an immediate corollary of~\cref{thm:canonical}.
This paper is structured as follows: in~\cref{sec:defs} we recall the relevant definitions of separation systems and profiles. In~\cref{sec:P-submodularity} we introduce the new definition required for~\cref{thm:canonical} and show that~\cref{thm:canonical_old} indeed is an immediate corollary of~\cref{thm:canonical}. In~\cref{sec:proof} we prove~\cref{thm:canonical_old,thm:canonical}.
\section{Separation Systems and Profiles}\label{sec:defs}
For a full introduction to abstract tangle theory and its terminology and notation we refer the reader to~\cite{AbstractSepSys,AbstractTangles}. In the remainder of this section we offer a brief introduction of only those terms and notation of tangle theory that are relevant to this paper.
A~\emph{separation system}~$ \vS=(\vS,\le,^*) $ is a poset~$ (\vS,\le) $ together with an order-reversing involution~$ ^* $ on~$ \vS $. Given an element~$ \vs $ of~$ \vS $ we denote its~\emph{inverse}, its image under~$ ^* $, by~$ \sv=(\vs)^* $. For~$ ^* $ to be order-reversing then means that~$ \vr\le\vs $ if and only if~$ \rv\ge\sv $ for all~$ \vr,\vs\in\vS $.
We refer to the elements of~$ \vS $ as~\emph{oriented separations}. Given such an oriented separation~$ \vs $ we call~$ \{\vs,\sv\} $ its underlying~\emph{unoriented separation} and denote it by~$ s $. Conversely, given an unoriented separation~$ s $, we call~$ \vs $ and~$ \sv $ the two~\emph{orientations} of~$ s=\{\vs,\sv\} $.
If the context is clear we often refer to both oriented and unoriented separations simply as~\emph{separations}. Moreover, when no confusion is possible, we may informally use terms defined for unoriented separations for oriented separations as well and vice-versa. For a set~$ \vS $ of oriented separations we write~$ S $ for the set of all~$ s $ with~$ \vs\in\vS $. Conversely, if~$ S $ is a set of unoriented separations, we write~$ \vS $ for the set of all~$ \vs $ and~$ \sv $ with~$ s\in S $.
Two separations~$ r $ and~$ s $ are~\emph{nested} if they admit orientations~$ \vr $ and~$ \vs $ with~$ \vr\le\vs $. If~$ r $ and~$ s $ are nested we also call all their respective orientations nested. Two separations that are not nested are said to~\emph{cross}. Note that two oriented separations~$ \vr $ and~$ \vs $ need not be themselves comparable in order to be nested: we may have for instance that~$ \vr\le\sv $. A set of separations is~\emph{nested} if its elements are pairwise nested.
We say that two oriented separations~$ \vr $ and~$ \vs $~\emph{point towards each other} if~$ \vr\le\sv $, and that they~\emph{point away from each other} if~$ \rv\le\vs $. Thus~$ \vr $ and~$ \vs $ are nested if and only if they are either comparable, or point towards each other, or point away from each other.
If~$ \vU=(\vU,\le,^*) $ is a separation system whose poset is a lattice with pairwise join and meet operations~$ \join $ and~$ \meet $ we call~$ (\vU,\le,^*,\join,\meet) $ a~\emph{universe of separations}. Given~$ r $ and~$ s $ in~$ U $ we call all separations of the form~$ \vr\join\vs $ as well as their inverses and underlying separations~\emph{corner separations} of~$ r $ and~$ s $. Note that in universes of separations DeMorgan's rule holds:
\[ (\vr\join\vs)^*=(\rv\meet\sv) \]
\noindent For a universe~$ \vU=(\vU,\le,^*,\join,\meet) $ of separations and a separation system~$ \vS\sub\vU $ we say that~$ \vS $ is~\emph{(structurally) submodular in $\vU$} if for all~$ \vr $ and~$ \vs $ in~$ \vS $ at least one of~$ \vr\join\vs $ and~$ \vr\meet\vs $ is also contained in~$ \vS $.
An~\emph{orientation} of a set~$ S $ of unoriented separations is a set~$ O\sub\vS $ containing exactly one orientation~$ \vs $ or~$ \sv $ of each~$ s=\{\vs,\sv\} $ in~$ S $. We call~$ O $~\emph{consistent} if there are no~$ \vr $ and~$ \vs $ in~$ O $ with~$ \rv\le\vs $ and~$ r\ne s $.
Given a universe~$ \vU $ of separations and a separation system~$ \vS\sub\vU $, a~\emph{profile} of~$ S $ is a consistent orientation of~$ S $ such that
\[ \forall \,\vr,\vs\in P \colon (\rv\meet\sv)\notin P \tag{P}\,.\label[property]{property:P} \]
This~\cref{property:P} is commonly referred to as the~\emph{profile property}.
A separation~$ s $~\emph{distinguishes} two profiles~$ P $ and~$ Q $ of~$ S $ if there is an orientation~$ \vs $ of~$ s $ with~$ \vs\in P $ and~$ \sv\in Q $.
A set~$ N $~\emph{distinguishes} a set~$ \cP $ of profiles of~$ S $ if every pair of distinct profiles in~$ \cP $ is distinguished by some~$ s $ in~$ N $.
A set~$ N\sub S $ of unoriented separations is a~\emph{tree set} if~$ N $ is nested and there are no~$ r\neq s\in N$ with orientations $\vr$ and $\vs$ such that~$ \vr\le\vs $ and~$ \vr\le\sv $. Note that if~$ N $ is nested and has the property that each separation in~$ N $ distinguishes some pair of profiles of~$ S $, then~$ N $ is a tree set.
Finally, an~\emph{isomorphism} of separation systems~$ \vS $ and~$ \vS' $ is a bijective map~$ \phi\colon\vS\to\vS' $ such that~$ \phi(\sv)=(\phi(\vs))^* $ for all~$ \vs $ in~$ \vS $ and furthermore~$ \vr\le\vs $ if and only if~$ \phi(\vr)\le\phi(\vs) $ for all~$ \vr $ and~$ \vs $ in~$ \vS $.
The most prominent occurrence of separation systems is in graphs: given a graph~${G=(V,E)}$, a separation of~$ G $ is a pair~$ (A,B) $ of vertex sets with~$ A\cup B=V $ such that there is no edge from~$ A\sm B $ to~$ B\sm A $. With involution~$ (A,B)^*=(B,A) $ and partial order~$ (A,B)\le(C,D) $ if and only if~$ A\sub C $ and~$ B\supseteq D $, the set of all separations of~$ G $ becomes a separation system -- in fact, a universe of separations.
Of special importance in the theory of graph separations are the separation systems~$ \vS_k $ of all separations~$ (A,B) $ of~$ G $ with~$ \abs{A\cap B}<k $. Each such~$ \vS_k $ constitutes a structurally submodular separation system by our definition. The~$ k $-tangle of a graph, whose introduction and study by Robertson and Seymour in~\cite{GMX} was the inception of tangle theory, is then a special type of consistent orientation of~$ S_k $. Each such~$ k $-tangle is a profile of~$ S_k $ in our sense. Profiles, both in graphs and in abstract separation systems, were first rigorously studied in~\cite{ProfilesNew}: that work also contains a plethora of examples and applications of profiles, including an example of a profile in a graph that is not a~$ k $-tangle~(\cite{ProfilesNew}*{Example~7}).
The separation systems considered in~\cite{ProfilesNew}, although a generalisation of graph separations, still come with some strong structural assumptions that are closely modelled on graph separations. The first paper to study profiles in universes of separations as defined here, without any additional structural assumptions, was~\cite{AbstractTangles}. That work, too, contains some examples of profiles in separation systems, as well as multiple applications of the theory of tangles and profiles both inside and outside of graph theory. The applications presented in~\cite{AbstractTangles} include profiles in matroids and using tangles for cluster analysis, as well as working with more exotic types of separations such as clique separations in graphs or circle separations outside of graphs.
Both~\cite{ProfilesNew} and~\cite{AbstractTangles} provide their own version of~\cref{thm:canonical_old}. The version given in~\cite{ProfilesNew}, like our~\cref{thm:canonical_old}, constructs a~\emph{canonical} tree set distinguishing a given set of profiles, but does so by leveraging a much stronger set of assumptions. Conversely, the version \cref{thm:Daniel} of~\cref{thm:canonical_old} given in~\cite{AbstractTangles} does not yield canonicity.
\section{Submodularity with respect to a set of profiles}\label{sec:P-submodularity}
The first hurdle to overcome when aiming for a canonical version of~\cref{thm:Daniel} is to pin down what exactly `canonical' ought to mean. At first glance this is obvious: the construction of the nested set~$ N $ shall use only invariants of~$ \vS $ and~$ \cP $, that is, properties which are preserved by isomorphisms of separation systems. This approach, however, runs into a subtle difficulty: the definitions of both structural submodularity and profiles depend on~$ \vS $ being embedded into an ambient universe of separations, whose existence~\cref{thm:Daniel} implicitly assumes. An isomorphism~$ \phi\colon\vS\to\vS' $ of separation systems, though, need not preserve such an embedding, which leads to the undesirable situation that a construction isomorphic to that of~$ N $ in~$ \vS $ could be carried out in~$ \vS' $, even though~\cref{thm:canonical_old} may not be directly applicable to~$ \vS' $ due to differences in their embeddings into ambient lattices.
To make our canonical version of~\cref{thm:Daniel} as widely applicable as possible, and to keep the definition of canonicity as straightforward and clean as possible, we must therefore tweak the assumptions of structural submodularity and profiles of~$ S $ in such a way that they no longer depend on any embedding into a universe of separations, and are themselves invariants of isomorphisms between separation systems. This is made possible by the following observation: the proof of~\cref{thm:Daniel} makes use of the assumptions that~$ \vS $ is submodular and~$ \cP $ a set of profiles solely to deduce that whenever some~$ \vr $ and~$ \vs $ in~$ \vS $ distinguish some two profiles in~$ \cP $, then either their meet or their join (as provided by the ambient lattice) is contained in~$ \vS $ and likewise distinguishes that pair of profiles.
For our canonical~\cref{thm:canonical} we will thus eliminate the need for an ambient universe by asking of~$ \vS $ and~$ \cP $ that they have this property, with the meets or joins now being taken directly in the poset~$ \vS $. Expressed solely in terms of~$ \vS $ and~$ \cP $, this `richness' property is then preserved by isomorphisms of separation system, independently of any embeddings into lattice structures. This solves the minor problem in the formulation of~\cref{thm:canonical_old} of~$ \vS' $ not meeting the assumption of the theorem despite being isomorphic to a separation system which does.~\cref{thm:canonical_old} is then obtained as a corollary of~\cref{thm:canonical}.
Let~$ \vS $ be a separation system and~$ \cP $ a set of consistent orientations of~$ \vS $. Given a set~$ M\sub\vS $ of oriented separations, an element~$ \vr\in\vS $ is an~\emph{infimum} of~$ M $ in~$ \vS $ if~$ \vr\le\vs $ for each~$ \vs\in M $ and additionally~$ \vr\ge\vt $ whenever~$ \vt\in\vS $ is such that~$ \vt\le\vs $ for all~$ \vs\in M $. Dually, an element~$ \vr\in\vS $ is a~\emph{supremum} of~$ M $ in~$ \vS $ if~$ \vr\ge\vs $ for each~$ \vs\in M $ and additionally~$ \vr\le\vt $ whenever~$ \vt\in\vS $ is such that~$ \vt\ge\vs $ for all~$ \vs\in M $. In general a set~$ M\sub\vS $ need not have such an infimum or supremum in~$ \vS $.
Given two separations~$ \vr $ and~$ \vs $ in~$ \vS $ we denote the infimum and supremum of~$ \menge{\vr,\vs} $ in~$ \vS $ by~$ \vr\meet\vs $ and~$ \vr\join\vs $, respectively, if those exist. Observe that~$ (\vr\join\vs)^\ast=\rv\meet\sv $.
If~$ \vr $ and~$ \vs $ have a supremum~$ \vr\join\vs $ in~$ \vS $, and every~$ P\in\cP $ containing both~$ \vr $ and~$ \vs $ also contains~$ \vr\join\vs $, then we call~$ \vr\join\vs $ a~\emph{$ \cP $-join} of~$ \vr $ and~$ \vs $ in~$ \vS $. Dually we call~$ \vr\meet\vs $ a~\emph{$ \cP $-meet} of~$ \vr $ and~$ \vs $ if~$ (\vr\meet\vs)^\ast\in P $ for each~$ P\in\cP $ containing both~$ \rv $ and~$ \sv $.
Finally, we say that~$ \vS $ is~\emph{$ \cP $-submodular} if every two crossing separations~$ \vr $ and~$ \vs $ in~$ \vS $ have a~$ \cP $-join or~$ \cP $-meet in~$ \vS $. For the remainder of this work we assume~$ \vS $ to be~$ \cP $-submodular.
Observe that~$ \cP $-submodularity is preserved by isomorphisms of separation systems: if~$ \phi\colon\vS\to\vS' $ is an isomorphism and~$ \cP'\coloneqq\menge{\phi(P)\mid P\in\cP} $, then~$ \vS' $ is~$ \cP' $-submodular. Using this notion of submodularity we can therefore meaningfully express canonicity in the context of~\cref{thm:Daniel}.
Note, however, that we are not assuming the elements of~$ \cP $ to be profiles of~$ \vS $: this is precisely because we prove~\cref{thm:canonical} without an ambient lattice structure, which would be necessary to define profiles. Therefore,~\cref{thm:canonical} improves on~\cref{thm:Daniel} not only by offering a canonical way of constructing~$ N $, but also by being applicable to an even larger number of separation systems.
Before getting to the proof of~\cref{thm:canonical} itself, let us demonstrate that it is in fact a strengthening of~\cref{thm:Daniel} by showing that~\cref{thm:canonical} implies~\cref{thm:canonical_old}. The following lemma does just that by proving that in the setting of~\cref{thm:Daniel} the assumptions of~\cref{thm:canonical} are satisfied:
\begin{LEM}\label{lem:Psub_strucsub}
Let~$ \vS $ be a structurally submodular separation system inside some universe~$ \vU $ of separations and~$ \cP $ a set of profiles of~$ \vS $. Then~$ \vS $ is~$ \cP $-submodular.
\end{LEM}
\begin{proof}
We must show that any~$ \vr $ and~$ \vs $ in~$ \vS $ have a~$ \cP $-meet or~$ \cP $-join. So let~$ \vr $ and~$ \vs $ in~$ \vS $ be given.
Since~$ \vS $ is structurally submodular it contains the infimum~$ \vr\meet\vs $ or the supremum~$ \vr\join\vs $ of~$ \vr $ and~$ \vs $ in~$ \vU $. Let us assume the latter; the other case is dual. Then~$ \vr\join\vs $ is also the supremum of~$ \vr $ and~$ \vs $ as taken in~$ \vS $. Moreover, by the profile property, every~$ P\in\cP $ containing both~$ \vr $ and~$ \vs $ also contains~$ \vr\join\vs $, making this a~$ \cP $-join of~$ \vr $ and~$ \vs $ in~$ \vS $.
\end{proof}
\section{\texorpdfstring{Proof of~\cref{thm:canonical_old,thm:canonical}}{Proof of Theorem 2 and 3}}\label{sec:proof}
In this section we will prove~Theorem~\ref{thm:canonical_old} and~Theorem~\ref{thm:canonical}. To this end, for the remainder of this paper let~$\vS$ be a separation system and let~$\cP$ be a set of consistent orientations of~$\vS$ such that~$\vS$ is~$\cP$-submodular.
A common tool in proving tree-of-tangles theorems is the so-called fish lemma \cite{AbstractSepSys}*{Lemma~3.2}. Since this lemma is usually formulated in the context of a separation system that is contained in a universe of separations, we need to prove our own version of this lemma.
\begin{LEM}[See also \cite{AbstractSepSys}*{Lemma 3.2}]\label{lem:fish}
Let~$r,s\in S$ be two crossing separations in~$ S $ and let~$t\in S$ be a separation that is nested with both~$r$ and~$s$. Given orientations~$\vr$ and~$\vs$ of~$r$ and~$s$ such that there exists a supremum~$\vr\join \vs$ of~$\vr$ and~$\vs$ in~$\vS$, the separation~$\vr\join \vs$ is nested with~$t$. The same is true for~$ \vr\meet\vs $.
\end{LEM}
\begin{proof}
If $t$ has an orientation $\vt$ such that $\vt\le \vr$ or $\vt\le \vs$ then clearly $\vt\le (\vr\join \vs)$. Otherwise, since $r$ and $s$ cross, there must be an orientation $\vt$ of $t$ such that $\vr\le \vt$ and $\vs\le \vt$. Thus, by the fact that $\vr\join \vs$ is the supremum of $\vr$ and $\vs$ in $\vS$, we have that~$(\vr\join \vs)\le \vt$.
\end{proof}
Let us say that a separation~$ \vs\in\vS $ is~\emph{exclusive (for~$ \cP $)} if it lies in exactly one orientation in~$ \cP $. If~$ P\in\cP $ is the orientation containing an exclusive separation~$ \vs $ then we might also say that~$ \vs $ is~\emph{$ P $-exclusive (for~$ \cP $)}. Observe that if~$ \vr $ is~$ P $-exclusive for~$ \cP $, then so is every~$ \vs\in P $ with~$ \vr\le\vs $.
For each~$ P\in\cP $ let~$ M_P $ consist of the maximal elements of the set of all~$ P $-exclusive separations. Equivalently,~$ M_P $ is the set of all maximal elements of~$ P $ that are exclusive for~$ \cP $.
Our strategy for proving~\cref{thm:canonical} will be to canonically pick nested $ P $-exclusive representatives of all orientations~$ P\in\cP $ that contain exclusive separations, then discard from~$ \cP $ and~$ \vS $ all those orientations~$ P $ for whom we selected a representative and all those separations not nested with these representatives, respectively. Iterating this procedure will yield the canonical nested set.
In order for this strategy to work we must ensure that the sets~$ M_P $ are not all empty. Our first lemma addresses this:
\begin{LEM}\label{lem:nonempty}
If~$ \cP $ and $S$ are non-empty, then some~$ M_P $ is non-empty.
\end{LEM}
In the case of $\vS$ being submodular in some universe of separation $\vU$ and $\cP$ being a set of profiles of $\vS$, the existence of exclusive separations and thus~\cref{lem:nonempty} is actually an immediate consequence of~\cref{thm:Daniel}: if~$ N\sub S $ is a nested set which distinguishes~$ \cP $, and each element of~$ N $ distinguishes some pair of profiles in~$ \cP $, then any maximal element of~$ \vN $ is exclusive for~$ \cP $. In other words, the separations labelling the incoming edges of leaves of the tree associated with~$ N $ are exclusive. (See~\cite{TreeSets} for the precise relationship between nested sets and trees.)
However, since we are working with the more general notion of $\vS$ being $\cP$-submodular, we give an independent proof of~\cref{lem:nonempty}.
\begin{proof}[Proof of~\cref{lem:nonempty}.]
If~$ \cP $ consists of only one orientation the assertion is trivial since $S$ is non-empty. For~$ \abs{\cP}\ge 2 $ we show the following stronger claim by induction on~$ \abs{\cP} $:
\begin{center}\em
If~$ \abs{\cP}\ge 2 $ there is for each~$ P\in\cP $ a separation that is exclusive but not~$ P $-exclusive for~$ \cP $.
\end{center}
For the base case~$ \abs{\cP}=2 $ observe that any separation distinguishing the two orientations in~$ \cP $ has two exclusive orientations, one in each element of $\cP$.
Suppose now that~$ \abs{\cP}>2 $ and that the claim holds for all non-singleton proper subsets of~$ \cP $. Let~$ P\in\cP $ be the given fixed orientation and set~$ \cP'\coloneqq\cP\sm\menge{P} $. By the induction hypothesis applied to~$ \cP' $ and an arbitrary orientation in $\cP'$ there is an exclusive separation~$ \vr $ for~$ \cP' $, contained in some~$ Q\in\cP' $. Applying the induction hypothesis again to~$ \cP' $ and~$ Q $ yields another separation~$ \vs $ that is exclusive for~$ \cP' $ and lies in some~$ Q'\in\cP' $ with~$ Q\ne Q' $.
If~$ \vr $ or~$ \vs $ is also exclusive for~$ \cP $ then we are done. So suppose not, that is, suppose we have~$ \vr,\vs\in P $. Then~$ r\ne s $, and hence~$ \vr $ and~$ \vs $ must be incomparable by the consistency of~$ Q $ and~$ Q' $. If~$ \vr\le\sv $ then~$ \sv $ is~$ Q $-exclusive for~$ \cP $. Moreover $\sv\le \vr$ is not possible by the consistency of $P$. Thus we may assume that~$ r $ and~$ s $ cross.
By $\cP$-submodularity of $\vS$, there is a $\cP$-join or a $\cP$-meet of $\vr$ and $\sv$ in $\vS$; by symmetry we may assume that there is a $\cP$-join~$ (\vr\join\sv)\in\vS $. Since~$ \vs $ is~$ Q' $-exclusive we have~$ \sv\in Q $ and hence~$ (\vr\join\sv)\in Q $ by the fact that $\vr\join \sv$ is the $\cP$-join of $\vr$ and $\sv$. From~$ (\vr\join\sv)\ge\vr $ we infer that~$ \vr\join\sv $ is~$ Q $-exclusive for~$ \cP' $. Moreover we cannot have~$ (\vr\join\sv)\in P $: it would be inconsistent with~$ \vs\in P $ as~$ r $ and~$ s $ cross.
Therefore~$ \vr\join\sv $ is exclusive but not~$ P $-exclusive for~$ \cP $.
\end{proof}
We remark that, in the case of a submodular separation system $\vS$ inside a universe of separations and a set $\cP$ of profiles, the stronger assertion used for the induction hypothesis in this proof, too, can be established immediately using~\cref{thm:Daniel}: for~$ \abs{\cP}\ge 2 $ the tree associated with the nested set~$ N\sub S $ distinguishing~$ \cP $ has at least two leaves, and hence some leaf for which the separation labelling its incoming edge does not lie in the fixed profile~$ P $.
Returning to the proof of~\cref{thm:canonical}, let us find a way to canonically pick representatives of those~$ P\in\cP $ with non-empty~$ M_P $ in such a way that these representatives are nested with each other. For the `canonically'-part of this we will make use of the fact that the sets~$ M_P $ themselves are invariants of~$ \cP $ and~$ \vS $. For the nestedness we start by showing that separations from different~$ M_P $'s cannot cross at all:
\begin{LEM}\label{lem:nestedbetween}
For~$ P\ne P' $ all~$ \vr\in M_P $ and~$ \vs\in M_{P'} $ are pairwise nested.
\end{LEM}
\begin{proof}
Suppose some~$ \vr\in M_P $ and~$ \vs\in M_{P'} $ cross. By $\cP$-submodularity of $\vS$, there is a $\cP$-join~$ \vr\join\sv $ or a $\cP$-meet~$ \vr\meet\sv $ in~$ \vS $; by symmetry we may suppose that~$ (\vr\join\sv)\in\vS $. Then~$ P $, too, contains this separation since~$ \sv\in P $. But~$ \vr\join\sv $ is also~$ P $-exclusive and strictly larger than~$ \vr $, a contradiction.
\end{proof}
It is possible, however, that the set~$ M_P $ itself is not nested. In fact the elements of~$ M_P $ all cross each other, unless~$ \cP=\menge{P} $: any~$ \vr $ and~$ \vs $ in~$ M_P $ that are nested must point towards each other by maximality. But every other orientation in~$ \cP $ contains both~$ \rv $ and~$ \sv $ and would then be inconsistent. If we want to represent a~$ P\in\cP $ with non-empty~$ M_P $ by an element of~$ M_P $, we are therefore limited to picking at most one element of~$ M_P $. However there is no canonical way of singling out an element of~$ M_P $ to be the representative of~$ P $; we must therefore find another way of choosing an invariant~$ P $-exclusive separation, using~$ M_P $ only as a starting point.
For this we will show that each $ M_P $ has an infimum in $\vS$ and that this infimum is again~$ P $-exclusive:
\begin{LEM}\label{lem:infimum}
Let~$ P\in\cP $ with~$ M_P\ne\emptyset $ and~$ \cP\ne\menge{P} $ be given. Then~$ M_P $ has an infimum~$ \vs_P $ in the poset~$ \vS $, and~$ \vs_P $ is~$ P $-exclusive for~$ \cP $. Moreover if some~$ t\in S $ is nested with~$ M_P $ then~$ t $ is also nested with~$ s_P $.
\end{LEM}
\begin{proof}
Fix an enumeration~$ M_P=\menge{\vr_1,\dots,\vr_n} $ and some~$ t\in S $ that is nested with~$ M_P $. We will show by induction on $i$ that there is an infimum $\vs_i\coloneqq \inf\{\vr_1,\dots,\vr_i\}$ in~$ \vS $; that this infimum is~$ P $-exclusive for~$ \cP $; and that it is nested with~$ t $. This then yields the claim for~$ i=n $.
The case~$ i=1 $ is trivially true, so suppose that~$ i>1 $ and that~$ \vs_{i-1}=\inf \{\vr_1,\dots,\vr_{i-1}\} $ is already known to be the infimum of $\vr_1,\dots, \vr_{i-1}$ in~$ \vS $, that it is~$ P $-exclusive, and that it is nested with~$ t $.
In the case that~$ s_{i-1}=r_i $ we have either~$\vs_{i-1}=\vr_i$ or~$ \vs_{i-1}=\rv_i $. The latter of these is impossible since~$ P $ contains both of the~$ P $-exclusive separations~$ \vs_{i-1} $ and~$ \vr_i $. The former, however, gives that~$ \vr_i $ is the infimum of~$ \vr_1,\dots,\vr_i $ and thus as claimed by~$ \vr_i\in M_P $.
So suppose that~$ s_{i-1}\ne r_i $. Let us first treat the case that~$ \vr_i $ and~$ \vs_{i-1} $ are nested. Clearly the two cannot point away from each other since~$ P $ is consistent. If~$ \vr_i $ and~$ \vs_{i-1} $ are comparable then one of the two is the infimum of~$\vr_i$ and~$\vs_{i-1}$ and thus the infimum of~$\vr_1, \dots, \vr_i$ in~$\vS$. Since both~$\vs_{i-1}$ and~$\vr_i$ are~$P$-exclusive and nested with~$t$, this infimum is thus as claimed. Finally, if~$ \vr_i $ and~$ \vs_{i-1} $ point towards each other, we obtain a contradiction: for then their inverses point away from each other, making every orientation in~$ \cP $ other than~$ P $ inconsistent. Thus if~$ \vr_i $ and~$ \vs_{i-1} $ are nested the induction hypothesis holds for~$ \vs_i $.
Let us now consider the case that~$ \vr_i $ and~$ \vs_{i-1} $ cross. Then there needs to be a $\cP$-join or a $\cP$-meet of $\vr_i$ and $\vs_{i-1}$. However we cannot have a $\cP$-join~$ \vr_i\join\vs_{i-1} $ in~$ \vS $ since this join would be~$ P $-exclusive and strictly larger than~$ \vr_i\in M_P $. Therefore there is a $\cP$-meet~$ (\vr\meet\vs_{i-1})\in\vS $. By consistency we have that~$ \vs_i\in P $. Every orientation in~$ \cP $ other than~$ P $ contains~$ \rv_i $ as well as~$ \sv_{i-1} $ and hence~$ \sv_{i} $ by the definition of $\cP$-meet, which shows that~$ \vs_i $ is~$ P $-exclusive. Finally, by~\cref{lem:fish},~$ \vs_i $ is also nested with~$ t $.
\end{proof}
It remains to show that after picking as a representative for each~$ P\in\cP $ with exclusive separations the infimum of~$ M_P $, the set of separations in~$ \vS $ that are nested with all these representatives is still rich enough to distinguish all orientations in~$ \cP $ for which we have not yet picked a representative.
For this let~$ \vS'\sub\vS $ be the system of all those separations that are nested with all~$ M_P $, and let~$ \cP'\sub\cP $ be the set of those orientations~$ Q $ that have empty~$ M_Q $. Our next lemma says that if we restrict ourselves to~$ \vS' $, we can still distinguish~$ \cP' $:
\begin{LEM}\label{lem:restriction}
The separation system~$ \vS' $ is $\cP'$-submodular and distinguishes~$ \cP' $.
\end{LEM}
\begin{proof}
The fact that~$ \vS' $ is $\cP'$-submodular is a direct consequence of~\cref{lem:fish}: it implies that for~$ \vr $ and~$ \vs $ in~$ \vS $ any~$ \cP $-meet or~$ \cP $-join of them in~$ \vS $ is contained in~$ \vS' $. Since~$ \cP'\sub\cP $ this~$ \cP $-join or~$ \cP $-meet is in fact a~$ \cP' $-join or~$ \cP' $-meet of~$ \vr $ and~$ \vs $ in~$ \vS' $.
To see that $\vS'$ distinguishes $\cP'$, let~$ Q $ and~$ Q' $ be distinct orientations in~$ \cP' $; we shall show that some~$ \vs'\in\vS' $ distinguishes them. For this choose a separation~$ s\in S $ which distinguishes~$ Q $ and~$ Q' $ and which is nested with~$ M_P $ for as many~$ P\in\cP $ as possible. If~$ s $ is nested with all~$ M_P $ we are done; otherwise there is some~$ P\in\cP $ for which~$ s $ crosses some separation in~$ M_P $.
So suppose that there is a~$ P\in\cP $ for which~$ s $ is not nested with~$ M_P $. Among all~$ \vs'\in\vS $ which distinguish~$ Q $ and~$ Q' $ and which are nested with each~$ M_{P'} $ with which~$ s $ is nested, pick a minimal~$ \vs' $ with~$ \vs'\in P $. We claim that this~$ \vs' $ is nested with~$ M_P $, contradicting the choice of~$ s $.
To see this, suppose that~$ \vs' $ crosses some~$ \vr\in M_P $. Then there cannot exist a~$\cP$-join of~$\vr$ and~$\vs'$ in~$\vS$ since this join would be a strictly larger~$ P $-exclusive separation than~$ \vr $. Hence there is a~$\cP$-meet~$\vr\meet\vs'$ of~$\vr$ and~$\vs'$ in~$\vS $. By~$ P\notin\menge{Q,Q'} $ we have that both~$ Q $ and~$ Q' $ contain~$ \rv $, and hence this separation $\vr\meet \vs'$ distinguishes~$ Q $ and~$ Q' $ as well: one of the two orientations contains $\vs'$ and thus also $\vr\meet \vs'$ by consistency. The other contains both~$\sv'$ and~$\rv$ and thus also~$(\rv\join \sv')=(\vr\meet \vs')^\ast$ by the fact that~$\vr\meet \vs'$ is the~$\cP$-meet of~$\vr$ and~$\vs'$.
However, by~\cref{lem:fish} and~\cref{lem:nestedbetween}, this~$ \vr\meet\vs' $ would be nested with each~$ M_{P'} $ with which~$ s $ was nested, while being strictly smaller than~$ \vs' $, a contradiction.
\end{proof}
If~$ M_P $ is non-empty let us write~$ \vs_P $ for its infimum in~$ \vS $ as in~\cref{lem:infimum}. We are now ready to prove~\cref{thm:canonical} by induction.
\begin{proof}[Proof of~\cref{thm:canonical}.]
We proceed by induction on~$ \abs{\cP} $. If~$ \abs{\cP}\le 1 $ there is nothing to show, so suppose that~$ \abs{\cP}>1 $ and that the assertion holds for all proper subsets of~$ \cP $.
Recall that~$ \vS'\sub\vS $ consists of all separations in~$ \vS $ that are nested with all sets~$ M_Q $ and that~$ \cP'\sub\cP $ is the set of all~$ Q\in\cP $ with empty~$ M_Q $. Clearly both~$ \vS' $ and~$ \cP' $ are invariants of~$ \vS $ and~$ \cP $ since the sets~$ M_Q $ themselves are invariants. For each non-empty~$ M_Q $ let~$ \vs_Q $ be its infimum in~$ \vS $ as described in~\cref{lem:infimum}. Then
\[ N_1\coloneqq\menge{s_Q\mid Q\in\cP\sm\cP'} \]
is clearly a canonical set. From~\cref{lem:infimum} we further know that~$ N_1 $ distinguishes all orientations in~$ \cP\sm\cP' $ from each other and from each orientation in~$ \cP' $.
By~\cref{lem:nestedbetween} every element of~$ M_P $ is nested with every element of~$ M_{P'} $ for all~$ P\ne P' $. Applying the `moreover'-part of~\cref{lem:infimum} twice thus implies that~$ s_P $ is nested with every element of~$ M_{P'} $ and subsequently with~$ s_{P'} $. Therefore~$ N_1 $ is a nested set. Likewise every separation in~$ \vS' $ is nested with~$ N_1 $.
Let us apply the induction hypothesis to~$ \cP' $ in~$ \vS' $, as made possible by~\cref{lem:nonempty,lem:restriction}, yielding a canonical nested set~$ N_2\sub S' $ which distinguishes~$ \cP' $. Since~$ \vS' $ and~$ \cP' $ themselves are invariants of~$ \vS $ and~$ \cP $ we have that the union~$ N_1\cup N_2 $ is the desired canonical nested set.
\end{proof}
\begin{proof}[Proof of \cref{thm:canonical_old}]
By \cref{lem:Psub_strucsub}, given a structurally submodular separation system $\vS$ and a set $\cP$ of profiles of $S$, we know that $\vS$ is $\cP$-submodular. Thus \cref{thm:canonical_old} follows from \cref{thm:canonical}.
\end{proof}
\section*{Acknowledgement}
We would like to thank one of the reviewers for suggesting the concept of~$\cP$-submodularity, which led to \cref{thm:canonical}.
\bibliography{collective}
\vspace{0.5cm}
\noindent
\begin{minipage}{\linewidth}
\raggedright\small
\textbf{Christian Elbracht},
\texttt{christian.elbracht@uni-hamburg.de}
\textbf{Jakob Kneip},
\texttt{jakob.kneip@uni-hamburg.de}
Universit\"at Hamburg,
Bundesstra\ss{}e 55,
20146 Hamburg, Germany
\end{minipage}
\end{document}
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Extensive data analysis allows businesses to extract meaningful facts about their consumers and marketing efforts, giving them an edge over their competitors. The goal is to find more effective ways of acquiring, engaging and keeping customers over the long term by understanding how their target consumers think and behave. Marketing and customer insights offer more than just data and research. They provide multiple perspectives on customers’ issues, needs, and desires.
What is the difference between a marketing and a consumer insight?
Brands can extract marketing insights from their business’s existing marketing analytics to improve their marketing efforts. This marketing analytics data tells them if campaigns and messages are working. Marketing analytics are available across all marketing channels, including social media, search advertising, email marketing, digital ads, TV ads, radio advertising and more.
Customer insights are crucial for brands to understand their existing customers. By providing important information on what customers like, how they behave, and any issues they have, customer insights allow brands to fine-tune their marketing operations and speed up marketing campaigns.
How insights can affect your business
They help to predict customer’s behaviour
Marketing and customer insights allow brands to identify patterns in the customer’s purchase behaviour. Using predictive analytics tools, a brand can predict a customer’s behaviour in advance. Different analysis methodologies can help brands divide customers based on their predicted lifetime value, which can help focus on retaining high-value customers.
They help keep customers satisfied
The best way to see if a business is effectively meeting customer expectations and discovering customer priorities is to engage them in conversations and surveys. Customer insights can also aid brands in building stronger relationships.
Insights help with better business decisions
Marketing and customer insights help businesses to make better decisions and allow businesses to resonate with their target customers. Many brands are under pressure to prove a return on investment. Customer insight can help by ensuring that they have effective messages and finely tuned targeting within their campaigns.
Turning data into contextual insight
Brands should start by separating out the marketing vanity metrics from those that truly show marketing performance. A Facebook follower is not worth much if they don’t buy! In this respect, analytics are only effective when a brand has fully mapped performance metrics and sets KPIS against them.
For example, what does it mean to have a 5.8% conversion rate? It means that, for every 100 clicks on the ad, about 5–6 of those who click convert into customers. Therefore, 5.8% of those who clicked found the product, copy or design useful for them. Brands should look into how conversion rates change from week to week and how they change based on tweaks to the customer journey and landing page. Which changes improve the conversion rates and which do not?
When companies are clear about these various metrics and how they affect growth, conversions and other key performance indicators, they are in a suitable position to optimise their marketing. Through insights, brands can make intelligent decisions and figure out where to invest their time and budget in the future.
What KPIs need measuring across digital marketing and customer engagement?
Key Performance Indicators or KPIs are vital in order to drive accountability in marketing programmes and should ideally apply across all marketing activities. When the results fall short of the required level, brands can then start analysing the cause in order to optimise (or cancel) the marketing activity.
Common KPIs
- Monthly number of unconverted leads–Number of leads who entered the sales funnel but did not purchase or subscribe to the service.
- Email conversion rate–Percentage of people who saw something in a brands email campaign, and then completed the desired action.
- Email click-through rate–The percent of users who clicked through on the email versus the total number of recipients. Some email marketing and automation tools that brands can use are EmailOctupus, OmniSend, and SendXio.
- Funnel drop-off rates–Measures the number of visits/visitors who left a sales/marketing funnel without completing it.
- Sales conversions–Measures the effectiveness of a brand’s sales tactics to convert leads into new customers.
- Average collection period–The average number of days between 1) the dates of credit sales, and 2) the date of money collection from the customers
- On-time delivery–Measure of process and supply chain efficiency. This measures the number of finished goods or services delivered to customers on time and in full.
Expanded KPIs
- Average customer lifetime value–The total revenue amount that a customer brings to a business over the period of the relationship.
- Customer churn rate–The number of lost customers divided by the number of customers at the start of the period. We typically express this as a percentage (%) of all customers and on a monthly/annual basis.
- Average customer retention period–The percentage of customers the company has kept over a period. It is the opposite side of the churn rate.
- Customer Acquisition Cost (CAC)–The cost associated with convincing a customer to buy a product/service.
- Share of wallet–The percentage share that a brand has of the customer’s total spend in their product or service category over a period.
How applying consumer insights helps improve brand loyalty
When brands have developed a robust set of marketing and consumer insights, the next step is to turn this into a strategy which improves customer experience.
1. Start by building and analysing the problem–Combining analytics data with customer feedback can bring powerful results.
2. Organise customer feedback into segments – To give the brand an idea of what needs to be tackled for each different type of customer that they serve.
3. Search for quick-wins–Actionable factors that brands can implement today, tomorrow or in a week.
4. Build segment specific strategies–Segmenting customers by value, purchase frequency or geography can enable brands to develop more tailored (and effective) marketing programmes. Building up segment-specific insights (their segment specific issues, needs and desires) can further strengthen this approach.
How can companies use insights to increase business growth?
Improving customer experience
According to the Customer Experience Strategy and Design Primer for 2018 by Gartner, 67% of companies compete mostly or completely on customer experience and 81% expect to be doing so within two years. Many companies cannot improve customer experience when they do not fully map out their customer’s journey or experience. Marketing and consumer insights are a vital component to mapping customer journeys and experiences accurately.
Selling more to your existing customers
According to the book, Marketing Metrics, by Paul Farris, “Brands have a 60-70% chance of selling to an existing customer, while the chances of selling to a new customer are only 5-20%”. If brands want fast growth, selling to existing customers should be a priority. Insights can be useful as they give brands an idea of why the customer bought from the brand originally.
Expanding into new markets with existing products
Insights can help brands adapt their marketing strategy to launch into new markets. Market insights can help brands mitigate risk – they will know the mistakes to avoid, the approach that works and be sensitive to cultural nuances.
Keeping more customers
Marketing and consumer insights can help businesses sharpen their customer acquisition strategy, understand why customers churn and optimise marketing programmes. Brands can keep customers for longer, accelerate marketing performance and mitigate risk by developing insight, analyzing data, and listening and engaging with their customers.
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Autozone's rebuilt engines are reliable when the person does a good job of rebuilding the engine, states Dodge Forum users. The quality of a rebuilt engine depends on the quality of the rebuilding work.Continue Reading
Check the return policy and any guarantee information when purchasing a rebuilt engine. Not everything on a rebuilt engine is new, so it is a matter of uncertainty. A rebuild is simply fixing whatever was wrong with an engine. It is the decision of the buyer if they want to take a minor risk on a rebuilt engine or purchase a new engine. Consider work quality, price and return policies when buying any rebuilt engine, says Dodge Forum.Learn more about Engine
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Hamilton and Hare: bespoke men’s underwear
Last night, The Upcoming attended the launch party of British brand Hamilton and Hare launching their new Lombard collection. The brand is like no other within the industry as their simple yet innovative idea is unique.
Olivia Francis, founder of the brand, believed there was a gap in the market for men’s fashion. A man can be suited and booted, with tailor-made suits, topped off with luxury accessories including watches, shoes, etc but when it comes to what is underneath the suits this is what seems to always be neglected.
That’s where Hamilton and Hare come in. The brand thinks a man should have luxury tailor-made underwear to complete an outfit and make you feel good both inside and out. And why not? You may look spiffin’ on the outside, but what’s underneath? Men’s underwear has no variety and is normally drab with materials that are not of a high quality, “nappy looking” and not sexy at all.
The new boxer shorts have been made to “define the bottom” as Olivia stated last night where the preview of the new collection was shown at the prestigious Wolf & Badger in Dover Street. Male models wearing only the boxers paraded and displayed the collection that was an instant success.
It seems to make sense and you have to question why this has not been done before. Women have a wide variety of underwear to choose from, from different shapes, styles, materials and for men there is hardly anything to choose from.
These luxury shorts are tailor-made from Savile Row’s finest suppliers and tailors. They have been made to enhance the lift of the bottom, made of high quality, mother of pearl buttons at the top and the brand’s label of boxing hares on the back to finish them off and feel incredibly comfortable.
Women’s underwear is sexy and it is about time that men’s underwear was too. Elegant, sexy and sophisticated underwear? Suits you Sir.
Leyla Nguyen
Photos: Marcus Dawes
For further information about Hamilton and Hare visit here.
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2Ioanna Konidari
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- Holly N Cukier, Nicole D Dueker, Susan H Slifer, Joycelyn M Lee, Patrice L Whitehead, Eminisha Lalanne +13 others
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The next major steps in human spaceflight include flyby, orbital, and landing missions to the Moon, Mars, and near earth asteroids. The first crewed deep space mission is expected to launch in 2022, which affords less than 7 years to address the complex question of whether and how to apply artificial gravity to counter the effects of prolonged… (More)
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We all know hydration is key to maintaining a healthy body and youthful skin.
Theses days mineral water is taking second stage to coconut water. The translucent fluid found inside young green coconuts is rich in detoxifying enzymes and is a great source of electrolytes to fuel those of us who are fitness-minded. Plus they are also great to add to smoothies for a twist of coconut flavour!
Here are some for you to try:
Naturally fat-free and low in calories, Jax Coco ($4.50 to $4.99/250ml at Pusateri’s and T&T Supermarkets) comes in a sleek glass bottle designed by Stella McCartney and Alasdhair Willis.
For us chocoholics, ZICO Chocolate Premium Hydrating Coconut Water ($2.99-4.99 at Metro, Shoppers Drug Mart, and Walmart) has a natural dark-chocolate flavour for perfect thirst-quenching, sweetness. Rich in potassium, ZICO helps prevent muscle cramps and post-workout recovery.
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Capsule Movie Review: “Shaun the Sheep Movie”
Despite not having children, we went to see the “Shaun the Sheep Movie.” We were looking for something lighthearted and silly.
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Capsule Movie Review: “Trainwreck”
We saw “Trainwreck” last night.
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You’ll see default settings already there which you’ll simply go away untouched if you want the administrator email to obtain the messages. For extra information on the means to arrange this mailer, make sure to check out our step-by-step setup guide. For extra information on tips on how to arrange this mailer, check out our step-by-step setup guide. Contact Form 7 is a great WordPress plugin that makes it straightforward to build your personal e mail forms.
By default, buttons assume your theme accent color as outlined in the Theme Customizer. This possibility lets you assign a customized textual content shade to the button in this module. Select your custom colour using the colour picker to vary the button’s color.
Implementing Spam Safety
However, the Other SMTP mailer normally isn’t as handy in the lengthy run. For instance, when you reset your email’s password at any time, you’d then must edit the password setting in WP Mail SMTP, too. This mailer can additionally be much more likely to have configuration issues with your server . Now we are prepared to place our newly created kind into the WordPress Page.
- The resolution ought to be fairly related for Gravity Forms.
- Form – Customize your HTML contact kind template with a wide selection of subject options like “text”, “email”, “checkboxes”, and so forth.
- Before moving into tips on how to let’s look at why it’s needed.
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Select the contact type you want to add and click theAdd Form button. So, create a new page or open to edit an current web page the place you need to add the contact type. In the search area, type WPForms to search out the plugin. To start, open your WordPress dashboard and install theWPForms Lite plugin by going to Plugins » Add New. With so many type plugin decisions obtainable, the process of discovering the best one in your wants may be an exhausting course of.
How Do I Link My Contact Kind To My E-mail In WordPress?
If your contact form includes importing a file, you’ll find a way to include the file with the notification email. In our contact form, we now have a file addContent kind tag named “[file-658]”. Thus, we can add this type tag under “File Attachments” to ensure the file might be included with the email notification. Here’s what this modification appears like on our contact type. In this case, the newly created label serves the identical purpose as the placeholder.
This is the strictest model of reCAPTCHA, and it’s the one which customers report essentially the most problems with. You can connect Contact Form 7 to Gmail with WP Mail SMTP. We recommend using the Gmail mailer option within the Setup Wizard for the best safety and features. When you’ve set every little thing up on your email service web site, come again to this information to enable e-mail logging on your web site. Don’t overlook that the Elite version of WP Mail SMTP comes with a full White Glove Setup service for Mailgun or SMTP.com.
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Hello, I’m Kingsley. I am a graduate communicator working and living in Germany.
My interest is in corporate communication, online content development, digital marketing, programmatic advertising, business development, branding and campaign management.
I am continually taking steps towards self-improvement and have completed certification courses in Google Analytics, Adwords, Inbound Methodology, Email Marketing, Content Marketing, Inbound Sales, Sales Enablement, Statistics, Big Data and Mobile Site Management.
Excellence is not a skill, it's an attitude
- Ralph Marston
I previously worked with Jaduda GmbH - A member of The Goldbach Group, and now part of the Global Communications Team at Sandoz - A Novartis Division in Holzkirchen, Germany.
Best regards,
K
| 277,999
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TITLE: Convergence of a sequence with assumption that exponential subsequences converge?
QUESTION [7 upvotes]: Problem
One of my best friends asked me to think about the following problem:
Suppose a sequence $\{a_n\}_{n=1}^\infty$ satisfies $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each $\alpha>1$. Is it true that $\lim_{n\to\infty}a_n=0$?
He told me that the preceding proposition (if true) implies Kolmogorov's strong law of large numbers in probability theory. I don't know how, but it's irrelevant.
Thoughts
Suppose $E\subseteq(1,+\infty)$ is countable (for example, $E=(1,+\infty)\cap\mathbb Q$) and $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each $\alpha\in E$, we cannot conclude that $\lim_{n\to\infty}a_n=0$. In fact, we can choose an infinite set $S\subseteq\mathbb Z_{>0}$ such that $S\cap\{\lfloor\alpha^n\rfloor\colon n\in\mathbb Z_{>0}\}$ is finite for each $\alpha\in E$ as follows:
Suppose $E=\{\alpha_1,\alpha_2,\dotsc\}$. We choose $S$ inductively. Suppose $T_0=\mathbb Z_{>0}$. Given $T_{n-1}$, we set $s_n=\min T_{n-1}$ and $T_n=(T_{n-1}\setminus\{\lfloor\alpha_n^k\rfloor\colon k\in\mathbb Z_{>0}\})\setminus\{s_n\}$. By a density argument, it's easy to see that $T_n$ are infinite therefore the process doesn't terminate. Let $S=\{s_n\colon n\in\mathbb Z_{>0}\}$. It's apparent that $\#(S\cap\{\lfloor\alpha_n^m\rfloor\colon m\in\mathbb Z_{>0}\})\le n$.
Given $S$, we set $a_n=1/n$ if $n\not\in S$, and $a_n=1$ if $n\in S$, then $\lim_{n\to\infty}a_{\lfloor\alpha^n\rfloor}=0$ for each $\alpha\in E$, but $\lim_{n\to\infty}a_n$ doesn't exist.
Here we choose $S$ by a diagonal process, therefore we cannot mimic the construction when $E$ is uncountable. In fact, the falsehood of the original statement is equivalent to the existence of $S$, therefore it's essential combinatorial, or related to some topological structure of $\mathbb Z_{>0}$ (say, compactness or Baire category, etc). I have no idea on the general case. Any idea? Thanks!
REPLY [6 votes]: Here is an attempt. Hopefully, this is correct.
Fix $\varepsilon >0$. We are looking for an integer $N$ such that $\vert a_n\vert<\varepsilon$ for every $n\geq N$.
For any $k\in\mathbb N$, set
$$ A_k:=\left\{ \alpha>1;\; \forall l\geq k\;:\; \vert a_{\lfloor \alpha^l\rfloor}\vert<\varepsilon \right\} .$$
Then $(1,\infty)=\bigcup_{ n\in\mathbb N} A_k$ by assumption. By the Baire category theorem, it follows that at least on $A_k$ is not nowhere dense. So one can fix $1<u<v$ and $k_0\in\mathbb N$ such that $A_{k_0}\cap (u,v)$ is dense in $(u,v)$.
Now, choose an integer $K\geq k_0$ such that $v^k> u^{k+1}$ for all $k\geq K$; this is possible since $v>u$. Then $\bigcup_{k\geq K} (u^k,v^k)$ is the interval $(u^K,\infty)$ (the intervals overlap by the definition of $K$).
Finally, choose an integer $N>u^K$. Let us check that this $N$ works.
Take any integer $n\geq N$. Then one can write $n=\beta^k$ for some $\beta\in (u,v)$ and some integer $k\geq K$. Now, since $A_{k_0}$ is dense in $(u,v)$, one can find a point $\alpha\in A_{k_0}$ such that $\alpha >\beta$ and $\alpha$ is very close to $\beta$. Then, since the floor function is upper semi-continuous and $n=\beta^k$, we have $n=\lfloor \alpha^k\rfloor$. By the definition of $A_{k_0}$ and since $k\geq K\geq k_0$, it follows that $\vert a_n\vert<\varepsilon$, as required.
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\begin{document}
\title{Schur finiteness and nilpotency}
\author{Alessio Del Padrone\\delpadro@dima.unige.it\\Carlo Mazza\\carlo@math.ias.edu}
\date{July 5th, 2005}
\maketitle
\begin{abstract}
Let $\mathcal{A}$ be a $\Q$-linear pseudo-abelian rigid tensor
category. A notion of finiteness due to Kimura and (independently)
O'Sullivan guarantees that the ideal of numerically trivial
endomorphism of an object is nilpotent. We generalize this result
to special Schur-finite objects.
In particular, in the category of Chow motives,
if $X$ is a smooth projective variety
which satisfies the homological sign conjecture, then
Kimura-finiteness, a special Schur-finiteness,
and the nilpotency
of $CH^{ni}(X^i\times X^i)_{num}$ for all $i$ (where $n=\dim X$)
are all equivalent.
\end{abstract}
Let $\mathcal{A}$ be a {\sem pseudo-abelian tensor category},
i.e., a ``$\tens$-cat\'egorie rigide sur $F$''
as in \cite[2.2.2]{andremotifs}
in which idempotents split.
We have
$F$-linear {\sem trace} maps ${\rm tr}\colon \End_{\c{A}}(A)\To \End_{\c{A}}(\I)$
compatible with
$\otimes$-functors,
and
$F$-submodules of {\sem numerically trivial morphisms}
$
\c{N}(A_1,A_2):=\{f\in\Hom_{\c{A}}(A_1,A_2)\mid {\rm tr}(f\circ g)=0,\;\; {\rm for\;\;all}\;\;
g\in\Hom_{\c{A}}(A_2,A_1)\}.$
We assume that $F=\End_{\c{A}}(\I)$ and it
contains $\QQ$. If $F$ is a field, $\c{N}$ is the
biggest non trivial $\otimes$-ideal of $\c{A}$,
and so it contains any morphism annihilated
by some $\otimes$-functor.\\
\begin{exmp}\label{examplemotives}\cite[Ch. 4]{andremotifs}
Assume $F$ is a field.
For any admissible equivalence $\sim$ on algebraic cycles, motives of
smooth projective varieties over a field $k$
with coefficients in $F$
form such a category $\c{A}:=\c{M}_{\sim}(k)_F$.
If $X$ is a variety, we write
$\mathfrak{h}(X)$ for its motive.
For any $f\in\mathrm{End}_{\mathcal{A}}(\mathfrak{h}(X))$,
$
{\rm tr}(f)={\rm deg}(\Gamma_f\cdot \Delta_X)
$
and therefore
$
\c{N}(\mathfrak{h}(X))=\c{Z}^{{\rm dim}(X)}_\sim(X\times X)_{F,\rm num}
$
(numerically trivial
correspondences of degree zero).
If $\sim$ is finer than homological equivalence
then any Weil cohomology $H$ factors through a $\tens$-functor on
$\mathcal{A}$, and
$
{\rm tr}(f)=
\sum_{j} (-1)^j{\rm Tr}(f|H^j(X))
$
by the Lefschetz
formula.\\
\end{exmp}
Recall that the partitions $\lambda$ of an integer $n$
give a complete set of mutually orthogonal central idempotents
$ d_\lambda:=\frac{\dim V_\lambda}{n!}\sum_{\sigma\in \Sigma_n}
\chi_\lambda(\sigma)\sigma$
in the group algebra $\Q\Sim_n$ (see \cite{FH}).
We define an endofunctor on $\c{A}$ by setting $S_\lambda(A)=d_\lambda(A^{\tens n})$. This is a multiple of the classical Schur functor corresponding to $\lambda$. In particular, we define $\mathrm{Sym}^n(A)=S_{(n)}(A)$
and $\Lambda^n(A)=S_{(1^n)}(A)$.
The following definitions are directly inspired by
\cite{delschur} and \cite{kimura} (see \cite{yveskahn},
\cite{gp2}, and \cite{maz} for further reference).\\
\begin{defn}
An object $A$ of $\calA$ is \textbf{Schur-finite} if there
is a partition $\lambda$ such that
$S_\lambda(A)=0$.
If $S_\lambda(A)=0$ with
$\lambda$ of the form $(n)$ (respectively, $\lambda=(1^n)$) then $A$ is called
\textbf{odd} (respectively, \textbf{even}).
We say that $A$ is \textbf{Kimura-finite}
if $A=A_+\oplus A_-$ with $A_+$ even and $A_-$ odd.\\
\end{defn}
Every Kimura-finite object is Schur-finite, but the converse
fails, for example, in the category of super-representations of $GL(p|q)$.
In \cite[7.5]{kimura} and \cite[9.1.14]{yveskahn}
it was proven that if $A$ is a Kimura-finite object then
the ideal $\mathcal{N}(A)$ is nilpotent.
In the case of example \ref{examplemotives}, an interesting
consequence of the nilpotence of $\mathcal{N}(M)$ is
that a summand $N$ of $M$ is zero if and only if its
cohomology is zero (the idempotent defining $N$ must then be nilpotent).
The nilpotency was used in \cite[Theorem 7]{gp2} to
show the equivalence of Bloch's conjecture for a smooth projective suface $X$ with $p_g=0$
and the Kimura-finiteness of the motive of $X$, improving \cite[7.7]{kimura}.
Albeit in general Schur-finiteness
is not sufficient to get the nilpotency of $\mathcal{N}(A)$
(see \cite[10.1.1]{yveskahn}),
we will identify additional conditions which imply the nilpotency.
In the category of motives we will show that for a motive
which is Kimura-finite modulo homological equivalence, the Kimura-finiteness
modulo rational equivalence is equivalent to the Schur-finiteness
for a particular rectangle.
\section{A technical result}
\begin{thm}\label{mainthm}
Suppose that $S_\lambda(A)=0$ for a partition
$\lambda$ of $n\geq 2$ with $a_\lambda$ rows and $b_\lambda$ columns. Let
$s:=a_\lambda+b_\lambda-1$ be the length of its biggest hook $\nu$, and $r:=n-s$.
Assume that either $\lambda$ is a hook or
that there is a $g\in\End_{\c{A}}(A)$ with trace
$t:=\mathrm{tr}(g)=\dots=\mathrm{tr}(g^{\circ r})$, and
$t\not\in \{-(b_\lambda-2),\dots,a_\lambda-2\}$. Then
$f^{\circ (s-1)}=0$ for each $f\in\mathcal{N}(A)$, and so
$\mathcal{N}(A)$ is nilpotent.
\end{thm}
\begin{proof}
The last statement follows from
\cite[7.2.8]{yveskahn}:
$\mathcal{N}(A)^{2^{s-1}-1}=0$.
For $\sigma\in \Sigma_n$, we index
the corresponding decomposition
of $\{1,\ldots,n\}$
into disjoint cycles $\gamma_1,\ldots,\gamma_n$ so that
the support of $\gamma_1$ contains $1$; moreover
we define $l_i$ to be the order
of the cycle $\gamma_i$, and $L=L(\sigma):={\rm max}_i\{l_i \}$ to be the
maximum length of the cycles of $\sigma$.
As $S_\lambda(A)=0$ we have
$\sum_\sigma \chi_\lambda(\sigma)\cdot
\sigma \circ f_1\tens \cdots \tens f_n=0
$
for any
$f_1,\dots,f_n\in \End_{\c{A}}(A)$.
By the Murnaghan-Nakayama rule (see \cite[Problem 4.45]{FH})
$\chi_\lambda(\sigma)=0$ if $L(\sigma)>s$.
Hence \cite[7.2.6]{yveskahn}
with $A_1=\cdots=A_n=A$, gives that in $\End_{\c{A}}(A)$
\[
\sum_{\sigma\in \Sigma_n:\;L(\sigma)\leq s} \chi_\lambda(\sigma)\cdot
t_\sigma\cdot f_{\gamma_1} = 0,
\]
where
$f_{\gamma_1}:=
f_{\gamma_1^{l_1-1}(1)}\circ \cdots \circ f_{\gamma_1(1)}\circ f_1$,
$t_\sigma:=\prod_{j=2}^q t_{\sigma,j}$,
and
$
t_{\sigma,j}:= {\rm tr}(f_{{\gamma_j}^{l_j-1}(k_j)}\circ \cdots \circ
f_{\gamma_j(k_j)}\circ f_{k_j})
$
with $k_j$ any element in the support of $\gamma_j$
(if $l_1=n$, i.e. $q=1$, then $t_\sigma=1$).
Set
$f_1:=\cid{A}$ and $f_2=\dots=f_s:=f$
(still no restrictions on $f_{s+1},\ldots, f_n$).
If ${\rm Supp}(\gamma_1)\subsetneq \{1,\dots, s\}$, not all
of the $f$'s are in the composition
$f_{\gamma_1}$, hence at least one of them must appear in a trace
${\rm tr}(f_{{\gamma_j}^{l_j-1}(k_j)}\circ \cdots \circ
f_{\gamma_j(k_j)}\circ f_{k_j})$. But $f$ is numerically trivial, so
$t_\sigma=0$
for any such $\sigma$, and
$$
0=\sum_{\sigma\in \Sigma_n:\;{\rm Supp}(\gamma_1)=\{1,\dots, s\}}
\chi_\lambda(\sigma)\cdot
t_\sigma \cdot f_{\gamma_1}
=
\left(\sum_{\sigma\in \Sigma_n:\;{\rm Supp}(\gamma_1)=\{1,\dots, s\}}
\chi_\lambda(\sigma)\cdot
t_\sigma\right)f^{\circ(s-1)}=x\cdot f^{\circ(s-1)},
$$
where
$
x:=\sum_{\sigma\in \Sigma_n:\;{\rm Supp}(\gamma_1)=\{1,\dots, s\}}
\chi_\lambda(\sigma)\cdot
t_\sigma\in F$.
It is enough to show $x\not=0$ for some choice
of the $f_i$'s.
If
$r=0$ then $\lambda=\nu=(n-j,1^j)$ is itself a hook,
$t_\sigma=1$ for any
$\sigma$ with $l_1=n$ and by
\cite[Exercise 4.16]{FH} $x$ is just $(n-1)!(-1)^j\not=0$,
hence $\c{N}(A)$ is nilpotent.
If $\lambda$ is not a hook let $\delta:=\lambda\setminus\nu$.
The element $x\in F$
is a sum over
$\sigma=\gamma_1\circ\sigma^\prime$
such that $\gamma_1$ is an $s$-cycle of $\{1,\dots,s\}$ and
$\sigma^\prime$ is a permutation of $\{s+1,\dots,n\}$, so
by Murnaghan-Nakayama
$
\chi_{\lambda}(\sigma)=
\chi_{\lambda\setminus \nu}(\sigma^\prime),
$
and $
x=(-1)^{a_\delta-1}\;
|\{s-{\rm cycles\;\, of\;\,} \Sigma_n\}|
\sum_{\sigma^\prime\in \Sigma_r}
\chi_{\delta}(\sigma^\prime)\cdot t_\sigma$.
Thus we are reduced to study elements of the form
\[
y(\delta;g_1,\dots,g_{r}):=\sum_{\sigma\in \Sigma_{r}}
\chi_{\delta}(\sigma)\cdot
\prod_{j=1}^q t_{\sigma,j},
\]
where we can choose freely $g_1,\dots,g_{r}\in\End_{\c{A}}(A)$.
Take $g\in\End_{\c{A}}(A)$ as in the hypothesis, then
$y(\delta;g,\dots,g)=
\sum_{\sigma\in \Sigma_{r}}
\chi_{\delta}(\sigma)\cdot t^{|{\rm cycles\;\; of \;\,} \sigma|}
$
is the polynomial in $t={\rm tr}(g)$ called the
{\sem content polynomial}
of $\delta$. It decomposes as $y(\delta;g)=
\chi_{\delta}(\cid{\Sigma_{r}})\cdot\prod_{(i,j)\in \delta}(t+j-i)$,
then $y(\delta;g)=0$
if and only if
${\rm tr}(g)\in \{-(b_\delta-1),\dots,a_\delta-1\}
\subseteq
\{-(b_\lambda-2),\dots,a_\lambda-2\}
$.
By hypothesis, there is a $g$ such that $y(\delta;g)\not=0$, which
implies that $x\not =0$, which in turn implies that $f$ is nilpotent.
Hence the theorem is proven.
\end{proof}
\begin{rem}[B. Kahn]
The existence of a
$g\in\End_{\c{A}}(A)$ with ${\rm tr}(g)\neq 0$ is not enough
to ensure the nilpotency of $\c{N}(A)$ with $A$ Schur-finite.
In \cite[10.1.1]{yveskahn} it is exhibited a
non-zero Schur-finite object $A^\prime$ with $\c{N}(A^\prime)=\End_{\c{A}}(A^\prime)$:
it suffices to look at $A:=A^\prime\oplus\I^n$.\\
\end{rem}
\begin{conj}\label{con}
From numerical evidence (\cite{GAP4}) we conjecture a stronger version of
Theorem \ref{mainthm}. Let
$A$ be an object with two
endomorphisms $\pi_1$ and $\pi_2$
such that $a:=\mathrm{tr}(\pi_1)=\mathrm{tr}(\pi_1^{\circ i})$
for all $i$,
$b:=\mathrm{tr}(\pi_2)=\mathrm{tr}(\pi_2^{\circ j})$ for all $j$,
and $\mathrm{tr}(\pi_1^{\circ i}\circ \pi_2^{\circ j})=0$ for all $i$ and $j$.
If $S_\lambda(A)=0$ where $\lambda\not\supset (b+2)^{a+2}$, then
$y(\lambda\setminus\nu;\alpha_1\pi_1+\alpha_2\pi_2)\not =0$ (as a polynomial
in $\alpha_1$ and $\alpha_2$) and hence $\mathcal{N}(A)$ is nilpotent.
\end{conj}
\section{Motives and nilpotency}
Let now $\c{A}$ be the category of Chow motives $\c{M}_{rat}(k)_\QQ$ (example \ref{examplemotives}),
let $H$ be any Weil cohomology, and
let $X$ be a smooth
projective variety.
The cohomology $H(X)$
is a super vector space
of dimension $(d_{ev},d_{odd})$, and
we set $\lambda_{H(X)}:=((d_{odd}+1)^{d_{ev}+1})$
(the rectangle with $d_{odd}+1$ columns and $d_{ev}+1$ rows).
By \cite[1.9]{delschur}, $S_\lambda(H(X))\not=0$ if and only if
$\lambda\not\supset\lambda_{H(X)}$.
Hence, $S_\lambda(\mathfrak{h}(X))\not=0$ if $\lambda\not\supset
\lambda_{H(X)}$. So $S_\lambda(\mathfrak{h}(X))=0$ implies that $\lambda\supset
\lambda_{H(X)}$.
Recall the ``homological sign conjecture''
(due to Jannsen, see \cite[5.1.3]{andremotifs}): we say that
$X$ satisfies the conjecture $C^+(X)$ if
the projections on the even
and the odd part of the cohomology are algebraic.
This conjecture is stable under products, and it holds true,
with respect to classical cohomologies,
for abelian varieties and
smooth projective varieties of dimension at most two.
It can be shown that $C^+(X)$ is equivalent to the Kimura-finiteness
of the motive of $X$ modulo
homological equivalence.\\
\begin{prop}\label{mainprop}
Let $X$ be a smooth projective variety,
and let $\lambda$ be a partition with at most
$d_{ev}+1$ rows or $d_{odd}+1$ columns.
If $S_{\lambda}(\mathfrak{h}(X))=0$
(and hence $\lambda\supset\lambda_{H(X)}$)
and $C^+(X)$ holds,
then $\c{N}(\mathfrak{h}(X))$ is nilpotent.
Moreover, if $X$ is a surface with $p_g=0$, Bloch's conjecture holds for $X$.
\end{prop}
\begin{proof}
By $C^+(X)$ there are two cycles $\pi_+$ and $\pi_-$ inducing
the projections on the even and odd cohomology. Then
$d_{ev}=\mathrm{tr}(\pi_+)=\mathrm{tr}(\pi_+^{\circ i})$ for all $i$,
and $-d_{odd}=\mathrm{tr}(\pi_-)=\mathrm{tr}(\pi_-^{\circ j})$ for all $j$.
Then either $\pi_+$ or $\pi_-$ satisfies the condition of
Theorem \ref{mainthm}, and therefore $\c{N}(\mathfrak{h}(X))$ is nilpotent.
Bloch's conjecture is now a formal consequence of \cite[7.6 and 7.7]{kimura}.
\end{proof}
\begin{thm}
Let $X$ be a smooth projective variety.
Under $C^+(X)$ the following are equivalent:
\[ 1)\ \text{$\mathfrak{h}(X)$ is Kimura-finite;}\qquad
2)\ \text{$S_{\lambda_{H(X)}}(\mathfrak{h}(X))=0$;}\qquad
3)\ \text{$\mathcal{N}(\mathfrak{h}(X^n))$ is nilpotent for all $n\geq 1$.}
\]
\end{thm}
\begin{proof}
It is easy to show that $1\Rightarrow 2$.
For $3\Rightarrow 1$ we proceed as follows. As $C^+(X)$ holds and $\mathcal{N}(\mathfrak{h}(X))$ is nilpotent,
then there exist two motives
$X_+$ and $X_-$ whose cohomologies are exactly the even and the odd
part of $H(X)$. It is now easy to prove
that $\mathfrak{h}(X)=M_+\oplus M_-$ with $M_+$ even and
$M_-$ odd
because it will be enough to check it in cohomology.
We need to verify $2\Rightarrow 3$.
Assume that $S_{\lambda_{H(X)}} (\mathfrak{h}(X))=0$.
From the proof of
\cite[Cor. 1.13]{delschur}, we find that $S_{\lambda_{H(X^n)}}(\mathfrak{h}(X^n))=
S_{\lambda_{H(X^n)}}(\mathfrak{h}(X)^{\tens n})=0$.
Since $C^+(X^n)$ holds true,
Proposition \ref{mainprop} gives that $\mathcal{N}(\mathfrak{h}(X^n))$ is nilpotent.
\end{proof}
If Conjecture \ref{con} is true, then Bloch's conjecture holds
for any smooth projective surface $X$ with $p_g=0$
such that $S_\lambda(\mathfrak{h}(X))=0$ for $\lambda\not\supset
(d_{odd}(X)+2)^{d_{ev}(X)+2}$.
\section*{Acknowledgments}
This article was inspired by the preprint \cite{Gul2} by V. Guletski{\u\i}.
Both authors would like to thank the Institut de Math\'ematiques de Jussieu
for hospitality while part of this manuscript was written.
For financial support,
the first author would like to thank the Fondazione Carige
and the second author would like to thank
Sergio Serapioni (Honorary President of the Societ\`a
Trentina Lieviti, Trento, Italy).
\bibliographystyle{amsalpha}
\bibliography{paper}
\end{document}
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\section{Pseudo-randomness of $X$}\label{sec:properties}
In this section, we prove that basically as long as the noise model is subgaussian and has variance 1(which generalizes the standard Bernoulli and Gaussian distributions), after normalizing the rows of the data matrix $X\sim H_0$, it satisfies the pseudorandomness condition~\ref{con:ps-random}.
\begin{theorem}\label{lem:check-ps}
Suppose independent random variables $X_1,\dots,X_p\in \mathbb{R}^n$ satisfy that for any $i$, $X_i$ has a i.i.d entries with mean zero, variance 1, and subgaussian variance proxy\footnote{A real random variable $X$ is subgaussian with variance proxy $\sigma^2$ if it has similar tail behavior as gaussian distribution with variance $\sigma^2$, and formally if for any $t \in \mathbb{R}$, $\Exp[\exp(tX)]\le \exp(t^2\sigma^2/2)$ } $O(1)$, then the matrix $\bar{X}$ with $\frac{X_i^T}{\|X_i\|}$ as rows satisfies the pseudorandomness condition~\ref{con:ps-random}.
\end{theorem}
The proof of the Theorem relies on the following Proposition~\label{con:randomvariable} and Theorem~\ref{lem:checking-ps-cond}. The proposition says that $\frac{X_i^T}{\|X_i\|}$ still satisfies {\em good} properties like symmetry and that each entries has a subgaussian tail, even though its entries are no longer independent due to normalization. These properties will be encapsulated in the definition of a {\em good} random variable following the proposition. Then we prove in Theorem~\ref{lem:checking-ps-cond} that these properties suffice for establishing the pseudorandomness Condition~\ref{con:ps-random} with high probability. We will heavily use the $\psi_{\alpha}$-Orlicz norm (denoted $\|\cdot\|_{\psi_{\alpha}}$) of a random variable, defined in Definition~\ref{def:orlicznorm}, and its properties, summarized in the next (toolbox) section. Intuitively, $\|\cdot\|_{\psi_2}$ norm is a succinct and convenient way to capture the ``subgaussianity'' of a random variable.
\begin{proposition}~\label{con:randomvariable}
Suppose $y\in \mathbb{R}^{n}$ has i.i.d entries with variance 1 and mean zero, and gaussian variance proxy $O(1)$, then random variable $x = \frac{y}{\|y\|}$ satisfies the following properties:
\begin{enumerate}
\item $\|x\|^2 = n$, almost surely.
\item for any vector $a\in \mathbb{R}^n$ with $\|a\|^2 \le 2n$, $\|\inner{x,a}\|^2_{\psi_{2}} \le O(n)$.
\item $\||x|_{\infty}\|_{\psi_{2}} \le \tilO(1)$
\item $\Exp[x_i^2] = 1$, $\Exp[x_i^4] = C_4$, and $\Exp[x_i^2x_j^2] = C_{2,2} $ for all $i$ and $j\neq i$, where $C_4, C_{2,2}
= O(1)$ are constants that don't depend on $i,j$
\item For any monomial $x^{\alpha}$ with an odd degree, $\Exp[x^{\alpha}] = 0$.
\end{enumerate}
\end{proposition}
\noindent For simplicity, we call a random variable {\em good }if it satisfies the five properties listed in the proposition above. Goodness will be invoked in most statements below.
\begin{definition}[goodness]
A random variable $x\in \mathbb{R}^n$ is called \textit{good}, if it satisfies the conclusion of Proposition~\ref{con:randomvariable}.
\end{definition}
\noindent We will show a random matrix $X$ with {\em good} rows satisfies the pseudo-randomness Condition~\ref{con:ps-random} with high probability.
\begin{theorem}\label{lem:checking-ps-cond}
Suppose independent $n$-dimensional random vectors $X_1,\dots,X_p$ with $p\ge n$ are all \textit{good}, then
$X_1,\dots,X_p$ satisfies the pseudorandomness condition~\ref{con:ps-random} with high probability.
\end{theorem}
The general approach to prove the theorem is just to use the concentration of measure. The only caveat here is that in most of cases, the random variables that we are dealing with are not bounded a.s. so we can't use Chernoff bound or Bernstein inequality directly. However, though these random variables are not bounded a.s., they typically have a light tail, that is, their $\psi_{\alpha}$ norms can be bounded. Then we are going to apply Theorem~\ref{thm:psi-bounded-rv-concentration} of Ledoux and Talagrand's, a extended version of Bernstein inequality with only $\psi_{\alpha}$ norm boundedness required. We will also use other known technical results listed in the toolbox Section~\ref{sec:toolbox}.
\begin{proof}[Proof of Theorem~\ref{lem:checking-ps-cond}]
Equation~\eqref{eqn:prop:normsqaure} and~\eqref{eqn:prop:inner} follows the assumptions on $X_i$'s and union bound. Equation~\eqref{eqn:prop:xixl3xjxl} is proved in Lemma~\ref{lem:xu3xv} by taking $u = X_s$ and $v= X_t$ and view the rest of $X_i$;s as $Z_j$'s in the statement of Lemma~\ref{lem:xu3xv}.
Equation~\eqref{eqn:prop:sumiljlsltl} is proved in Lemma~\ref{lem:prop:sumiljlsltl}, ~\eqref{eqn:prop:sumxixsxixt} in Lemma~\ref{lem:prop:sumxixsxixt-q},
~\eqref{eqn:prop:sumxixl2xtxlxsxl} in Lemma~\ref{lem:sumilil2sltl}, and equation~\ref{eqn:prop:obj} is proved in Lemma~\ref{lem:xxtfro}.
\end{proof}
\begin{lemma}\label{lem:xu3xv}
For any \textit{good} random variable $x$,
we have that for fixed $u,v$ with $\|u\|^2 = \|v\|^2 = n$
, $|u|_{\infty}\le \tilO(1)$, $|v|_{\infty}\le \tilO(1)$, and $\inner{u,v}\le \tilO(\sqrt{n})$,
$$\left|\Exp\left[\inner{x,u}^3\inner{x,v}\right] \right|\le \tilO(n^{1.5})$$
and moreover, for $p\ge n$ and a sequence of \textit{good} independent random variables $Z_1,\dots,Z_p$,
we have that with high probability,
$$\left|\sum_{i=1}^p \inner{Z_i,u}^3\inner{Z_i,v} \right|\le \tilO(n^{1.5}p)$$
\end{lemma}
\begin{proof}
We calculate the expectation as follows
\begin{align*}
\Exp\left[\inner{x,u}^3\inner{x,v}\right] & = \Exp\left[\left(\sum_{i}u_i^2x_i^2 + 2\sum_{i< j}u_iu_jx_ix_j\right)\left(\sum_{i}v_iu_ix_i^2 + \sum_{i\neq j}u_iv_jx_ix_j\right)\right] \\
&= \Exp\left[\sum_i u_i^3v_ix_i^4\right] + \Exp\left[\sum_{i\neq j}u_i^2u_jv_jx_i^2x_j^2\right] + \Exp\left[\sum_{i \neq j}u_iu_j(u_iv_j+v_iu_j)x_i^2x_j^2\right]\\
& = \left(C_4-C_{2,2}\right)\sum_i u_i^3v_i + C_{2,2}n\sum_{i}u_iv_i + C_{2,2}\sum_{i \neq j}(u_i^2u_jv_j+u_j^2u_iv_i)\\
& = \left(C_4-3C_{2,2}\right)\sum_i u_i^3v_i +3 C_{2,2}n\sum_{i}u_iv_i
\end{align*}
Therefore by our assumption on $u$ and $v$ we obtain that
\begin{align*}
\left|\Exp\left[\inner{x,u}^3\inner{x,v}\right] \right|\le \tilO(n)+ O(n)|\inner{u,v}|\le \tilO(n^{1.5})
\end{align*}
Now we prove the second statement.
Since $\orlicznorm{\inner{Z_i,u}}{2}\le O(\sqrt{n})$, by Lemma~\ref{lem:psi-norm-product} we have that $\orlicznorm{\inner{Z_i,u}^3\inner{Z_i,v}}{1/2}\le O(n^2)$, and it follows Lemma~\ref{lem:psi-norm-mean-shift} that $\orlicznorm{\inner{Z_i,u}^3\inner{Z_i,v}-\Exp\left[\inner{Z_i,u}^3\inner{Z_i,v}\right]}{1/2}\le O(n^2)$
Then by Lemma~\ref{thm:psi-bounded-rv-concentration} we obtain that with high probability,
$$\sum_{i=1}^p \inner{Z_i,u}^3\inner{Z_i,v} - \Exp\left[\sum_{i=1}^p \inner{Z_i,u}^3\inner{Z_i,v} \right] \le \tilO(n^2\sqrt{p})$$
Note that we have proved that $\left|\Exp\left[\sum_{i=1}^p \inner{Z_i,u}^3\inner{Z_i,v}\right] \right|=\tilO(n^{1.5})$, therefore we obtain the desired result.
\end{proof}
\begin{lemma}\label{lem:prop:sumiljlsltl}
Suppose $p\ge n$ and $X_1,\dots,X_p$ are \textit{good} independent random variables,
then with high probability, for any distinct $i,j,s,t$,
$$ \left|\sum_{\ell\in [p]}\inner{X_i,X_{\ell}}\inner{X_j,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}}\right| \le \tilO( n^2\sqrt{p})$$
\end{lemma}
\begin{proof}
Fixing $i,j,s,t$, we can write
\begin{align*}
\sum_{\ell\in [p]}\inner{X_i,X_{\ell}}\inner{X_j,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} & = \sum_{\ell\in [p]\backslash \{i,j,s,t\}}\inner{X_i,X_{\ell}}\inner{X_j,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} \\
& + n\inner{X_j,X_{i}}\inner{X_s,X_{i}}\inner{X_t,X_{i}} + n\inner{X_i,X_{j}}\inner{X_s,X_{j}}\inner{X_t,X_{j}} \\
&+ n\inner{X_i,X_{s}}\inner{X_j,X_{s}}\inner{X_t,X_{s}} + n\inner{X_i,X_{t}}\inner{X_j,X_{t}}\inner{X_s,X_{t}}
\end{align*}
Using Lemma~\ref{lem:xaxbxcxd}, the first term on RHS is bounded by $\tilO(n^2\sqrt{p})$ with high probability over the randomness of $X_{\ell}, \ell\in [p]\backslash \{i,j,s,t\}$. The rest of the four terms are bounded by $\tilO(n^{2.5})$. Therefore putting together $\|\sum_{\ell\in [p]}\inner{X_i,X_{\ell}}\inner{X_j,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}}\| \le \widetilde{O}(n^2\sqrt{p})$ for any fixed $i,j,s,t$ with high probability and taking union bound we get the result.
\end{proof}
\begin{lemma}\label{lem:xaxbxcxd}
For any good random variable $x$,
and for fixed $a,b,c, d$ such that $\max \{|a|_{\infty}, |b|_{\infty}, |c|_{\infty}, |d|_{\infty}\} = \tilO(1)$, and all the pair-wise inner products between $a,b,c,d$ have magnitude at most $\tilO(\sqrt{n})$, we have that
$$\left|\Exp\left[\inner{x,a}\inner{x,b}\inner{x,c}\inner{x,d}\right]\right| = \widetilde{O}(n)$$
and moreover, for $p\ge n$ and a sequence independent random variable $Z_1,\dots,Z_p$ such that each $Z_i$ satisfies the conclusion of proposition~\ref{con:randomvariable},
we have that with high probability,
$$\left|\sum_{i=1}^p \inner{Z_i,a}\inner{Z_i,v}\inner{Z_i,c}\inner{Z_i,d} \right| \le \tilO(n^2\sqrt{p})$$
\end{lemma}
\begin{proof}
We calculate the mean
\begin{align*}
\Exp\left[\inner{x,a}\inner{x,b}\inner{x,c}\inner{x,d}\right] &= \Exp\left[\sum_{i\in [p]}a_ib_ic_id_ix_i^4 + \left\{\sum_{i\neq j}a_ib_ic_jd_jx_i^2x_j^2 \right\} \right]
\end{align*}
where we use $\left\{\sum_{i\neq j}a_ib_ic_jd_jx_i^2x_j^2 \right\}$ to denote the sum of $a_ib_ic_jd_jx_i^2x_j^2$ and all its permutations with repect to $a,b,c,d$.
Note that $$\left|\Exp\left[\sum_{i\neq j}a_ib_ic_jd_jx_i^2x_j^2 \right]\right|= C_{2,2}\left|\inner{a,b}\inner{c,d} - \sum_{i\in [p]}a_ib_ic_id_i\right| \le \widetilde{O}(n)$$
and
$$ \left|\Exp\left[\sum_{i\in [p]}a_ib_ic_id_ix_i^4\right]\right| = \left|C_4 \sum_{i\in [p]}a_ib_ic_id_i \right|\le\tilO(n)$$
and therefore we have $ \left|\Exp\left[\inner{x,a}\inner{x,b}\inner{x,c}\inner{x,d}\right] \right| \le \tilO(n)$.
Since $\inner{x,a}$ has $\psi_2$ norm $\sqrt{n}$ and similar for the other three terms, we have that by Lemma~\ref{lem:psi-norm-product} that
$\|\inner{x,a}\inner{x,b}\inner{x,c}\inner{x,d}\|_{\psi_{1/2}}\le O(n^2)$. Therefore using Theorem~\ref{thm:psi-bounded-rv-concentration} we have that
$$\left\|\sum_{i=1}^p \inner{Z_i,a}\inner{Z_i,v}\inner{Z_i,c}\inner{Z_i,d} - \Exp[\sum_{i=1}^p \inner{Z_i,a}\inner{Z_i,v}\inner{Z_i,c}\inner{Z_i,d}]\right\|_{\psi_{1/2}} \le \tilO(n^2\sqrt{p})$$
\end{proof}
\begin{lemma}\label{lem:prop:sumxixsxixt-q}
Suppose $p\ge n$ and $X_1,\dots,X_p$ are \textit{good} independent random variables,
then with high probability, for any distinct $s,t$,
$$\left|\sum_{i\in[p]}\inner{X_i,X_s}\inner{X_i,X_t}\right|\le \tilO(p\sqrt{n})$$
\end{lemma}
\begin{proof}
With high probability over the randomness of $X_i, i\in[p]\backslash \{s,t\}$,
$$\sum_{i\in[p]}\inner{X_i,X_s}\inner{X_i,X_t} = \sum_{i\in[p]\backslash \{s,t\}}\inner{X_i,X_s}\inner{X_i,X_t} + 2\inner{X_s,X_t} \le \widetilde{O}(p\sqrt{n}) + \widetilde{O}(\sqrt{n})$$
where the last inequality is by Lemma~\ref{lem:xuxv}. Taking union bound we complete the proof.
\end{proof}
\begin{lemma}\label{lem:xuxv}
For $p\ge n$ and a sequence of \textit{good} independent random variable $Z_1,\dots,Z_p$,
and any two fixed vectors $u,v $ with $|u|_{\infty}\le \tilO(1)$ and $|v|_{\infty}\le \tilO(1)$, and $\inner{u,v}\le \tilO(\sqrt{n})$,
we have that with high probability,
$$\left|\sum_{i\in[p]}\inner{Z_i,u}\inner{Z_i,v}\right|\le \tilO(p\sqrt{n})$$
\end{lemma}
\begin{proof}
$\Exp[\inner{Z_i,u}\inner{Z_i,v}] = \inner{u,v} \le \tilO(\sqrt{n})$, and therefore$ \left|\Exp\left[\sum_{i\in [p]}\inner{Z_i,u}\inner{Z_i,v} \right]\right|\le \tilO(p\sqrt{n})$. Note that $\orlicznorm{\inner{Z_i,u}}{2}\le O(\sqrt{n})$
and therefore $\orlicznorm{\inner{Z_i,u}\inner{Z_i,v}}{1}\le O(n)$.
By Theorem~\ref{thm:psi-bounded-rv-concentration}, we have the desired result.
\end{proof}
\begin{lemma}\label{lem:sumilil2sltl}
Suppose $p\ge n$ and $X_1,\dots,X_p$ are \textit{good} independent random variables,
then with high probability, for any distinct $s,t$,
$$\sum_{i,\ell\in [p]}\inner{X_i,X_{\ell}}\inner{X_i,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} \le \tilO(p^2n^{1.5})$$
\end{lemma}
\begin{proof}
We expand the target as follows:
\begin{align*}
\sum_{i,\ell\in [p]}\inner{X_i,X_{\ell}}\inner{X_i,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} &= \sum_{i\in [p], \ell\in [p]\backslash s\cup t} \inner{X_i,X_{\ell}}\inner{X_i,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} \\
& + \sum_{i}\inner{X_i,X_{s}}^2\inner{X_s,X_s}\inner{X_t,X_{s}} + \sum_{i}\inner{X_i,X_{t}}^2\inner{X_t,X_t}\inner{X_s,X_{t}} \\
&= \sum_{i\in [p]\backslash s\cup t, \ell\in [p]\backslash s\cup t} \inner{X_i,X_{\ell}}\inner{X_i,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}} \\
& + \sum_{i}\inner{X_i,X_{s}}^2\inner{X_s,X_s}\inner{X_t,X_{s}} + \sum_{i}\inner{X_i,X_{t}}^2\inner{X_t,X_t}\inner{X_s,X_{t}} \\
& + \sum_{\ell\in [p]\backslash s\cup t}\inner{X_s,X_{\ell}}^3\inner{X_{\ell},X_t} + \sum_{\ell\in [p]\backslash s\cup t}\inner{X_t,X_{\ell}}^3\inner{X_{\ell},X_s}
\end{align*}
By equation~\eqref{eqn:prop:xixl3xjxl}, we have that
$$\sum_{\ell\in [p]\backslash s\cup t}\inner{X_s,X_{\ell}}^3\inner{X_{\ell},X_t}\le \tilO(pn^{1.5})$$
Since $\inner{X_s,X_t}\le \tilO(\sqrt{n})$ and $\sum_{i\in [p]}\inner{X_i,X_s} ^2= n^2 + \sum_{i\neq s}\inner{X_i,X_s}^2 \le \tilO(np)$, we have that
$$\sum_{i}\inner{X_i,X_{s}}^2\inner{X_s,X_s}\inner{X_t,X_{s}}\le \tilO(pn^{2.5})$$
Invoking Lemma~\ref{lem:zizj2zjuzjv} with $u = X_s$ and $v =X_t$ fixed and view $X_{\ell}, \ell\in [p]\backslash s\cup t$ as random variables $Z_i$'s, we have that with high probability,
$$\sum_{i\in [p]\backslash s\cup t, \ell\in [p]\backslash s\cup t} \inner{X_i,X_{\ell}}\inner{X_i,X_{\ell}}\inner{X_s,X_{\ell}}\inner{X_t,X_{\ell}}\le \widetilde{O}(p^2n^{1.5})$$
Hence combining the three equations above, taking union bound over all choices of $s,t$, we obtain the desired result.
\end{proof}
\begin{lemma}\label{lem:zizj2zjuzjv}
For $p\ge n$ and a sequence of \textit{good} independent random variables $Z_1,\dots,Z_p$,
and any two fixed vectors $u,v $ with $|u|_{\infty}\le \tilO(1)$ and $|v|_{\infty}\le \tilO(1)$, and $\inner{u,v}\le \tilO(\sqrt{n})$,
we have that with high probability,
\begin{equation*}
\sum_{i\in [p]}\sum_{j\in [p]}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} \le \widetilde{O}(p^2n^{1.5})
\end{equation*}
\end{lemma}
\begin{proof}
We first extract the consider those cases with $i = j$ separately by expanding
\begin{eqnarray}
\sum_{i\in [p]}\sum_{j\in [p]}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} &=& \sum_{i\neq j}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} +\sum_{i}\inner{Z_i,Z_{i}}^2 \inner{Z_i,u}\inner{Z_i,v} \nonumber\\
&=& \sum_{i\neq j}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} +\widetilde{O}(pn^{2.5})\label{eqn:inter1}
\end{eqnarray}
where the the last line uses Lemma~\ref{lem:xuxv}.
Let $Y_1,\dots,Y_p$ are independent random variables that have the same distribution as $Z_1,\dots,Z_p$, respectively, then by Theorem~\ref{thm:decoupling}, we can decouple the sum of functions of $Z_i,jZ_j$ into a sum that of functions of $Z_i$ and $Y_j$,
\begin{eqnarray*}
\Pr\left[\sum_{i\neq j}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} \ge t\right] \le C \Pr\left[\sum_{i\neq j}\inner{Y_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} \ge t/C\right]
\end{eqnarray*}
Now we can invoke Lemma~\ref{lem:yz2zuzv} which deals with RHS of the equation above, and obtain that with high probability $$\sum_{i\neq j}\inner{Y_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v}\le \tilO(p^2n^{1.5})$$
Therefore, with high probability, $$\sum_{i\neq j}\inner{Z_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} \le \tilO(p^2n^{1.5})$$Then combine with equation~\eqref{eqn:inter1} we obtain the desired result.
\end{proof}
\begin{lemma}\label{lem:yz2zuzv}
For $p\ge n$ and a sequence of \textit{good} independent random variables $Z_1,\dots,Z_p$,
let $Y_1,\dots,Y_p$ be independent random variables which have the same distribution as $Z_1,\dots,Z_p$, respectively,
then for any two fixed vectors $u,v $ with $|u|_{\infty}\le \tilO(1)$ and $|v|_{\infty}\le \tilO(1)$, and $\inner{u,v}\le \tilO(\sqrt{n})$, with high probability,
\begin{equation*}
\sum_{i\in [p]}\sum_{j\in [p]}\inner{Y_i,Z_{j}}^2 \inner{Z_j,u}\inner{Z_j,v} \le \widetilde{O}(p^2n^{1.5})
\end{equation*}
\end{lemma}
\begin{proof}
Let $B = \sum_{i\in [p]}Y_iY_i^T$. Therefore by Lemma~\ref{lem:spectralyy}, we have that with high probability over the randomness of $Y$, $\|B\|_2\le \tilO(p)$, $\textrm{tr}(B)= pn$.
Moreover, by Lemma~\ref{lem:xuxv}, we have that with high probability, $|u^TBv|\le \widetilde{O}(p\sqrt{n})$.
Note that these bounds only depend on the randomness of $Y$, and conditioning on all these bounds are true, we can still use the randomness of $Z_i$'s for concentration. We invoke Lemma~\ref{lem:sumzbzzuzv} and obtain that
\begin{align*}
\left|\sum_{i,j\in [p]} \inner{Z_j,Y_i}^2\inner{Z_j,u}\inner{Z_j,v} \right| &=
\left|\sum_{j=1}^p Z_j^TBZ_i\inner{Z_i,u}\inner{Z_i,v} \right|
\le \widetilde{O}(p^2n^{1.5})
\end{align*}
\end{proof}
\begin{lemma}\label{lem:spectralyy}
For $p\ge n$ and a sequence of \textit{good} independent random variables $Z_1,\dots,Z_p$,
we have that with high probability,
$$\left\|\sum_{i\in [p]}Z_iZ_i^T \right\|\le \tilO(p)$$
\end{lemma}
\begin{proof}
We use matrix Bernstein inequality. First of all, we have that $\Exp[Z_iZ_i^T ] = I_{n\times n}$, and therefore $\Exp\left[\sum_{i\in [p]}Z_iZ_i^T \right] = pI_{n\times n}$. Moreover, we check the variance of the $Z_iZ_i^T$:
\begin{align*}
\Exp[Z_iZ_i^TZ_iZ_i^T] & = n\Exp[Z_iZ_i^T] = nI_{n\times n}
\end{align*}
Finally we observe that $\|Z_iZ_i^T\|\le n$. Thus applying matrix Bernstein inequality we obtain that with high probability,
$$\left\|\sum_{i\in [p]}Z_iZ_i^T - p I_{n\times n}\right\|\le \tilO(\sqrt{np}+ n) = \tilO(\sqrt{np}) $$
\end{proof}
\begin{lemma}\label{lem:sumzbzzuzv}
For $p\ge n$ and a sequence of \textit{good} independent random variables $Z_1,\dots,Z_p$,
and for any fixed symmetric PSD matrix $B\in \mathbb{R}^{n\times n}$ with $\|B\|\le \tilO(p)$, $\textrm{tr}(B)\le 2pn$, and any two fixed vectors $u,v $ with $|u|_{\infty}\le \tilO(1)$ and $|v|_{\infty}\le \tilO(1)$, and $\inner{u,v}\le \tilO(\sqrt{n})$,
we have that with high probability over the randomness of $Z_i$'s,
$$\left|\sum_{i=1}^p Z_i^TBZ_i\inner{Z_i,u}\inner{Z_i,v} \right|\le \tilO(p^2n^{1.5})$$
\end{lemma}
\begin{proof}
Let $W = x^TBx\inner{x,u}\inner{x,v}$, where $x$ is a random variable that satisfies the conclusion of Proposition~\ref{con:randomvariable}. We first calculate the expectation of $W$,
\begin{eqnarray*}
\Exp\left[W\right] &=& \Exp\left[\left(\sum_{i}B_{ii}x_i^2 + \sum_{i\neq j}x_ix_j B_{ij}\right)\left(\sum_{i}u_iv_ix_i^2 + \sum_{i\neq j}x_ix_ju_iv_j\right)\right] \\
&=& (C_4-C_{2,2})\sum_i B_{ii}u_iv_i + C_{2,2}\textrm{tr}(B) \inner{u,v}+ \Exp\left[ \sum_{i\neq j}B_{ij}(u_iv_j+u_jv_i)x_i^2x_j^2\right] \\
&=& (C_4-3C_{2,2})\sum_i B_{ii}u_iv_i+ \textrm{tr}(B) \inner{u,v}\\
\end{eqnarray*}
Therefore by the fact that $|u|_{\infty}\le \tilO(1)$ and $\textrm{tr}(B)\le 2pn$, we obtain that $\left|\Exp[W]\right|\le \tilO(pn^{1.5})$.
Observe that $Z_j^TBZ_j\le \tilO(pn)$ a.s. (with respect to the randomness of $Z_j$), and $\orlicznorm{\inner{Z_i,u}\inner{Z_i,v}}{1}\le O(n)$, therefore we have that $\orlicznorm{Z_j^TBZ_j\inner{Z_i,u}\inner{Z_i,v}}{1}\le O(pn^2)$. Using Theorem~\ref{thm:psi-bounded-rv-concentration}, we obtain that with high probability,
$$\left|\sum_{i=1}^p Z_i^TBZ_i\inner{Z_i,u}\inner{Z_i,v} - \Exp\left[\sum_{i=1}^p Z_i^TBZ_i\inner{Z_i,u}\inner{Z_i,v} \right]\right|\le \widetilde{O}(n^2p^{1.5})$$
Using the fact that $\Exp\left[ Z_i^TBZ_i\inner{Z_i,u}\inner{Z_i,v} \right]\le \tilO(pn^{1.5})$ we obtain that with high probability
$$\left|\sum_{i=1}^p Z_i^TBZ_i\inner{Z_i,u}\inner{Z_i,v} \right|\le \tilO(p^2n^{1.5})$$
\end{proof}
\begin{lemma}\label{lem:xxtfro}
Suppose $p\ge n$ and $X_1,\dots,X_p$ are \textit{good} independent random variables,
then with high probability,
$$\|XX^T\|_F^2\ge (1-o(1))p^2n$$
\end{lemma}
\begin{proof}
We first $i$ and examine $\sum_{j\neq i}\inner{X_j,X_i}^2$ first. We have that $\Exp[\sum_{j\neq i}\inner{X_j,X_i}^2] = (p-1)\|X_i\|^2 = (p-1)n$. Moreover, $\orlicznorm{\inner{X_j,X_i}^2}{1}\le O(n)$ (where $X_j$ is viewed as random and $X_i$ is viewed as fixed). Therefore by Theorem~\ref{thm:psi-bounded-rv-concentration}, we obtain that with high probability over the randomness of $X_j$'s, ($j\ne i$),
$\sum_{j\neq i}\inner{X_j,X_i}^2 = (p-1)n \pm \tilO(n\sqrt{p}) = (1\pm o(1))pn$. Therefore taking union bound over all $i$, and taking the sum we obtain that $$\|XX^T\|_F^2 \ge \sum_i \sum_{j\neq i}\inner{X_j,X_i}^2\ge (1-o(1))p^2n$$
\end{proof}
\iffalse
\begin{definition}
We say a random variable $Z$ is $(R,c)$-bounded with respect to norm $\|\cdot \|$ if for universal constant $C$, for any $\rho < 1$,
\begin{equation}
\Pr\left[\|Z\| \ge R (\log (1/\rho))^c\right] \le C\rho \label{eqn:cond:soft-max}
\end{equation}
\end{definition}
\begin{lemma}\label{lem:product-boundedness}
If real random variable $Z_1$ and $Z_2$ are both $(R,c)$-bounded, then $Z_1Z_2$ are $(R^2,2c)$-bounded.
\end{lemma}
\begin{proof}
By union bound and the definition of $(R,c)$-boundedness, we have that
\begin{eqnarray}
\Pr[|Z_1Z_2|\ge R\log^{2c}(1/\rho)] & \le& \Pr[Z_1\ge R\log^c(1/\rho) \textrm{ or } Z_2\ge R\log^c(1/\rho)]\\
&\le& \Pr[Z_1\ge R\log^c(1/\rho)] + \Pr[Z_2\ge R\log^c(1/\rho)] \le O(\rho)
\end{eqnarray}
\end{proof}
\begin{lemma}[c.f. \cite{AGMM15}]\label{lem:general_bernstein}
Suppose independent random variable $Z_1,\dots,Z_n$ satisfies that for every $i\in [n]$ $Z_i$ is $(R,c)$-bounded as defined in~\eqref{eqn:cond:soft-max} , the we have
\begin{itemize}
\item[(a)]
for some fixed $\widetilde{R}= O(R\log^{2c+1} n)$,
$$\Pr\left[\forall i\in [n], \|Z_i\| \le R\right] \ge 1- n^{-10\log n}$$
\item[(b)] and when $R \le n^{c_1}$ for a fixed universal constant $c_1$,
$$\left\|\Exp\left[Z \cdot\mathbf{1}_{\|Z\| \ge \widetilde{R}}\right]\right\| \ge n^{-10\log n}$$
\end{itemize}
\end{lemma}
\begin{proof}
The first part of the lemma follows from choosing $\rho = n^{-11\log n}$ and applying a union bound. The second part of the lemma follows from
\begin{align*}
\Exp[Z\mathbf{1}_{\|Z\| \ge \widetilde{R}}]& \le \Exp[\|Z\|\mathbf{1}_{\|Z\| \ge \widetilde{R}}]\\
& = \widetilde{R}\Pr[\|Z\|\ge \widetilde{R}] + \int_{\widetilde{R}}^\infty \Pr[\|Z\|\ge t] dt = n^{-\log n}.
\end{align*}
and this completes the proof.
\end{proof}
\begin{lemma}\label{lem:extended-Berstein}
Suppose a sequence of independent real random variable $Z_1,\dots, Z_n$ satisfies that for every $i\in [n]$ $Z_i$ is $(R,c)$-bounded as defined in~\eqref{eqn:cond:soft-max},
then we have that
\begin{equation}
\Pr\left[\left|\sum_{i=1}^n Z_i - \Exp\left[\sum_{i=1}^n Z_i\right]\right|\ge 4\widetilde{R}\sqrt{n}\log n + n^{-10\log n}\right] \le n^{-\log n}
\end{equation}
for some $\widetilde{R} = O(R\log^{2c+2} n)$.
Moreover, if $\Exp[Z_i^2]\le \sigma^2$, then we have
\begin{equation}
\Pr\left[\left|\sum_{i=1}^n Z_i - \Exp\left[\sum_{i=1}^n Z_i\right]\right|\ge 3\sigma\sqrt{n}\log n + 3\widetilde{R}\log^2 n + n^{-10\log n}\right] \le n^{-\log n}
\end{equation}
\end{lemma}
\begin{proof}
We consider the random variable $X_i = Z_i \cdot \mathbf{1}_{|Z|_i \ge \widetilde{R}}$, these are independent random variables with $\Exp[X_i^2] \le \Exp[Z_i^2] \le \sigma^2$, and $|X_i|\le \widetilde{R}$ almost surely. Therefore by Bernstein inequality, we have that
\begin{equation}
\Pr\left[\left|\sum_{i=1}^n X_i - \Exp\left[\sum_{i=1}^n X_i\right]\right|\ge 3\sigma\sqrt{n}\log n + 3\widetilde{R}\log^2 n \right] \le n^{-\log n}
\end{equation}
Moreover, note that by Lemma~\ref{lem:extended-Berstein}, we have that $\left|\Exp[X_i] - \Exp[Z_i]\right| \le n^{-10\log n}$. Also note that
$$\Pr\left[\sum_{i=1}X_i = \sum_{i=1}^n Z_i\right]\ge 1-n^{-10\log n}$$ and therefore we obtain that
\begin{equation*}
\Pr\left[\left|\sum_{i=1}^n Z_i - \Exp\left[\sum_{i=1}^n Z_i\right]\right|\ge 3\sigma\sqrt{n}\log n + 3\widetilde{R}\log^2 n + n^{-10\log n}\right] \le n^{-\log n}
\end{equation*}
\end{proof}
\fi
| 9,222
|
Nevada Health Care Providers Must Disclose Interest in Physical Therapy Business
Posted by Steven J. Klearman
June 1, 2007 1:41 PM
June 1, 2007 1:41 Nevada legislative session has come and is almost gone.
I will report on interesting legislative developments over the next few months.
AB 468, effective October 1, 2007, requires providers of health care who refer patients to or recommend
physical therapy to a patient to provide a written disclosure to the patient of any financial interests that the provider of health care has in a facility recommended or to which a patient is referred. This bill also clarifies that this new requirement does not authorize a referral or recommendation which is otherwise prohibited.
| 414,829
|
TITLE: Cyclotomic Integers?
QUESTION [3 upvotes]: Let $K=\mathbb{Q}(\zeta_{p^\infty})$ be the extension of $\mathbb{Q}$ obtained by adjoining all $p-$power roots of unity.
My question is : how to show that the the ring of integes of $K,$ $O_K$ is not a Dedekind domain ?
Thanks.
REPLY [2 votes]: At this point, I think there's been enough discussion in the comments to turn it into a proper answer.
As Alex noted, our goal is really to prove that $\mathcal O_K$ is not Noetherian. To do so, we will prove that some rational prime $\ell$ "factors into infinitely many factors" in $\mathcal O_K$.
We will do this for $p=\ell$. Recall that
$$(p)=(1-\zeta_p)^{p-1}=(1-\zeta_{p^2})^{p^2-p}=\ldots=(1-\zeta_{p^n})^{\varphi(p^n)}=\ldots$$
Then we have an increasing chain
$$(p)\subset(1-\zeta_p)\subset(1-\zeta_{p^2})\subset\ldots\subset(1-\zeta_{p^n})\subset\ldots$$
and thus the ring is not Noetherian. One can think of this as a failure of unique factorization into prime ideals, which characterizes Dedekind rings. In this case, the uniqueness is not the problem, but rather the existence of a factorization in the first place--we can keep on factoring and factoring $(p)$ but will never get all the way down to prime ideals.
It turns out that the only possible choice for $\ell$ was $p$: Suppose $\ell\neq p$ and let $n$ be the greatest integer so that $\zeta_{p^n}\in\mathbb Q_\ell$. Then $[\mathbb Q_\ell(\zeta_{p^{n+m}}):\mathbb Q_\ell]=\frac{\varphi(p^{n+m})}{\varphi(p^n)}=[\mathbb Q(\zeta_{p^{n+m}}):\mathbb Q(\zeta_{p^n})]$, so primes lying above $\ell$ in $\mathbb Q(\zeta_{p^n})$ remain prime in larger subextensions.
| 105,093
|
\begin{document}
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\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{givendata}[theorem]{Given Data}
\newtheorem{claim}[theorem]{Claim}
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\numberwithin{equation}{section}
\title[Asymptotic Density and the Coarse Computability
Bound]{Asymptotic Density and \\the Coarse Computability Bound}
\author[D. R. Hirschfeldt]{Denis R. Hirschfeldt}
\address{Department of Mathematics\\University of Chicago}
\email{drh@math.uchicago.edu}
\thanks{Hirschfeldt was partially supported by grant
DMS-1101458 from the National Science Foundation of the United
States.}
\author[C. G. Jockusch, Jr.]{Carl G. Jockusch, Jr.}
\address{Department of Mathematics\\University of Illinois at
Urbana-Cham\-paign}
\email{jockusch@math.uiuc.edu}
\author[T. H. McNicholl]{Timothy H. McNicholl}
\address{Department of Mathematics\\Iowa State University}
\email{mcnichol@iastate.edu}
\thanks{McNicholl was partially supported by a Simons Foundation Collaboration Grant for Math\-e\-ma\-ti\-cians}
\author[P. E. Schupp]{Paul E. Schupp}
\address{Department of Mathematics\\University of Illinois at
Urbana-Cham\-paign}
\email{schupp@illinois.edu}
\keywords{Asymptotic density, Coarse computability, Turing degrees}
\subjclass[2010]{Primary 03D28; Secondary 03D25}
\begin{abstract}
For $r \in [0,1]$ we say that a set $A \subseteq \omega$ is
\emph{coarsely computable at density} $r$ if there is a computable
set $C$ such that $\{n : C(n) = A(n)\}$ has lower density at least
$r$. Let $\gamma(A) = \sup \{r : A \hbox{ is coarsely computable at
density } r\}$. We study the interactions of these concepts with
Turing reducibility. For example, we show that if $r \in (0,1]$
there are sets $A_0, A_1$ such that $\gamma(A_0) = \gamma(A_1) = r$
where $A_0$ is coarsely computable at density $r$ while $A_1$ is not
coarsely computable at density $r$. We show that a real $r \in
[0,1]$ is equal to $\gamma(A)$ for some c.e.\ set $A$ if and only if
$r$ is left-$\Sigma^0_3$. A surprising result is that if $G$ is a
$\Delta^0_2$ $1$-generic set, and $A \leq\sub{T} G$ with $\gamma(A)
= 1$, then $A$ is coarsely computable at density $1$.
\end{abstract}
\maketitle
\section{Introduction}
There are two natural models of ``imperfect computability''
defined in terms of the standard notion of asymptotic density, which we
now review. For $A \subseteq \omega$ and $n \in \omega \setminus
\{0\}$, define $\rho_n(A)$, the density of $A$ below $n$, by
$\rho_n(A) = \frac{|A \upharpoonright n|}{n}$, where $A
\upharpoonright n = A \cap \{0, 1, \dots, n-1\}$. Then
\[ \underline{\rho}(A) = \liminf_n \rho_n(A)\qquad \mbox{and}\qquad \overline{\rho}(A) = \limsup_n \rho_n(A) \]
are respectively the \emph{lower density} of $A$
and the \emph{upper density} of $A$.
The
\emph{(asymptotic) density} of $A$ is $\rho(A) = \lim_n \rho_n(A)$
provided the limit exists.
The idea of generic computability was introduced and studied in
connection with group theory in \cite{KMSS} and then studied in
connection with arbitrary subsets of $\omega$ in \cite{JS}. In
generic computability we have a partial algorithm that is always
correct when it gives an answer but may fail to answer on a set of
density $0$. The paper \cite{DJS} began studying computability at
densities less than $1$ and introduced the following definitions.
\begin{definition}[{\cite[Definition 5.9]{DJS}}]
\label{def: Computable.at. r}
Let $A$ be a set of natural numbers and let $r$ be a real number in
the unit interval $[0,1]$. The set $A$ is \emph{partially computable at density
$r$} if there is a partial computable function $\phi$ such that
$\phi(n) = A(n)$ for all $n$ in the domain of $\phi$ and the
domain of $\phi$ has lower density at least $r$.
\end{definition}
Thus $A$ is \emph{generically computable} if and only if $A$ is
partially computable at density $1$.
\begin{definition}[{\cite[Definition 6.9]{DJS}}]
\label{def:ALPHA}
If $A \subseteq \omega$, the \emph{partial computability bound} of $A$ is
\[ \alpha(A) = \sup\{r : \mbox{$A$ is partially computable at density $r$}\}. \]
\end{definition}
In the paper \cite{DJS} the term ``partially computable at density $r$'' was
simply called ``computable at density $r$'' and the ``partial computability bound''
was called the ``asymptotic computability bound''. That paper considered only
partial computability at densities less than $1$, but since we are here comparing the
partial computability concepts with their coarse analogs, the present terminology
is more exact.
If $A$ is generically computable, then $\alpha(A) = 1$. The converse
fails by \cite[Observation 5.10]{DJS}. There are sets that are partially
computable at every density less than $1$ but are not generically
computable.
\begin{definition}\label{def:COARSESIM}
If $A, B \subseteq \N$, then $A$ and $B$ are \emph{coarsely
similar}, written $A \backsim\sub{c} B$, if the density of the
symmetric difference of $A$ and $B$ is $0$, that is, $ \rho(A
\triangle B) = 0$. Given $A$, any set $B$ such that $B \backsim\sub{c} A$
is called a \emph{coarse description} of $A$.
\end{definition}
It is easy to check that coarse similarity is indeed an equivalence
relation. Coarse similarity was called \emph{generic similarity} in
\cite{JS}, but the current terminology seems better.
Coarse computability considers algorithms that
always give an answer, but may give an incorrect answer on a set of
density $0$. We have the following definition.
\begin{definition}[{\cite[Definition 2.13]{JS}}]
\label{def:COARSE}
The set $A$ is \emph{coarsely computable} if there is a computable
set $C$ such that the density of $\{n : A(n) = C(n)\}$ is $1$.
That is, $A$ is coarsely computable if it has a computable coarse description $C$.
\end{definition}
The following definitions are similar to those for partial computability.
\begin{definition}\label{def:COARSE.AT.r}
If $A \subseteq \omega$ and $r \in [0,1]$, an \emph{$r$-description} of $A$
is any set $B$ such that the lower density of $\{n : A(n) = B(n)\}$ is
at least $r$. A set $A$ is \emph{coarsely computable at density $r$} if there is a
computable $r$-description $B$ of $A$.
\end{definition}
Note that $A$ is coarsely computable if and only $A$ is coarsely computable
at density $1$.
\begin{definition}\label{def:GAMMA}
If $A \subseteq \omega$, the \emph{coarse computability bound} of
$A$ is
\[ \gamma(A) = \sup\{r : \mbox{$A$ is coarsely computable at density $r$}\}. \]
\end{definition}
If $A$ is coarsely computable, then $\gamma(A) = 1$, but the next
lemma implies that the converse fails.
It is shown in \cite[Proposition 2.15 and Theorem 2.26]{JS} that
neither of generic computability and coarse computability implies the
other, even for c.e.\ sets. Nonetheless, the following lemma gives an
inequality between $\alpha$ and $\gamma$.
\begin{lemma} \label{ineq} For any $A \subseteq \omega$, $\alpha(A)
\le \gamma(A)$. In particular, if $A$ is generically computable
then $\gamma(A) = 1$.
\end{lemma}
\begin{proof} Fix $\epsilon > 0$. If $\alpha(A) = r$ then there is a
partial algorithm $\phi$ for $A$ such that the lower density of
the c.e.\ set $ D = \dom\phi$ is greater than or equal to
$r - \epsilon$. Theorem 3.9 of \cite{DJS} shows that if $D$ is a
c.e.\ set there is a computable set $C \subseteq D$ such that
$\underline{\rho}(C) > \underline{\rho}(D) - \epsilon$. Let $C_1 =
\{n \in C : \phi(n) = 1\}$. Then $C_1$ is a computable set and
$\{n : A(n) = C_1(n)\} \supseteq C$. It follows that
$\underline{\rho}(\{n : A(n) = C_1(n)\}) \geq \underline{\rho}(C) >
\underline{\rho}(D) - \epsilon \geq r - 2\epsilon$, and hence $A$ is
coarsely computable at density $r - 2\epsilon$. Since $\epsilon > 0$
was arbitrary, it follows that $\gamma(A) \geq r = \alpha(A)$.
\end{proof}
One consequence of this lemma is that any set that is generically
computable but not coarsely computable is an example of a set $A$ such
that $\gamma(A)=1$ but $A$ is not coarsely computable.
\begin{definition}\label{def:METRIC}
If $A, B \subseteq \N$, let $D(A,B) = \overline{\rho}(A \triangle B )$.
\end{definition}
It is shown in \cite[remarks after Proposition 3.2]{DJS} that $D$ is
a pseudometric on subsets of $\omega$ and, since $D(A,B) = 0$ exactly
when $A$ and $B$ are coarsely similar, $D$ is actually a metric on the
space of coarse similarity classes. Note that $\gamma$ is an
invariant of coarse similarity classes.
Although easy, the following is useful enough to be stated as a lemma.
\begin{lemma} \label{comp} If $A \subseteq \omega$ then $
\underline{\rho} (A) = 1 - \overline{\rho}(\overline A) $.
\end{lemma}
\begin{proof} Note that $\rho_n(A) = 1 - \rho_n(\overline{A})$ for all $n \ge 1$.
The lemma follows by taking the lim inf of both sides of this
equation.
\end{proof}
Since we have a pseudometric space, we can consider the distance from
a single point to a subset of the space in the usual way.
\begin{definition}\label{SETDISTANCE}
If $A \subseteq \omega$ and $\mathcal{S} \subseteq \mathcal{P}(\N)$, let
\[
\delta(A, \mathcal{S}) = \inf\{D(A, S) : S \in \mathcal{S}\}.
\]
\end{definition}
The above lemma shows that
\[
\gamma(A) = 1 - \delta(A, \mathcal{C}),
\]
where $\mathcal{C}$ is the class of computable sets. Thus $\gamma(A) =
1 $ if and only if $A$ is a limit of computable sets in the
pseudometric. A set $A$ is coarsely computable at density $r$ if and only
if $\delta(A,\mathcal{C}) \leq 1 - r$.
The symmetric difference $ A \triangle B = \{ n: A(n) \ne B(n) \} $ is
the subset of $\omega$ where $A$ and $B$ disagree. There does not seem to be
a standard notation
for the complement of $ A \triangle B $, which is $ \{ n: A(n) = B(n)\} $,
the ``symmetric agreement'' of $A$ and $B$. We find it useful
to use $ A \triangledown B $ to denote $\{n : A(n) = B(n)\}$.
We assume that the reader is familiar with basic computability theory.
See, for example, \cite{S}. If $S$ is a set of finite binary strings and $A
\subseteq \omega$ we say that $A$ \emph{meets} $S$ if $A$ extends some
string in $S$ and that $A$ \emph{avoids} $S$ if $A$ extends a string
that has no extension in $S$.
\section{Turing degrees, coarse computability, and $\gamma$}
It is easily seen that every Turing degree contains a set that is
both coarsely and generically computable and hence a set
$A$ with $\alpha(A) = \gamma(A) = 1$. In the other direction
it is shown in Theorem 2.20 of \cite{JS} that every nonzero Turing
degree contains a set that is neither generically computable
nor coarsely computable. The same construction now yields a
quantitative version of that result.
\begin{theorem} \label{I(A)} Every nonzero Turing degree contains a
set whose partial computability bound is $0$ but whose coarse computability bound is $1/2$.
\end{theorem}
\begin{proof} Let $I_n = [n!,(n+1)!)$. Suppose that $A$ is not
computable, and let $\mathcal{I}(A) = \bigcup_{n \in A} I_n$. It is
clear that $\mathcal{I}(A)$ is Turing equivalent to $A$. We prove
first that $\gamma(\mathcal{I}(A)) \le \frac{1}{2}$. If there is a
computable $C$ with $\underline{\rho}(\mathcal{I} (A) \triangledown
C) > \frac{1}{2}$ we can compute $A$ by ``majority vote''. That is, for all
sufficiently large $n$, we have that $n$ is in $A$ if and only if
more than half
of the elements of $I_n$ are in $C$. (For any $n$ for which this
equivalence fails, we have $\rho_{(n+1)!}(\mathcal{I}(A)
\triangledown C) \leq (1 + (n+1)^{-1})/2$.) It follows that $A$ is
computable, a contradiction. If $C$ is the set of even numbers,
then it is easily seen that $\rho(C \triangledown \mathcal{I}(A)) =
\frac{1}{2}$, so $\gamma(\mathcal{I}(A)) \geq \frac{1}{2}$. It follows that
$\gamma(\mathcal{I}(A)) = \frac{1}{2}$. To see that $\alpha(\mathcal{I}(A))
= 0$, note that any set of positive lower density intersects $I_n$
for all but finitely many $n$, and apply this observation to the
domain of any partial computable function that agrees with
$\mathcal{I}(A)$ on its domain.
\end{proof}
We next observe that a large class of degrees contain sets $A$ with
$\gamma(A) = 0$.
\begin{theorem} \label{hi}
Every hyperimmune degree contains a set whose coarse computability bound is $0$.
\end{theorem}
\begin{proof}
A set $S \subseteq 2^{ < \omega}$ of finite binary strings is
\emph{dense} if every string has some extension in $S$. Stuart
Kurtz \cite{K} defined a set $A$ to be \emph{weakly $1$-generic} if
$A$ meets every dense c.e.\ set $S$ of finite binary strings
and proved that the weakly $1$-generic degrees coincide with
the hyperimmune degrees. Hence, it suffices to show that every
weakly $1$-generic set $A$ satisfies $\gamma(A) = 0$. Assume that
$A$ is weakly $1$-generic.
If $f$ is a computable function then, for each $n,j > 0$, define
\[
S_{n,j} = \left\{ \sigma \in 2^{< \omega} : |\sigma| \ge j \enspace \&
\enspace \rho_{|\sigma|}(\{k < |\sigma| : \sigma(k) = f(k)\}) <
\frac{1}{n} \right\}.
\]
Each set $S_{n,j}$ is computable and dense so $A$ meets each
$S_{n,j}$. Thus $\{ k: f(k) = A(k)\}$ has lower density $0$.
\end{proof}
In view of the preceding result, it is natural to ask whether \emph{every}
nonzero degree contains a set $A$ such that $\gamma(A) = 0$.
This question is answered in the negative in \cite{ACDJL} where it is
shown that that every computably traceable set is coarsely computable
at density $\frac{1}{2}$, and also that every set computable from a $1$-random
set of hyperimmune-free degree is coarsely computable at density
$\frac{1}{2}$. Each of these results implies that there is a nonzero degree
$\mathbf{a \leq 0''}$ such that every $\mathbf{a}$-computable set is
coarsely computable at density $\frac{1}{2}$. Here it is not possible to
replace $\frac{1}{2}$ by any larger number, by Theorem \ref{I(A)}. In
\cite{ACDJL}, the following definition is made for Turing degrees
$\mathbf{a}$:
$$\Gamma(\mathbf{a}) = \inf \{\gamma(A) : A \mbox{ is
$\mathbf{a}$-computable} \}.$$
By the above, $\Gamma$ takes on the values $0$ and $\frac{1}{2}$, and of
course $\Gamma(\mathbf{0}) = 1$. By
Theorem \ref{I(A)}, $\Gamma$ does not take on any values in the open
interval $(\frac{1}{2}, 1)$. An open question posed in \cite{ACDJL}
is whether $\Gamma$ takes on any values other than $0, \frac{1}{2}$, and $1$.
\section{Coarse computability at density $\gamma(A)$}
If $A$ is any set, it follows from the definition of $\gamma(A)$ that
$A$ is coarsely computable at every density less than $\gamma(A)$
and at no density greater than $\gamma(A)$.
What happens at $\gamma(A)$? Let us say that $A$ is \emph{extremal for coarse computability} if it is coarsely computable at density $\gamma(A)$.
In this section, we show that extremal and non-extremal sets exist. Moreover, we also show that every real in $(0,1]$ is the coarse computability bound of an extremal set and of a non-extremal set. We also explore the distribution of these cases in the Turing degrees. Roughly speaking, we show that hyperimmune degrees yield extremal sets and high degrees yield non-extremal sets.
\begin{theorem} \label{prescribed}
Every real in $[0,1]$ is the coarse computability bound of a set that is extremal for coarse computability.
\end{theorem}
\begin{proof} Suppose $0 \leq r \leq 1$. By Corollary 2.9 of \cite{JS} there is a set $A_1$ such
that $\rho(A_1) = r$. Let $Z$ be a set with $\gamma(Z) = 0$, which
exists by Theorem \ref{hi}, and let $A = A_1 \cup Z$. Note
first that that $A$ is coarsely computable at density $r$ via the
computable set $\omega$ since
$$\underline{\rho}(A \triangledown \omega) = \underline{\rho}(A)
\geq \underline{\rho}(A_1) = r.$$
It follows that $\gamma(A) \geq r$,
so it remains only to show that $\gamma(A) \leq r$.
Suppose for a contradiction that $\gamma(A) > r$, so $A$ is coarsely
computable at some density $r' > r$. Let $C$ be a computable set such
that $\underline{\rho}(A \triangledown C) \ge r'$. Let:
\begin{eqnarray*}
S_1 & = & A_1 \cap C\\
S_2 & = & (Z \setminus A_1) \cap
C\\
S_3 & = & \overline{A} \cap \overline{C}.
\end{eqnarray*}
Note that $A
\triangledown C$ is the disjoint union of $S_1$, $S_2$, and $S_3$ so
\[ \rho_n(A \triangledown C) = \rho_n(S_1) + \rho_n(S_2) + \rho_n(S_3) \]
for all $n$.
Let $\epsilon = r' - r$. For all sufficiently large $n$ we have
$\rho_n(A \triangledown C) > r + \frac{\epsilon}{2}$. Since $S_1
\subseteq A_1$ and $\rho_n(A_1) < r + \frac{\epsilon}{3}$ for all
sufficiently large $n$, we have
$\rho_n(S_2) + \rho_n(S_3) > \frac{\epsilon}{6}$ for all sufficiently
large $n$. Hence $\underline{\rho}(S_2 \cup S_3) > 0$. But $S_2
\cup S_3 \subseteq C \triangledown Z$ so $\underline{\rho}(C
\triangledown Z) > 0$, contradicting $\gamma(Z) = 0$. This
contradiction shows that $\gamma(A) \leq r$, and the proof is complete.
\end{proof}
\begin{corollary}[to proof]
Suppose $\mathbf{a}$ is a hyperimmune degree. Then, every $\Delta^0_2$ real in $[0,1]$ is the coarse computability bound of a set in $\mathbf{a}$ that is extremal for coarse computability.
\end{corollary}
\begin{proof} Just note that the proof of the theorem can be carried
out effectively in $\mathbf{a}$. In more detail, by Theorem 2.21 of
\cite{JS} there is a computable set $A_1$ of density $r$. Further,
by Theorem \ref{hi} there is an $\mathbf{a}$-computable set $Z$
such that $\gamma(Z) = 0$. Then $A = A_1 \cup Z$ satisfies the
theorem and is $\mathbf{a}$-computable. We can ensure that $A \in \mathbf{a}$ by coding a set in $\mathbf{a}$ into $A$ on a set of density $0$.
\end{proof}
We now consider sets that are not extremal for coarse computability. We first consider the degrees
of the sets $A$ such that $\gamma(A) = 1$ but $A$ is not coarsely
computable.
Define
$$R_n = \{k : 2^n \mid k \enspace \& \enspace 2^{n+1} \nmid k\}.$$
The sets $R_n$ were heavily used in \cite{JS} and \cite{DJS}. Note
that they are uniformly computable and pairwise disjoint, and
$\rho(R_n) = 2^{-(n+1)}$. As in \cite{JS} and \cite{DJS}, define
$$\mathcal{R}(A) = \bigcup_{n \in A} R_n.$$
Note that, for all $A$, we have that $A \equiv\sub{T} \mathcal{R}(A)$ and
$\alpha(\mathcal{R}(A)) = \gamma(\mathcal{R}(A)) = 1$. To see the
latter (which was pointed out by Asher Kach), note that if $C_k =
\bigcup \{R_n : n \in A \enspace \& \enspace n < k\}$, then $C_k$ is computable and agrees
with $\mathcal{R}(A)$ on $\bigcup_{n < k} R_n$, and the latter has density
$1 - 2^{-k}$.
\begin{theorem} \label{degreeobs}
\begin{itemize}
\item[(i)]
If $\boldsymbol{a}$ is a degree such that $\mathbf{a \nleq 0'}$,
then $\boldsymbol{a}$ contains a set that is not coarsely computable but whose coarse computability bound is $1$.
\item[(ii)]
If $\boldsymbol{a}$ is a nonzero c.e.\ degree, then
$\boldsymbol{a}$ contains a c.e.\ set that is not coarsely computable but whose coarse computability bound is $1$.
\end{itemize}
\end{theorem}
\begin{proof} It is shown in Theorem 2.19 of \cite{JS} that
$\mathcal{R}(B)$ is coarsely computable if and only if $B$ is
$\Delta^0_2$. If $\mathbf{a \nleq 0'}$ and $B$ has degree
$\mathbf{a}$, then $\mathcal{R}(B)$ is a set of degree
$\mathbf{a}$ that is not coarsely computable even though its coarse computability bound is $1$. Part (i) follows.
Theorem 4.5 of \cite{DJS} shows that every nonzero c.e.\
degree contains a c.e.\ set $A$ that is generically computable but not
coarsely computable. Then $\alpha(A) = 1$, so by Lemma \ref{ineq},
$\gamma(A) = 1$. This proves part (ii).
\end{proof}
This result raises the natural question: Does \emph{every} nonzero Turing
degree contain a set $A$ such
that $\gamma(A) = 1$ but $A$ is not coarsely computable? We will
obtain a negative answer in Theorem
\ref{1G} in the next section. In fact, we will show that if $G$ is
$1$-generic and $\Delta^0_2$, and $A \leq\sub{T} G$ has $\gamma(A) = 1$, then
$A$ is coarsely computable.
We now consider the coarse computability bounds of non-extremal sets.
\begin{theorem} \label{notcc}
Every real in $(0,1]$ is the coarse computability bound of a set that is not extremal for coarse computability.
\end{theorem}
\begin{proof}
Suppose $0 < r \leq 1$. We construct a set $A$ so that $\gamma(A) = r$ but $A$ is not coarsely computable at density $r$.
As an auxiliary for defining $A$, we first use the technique of Corollary 2.9 of \cite{JS} to define a set $S$ of density $r$.
To this end, we turn $r$ into a set $B$ in the natural way.
That is, since $r > 0$, it has a non-terminating binary expansion $r = 0.b_0b_1 \dots$.
We then set $B = \{i : b_i = 1\}$. By restricted countable additivity (Lemma 2.6 of \cite{JS}), $\mathcal{R}(B)$ has density $r$. Set $S = \mathcal{R}(B)$.
We now divide $S$ into ``slices'' $S_0, S_1, \ldots$ as follows. Let $c_0 < c_1 < \cdots$ be the increasing enumeration of $B$. Set $S_e = R_{c_e}$. Note that the $S_e$'s are pairwise disjoint and that $S = \bigcup_e S_e$. Note also that each $S_e$ is computable (though not necessarily computable uniformly in $e$).
We now define $A$. We first choose a set $Z$ so that $\gamma(Z) = 0$. Such a set exists by Theorem \ref{hi}. Let $C_0, C_1, \ldots$ be an enumeration of the computable sets. We then set
\[
A = (\overline{S} \cap Z) \cup \bigcup_e (S_e \cap \overline{C_e}).
\]
We now claim that $A$ is coarsely computable at density $q$ whenever $0 \leq q < r$. For, suppose $0 \leq q < r$. Since the density of $S$ is $r$, there is a number $n$ so
that $\rho(\bigcup_{e < n} S_e) \geq q$. Let $C = \bigcup_{e < n} (S_e \cap
\overline{C_e})$. Then, $C$ is a computable set. Also $A$ and $C$ agree on each $S_e$ for $e
< n$, so $\underline{\rho}(A \triangledown C) \geq
\underline{\rho}(\bigcup_{e < n} S_e) \geq q$. Hence, $C$ witnesses that $A$
is coarsely computable at density $q$.
To complete the proof, it suffices to show that $A$ is not coarsely computable at density $r$. To this end, it suffices to show that the lower density of $A \triangledown C_e$ is smaller than $r$ for each $e$. Fix $e \in \N$. By construction, $(C_e
\triangledown A) \cap S$ is disjoint from $S_e$ and so has
\emph{upper} density less than $r$. At the same time, note that $(A \triangledown C_e) \cap \overline{S} \subseteq C_e \triangledown Z$. Let $r_0 =
\overline{\rho}((C_e \triangledown A) \cap S)$, and let $\epsilon = r
- r_0$. Then for infinitely many $n$ we have
$$\rho_n(A \triangledown C_e) = \rho_n((A \triangledown C_e) \cap S) +
\rho_n((A \triangledown C_e) \cap \overline{S}) < \left(r_0 + \frac{\epsilon}{2}\right) +
\frac{\epsilon}{3} < r.$$ It follows that $\underline{\rho}(A \triangledown C_e) < r$.
Hence $A$ is not coarsely computable at density $r$, which completes
the proof.
\end{proof}
We note that the proof of Theorem \ref{notcc} shows that if $A$ is any set so that $A \cap S_e = \overline{C_e} \cap S_e$ for all $e$, then $A$ is computable at density $q$ whenever $0 \leq q < r$. That is, the construction of $A \cap S$ ensures that $\gamma(A) \geq r$.
\begin{corollary}[to proof]
Suppose $\mathbf{a}$ is a high degree. Then, every computable real in $(0,1]$ is the coarse computability bound of a set in $\mathbf{a}$ that is not extremal for coarse computability.
\end{corollary}
\begin{proof}
We just observe that the preceding proof can be carried out in an
$\mathbf{a}$-com\-put\-able fashion. By Theorem 1 of \cite{J2}, there
is a listing $C_0, C_1, \dots$ of the computable sets that is
uniformly $\mathbf{a}$-computable. Also, since $r$ is computable,
the sequence $S_0, S_1, \dots$ in the proof of the theorem is also
uniformly $\mathbf{a}$-computable. It does not affect the proof to
modify each $S_e$ so that it contains no numbers less than $e$, and
then $S = \bigcup_e S_e$ is $\mathbf{a}$-computable. Finally, every
high degree is hyperimmune by a result of D. A. Martin \cite{M},
and so every high degree computes a set $Z$ with $\gamma(Z) = 0$ by
Theorem \ref{hi}. Hence the set $A$ defined in the proof of the
theorem can be chosen to be $\mathbf{a}$-computable. By coding a set in $\mathbf{a}$ into $A$ on a set of density $0$ we can ensure that $A \in \mathbf{a}$.
\end{proof}
By using suitable computable approximations, the previous corollary
can be extended from computable reals to
$\Delta^0_2$ reals. We omit the details.
It was shown in Theorem 4.5 of \cite{DJS} that every nonzero
c.e.\ degree contains a c.e.\ set that is generically computable but
not coarsely computable. It follows at once from Lemma \ref{ineq}
that every nonzero c.e.\ degree contains a c.e.\ set $A$ such that
$\gamma(A) = 1$ but $A$ is not coarsely computable. We now use the
method of Theorem \ref{notcc} to extend this result to the case where
$\gamma(A)$ is a given computable real.
\begin{theorem}
Suppose $\mathbf{a}$ is a nonzero c.e.\ degree. Then, every computable real in $(0,1]$ is the coarse computability bound and the partial computability bound of a c.e.\ set in $\mathbf{a}$ that is not extremal for coarse computability.
\end{theorem}
\begin{proof}
Define the sets $S, S_0, S_1, \ldots$ as in the proof of Theorem \ref{notcc} so that $S = \bigcup_e S_e$ and so that $\rho(S) = r$. Let $B$ be
a c.e.\ set of degree $\mathbf{a}$, and let $\{B_s\}$ be a computable
enumeration of $B$. We construct the desired set $A \leq\sub{T} B$ using
ordinary permitting; i.e.\ if $x \in A_{s+1} \setminus A_s$, then there exists $y \leq x$ such that $y \in B_{s+1} \setminus B_s$. To ensure that $B \leq\sub{T} A$, we code $B$ into $A$ on a set of density zero.
Let the requirement $N_e$ assert that if $\Phi_e$ is total, then the lower density of the
set on which it agrees with $A$ is smaller than $r$. Thus, if $N_e$ is met for every $e$, then $A$ is not coarsely computable at density $r$. We
meet $N_e$ by appropriately defining $A$ on $S_e$ and on
$\overline{S}$. If $\Phi_e$ is total, we meet $N_e$ by making $A$ completely
disagree with $\Phi_e$ on infinitely many large finite sets $I \subseteq
S_e \cup \overline{S}$. To this end, we effectively choose finite sets $I_{e,i}$ such that the following hold for all $e$, $i$, $e'$, and $i'$:
\begin{enumerate}
\item[(i)] $I_{e,i} \subseteq (S_e \cup \overline{S})$.
\item[(ii)] $\min I_{e, i+1} > \max I_{e,i}$.
\item[(iii)] $\rho_m(I_{e,i}) \geq \frac{i}{i+1} \rho_m(S_e \cup
\overline{S})$ where $m = \max I_{e,i} + 1$.
\item[(iv)] If $(e,i) \neq (e', i')$, then $I_{e,i} \cap I_{e', i'}
= \emptyset$.
\end{enumerate}
The sets $I_{e,i}$ may be obtained by intersecting appropriately large
intervals with $S_e \cup \overline{S}$ while preserving pairwise
disjointness, and we will call the sets $I_{e,i}$ ``intervals''.
During the construction we will designate an interval $I_{e,i}$ as
``successful'' if we have ensured that $\Phi_e$ and $A$ totally
disagree on $I_{e,i}$. The construction is as follows:
\emph{Stage} $0$. Let $A_0 = \emptyset$.
\emph{Stage} $s + 1$. For each $e, i \leq s$, declare $I_{e,i}$ to be successful if it has not yet been declared successful and if the following conditions are met.
\begin{enumerate}
\item $\Phi_{e,s}$ is defined on all elements
of $I_{e,i}$.
\item $\min(I_{e,i})$ exceeds all elements of $A_s \cap S_e$.\label{itm:inc}
\item At least one number in $B_{s+1} \setminus B_s$ is less than or equal to $\min(I_{e,i})$.
\end{enumerate}
If $I_{e,i}$ is declared to be successful at stage $s + 1$, then enumerate into $A$ all $x \in I_{e,i}$ with $\Phi_e(x) = 0$.
The set $A$ is clearly c.e., and $A \leq\sub{T} B$ by ordinary permitting.
If the interval $I_{e,i}$ is ever declared to be successful, then $A$
and $\Phi_e$ totally disagree on $I_{e,i}$, by the action taken
when it is declared successful and the disjointness condition (iv),
which ensures that no elements of $I_{e,i}$ are enumerated into $A$
except by this action.
Note that (\ref{itm:inc}) ensures that $A \cap S_e$ is computable for each $e$. It follows that $\gamma(A) \geq \alpha(A) \geq r$ as in the
proof of Theorem \ref{notcc}.
It remains to show that every requirement $N_e$ is met. Suppose that $\Phi_e$ is
total. We claim first that the interval $I_{e,i}$ is declared
successful for infinitely many $i$. Suppose not. Then $A \cap S_e$
is finite. It follows that $B$ is computable, since, for all
sufficiently large $i$, if $s \geq i$ and $\Phi_{e, s}$ is defined on
all elements of $I_{e,i}$, then no number less than $\min(I_{e,i})$
enters $B$ after stage $s$. Since we assumed that $B$ is
noncomputable, the claim follows.
Suppose $I_{e,i}$ is successful. Set $I = I_{e,i}$. Then $A \triangle \Phi_e \supseteq I$, so
$$\rho_m (A \triangle \Phi_e) \geq \rho_{m}(I) \geq
\frac{i}{i+1} \rho_{m}(S_e \cup \overline{S}),$$ where $m = \max
I_{e,i} +1$. There are infinitely many such $i$'s, and as $i$ tends
to infinity, the right hand side of the above inequality tends to
$\rho(S_e) + \rho(\overline{S})$. It follows that $\overline{\rho}(A
\triangle \Phi_e) \geq \rho(S_e) + (1 - r)$, and so by Lemma \ref{comp},
$\underline{\rho}(A \triangledown \Phi_e) \leq r - \rho(S_e) < r$, as needed to
complete the proof.
\end{proof}
\section{Coarse Computability and Lowness}
We now consider the coarse computability properties of c.e.\ sets that
have a density.
\begin{proposition} \label{c.e.density} Every low c.e.\ set having a
density is coarsely computable. Every c.e.\ set having a density
has coarse computability bound $1$.
\end{proposition}
\begin{proof} The first statement is Corollary 3.16 of \cite{DJS}.
Let $A$ be a c.e.\ set that has a density and let $\epsilon > 0$.
Theorem 3.9 of \cite{DJS} shows that $A$ has a computable subset $C$
such that $\underline{\rho}(C) > \rho(A) - \epsilon$. Then $C
\triangle A = A \setminus C$. Hence, by Lemma \ref{comp},
$\underline{\rho}(A \triangledown C) = 1 - \overline{\rho}(A
\setminus C)$. But by Lemma 3.3 (iii) of \cite{DJS},
$$\overline{\rho}(A \setminus C) \leq \rho(A) - \underline{\rho}(C) <
\epsilon.$$
Hence $\underline{\rho}(A \triangledown C) > 1 - \epsilon$. Since
$\epsilon > 0$ was arbitrary, we conclude that $\gamma(A) = 1$.
\end{proof}
The next result shows that the lowness
assumption is strongly required in the first part of Proposition \ref{c.e.density}.
\begin{theorem}\label{thm:nonlow}
Every nonlow c.e.\ Turing degree $\mathbf{a}$ contains
a c.e.\ set of density $1/2$ that is not coarsely computable.
\end{theorem}
\begin{proof} The proof of the theorem is similar to the proof in
Theorem 4.3 of \cite{DJS} that every nonlow c.e.\ degree contains a
c.e.\ set $A$ such that $\rho(A) = 1$ but $A$ has no computable
subset of density $1$. Hence we give only a sketch. Let $C$ be a
c.e.\ set of degree $\mathbf{a}$. We ensure that $A \leq\sub{T} C$ by a
slight variation of ordinary permitting: If $x$ enters $A$ at stage
$s$, then either some number $y \leq x$ enters $C$ at $s$ or $x =
s$. This implies that $A \leq\sub{T} C$, and by coding $C$ into $A$ on a
set of density $0$ we can ensure that $A \equiv\sub{T} C$ without
disturbing the other desired properties of $A$.
To ensure that $\rho(A) = \frac{1}{2}$, we arrange that $\rho(A \cap R_n) =
\frac{\rho(R_n)}{2}$ for all $n$. Then by restricted countable additivity
(Lemma 2.6 of \cite{JS}),
$$\rho(A) = \sum_n \rho(A \cap R_n) = \sum_n \frac{\rho(R_n)}{2} = \frac{\sum_n \rho(R_n)}{2} = \frac{1}{2}.$$
Let $R_n$ be listed in increasing order as $r_{n,0}, r_{n,1}, \dots$.
We require that, for all $n$ and all sufficiently large $k$, exactly
one of $r_{n, 2k}$ and $r_{n,2k+1}$ is in $A$. This clearly implies
that $\rho(A \cap R_n) = \frac{\rho(R_n)}{2}$.
Let $N_e$ be the requirement that $\overline{\rho}(A \triangle \Phi_e)
> 0$ if $\Phi_e$ is total. So, if $N_e$ is met, then $A$ is not
coarsely computable via $\Phi_e$. We will define a ternary computable
function $g(e,i,s)$ to help us meet this requirement by
``threatening'' to witness that $C$ is low. Let $N_{e,i}$ be the
requirement that either $N_e$ is met or $C'(i) = \lim_s g(e,i,s)$.
Since $C$ is not low, to meet $N_e$ it suffices to meet all of its
subrequirements $N_{e,i}$. Let $R_{e,i}$ denote $R_{\langle e, i
\rangle}$. We use $R_{e,i}$ to meet $N_{e,i}$.
Fix $e,i$. Our module for satisfying $N_{e,i}$ proceeds as follows.
Let $s_0$ be the least number so that $\Phi_{i,s_0}(C_{s_0}; i)
\converges$; if there is no such number, then let $s_0 = \infty$. For
each $s < s_0$, let $g(e,i,s) = 0$ and put $s$ into $A$ if $s$ is of
the form $r_{\langle e,i \rangle, 2k}$ for some $k$. If $s_0$ is
infinite, that is if the search for $s_0$ fails then no other work is done
on $N_{e,i}$. (Note that in this case $\lim_s g(e,i,s) = 0 = C'(i)$,
so $N_{e,i}$ is met.) Suppose $s_0$ is finite (that is, the search for
$s_0$ succeeds). We choose an interval $I_0 \subseteq R_{e,i}$ as
follows. Let $I_0$ be of the form $\{r_{\langle
e,i\rangle, 2j}, \ldots, r_{\langle e,i \rangle, 2k+1} \}$ so that
$\min(I_0) > s_0$ and so that $\rho_m(I_0) \geq \rho_m(R_{e,i})/2$
where $m = r_{\langle e,i \rangle, 2k + 1} + 1$. Let $u_0$ be the use
of the computation $\Phi_{i,s_0}^{C_{s_0}}(i)$. Note that $u_0 < s_0$
by a standard convention and that no element of $I_0$ has been
enumerated in $A$. We then restrain all elements of $I_0$ from
entering $A$ but continue putting alternate elements of $R_{e,i}$
above $\max I_0$ into $A$ as before.
We then continue by searching for the least number $s_1 > s_0$ so that
$\Phi_{e,s_1}(x)\converges$ for every $x \in I_0$ or some number less
than $u_0$ is enumerated into $C$ at stage $s_1$. If no such number
$s_1$ exists, then let $s_1 = \infty$. Set $g(e,i,s) = 0$ whenever $s_0
\leq s < s_1$. If $s_1$ is infinite, then no other work is done on
$N_{e,i}$. (In this case, $N_e$ is met because $\Phi_e$ is not total.)
Suppose $s_1$ is finite (that is, this search succeeds). There are
two cases. First, suppose some number less than $u_0$ is enumerated
in $C$ at stage $s_1$. We then have permission from $C$ to enumerate
numbers in $I_0$ into $A$. Accordingly, we cancel the restraint on
$I_0$ and put $r_{\langle e,i \rangle, 2j'}$ into $A$ whenever $j \leq
j' \leq k$. In this case the interval $I_0$ has become useless to us,
and we go back to our first step but now starting at stage $s_1$. If
we find a stage $s_2 \geq s_1$ with $\Phi_{i,s_2}^{C_{s_2}}(i)
\converges$, say with use $u_1$, we choose a new interval $I_1$ of the
same form as before, but now with $\min(I_1) > s_2$ and proceed as
before with $I_1$ in place of $I_0$, and setting $g(e,i,s) = 0$ for
$s_1 \leq s < s_2$.
Now, suppose no number smaller than $u_0$ is enumerated into $C$ at $s_1$. Then, $\Phi_{e,s_1}(x)\converges$ for all $x \in I_0$. We are
now in a position to make progress on $N_e$ provided that $C$ later
permits us to change $A$ on $I_0$. We then search for the least
number $s_2 \geq s_1$ so that some number less than $u_0$ is
enumerated in $C$ at stage $s_2$. If there is no such number then let
$s_2 = \infty$. We set $g(e,i,s) = 1$ whenever $s_1 \leq s < s_2$ in
order to force $C$ to give us the desired permission. If $s_2$ is
infinite, then no other work is done on $N_{e,i}$. (In this case, we
have $\lim_s g(e,i,s) = 1 = C'(i)$.) Suppose $s_2$ is finite (that is,
this search succeeds). We then declare the interval $I_0$ to be
\emph{successful} and cancel the restraint on $I_0$. Since a number
smaller than $u_0 < \min(I_0)$ has now entered $C$, we have permission
to enumerate elements of $I_0$ into $A$. So, for each $j \leq j' \leq
k$ put exactly one of $r_{\langle e, i \rangle, 2j'}$, $r_{\langle e,i
\rangle, 2j' + 1}$ into $A$ so that $A$ and $\Phi_e$ differ on at
least one of these numbers. (This ensures that at least half of the
elements of $I_0$ are in $A \triangle \Phi_e$ and hence that $\rho_m
(A \triangle \Phi_e) > \frac{\rho_m (R_{e,i})}{4}$ where $m = \max I_0
+1$.) We now restart our process as above. We continue in this
fashion, defining a sequence of intervals. Note that, in general,
$g(e,i,s) = 1$ if at stage $s$ the most recently chosen interval has
been declared successful and we are awaiting a $C$-change below it,
and otherwise $g(e,i,s) = 0$.
This strategy clearly succeeds if any of its searches fail, by the
parenthetical remarks in the construction. Also, if there are
infinitely many successful intervals, it ensures that
$\overline{\rho}(A \triangle \Phi_e) \geq \frac{\rho(R_{e,i})}{4} >
0$, so $N_e$ is met. If all searches are successful but there are
only finitely many successful intervals, then $C'(i) = 0 = \lim_s
g(e,i,s)$ and $N_{e,i}$ is met. Only finitely many elements of $R_{e,i}$ are
permanently restrained from entering $A$ (namely the elements of the final
interval, if any), so $\rho(A) = \frac{1}{2}$ for reasons already given.
\end{proof}
We now obtain the following from Proposition \ref{c.e.density} and Theorem \ref{thm:nonlow}.
\begin{corollary}
If $\mathbf{a}$ is a c.e. degree, then $\mathbf{a}$ is low if and only if every c.e. set in $\mathbf{a}$ that has a density is coarsely computable.
\end{corollary}
For an application of this result to a degree structure arising from
the notion of coarse computability, see Hirschfeldt, Jockusch, Kuyper,
and Schupp \cite{HJKS}.
\section{Density, $1$-genericity, and randomness}
As we have already mentioned, it is easily seen that every degree
contains a set that is both coarsely computable and generically
computable, and every nonzero degree contains a set with neither of
these properties. On the other hand, the next two results show that
for ``most'' degrees $\mathbf{a}$, every $\mathbf{a}$-computable
set that is generically computable is also coarsely computable.
A set $A$ is called \emph{$1$-generic} if for every c.e.\ set $S$ of binary strings,
$A$ either meets or avoids $S$.
\begin{theorem} Let $A$ be a $1$-generic set and let $r \in [0,1]$.
Suppose that $B \leq\sub{T} A$ and $B$ is partially computable at density $r$.
Then $B$ is coarsely computable at density $r$.
\end{theorem}
\begin{proof}
Fix a Turing functional $\Phi$ with $B = \Phi^A$ and a partial computable
function $\phi$ such that $\phi(n) = B(n)$ for all $n$ in the domain of $\phi$, and $\underline{\rho}(\dom\phi) \geq r$. Let
$$S = \{\sigma \in 2^{<\omega} : \Phi^\sigma \hbox{ is incompatible with } \phi\}.$$
Then $S$ is a c.e.\ set of strings so $A$ either meets or avoids $S$.
If $A$ meets $S$, then $B$ disagrees with $\phi$ on some argument, a
contradiction. Hence $A$ avoids $S$. Fix a string $\gamma \prec
A$ such that no string extending $\gamma$ is in $S$. Now define a
computable set $C$ as follows. Given $n$, search for a string
$\sigma$ extending $\gamma$ such that $\Phi^\sigma(n) \converges$ and
put $C(n) = \Phi^\sigma(n)$ for the first such $\sigma$ that is
found. Then $C$ is total because $A$ extends $\gamma$ and $\Phi^A$ is
total. Hence $C$ is a computable set. Further, if $\phi(n)
\converges$ then $B(n) = C(n)$ since no extension of $\gamma$ is in
$S$. Hence $C \triangledown B \supseteq \dom\phi$, so
$\underline{\rho}(C \triangledown B) \geq r$, and hence $B$ is
coarsely computable at density $r$.
\end{proof}
\begin{corollary} If $A$ is $1$-generic and $B \leq\sub{T} A$ is
generically computable, then $B$ is coarsely computable.
\end{corollary}
We do not need the definition of $n$-randomness here, but we simply point
out the easy result that if $A$ is $1$-random, then $\gamma(A) = \frac{1}{2}$.
A set $A$ is called \emph{weakly $n$-random} if $A$ does not belong to
any $\Pi^0_n$ class of measure $0$.
\begin{theorem}
\begin{itemize}
\item[(i)] If $A$ is weakly $1$-random, $B \leq\sub{tt} A$, and $B$ is
partially computable at density $r$, then $B$ is coarsely computable at
density $r$.
\item[(ii)] If $A$ is weakly $2$-random, $B \leq\sub{T} A$, and $B$ is
partially computable at density $r$, then $B$ is coarsely computable at density $r$.
\end{itemize}
\end{theorem}
\begin{proof}
To prove (i), fix a Turing functional $\Phi$ such that $B = \Phi^A$
and $\Phi^X$ is total for all $X \subseteq \omega$. Let $\phi$
be a partial computable function that witnesses that $B$ is
partially computable at density $r$, and define
$$P = \{X : \Phi^X \hbox{ is compatible with } \phi\}.$$
Then $P$ is a $\Pi^0_1$ class and $A \in P$, so $\mu(P) > 0$, where
$\mu$ is Lebesgue measure. By the Lebesgue density theorem, there is
a string $\gamma$ such that $\frac{\mu(P \cap
[\gamma])}{\mu([\gamma])} > .6$, where $[\gamma]=\{X \in 2^\omega :
\gamma \prec X\}$. Define
$$C = \left\{n : \frac{\mu(\{Z \succ \gamma : \Phi^Z(n) = 1\})}{\mu([\gamma])} \geq .5\right\}.$$
Then it is easily seen that $C$ is a computable set and $C
\triangledown B$ contains the domain of $\phi$, so $B$ is coarsely
computable at density $r$.
To prove (ii), fix a Turing functional $\Phi$ with $B = \Phi^A$ and
fix a partial computable function $\phi$ that witnesses that $B$
is partially computable at density $r$. Define
$$P = \{ X : \Phi^X \hbox{ is total and compatible with }
\phi \}.$$ Then $P$ is a $\Pi^0_2$ class and $A \in P$, so $\mu(P) >
0$. Then for notational convenience assume that $\mu(P) > .8$,
applying the Lebesgue density theorem as in part (a). It follows that
for every $n$ there exists $i \leq 1$ such that $\mu(\{X : \Phi^{X}(n)
= i\}) \geq .4$. Given $n$, one can compute such an $i$ effectively,
and then put $n$ into $C$ if and only if $i = 1$. One can easily
check that $C$ is computable and $C \triangledown B \supseteq
\dom\phi$, so $\underline{\rho}(C \triangledown B) \geq
\underline{\rho}(\dom\phi) \geq r$. Hence $B$ is coarsely computable
at density $r$.
\end{proof}
Note that $1$-randomness does not suffice in part (ii) of the above
theorem, since every set is computable from some $1$-random set.
Since the $1$-generic sets are comeager and the weakly $2$-generic
sets have measure $1$, it follows from the last two theorems that
generic computability implies coarse computability below almost every
set, both in the sense of Baire category and in the sense of measure.
The next result contrasts with this fact.
\begin{theorem}
If the degree $\mathbf{a}$ is hyperimmune, there is a set $B \leq\sub{T}
A$ such that $B$ is bi-immune and of density $0$.
\end{theorem}
We omit the proof, which is an easy variation of Jockusch's proof in
\cite{J1}, Theorem 3, that every hyperimmune set computes a bi-immune set.
Bienvenu, Day, and H\"olzl \cite{BDH} proved the beautiful theorem that
every nonzero Turing degree contains an absolutely undecidable set
$A$; that is, a set such that every partial computable function that
agrees with $A$ on its domain has a domain of density $0$. We now
consider the degrees of sets that are both absolutely undecidable and
coarsely computable.
\begin{corollary}
In the sense of Lebesgue measure, almost every set $A$ computes a
set $B$ that is absolutely undecidable and coarsely computable.
\end{corollary}
\begin{proof} D. A. Martin (see \cite[Theorem 8.21.1]{DH}) proved
that almost every set has hyperimmune degree. It is obvious that
every bi-immune set is absolutely undecidable.
\end{proof}
On the other hand, Gregory Igusa has proved the following theorem
using forcing with computable perfect trees.
\begin{theorem}[Igusa, to appear] There is a noncomputable set $A$
such that no set $B \leq\sub{T} A$ is both coarsely computable and
absolutely undecidable.
\end{theorem}
We now turn to studying the degrees of sets $A$ such that $\gamma(A) =
1$ but $A$ is not coarsely computable. As shown in Theorem
\ref{degreeobs}, if either $\mathbf{a \nleq 0'}$ or $\mathbf{a}$ is a
nonzero c.e.\ degree, then $\mathbf{a}$ contains such a set. This
observation might lead one to conjecture that every nonzero degree
computes such a set, but we shall prove the opposite for $\Delta^0_2$
$1$-generic degrees. We will reach this result by first considering sets for which $\gamma(A) = 1$ is witnessed constructively.
\begin{definition} We say that $\gamma(A) = 1$ \emph{constructively}
if there is a uniformly computable sequence of computable sets $C_0, C_1,
\dots$ such that $\overline{\rho}(A \triangle C_n) < 2^{-n}$ for all $n$.
\end{definition}
Of course, if $A$ is coarsely computable, then $\gamma(A) = 1$
constructively. Although the converse appears unlikely, it
was proved by Joe Miller.
\begin{theorem}[Joe Miller, private communication]
\label{constr}
If
$\gamma(A) = 1$ constructively, then $A$ is coarsely computable.
\end{theorem}
\begin{proof}
We present Miller's proof in essentially the form in which he gave it.
Let $I_k$ be the interval $[2^k - 1, 2^{k+1} - 1)$. For
any set $C$, let $d_k(C)$ be the density of $C$ on $I_k$, so $d_k(C)
= \frac{|C \cap I_k|}{2^k}$. The following lemma, which will also
be useful in the proof of Theorem \ref{1G}, relates $\overline{\rho}(C)$
to $\overline{d}(C)$, where $\overline{d}(C) = \limsup_k d_k(C)$.
\begin{lemma}\label{factor2} For every set $C$,
$$\frac{\overline{d}(C)}{2} \leq
\overline{\rho}(C) \leq 2 \overline{d}(C).$$
\end{lemma}
\begin{proof}
For all $k$,
$$d_k(C) = \frac{|C \cap I_k|}{2^k} \leq \frac{|C \upharpoonright(2^{k+1} -1)|}
{2^k} \leq 2 \rho_{2^{k+1} - 1} (C).$$
Dividing both sides of this inequality by $2$ and then taking the lim
sup of both sides yields that $\frac{\overline{d}(C)}{2} \leq \overline{\rho}(C)$.
To prove that $\overline{\rho}(C) \leq 2 \overline{d}(C)$, assume that
$k - 1 \in I_n$, so $2^n \leq k < 2^{n+1}$. Then
$$\rho_k(C) = \frac{|C \upharpoonright k|}{k} \leq
\frac{|C \upharpoonright (2^{n+1} - 1)|}{2^n} = \frac{\sum_{0 \leq i
\leq n} 2^i d_i(C)}{2^n} < 2 \max_{i \leq n} d_i(C).$$
Let $\epsilon > 0$ be given. Then $d_i(C) < \overline{d}(C) +
\epsilon$ for all sufficiently large $i$. Hence there is a finite set
$F$ such that $d_i(C \setminus F) < \overline{d}(C \setminus F) + \epsilon$
for \emph{all} $i$. Then, by the above inequality applied to $C\setminus F$,
we have $\rho_k(C\setminus F) < 2 (\overline{d}(C\setminus F) + \epsilon)$ for all $k$,
so $\overline{\rho}(C\setminus F) \leq 2\overline{d}(C\setminus F)$. As
$\overline{\rho}$ and $\overline{d}$ are invariant under finite
changes of their arguments and $\epsilon > 0$ is arbitrary, it follows
that $\overline{\rho}(C) \leq 2 \overline{d}(C)$.
\end{proof}
We now complete the proof of Theorem \ref{constr}. Let the sequence
$C_n$ witness that $\gamma(A) = 1$ constructively, so $\{C_n\}$ is
uniformly computable and $\overline{\rho}(A \triangle C_n) < 2^{-n}$
for all $n$. It follows from the lemma that $\overline{d}(A \triangle
C_n) < 2^{-n +1}$. Hence, for each $n$, if $k$ is sufficiently large,
we have $d_k (A \triangle C_n) < 2^{-n + 1}$.
For $m < n$, we say that $C_m$ \emph{trusts} $C_n$ on $I_k$ if
$d_k(C_n \triangle C_m) < 2^{-m + 2}$. We say that $C_n$ is
\emph{trusted} on $I_k$ if $C_m$ trusts $C_n$ for all $m < n$. Note
that $C_0$ is trusted on every interval $I_k$. We now define a
computable set $C$ that will witness that $A$ is coarsely computable.
For each $k$, let $N \leq k$ be maximal such that $C_N$ is trusted on
$I_k$, and let $C \upharpoonright I_k = C_N \upharpoonright I_k$.
We claim that $\rho(A \triangle C) = 0$. Fix $n$. Let $k \geq n$ be
large enough that $d_k(A \triangle C_m) < 2^{-m + 1}$ for all $m \leq
n$. Then $d_k (C_n \triangle C_m) \leq d_k(A \triangle C_n) + d_k(A
\triangle C_m) < 2^{-m +1} + 2^{-n +1} < 2^{-m + 2}$ for all $m < n$.
Therefore, $C_n$ is trusted on $I_k$. Hence $C \upharpoonright I_k =
C_N \upharpoonright I_k$ for some $N \geq n$ such that $C_N$ is
trusted on $I_k$. Therefore, $C_n$ trusts $C_N$ on $I_k$, so $d_k
(C_n \triangle C_N) < 2^{-n + 2}$. It follows that $d_k(A \triangle
C) = d_k(A \triangle C_N) \leq d_k(A \triangle C_n) + d_k(C_n
\triangle C_N) < 2^{-n+1} + 2^{-n+2} < 2^{-n + 3}$. Because this is
true for every sufficiently large $k$, we have $\overline{d}(A
\triangle C) \leq 2^{-n+3}$. Since $n$ was arbitrary, it follows that
$\overline{d}(A \triangle C) = 0$ and hence, by the lemma, $\rho(A
\triangle C) = 0$. Thus $A$ is coarsely computable.
\end{proof}
\begin{corollary} \label{0'approx}
Suppose there is a $0'$-computable function $f$ such that, for all
$e$, we have that $\Phi_{f(e)}$ is total and $\{0,1\}$-valued, and
$\overline{\rho}(A \triangle \Phi_{f(e)}) \leq 2^{-e}$.
Then $A$ is coarsely computable.
\end{corollary}
\begin{proof}
By the theorem, it suffices to show that $\gamma(A) = 1$
constructively. Let $g$ be a computable function such that $f(e) =
\lim_s g(e,s)$. We now define a computable function $h$ such that,
for all $e$, we have that $\Phi_{h(e)}$ is total and differs on only finitely
many arguments from $\Phi_{f(e)}$, so that $\Phi_{h(0)},
\Phi_{h(1)}, \dots$ witnesses that $\gamma(A) = 1$ constructively.
To compute $\Phi_{h(e)}(n)$, search for $s \geq n$ such that
$\Phi_{g(e,s)}(n)$ converges in at most $s$ many steps, and let
$\Phi_{h(e)}(n) = \Phi_{g(e,s)}(n)$. The $s$-$m$-$n$ theorem gives us
such an $h$, and clearly $h$ has the desired properties.
\end{proof}
We now have the tools to prove the following result, which we
did not initially expect to be true.
\begin{theorem} \label{1G} Let $G$ be a $\Delta^0_2$ $1$-generic set,
and suppose that $A \leq\sub{T} G$ and $\gamma(A) = 1$. Then $A$ is
coarsely computable.
\end{theorem}
\begin{proof}
Fix $\Phi$ such that $A = \Phi^G$. As in the proof of Theorem
\ref{constr} let $I_k$ be the interval $[2^k - 1, 2^{k+1} - 1)$ and
define $d_k(C) = \frac{|C \upharpoonright I_k|}{2^k}$ and
$\overline{d}(C) = \limsup_k d_k(C)$.
Consider first the case that for some $\epsilon > 0$ and for every
computable set $C$ and every number $k$, we have that $G$ meets the
set $S_{\epsilon, C, k}$ of strings defined below:
$$S_{\epsilon, C, k} = \{\nu : (\exists l > k)
[d_l (\Phi^\nu \triangle C) \geq \epsilon]\}.$$
Of course, $\nu$ must be such that $\Phi^\nu(j) \converges$ for all $j
\in I_l$ for the above to make sense. We claim that $\gamma(A) < 1$
in this case, so that this case cannot arise. Let $C$ be a computable
set and fix $\epsilon$ as in the case hypothesis. Then, for every $k$
there exists $l > k$ such that $d_l (A \triangle C) \geq \epsilon$ by
the choice of $\epsilon$. It follows that $\overline{d}(A \triangle
C) \geq \epsilon$, so $\overline{\rho}(A \triangle C) \geq
\frac{\epsilon}{2}$ by Lemma \ref{factor2}. By Lemma \ref{comp} it
follows that $\underline{\rho}(A \triangledown C) \leq 1 -
\frac{\epsilon}{2}$. Hence $\gamma(A) \leq 1 - \frac{\epsilon}{2} <
1$. Since $\gamma(A) = 1$ by assumption, this case cannot arise.
Since $G$ is $1$-generic, it follows that for every $n$ there is a
computable set $C$ and a number $k$ such that $G$ avoids
$S_{2^{-(n+2)}, C, k}$; i.e., there exists $\gamma \prec G$ such
that $\gamma$ has no extension in $S_{2^{-(n+2)}, C, k}$. Given $l
\geq k$, let $\nu_0$ and $\nu_1$ be strings extending $\gamma$ such
that $\Phi^{\nu_i}(x)\converges$ for all $x \in I_l$ and $i \leq
1$. Then
$$d_l (\Phi^{\nu_0} \triangle \Phi^{\nu_1}) \leq
d_l (\Phi^{\nu_0} \triangle C) + d_l(C \triangle \Phi^{\nu_1}) <
2^{-(n+2)} + 2^{-(n+2)} = 2^{-(n+1)}.$$ Since $G$ is $\Delta^0_2$,
using an oracle for $0'$ we can find $\gamma_n$ and $k_n$ such that
for all $\nu_0,\nu_1$ extending $\gamma_n$ and all $l \geq k_n$, if
$\Phi^{\nu_i}(x)\converges$ for all $x \in I_l$ and $i \leq 1$ then
$d_l (\Phi^{\nu_0} \triangle \Phi^{\nu_1}) \leq 2^{-(n+1)}$. Note that
if we take $\nu_0 \prec G$ then $d_l(\Phi^{\nu_1} \triangle A) <
2^{-(n+1)}$.
For each $n$, define a computable set $B_n$ as follows. On each
interval $I_k$ search for $\nu_1 \succcurlyeq \gamma_n$ such that
$\Phi^{\nu_1}$ converges on $I_k$. Note that such a $\nu_1$ exists
because $\gamma_n \prec G$ and $\Phi^G$ is total. Let $B_n
\upharpoonright I_k = \Phi^{\nu_1} \upharpoonright I_k$. Then $B_n$ is
a computable set, since the only non-effective part of its definition
is the use of the \emph{single} string $\gamma_n$. Furthermore, an
index for $B_n$ as a computable set can be effectively computed from $\gamma_n$ and hence from $0'$.
We claim that $\overline{\rho}(B_n \triangle A) \leq 2^{-n}$. Fix
$n$. By Lemma \ref{factor2}, it suffices to show that
$\overline{d}(B_n \triangle A) \leq 2^{-(n+1)}$. For all $k$, we have
that $d_k(B_n \triangle A) = d_k(\Phi^{\nu_1} \triangle A)$ for some
string $\nu_1$ extending $\gamma_n$. Hence, if $k$ is sufficiently
large, it follows that $d_k(B_n \triangle A) \leq 2^{-n+1}$, and hence
$\overline{d}(B_n \triangle A) \leq 2^{-(n+1)}$, so
$\overline{\rho}(B_n \triangle A) \leq 2^{-n}$. It now follows from
Corollary \ref{0'approx} with $\Phi_{f(e)} = B_e$ that $A$ is coarsely
computable.
\end{proof}
\section{Further results}
In this section we investigate the complexity of $\gamma(A)$ as a real
number when $A$ is c.e.\ and look at the distribution of values of
$\gamma(B)$ as $B$ ranges over all sets computable from a given set
$A$. A real is \emph{left-$\Sigma^0_3$} if $\{q\in \mathbb Q:q<r\}$ is
$\Sigma^0_3$.
\begin{proposition} \label{sigma3} If $A$ is a c.e.\ set, then
$\gamma(A)$ is a left-$\Sigma^0_3$ real.
\end{proposition}
\begin{proof}
Let $A$ be a c.e.\ set, and let $q$ be a rational number with $q
\neq \gamma(A)$. Then the following two statements are equivalent:
\begin{enumerate}
\item[(i)] $q < \gamma(A)$.
\item[(ii)] There is a computable set $C$ such that $\rho_n(A
\triangledown C) \geq q$ for \emph{all} $n$.
\end{enumerate}
It is immediate that (ii) implies (i) since (ii)
implies that $A$ is coarsely computable at density $q$ and hence $q
\leq \gamma(A)$.
Now assume (i) in order to prove (ii). Let $r$ be a real number with
$q < r < \gamma(A)$. Then $A$ is coarsely computable at density $r$,
so there is a computable set $C$ such that $A \triangledown C$ has
lower density at least $r$. Since $q < r$, it follows that $\rho_n(A
\triangledown C) \geq r$ for all but finitely many $n$. By making a
finite change in $C$, we can ensure that this inequality holds for
\emph{all} $n$.
Routine expansion shows that the set of rational numbers $q$ satisfying (ii) is
$\Sigma^0_3$, so $A$ is left-$\Sigma^0_3$ by definition.
\emph{Note:} The formulation of (ii) was chosen in order to minimize
the number of quantifiers when it is expanded. If we proceeded
by simply using the fact that, for $q \neq \gamma(A)$, we have that $q
< \gamma(A)$ if and
only if $A$ is coarsely computable at density $q$ and used a routine
expansion of the latter, we could conclude only that $\gamma(A)$ is
left-$\Sigma^0_5$.
\end{proof}
In the next result, we prove the converse and thus characterize the
reals of the form $\gamma(A)$ for $A$ c.e.
\begin{theorem}
Suppose $0 \leq r \leq 1$. Then the
following are equivalent:
\begin{itemize}
\item[(i)] $r = \gamma(A)$ for some c.e.\ set $A$.
\item[(ii)] $r$ is left-$\Sigma^0_3$.
\end{itemize}
\end{theorem}
\begin{proof}
It was shown in the previous proposition that (i) implies (ii), so it
remains to be shown that (ii) implies (i). Let $r$ be
left-$\Sigma^0_3$. Our proof is based on that of Theorem 5.7 of
\cite{DJS}, which shows that $r$ is the lower density of some
c.e.\ set. That proof consists in taking a $\Delta^0_2$ set $B$ such
that $\underline{\rho}(B)=r$ (which exists by the relativized form of
Theorem 5.1 of \cite{DJS}) and constructing a strictly increasing
$\Delta^0_2$ function $t$ and a c.e.\ set $A$ such that for each $n$,
\begin{enumerate}
\item $\rho_{t(n)}(A)=\rho_n(B)$
\item $A \cap [t(n),t(n+1))$ is an initial segment of $[t(n),t(n+1))$.
\end{enumerate}
It then follows easily that $\underline{\rho}(A)=\underline{\rho}(B)=r$.
Let $S$ be the set of all pairs $(k,e)$ such that $e \leq k$. Let $f$
be a computable bijection between $S$ and $\omega$. We can easily adapt the proof
of Theorem 5.7 of \cite{DJS} to replace (1) by
\begin{itemize}
\item[(1$^\prime$)] $\rho_{t(f(k,e))}(A)=\rho_k(B)$ for each $k$ and $e
\leq k$,
\end{itemize}
while still having (2) hold for each $n$. Furthermore, we can also
ensure that when a new approximation $t(n,s+1)$ to $t(n)$ is defined,
it is chosen to be greater than both $2^{t(n-1,s+1)}$ and $2^{t(s,s)}$
(because for each instance of Lemma 5.8 of \cite{DJS}, there are
infinitely many $c$ witnessing the truth of the lemma).
We now define a c.e.\ set $C$. At stage $s$, proceed as
follows for each pair $(k,e)$ with $f(k,e) \leq s$. Let $n=f(k,e)$. If
$\Phi_{e,s}(x)\converges$ for all $x \in [t(n-1,s),t(n,s))$, then for
each such $x$ for which $\Phi_e(x)=0$, enumerate $x$ into $C$ (if
$x$ is not already in $C$). We say that $x$ is put into $C$ for the
sake of $(k,e)$.
Let $D=A \cup C$. Then $D$ is a c.e.\ set, and $\underline{\rho}(D)
\geq \underline{\rho}(A) = r$. By Theorem 3.9 of \cite{DJS}, for each
$\epsilon>0$, there is a computable subset of $D$ with lower density
greater than $r-\epsilon$. It follows that $\gamma(D) \geq r$.
Now let $e$ be such that $\Phi_e$ is total. Fix a $k$ and let
$n=f(k,e)$. Let $s$ be least such that $t(n,s+1)=t(n)$. Every number
put into $C$ by the end of stage $s$ is less than $t(s,s)$. Every
number put into $C$ after stage $s$ for the sake of any pair other
than $(k,e)$ is either less than $t(n-1)=t(n-1,s+1)$ or greater than
or equal to $t(n)$. By our assumption on the size of $t(n)$, it
follows that $C(x) \neq \Phi_e(x)$ for every $x \in [\log_2
t(n),t(n))$, so $\rho_{t(n)}(C \triangledown \Phi_e) \leq
\frac{\log_2 t(n)}{t(n)}$, and hence
\begin{multline*}\rho_{t(n)}(D \triangledown \Phi_e) \leq
\rho_{t(n)}(C \triangledown
\Phi_e) + \rho_{t(n)}(D \triangledown C) \\ \leq \frac{\log_2
t(n)}{t(n)} + \rho_{t(n)}(A)=\frac{\log_2
t(n)}{t(n)} + \rho_k(B).
\end{multline*}
Since $\lim_n \frac{\log_2 t(n)}{t(n)} =0$, we have
$\underline{\rho}(D \triangledown \Phi_e) \leq
\underline{\rho}(B)=r$. Since $e$ is arbitrary, $\gamma(D) \leq r$.
\end{proof}
\begin{definition}
If $A \subseteq \N$ we call
\[S(A) = \{\gamma(B) : B \leq\sub{T} A\} \subseteq [0,1]\]
the \emph{coarse spectrum} of $A$.
\end{definition}
\begin{theorem} For any set $A$ and any $\Delta^0_2$ real $s \in
[0,1]$, we have that $s \cdot \gamma(A) + (1-s) \in S(A)$. It follows that $S(A)$ is
dense in the interval $[\gamma(A), 1]$.
\end{theorem}
\begin{proof}
We may assume that $s>0$, since any computable $B \leq\sub{T} A$
witnesses the fact that $1 \in S(A)$.
By Theorem 2.21 of \cite{JS} there is a computable set
$R$ of density $s$. Note that $R$ is infinite.
Let $h$ be an increasing computable function with range $R$, and let
$B = h(A)$. Then $B \leq\sub{T} A$, so it suffices to prove that
$\gamma(B) = s \cdot \gamma(A) + (1-s)$. For this, we need the
following lemma, which relates the lower density of $h(X)$ to that
of $X$. The corresponding lemma for density was proved as Lemma 3.4
of \cite{DJMS}, and the proof here is almost the same.
\begin{lemma} \label{prod} Let $h$ be a strictly increasing function
and let $X \subseteq \omega$. Then $\underline{\rho}(h(X)) =
\rho(\mbox{range}(h)) \underline{\rho}(X)$, provided that the range
of $h$ has a density.
\end{lemma}
\begin{proof}
Let $Y$ be the range of $h$, and for each $u$, let $g(u)$ be the
least $k$ such that $h(k) \geq u$. As shown in the proof of Lemma
3.4 of \cite{DJMS}, $\rho_u (h(X)) = \rho_u (Y) \rho_{g(u)}(X) $ for
all $u$, via bijections induced by $h$. Taking the lim inf of both
sides and using the fact that $\rho(Y)$ exists, we see that
$$\underline{\rho}(h(X))
= \rho(Y)(\liminf \langle \rho_{g(0)}(X), \rho_{g(1)}(X) , \dots
\rangle).$$ It is easily seen that the function $g$ is finite-one and
$g(h(x)) = x$ for all $x$, and $g(u+1) \leq g(u) + 1$ for all $u$.
Hence the sequence on the right-hand side of the above equation can be
obtained from the sequence $\rho_0 (X), \rho_1(X), \dots$ by replacing
each term by a finite, nonempty sequence of terms with the same value.
Thus the two sequences have the same lim inf, and we obtain
$\underline{\rho}(h(X)) = \rho(Y) \underline{\rho}(X)$, as needed
to prove the lemma.
\end{proof}
To prove that $\gamma(B) = s \cdot \gamma(A) + (1-s)$, it suffices to
show that for each $t \in [0,1]$, $A$ is coarsely computable at
density $t$ if and only if $B$ is coarsely computable at density $st +
1 - s$. Suppose first that $A$ is coarsely computable at density $t$,
and let $C$ be a computable set such that $\underline{\rho}(A
\triangledown C) \geq t$. Let $\widehat{C} = h(C) \cup \overline{R}$. Then
$\widehat{C}$ is a computable set and
$$\underline{\rho}(\widehat{C} \triangledown B) = \underline{\rho}(h(C \triangledown A) \cup \overline{R}) \geq \underline{\rho}(h(C \triangledown A)) +
\underline{\rho}(\overline{R}) = s \underline{\rho}(C \triangledown A)
+ 1 - s \geq s \cdot t + (1-s).$$
It follows that $B$ is coarsely computable at density $st + (1-s)$.
Conversely, if a computable set $\widehat{C}$ witnesses that $B$ is
coarsely computable at density $st + (1-s)$, let $C =
h^{-1}(\widehat{C})$, and check by a similar argument that $C$ witnesses
that $A$ is coarsely computable at density $t$ since $s > 0$.
\end{proof}
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Gary Neville has admitted that he is concerned for Harry Kane’s welfare amid fears that the Spurs and England striker is suffering from burnout.
Since establishing himself in Spurs’ starting line-up at the start of the 2014-15 season, Kane has barely missed a game for club or country, clocking up 232 appearances for his club as well as for England’s senior team and the U21s.
The 25-year-old has also had only one summer off over the past four years having featured for England at the 2016 European Championships and the 2018 World Cup as well as for the U21s in the Euro’s back in 2015.
Kane has appeared sluggish and off the pace since returning from an ankle injury sustained against Bournemouth in March and Neville expressed his surprise that Gareth Southgate decided to use him as a second half substitute during England’s 1-0 win over Switzerland on Tuesday.
‘I am concerned about Harry Kane. Since 2015 he has played 175 matches and he’s struggling. Forget the physical side, he needs a mental break as well,’ Neville said on Sky Sports.
‘I don’t think he should have been here for the two weeks. I think there is a welfare thing in terms of looking after players. He has not had enough of a break since 2015.’
Such is his importance to both Spurs and England, Kane has generally played for both whenever he’s fit with only a few separate ankle injuries keeping him out of either team for a prolonged period/
Although Kane top-scored at the World Cup with six goals and has two from four games in the Premier League this season, there have been question marks over his overall levels of performance with some pundits suggesting that his drop in shot attempts per game highlights his fatigue.
However, Southgate insisted there was ‘no big issue’ in using Kane for half an hour against the Swiss and suggested that his captain is fine with the workload.
Speaking after the game, Southgate said: ‘I think he’s going to get matches at his club. He’s had three days of little training. We needed to keep him ticking over.
‘If he didn’t play today, he would have gone out and done some running on the pitch. Playing him for 30 minutes was no big issue. To start the game is a different sort of feel, mentally it’s different.’
Kane will be back in action from the start when Spurs take on Liverpool at Wembley in the Premier League on Saturday.
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TITLE: Name for a particular subgroup of parabolic subgroups of the general linear groups.
QUESTION [0 upvotes]: Possible Duplicate:
Name for a particular subgroup of parabolic subgroups of the general linear groups.
Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms stabilizing a flag $V=V_1\supset V_2\supset\cdots\supset V_1$ is called a parabolic subgroup of $GL(V)$. I am interested in the subgroup $Q$ of $P$ consisting of all automorphisms $h$ such that the induced automorphism $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the identity for every $i$. My question is:
Is this subgroup $Q$ named somewhere yet? If not, can you recommend a name?
Similarly, I am also interested in the subgroup $T$ of $P$ consisting of automorphisms $h$ such that the group of induced automorphisms $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the symmetric group (on a basis of $V_i/V_{i+1}$). Is this subgroup $T$ named somewhere yet? If not, can you recommend a name? Is there anyway to realize that $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is a symmetric group without looking a specific basis of $V_i/V_{i+1}$)?
Finally, I would appreciate very much if you have any reference on the study of these subgroups.
REPLY [4 votes]: As David says $Q$ is the unipotent radical of $P$. The subgroup $T$ is a preimage of the Weyl group $W$ of the group $G_i\cong GL(V_i/ V_{i+1})$. This group $T$ looks a direct product of $Q$ with a big chunk of a Levi complement of $P$. The Levi complement is a direct product isomorphic to $G_1\times\cdots \times G_k$; to obtain the group $T$, you replace the $i$-th factor by the normalizer $N$ of a maximal split torus $T_0$ of $G_i$.
This is, in fact, the typical way to realize the Weyl group of $G_i$ -- $W$ is isomorphic to the quotient $N/T_0$ -- but this is effectively the same thing as your method of fixing a specific basis of $V_i/V_{i+1}$. The Weyl group rears its head in lots of different ways (most especially as a Coxeter group related to the Dynkin diagram of $G_i$) so this is certainly not the only way to realise it. I don't, however, see any other way to realise your group $T$ (although it depends what you mean by `realise'!).
As for references, it depends on what kind of approach you want. If you want a treatment of $GL_n$ as an algebraic group then I recommend anything by Carter or Humphreys, or else there is the book by Borel. All of these people work in much greater generality than $GL_n$ though. If you just want to understand $GL_n$, then standard algebra texts like the one of Jacobson might be your best bet. (I have e-copies of some of these. If you want them, email me.)
| 183,545
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Father/Daughter Wedding Dance
Ah, the Father/Daughter Wedding Dance ... the loving glance of a parent and child, the awkward swaying to cheesy music ...
Well, not at Brooke Lavin's wedding! At her wedding, her pop Bill showed some of his smooth moves: Hit play or go to Link [YouTube] - via The Frisky
In fact, during the long intro, it was creepy, with him mouthing romanticisms (not father-daughter love phrases). The choreography, following that, reinforced the creepy.
| 272,843
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\begin{document}
\begin{abstract}
In this paper we discuss different properties of noncommutative schemes over a field.
We define a noncommutative scheme as a differential graded category of a special type.
We study regularity, smoothness and properness for noncommutative schemes.
Admissible subcategories
of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes
that are called geometric noncommutative schemes.
In this paper we show that the world of all geometric noncommutative schemes is closed
under an operation of a gluing of differential graded categories via bimodules.
As a consequence of the main theorem we obtain that
for any finite dimensional algebra with separable semisimple part
the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on
a smooth projective scheme.
Moreover, if the algebra has finite global dimension, then
the full subcategory is admissible.
We also provide a construction of a smooth projective scheme
that admits a full exceptional collection and contains
as a subcollection an exceptional collection given in advance.
As another application of the main theorem we obtain, in characteristic 0, an existence of a full embedding for the category of perfect complexes
on any proper scheme to the category of perfect complexes on a smooth projective scheme.
\end{abstract}
\maketitle
\section*{Introduction}
One of the main approaches to noncommutative geometry is to consider categories of sheaves on varieties instead of varieties
themselves. In algebraic geometry only quasi-coherent sheaves represent well the algebraic structure of a variety and
do not depend on a topology. Besides, homological algebra convinces to consider derived categories whenever we meet an abelian category.
Thus, instead of schemes one may consider derived versions of categories of quasi-coherent sheaves and triangulated category
of perfect complexes that are compact objects therein. This approach is very powerful.
Let $X$ be a scheme over a field $\kk.$ Studying schemes it is natural to consider the unbounded
derived category of quasi-coherent sheaves $\D(\Qcoh X)$ and the unbounded derived category of complexes of $\cO_X$\!--modules with
quasi-coherent cohomology $\D_{\Qcoh}(X).$ Fortunately, and it is well known and proved in \cite{BN}, for a quasi-compact and separated scheme $X$
the canonical functor $\D(\Qcoh X)\to \D_{\Qcoh}(X)$ is an equivalence. Moreover, it was shown in \cite{Ne}
that in this case the derived category $\D(\Qcoh X)$ has enough compact objects and the subcategory
of compact objects is nothing else but the subcategory of perfect complexes $\prf X.$
Recall that a complex is called perfect if it is locally quasi-isomorphic to a bounded complex
of locally free sheaves of finite type.
Furthermore, it was proved in \cite{Ne, BVdB} that the category $\prf X$ admits a classical generator $\mE,$ i.e.
the minimal full triangulated subcategory of $\prf X$ that contains $\mE$ and is closed under direct summands
coincides with the whole $\prf X.$ As a consequence, such a perfect complex $\mE$ is
a compact generator of the whole $\D(\Qcoh X).$
It is very useful to consider a triangulated category $\T$ together with
an {\em enhancement} $\dA,$ which is a differential graded (DG) category
with the same objects as in $\T,$ but
the set of morphisms between two objects in $\dA$ is a complex of vector spaces.
One recovers morphisms in $\T$ by taking the cohomology $H^0$ of the
corresponding morphism complex in $\dA.$ Thus,
$\T$ is the {\em homotopy} category $\Ho (\dA)$ of the DG category $\dA.$
Such an $\dA$ is called an {\em enhancement} of $\T.$
The triangulated category $\D(\Qcoh X)$ has several natural DG enhancements:
the category of h-injective complexes, the DG quotient of all complexes by acyclic complexes, the DG quotient
of h-flat complexes by acyclic h-flat complexes. They all are quasi-equivalent and we can work with any of them.
Denote by $\dD(\Qcoh X)$ a DG enhancement of $\D(\Qcoh X)$ and by $\prfdg X$ the induced
DG enhancement of the category of perfect complexes $\prf X.$
Let us take a generator $\mE\in\prfdg X.$ Denote by $\dE$ its DG algebra of endomorphisms,
i.e. $\dE=\dHom(\mE, \mE).$ Since $\mE$ is perfect the DG algebra $\dE$ has only finitely many nonzero cohomology groups.
Keller's results from \cite{Ke} imply that
the DG category $\prfdg X$ is quasi-equivalent to
$\prfdg \dE,$ where $\dE$ is a cohomologically bounded DG algebra and $\D(\Qcoh X)$ is equivalent
to the derived category of DG $\dE$\!--modules $\D(\dE).$
The facts described above allow us to suggest a definition of a {\em (derived) noncommutative scheme} over $\kk$
as a $\kk$\!--linear DG category of the form $\prfdg\dE,$
where $\dE$ is a cohomologically bounded DG algebra over $\kk.$
In this case the derived category
$\D(\dE)$ will be called the derived category of quasi-coherent sheaves on this noncommutative scheme.
The simplest and, apparently, the most important class of schemes is the class of smooth and projective schemes
or, more generally, the class of schemes that are regular and proper.
The properties of regularity, smoothness and properness can be interpreted in categorical terms and
can be extended to noncommutative schemes.
We say that a $\kk$\!--linear triangulated category $\T$ is {\em proper} if
$\bigoplus_{m\in\ZZ}\Hom(X, Y[m])$ is finite dimensional for any two objects $X, Y\in\T.$
It will be called {\em regular} if it has a strong generator, i.e. such an object that generates the whole $\T$
in a finite number of steps
(Definitions \ref{strong_gen} and \ref{reg_and_prop}).
The notion of smoothness is well-defined for a DG category. A $\kk$\!--linear DG category $\dA$ is called {\em $\kk$\!-smooth} if it is perfect as a module over $\dA^{\op}\otimes_{\kk}\dA.$
Application of these definitions to a DG category of perfect objects $\prfdg\dE$ and its homotopy category
$\prf\dE$ leads us to well-defined notions of a
regular, smooth, and proper noncommutative schemes.
It can be proved that a usual separated noetherian scheme $X$ is regular if and only if the category
$\prf X$ is regular (Theorem \ref{regular}). Furthermore, for separated schemes of finite type
properness of $X$ is equivalent to properness of $\prf X$
(Propositions \ref{prop_scheme}) and
smoothness of $X$ is equivalent of $\prfdg X$
(\cite{Lu} and Propositions \ref{smooth_proper}).
These facts imply that smooth and projective schemes form a subclass of
the class of smooth and proper noncommutative schemes. It is not difficult to give an example of a noncommutative
smooth and proper scheme $\prfdg\dE$ that is not quasi-equivalent to $\prfdg X$ of a usual commutative scheme.
Let us consider the world of all smooth projective schemes.
As in the theory of motives, an important step is adding direct summands,
in our situation it is natural to extend the world of smooth projective schemes to the world of all admissible subcategories
$\N\subset \prf X,$ where $X$ is a smooth and projective scheme.
(Recall that a full triangulated subcategory $\N\subset\prf X$ is called admissible if the inclusion
functor has right and left adjoint functors and, hence, $\N$ is a semi-orthogonal summand of $\prf X.$)
Such admissible subcategories are also smooth and proper, and they give natural examples of smooth and proper noncommutative schemes, which will be called {\em geometric noncommutative schemes}.
One may consider the 2-category of smooth and proper noncommutative schemes
$\RPNS$ over a field $\kk.$
Objects of $\RPNS$ are DG categories $\dA$ of the form $\prfdg\dE,$ where
$\dE$ is a smooth and proper DG algebra; 1-morphisms are
quasi-functors, i.e. DG functors modulo inverting quasi-equivalences; and 2-morphisms are morphisms of quasi-functors.
The 2-category $\RPNS$ has a full 2-subcategory of geometric smooth and proper noncommutative schemes $\GNS.$
Evidently, $\GNS$ contains all smooth and projective commutative schemes. Moreover, a To\"en's theorem \cite{To}
says that quasi-functors from $\prfdg X$ to $\prfdg Y$ correspond bijectively to perfect complexes on the product, i.e.
$\prf(X\times Y)$ is the category of morphisms
from $X$ to $Y$ in $\RPNS.$
The world of smooth and proper noncommutative schemes is plentiful and multiform in the sense of different constructions and operations.
For instance, it contains all $\prfdg \Lambda$ for all finite dimensional algebras $\Lambda$ of finite global dimension.
Besides, for any two noncommutative schemes $\dA$ and $\dB$ every perfect $\dB^{\op}\hy\dA$\!--bimodule
$\mS$ produces a new noncommutative scheme $\dC=\dA\underset{\mS}{\oright}\dB,$ which we call the gluing of
$\dA$ and $\dB$ via $\mS.$ The resulting DG category $\dC$ is also smooth and proper (Definition
\ref{gluing_cat} and Section \ref{reg_proper_noncommutative}).
Its homotopy category has a semi-orthogonal decomposition of the form $\Ho(\dC)=\langle\Ho(\dA),\, \Ho(\dB)\rangle.$
Of course, this procedure can be iterated and it allows to reproduce new and new noncommutative schemes.
If we glue commutative schemes like $\prfdg X$ and $\prfdg Y,$ then the result is almost never a commutative scheme, except
for a few special examples (Examples \ref{ex1} and \ref{ex2}).
The main purpose of this paper is to show that the world of all geometric noncommutative schemes is closed
under operation of the gluing. More precisely, we prove that for any smooth and projective
$X$ and $Y$ the gluing $\prfdg X\underset{\mS}{\oright}\prfdg Y$
of two DG categories of the form $\prfdg X$ and $\prfdg Y$ via arbitrary perfect bimodule $\mS$
is a geometric noncommutative scheme, i.e. it is quasi-equivalent to an admissible full DG subcategory in $\prfdg V$
for some smooth projective scheme $V$ (Theorem \ref{main}).
This result implies that the subcategory of smooth proper geometric noncommutative schemes is closed under gluing
via any bimodules (Theorem \ref{main2}).
These theorems have useful applications. Using results of \cite{KL} we obtain that
for any proper scheme
$Y$ over a field of characteristic $0$ there is a full embedding of $\prf Y$ into $\prf V,$ where $V$ is smooth and projective (Corollary \ref{emb}).
In Section \ref{applications} we show that for any finite dimensional algebra $\Lambda$
with the semisimple part $S=\Lambda/\rd$ separable over the base field $\kk,$ there exist a smooth projective scheme
$V$ and a perfect complex $\E^{\cdot}$ on $X$ such that $\End(\E^{\cdot})=\Lambda$ and
$\Hom(\E^{\cdot}, \E^{\cdot}[l])=0$ when $l\ne 0$ (Theorem \ref{algebra}).
As a consequence of this theorem we obtain that
for any finite dimensional algebra $\Lambda$ over $\kk$ with separable semisimple part $S=\Lambda/\rd$
there is a smooth projective scheme $V$ such that
the DG category $\prfdg\Lambda$ is quasi-equivalent to a full DG subcategory of $\prfdg V.$
Moreover, if $\Lambda$ has finite global dimension, then
$\prf\Lambda$ is admissible in $\prf V$ (Corollary \ref{algebra_inclusion}).
Note that over a perfect field all algebras a separable.
In Section \ref{exceptional_coll} we give an alternative and more useful procedure of constructing a smooth projective scheme
that admits a full exceptional collection and contains
as a subcollection an exceptional collection given in advance.
More precisely, for any DG category $\dA,$ for which the homotopy category
$\Ho(\dA)$ has a full exceptional collection,
we give an explicit construction of a smooth projective scheme $X$ and an exceptional collection of line bundles
$\sigma=(\L_1,\dots, \L_n)$ in $\prf X$ such that the DG subcategory $\dN\subset\prfdg X$
generated by $\sigma$ is quasi-equivalent to $\dA.$
Moreover, by construction $X$ is rational and has a full exceptional collection (Theorem \ref{exc_col}).
In the last section we illustrate this theorem considering the case
of noncommutative projective planes, in the sense of noncommutative deformations of the usual projective plane,
which have been introduced and described by M.~Artin, J.~Tate, and M.~Van den Bergh in \cite{ATV}.
The author is very grateful to Valery Alexeev, Alexei Bondal, Sergei Gorchinskiy,
Anton Fonarev, Alexander Kuznetsov, Valery Lunts, Amnon Neeman, Stefan Nemirovski, Yuri Prokhorov,
Constantin Shramov for useful and valuable discussions.
The author would like to thank anonymous referee and
Aise Johan de Jong for pointing out the results of the publication \cite{LN},
which allowed to prove Proposition \ref{prop_scheme} in full generality.
The author wishes to express his gratitude to Theo Raedschelders who drew attention
to the results of Dieter Happel's paper \cite{Hap}.
\section{Preliminaries on triangulated categories, generators, and semi-orthogonal decompositions }
\subsection{Generators in triangulated categories}
In this section we discuss different notions of generators in triangulated categories.
Let $\T$ be a triangulated category and $S$ be a set of objects.
\begin{definition}
A set of objects $S\subset \Ob\T$
{\em generates} the triangulated category $\T$ if $\T$ coincides with the smallest strictly full triangulated subcategory of
$\T$ which contains $S.$ (Strictly full means it is full and closed under isomorphisms).
\end{definition}
The notion of generating a triangulated category is very rigid, because a triangulated subcategory
that is generated by a set of objects is not necessarily idempotent complete. A much more useful notion of generating
a triangulated category is the notion of a set of classical generators.
\begin{definition}
A set of objects $S\subset \Ob\T$
forms a {\em set of classical generators} for $\T$ if the category $\T$ coincides with the smallest triangulated subcategory of
$\T$ which contains $S$ and is closed under taking direct summands.
When $S$ consists of a single object we obtain the notion of a {\em classical generator}.
\end{definition}
If a classical generator $X$ generates the whole category in a finite number of steps, then it is called
a {\em strong generator}.
More precisely, let $\I_1, \I_2 \subset
\mathcal{T}$ be two full subcategories. Define their product as
\[ \I_1 * \I_2 = \bigl\{ \text{the full subcategory, consisting on all objects $Y$ of the form} : Y_1 \to Y \to Y_2 \text{ , } Y_i \in \I_i \bigr\} \text{ .} \]
Let $\langle \I \rangle$ be the smallest full subcategory
that contains $\I \subset \langle \I\rangle$ and that is
closed under shifts, finite direct sums, and direct summands. We call
$\langle \I \rangle$ the \emph{envelope} of $\I.$
Put
$\I_1 \diamond \I_2 := \langle \I_1 *
\I_2 \rangle$
and we define by induction
$\langle \I\rangle_k=\langle\I\rangle_{k-1}\diamond\langle \I\rangle.$ If $\I$ consists of a single object $X$ we denote $\langle \I\rangle$ by
$\langle X\rangle_1$ and put by induction $\langle X\rangle _k=\langle X\rangle_{k-1}\diamond\langle X\rangle_1.$
\begin{definition}\label{strong_gen}
An object $X$ is called a {\em strong generator} if $\langle X\rangle_n=\T$ for some $n\in\NN.$
\end{definition}
\begin{remark}{\rm
If $X \in \mathcal{T}$ is a classical generator, then $\mathcal{T} =
\bigcup_{i=1}^\infty \langle X \rangle_i.$
It is also easy to see that if $\mathcal T$ has a strong generator,
then any classical generator is strong as well.}
\end{remark}
Following \cite{Ro} we define the dimension of a triangulated
category.
\begin{definition}
The {\em dimension} of a triangulated category $\T,$ denoted by $\dim \T,$ is the smallest integer $d\ge 0$ such that there exists an object $X\in \T$
for which $\langle X\rangle_{d+1}=\T.$
\end{definition}
Let now $\T$ be a triangulated category which admits arbitrary small coproducts (direct sums).
Such a category is called {\em cocomplete.}
\begin{definition} Let $\T$ be a cocomplete triangulated category.
An object $X\in \T$ is called {\em compact} in $\T$ if $\Hom(X,-)$ commutes with arbitrary small coproducts, i.e. for any
set of objects $\{ Y_i\}\subset \T$ the canonical map
$
\bigoplus_i\Hom (X, Y_i){\longrightarrow}
\Hom (X, \bigoplus_i Y_i)
$ is an isomorphism.
\end{definition}
Compact objects in $\T$ form a triangulated subcategory denoted by $\T^c\subset \T.$
\begin{definition}\label{van}
Let $\T$ be a cocomplete triangulated category. A set $S \subset \Ob\T^c$ is
called a {\em set of compact generators} if
any object $Y\in \T$ for which $\Hom(X, Y[n])=0$ for all $X\in S$ and all $n\in \ZZ$ is a zero object.
\end{definition}
\begin{remark}
{\rm
Since $\T$ is cocomplete, it can be proved that the property of $S \subset \Ob\T^c$ to be
a set of compact generators is equivalent to the following property:
the category $\T$ coincides with the smallest full
triangulated subcategory containing $S$ and closed under
small coproducts \cite{Ne1}.
}
\end{remark}
\begin{remark}\label{twodef}{\rm
The definition of compact generators is closely related to the definition of classical generators.
Assume that a cocomplete triangulated category $\T$ is compactly generated by the set
of compact objects $\T^c.$ In this case a set $S\subset \T^c$ is a set of compact generators of $\T$ if and only if
the set $S$ is a set of classical generators of the subcategory of compact objects $\T^c$ \cite{Ne1}.
}
\end{remark}
Let $\T$ be a cocomplete triangulated category and let
$X\in\T^c$ be a compact object. If on each step we add not only finite sums but also all arbitrary direct sums
one can define full subcategories $\widebar{\langle X\rangle}_k\subset\T.$
The following proposition is proved in \cite[2.2.4]{BVdB}.
\begin{proposition}\label{compact_gen}
If $X$ is a compact object in a cocomplete triangulated category $\T,$ then
$\widebar{\langle X\rangle}_k\bigcap\T^c=\langle X\rangle_k.$
\end{proposition}
In particular, if $\widebar{\langle X\rangle}_k=\T$ for some $k,$ then $\langle X\rangle_k=\T^c$ and
$X$ is a strong generator of $\T^c.$
\subsection{Semi-orthogonal decompositions}
Let $\T$ be a $\kk$\!--linear triangulated category, where $\kk$ is a base field.
Recall some definitions and facts concerning admissible
subcategories and semi-orthogonal decompositions.
Let ${\N\subset\T}$ be a full triangulated subcategory. The {\em
right orthogonal} to ${\N}$ is the full subcategory
${\N}^{\perp}\subset {\T}$ consisting of all objects $X$ such that
${\Hom(Y, X)}=0$ for any $Y\in{\N}.$ The {\em left orthogonal}
${}^{\perp}{\N}$ is defined analogously. The orthogonals are also
triangulated subcategories.
\begin{definition}\label{adm}
Let $I\colon\N\hookrightarrow\T$ be a full embedding of triangulated
categories. We say that ${\N}$
is {\em right admissible} (respectively {\em left admissible}) if
there is a right (respectively left) adjoint functor $Q\colon\T\to \N.$ The
subcategory $\N$ will be called {\em admissible} if it is both right and left
admissible.
\end{definition}
\begin{remark}\label{semad}
{\rm The subcategory $\N$ is right admissible
if and only if for each object $Z\in{\T}$ there is an
exact triangle $Y\to Z\to X,$ with $Y\in{\N},\, X\in{\N}^{\perp}.$ }
\end{remark}
Let $\N$ be a full triangulated subcategory in a triangulated category
$\T.$ If $\N$ is right (respectively left) admissible, then the
quotient category $\T/\N$ is equivalent to $\N^{\perp}$ (respectively~${}^{\perp}\N$). Conversely, if the quotient functor $Q\colon\T\lto\T/\N$
has a left (respectively right) adjoint, then~$\T/\N$ is equivalent to
$\N^{\perp}$ (respectively~${}^{\perp}\N$).
\begin{definition}\label{sd}
A {\em semi-orthogonal decomposition} of a triangulated category $\T$
is a sequence of full triangulated subcategories ${\N}_1, \dots, {\N}_n$ in ${\T}$
such that there is an increasing filtration $0=\T_0\subset\T_1\subset\cdots\subset\T_n=\T$
by left admissible subcategories for which the left orthogonals ${}^{\perp}\T_{i-1}$
in $\T_{i}$ coincide with $\N_i.$ In particular, $\N_i\cong\T_i/\T_{i-1}.$
We write
$
{\T}=\left\langle{\N}_1, \dots, {\N}_n\right\rangle.
$
\end{definition}
If we have a semi-orthogonal decomposition ${\T}=\left\langle{\N}_1, \dots, {\N}_n\right\rangle,$ then the inclusion functors induce an isomorphism of the Grothendieck groups
\[
K_0(\N_1)\oplus K_0(\N_2)\oplus\cdots\oplus K_0(\N_n)\cong K_0(\T).
\]
It is more convenient to consider so called enhanced triangulated categories, i.e. triangulated categories
that are homotopy categories of pretriangulated DG categories (see Section \ref{enhancements}).
An enhancement of a triangulated category $\T$ induces an enhancement of any full triangulated subcategory
$\N\subset\T.$
Using an enhancement of a triangulated category $\T$ we can define the K--theory spectrum $K(\T)$ of $\T$
(see \cite{Ke2}). It also gives us an additive invariant (see, for example, \cite[5.1]{Ke2}), i.e for any
semi-orthogonal decomposition we have an isomorphism
\[
K_*(\N_1)\oplus K_*(\N_2)\oplus\cdots\oplus K_*(\N_n)\cong K_*(\T).
\]
\subsection{Exceptional, w-exceptional, and semi-exceptional collections}
The existence of a semi-orthogonal decomposition on a triangulated
category $\T$ clarifies the structure of $\T.$ In the best scenario,
one can hope that $\T$ has a semi-orthogonal decomposition
${\T}=\left\langle{\N}_1, \dots, {\N}_n\right\rangle$ in which each
$\N_p$ is as simple as possible, i.e. is
equivalent to the bounded derived category of finite-dimensional
vector spaces.
\begin{definition}\label{exc}
An object $E$ of a $\kk$\!--linear triangulated category ${\T}$ is
called {\em exceptional} if ${\Hom}(E, E[l])=0$ whenever $l\ne 0,$ and
${\Hom}(E, E)=\kk.$ An {\em exceptional collection} in ${\T}$ is a
sequence of exceptional objects $\sigma=(E_1,\dots, E_n)$ satisfying the
semi-orthogonality condition ${\Hom}(E_i, E_j[l])=0$ for all $l$ whenever
$i>j.$
\end{definition}
If a triangulated category $\T$ has an exceptional collection
$\sigma=(E_1,\dots, E_n)$ that generates the whole of $\T,$ then
this collection is called {\em full}. In this case $\T$ has a
semi-orthogonal decomposition with $\N_p=\langle E_p\rangle.$ Since
$E_{p}$ is exceptional, each of these categories is equivalent to the
bounded derived category of finite dimensional $\kk$\!-vector spaces. In
this case we write $ \T=\langle E_1,\dots, E_n \rangle.$
\begin{definition}\label{strong}
An exceptional collection $\sigma=(E_1,\dots, E_n)$ is called {\em strong} if, in addition,
${\Hom}(E_i, E_j[l])=0$ for all $i$ and $j$ when $l\ne 0.$
\end{definition}
The best known example of an exceptional collection is the sequence
of invertible sheaves
$\left(\mathcal{O}_{\mathbb{P}^n},\dots,\mathcal{O}_{\mathbb{P}^n}(n)\right)$
on the projective space $\mathbb{P}^n.$ This exceptional collection is
full and strong.
When the field $\kk$ is not algebraically closed it is reasonable to weaken the notions of an exceptional
object and an exceptional collection.
\begin{definition}\label{semi-exc}
An object $E$ of a $\kk$\!--linear triangulated category ${\T}$ is
called {\em w-exceptional (weak exceptional)} if ${\Hom}(E, E[l])=0$ when $l\ne 0,$ and
${\Hom}(E, E)=D,$ where $D$ is a finite dimensional division algebra over $\kk.$
It is called {\em semi-exceptional} if ${\Hom}(E, E[l])=0$ when $l\ne 0$ and
${\Hom}(E, E)=S,$ where $S$ is a finite dimensional semisimple algebra over $\kk.$
\end{definition}
It is evident that exceptional and semi-exceptional objects are stable under base field change while
w-exceptional objects are not.
A {\em w-exceptional (semi-exceptional) collection} in ${\T}$ is a
sequence of w-exceptional (semi-exceptional) objects $(E_1,\dots, E_n)$ with
semi-orthogonality conditions ${\Hom}(E_i, E_j[l])=0$ for all $l$ whenever
$i>j.$
\begin{example}\label{Severi-Brauer}
{\rm
Let $\kk$ be a field and $D$ be a central simple algebra over $\kk.$
Consider a Severi-Brauer variety $SB(D).$ There is a full semi-exceptional collection
$(S_0, S_1,\dots S_n)$ on $SB(D)$ such that $S_0=\cO_{SB}$ and
$\End(S_i)\cong D^{\otimes i},$ where $n+1$ is the order of the class of $D$ in the Brauer group of $\kk.$
Since each $D^{\otimes i}$ is a matrix algebra over a central division algebra $D_i,$ there is a w-exceptional collection
$(E_0, E_1,\dots, E_n)$ such that $\End(E_i)\cong D_i$ (see \cite{Be} for a proof).
In this situation $S_i$ are isomorphic to $E_i^{\oplus k_i}$ for some integers $k_i.$
These collections are also strong.
}
\end{example}
\section{Preliminaries on differential graded categories}
\subsection{Differential graded categories}
Our main references for differential graded (DG) categories are \cite{Ke,Dr,To, TV}.
Here we only recall some points and introduce notation.
Let $\kk$ be a field.
All categories, DG categories, functors, DG
functors and so on are assumed to be $\kk$\!--linear.
A {\it differential graded or DG category} is a $\kk$\!--linear category $\dA$ whose morphism spaces $\dHom (\mX, \mY)$
are complexes of $\kk$\!-vector spaces (DG $\kk$\!--modules), so that for any $\mX, \mY, \mZ\in
\Ob\dC$ the composition $\dHom (\mY, \mZ)\otimes \dHom (\mX, \mY)\to
\dHom (\mX, \mZ)$ is a morphism of DG $\kk$\!--modules. The identity morphism $1_\mX\in \dHom (\mX, \mX)$ is closed of
degree zero.
Using the supercommutativity isomorphism $\mU\otimes \mV\simeq \mV\otimes
\mU$ in the category of DG $\kk$\!--modules one defines for every DG
category $\dA$ the {\it opposite DG category} $\dA^{\op}$ with $\Ob\dA
^{\op}=\Ob\dA$ and $\dHom_{\dA^{\op}}(\mX, \mY)=\dHom_{\dA}(\mY, \mX).$
For a DG category $\dA$ we denote by $\Ho(\dA)$
its homotopy category.
The {\it homotopy category} $\Ho(\dA)$ has the same objects as the DG category $\dA$ and its
morphisms are defined by taking the $0$\!-th cohomology
$H^0(\dHom_{\dA} (\mX, \mY))$
of the complex $\dHom_{\dA} (\mX, \mY).$
As usual, a {\it DG functor}
$\mF:\dA\to\dB$ is given by a map $\mF:\Ob(\dA)\to\Ob(\dB)$ and
by morphisms of DG $\kk$\!--modules
$$
\mF_{\mX, \mY}: \dHom_{\dA}(\mX, \mY) \lto \dHom_{\dB}(\mF \mX,\mF \mY),\quad \mX, \mY\in\Ob(\dA)
$$
compatible with the composition and the units.
A DG functor $\mF: \dA\to\dB$ is called a {\it quasi-equivalence} if
$\mF_{\mX, \mY}$ is a quasi-isomorphism for all pairs of objects $\mX, \mY$ of $\dA$
and the induced functor $H^0(\mF): \Ho(\dA)\to \Ho(\dB)$ is an
equivalence. DG categories $\dA$ and $\dB$ are called {\it quasi-equivalent} if there exist DG
categories $\dC_1,\dots, \dC_n$ and a chain of quasi-equivalences
$\dA\stackrel{\sim}{\leftarrow} \dC_1 \stackrel{\sim}{\rightarrow} \cdots \stackrel{\sim}{\leftarrow} \dC_n
\stackrel{\sim}{\rightarrow} \dB.$
\subsection{Differential graded modules}
Given a small DG category $\dA$ we define a {\it right DG $\dA$\!--module} as a DG functor
$\mM: \dA^{op}\to \Mod \kk,$ where $\Mod \kk$ is the DG category of DG $\kk$\!--modules. We denote by $\Mod \dA$ the DG
category of right DG $\dA$\!--modules.
Each object $\mY$ of $\dA$ produces a right module
represented by $\mY$
\[
\mh^\mY(-):=\dHom_{\dA}(-, \mY)
\]
which is called a {\it representable} DG module. This
gives the Yoneda DG functor
$\mh^\bullet :\dA \to
\Mod\dA$ that is full and faithful.
A DG $\dA$\!--module is called {\it free} if it is isomorphic to a direct sum of DG modules of the form
$\mh^\mY[n],$ where $\mY\in\dA,\; n\in\ZZ.$
A DG $\dA$\!--module
$\mP$ is called {\it semi-free} if it has a filtration
$0=\mPhi_0\subset \mPhi_1\subset ...=\mP$
such that each quotient $\mPhi_{i+1}/\mPhi_i$ is free. The full
DG subcategory of semi-free DG modules is denoted by $\SF\dA.$
We denote by $\SFf\dA\subset \SF\dA$ the full DG subcategory of finitely generated semi-free
DG modules, i.e. such that $\mPhi_m=\mP$ for some $m$ and $\mPhi_{i+1}/\mPhi_i$ is a finite direct sum of DG modules of the form
$\mh^Y[n].$
For every DG
$\dA$\!--module $\mM$ there is a quasi-isomorphism $\bp \mM\to \mM$ such that $\bp M$ is a semi-free
DG $\dA$\!--module (see \cite{Ke} 3.1, \cite{Hi} 2.2, \cite{Dr} 13.2).
Denote by $\Ac\dA$ the full
DG subcategory of $\Mod\dA$ consisting of all acyclic DG modules, i.e. DG modules $\mM$
for which the complex $\mM(\mX)$ is acyclic for all $X\in\dA.$
The
homotopy category of DG modules $\Ho(\Mod\dA)$ has a natural structure of a triangulated category
and the homotopy subcategory of acyclic complexes $\Ho (\Ac\dA)$ forms a full triangulated subcategory in it.
The {\it derived
category} $\D(\dA)$ is defined as the Verdier quotient
\[
\D(\dA):=\Ho(\Mod\dA)/\Ho (\Ac\dA).
\]
It is also natural to consider the category of h-projective DG modules.
We call a DG $\dA$\!--module $\mP$ {\it h-projective (homotopically projective)} if
$$\dHom_{\Ho(\Mod\dA)}(\mP, \mN)=0$$
for every acyclic DG module $\mN$ (dually, we can define {\it h-injective} DG modules).
Let $\dP(\dA)\subset \Mod\dA$ denote the full
DG subcategory of h-projective objects. It can be easily checked that a semi-free
DG-module is h-projective and the natural embedding $\SF\dA\hookrightarrow\dP(\dA)$
is a quasi-equivalence.
Moreover, the canonical DG functors $\SF\dA\hookrightarrow\dP(\dA)\hookrightarrow\Mod\dA$ induce equivalences
$\Ho(\SF\dA)\stackrel{\sim}{\to} \Ho(\dP(\dA))\stackrel{\sim}{\to} \D(\dA)$ of triangulated categories.
Let $\mF:\dA \to \dB$ be a DG functor between small DG categories.
It induces the restriction DG functor
\[
\mF_*:\Mod\dB\lto \Mod\dA
\]
which sends a DG $\dB$\!--module $\mN$ to $\mN\circ\mF.$
The restriction functor $\mF_*$ has left and right adjoint functors $\mF^*, \mF^{!}$ that are defined as follows
\[
\mF^*M(Y)=\mM\otimes_{\dA} \mF_* \mh_Y,\quad \mF^{!}M(Y)=\dHom(\mF_*\mh_Y, \mM),\quad \text{where}\quad Y\in \dB \; \text{and}\; \mM\in\Mod\dA.
\]
The DG functor $\mF^*$ is called the induction functor and it is an extension of $\mF$ on the category of DG modules, i.e there is an isomorphism of DG functors
$\mF^* \mh^\bullet_{\dA}\cong \mh^\bullet_{\dB}\mF.$
The DG functor $\mF_*$ preserves acyclic DG modules and induces a derived functor $F_*: \D(\dB)\to \D(\dA).$
Existence of h-projective and h-injective resolutions allows us to define derived functors
$\bL F^*$ and $\bR F^!$ from $\D(\dA)$ to $\D(\dB).$
More generally, let $\mT$ be an
$\dA\hy\dB$\!--bimodule that is, by definition, a DG-module over $\dA^{op}\otimes\dB.$
For each DG $\dA$\!--module $\mM$ we obtain a DG $\dB$\!--module
$\mM\otimes_{\dA} \mT.$
The DG functor $(-)\otimes_{\dA} \mT: \Mod\dA \to \Mod\dB$ admits a right adjoint
$\dHom_{\dB} (\mT, -).$
These functors do not respect
quasi-isomorphisms in general, but they form a Quillen adjunction
and the derived functors
$(-)\stackrel{\bL}{\otimes}_{\dA}\mT$ and
$\bR \Hom_{\dB} (\mT, -)$
form an adjoint pair of functors between derived categories $\D(\dA)$ and
$\D(\dB).$
\subsection{Pretriangulated DG categories, categories of perfect DG modules, and enhancements}\label{enhancements}
For any DG category $\dA$ there exist a DG category $\dA^{\ptr}$ that is called the {\it pretriangulated hull}
and a canonical fully faithful DG
functor $\dA\hookrightarrow\dA^{\ptr}.$
The idea of the definition of $\dA^{\ptr}$ is to formally add to $\dA$
all shifts, all cones, cones of morphisms between cones and etc.
There is a canonical fully faithful DG functor
(the Yoneda embedding) $\dA^{\ptr}\to \Mod\dA,$ and under this embedding
$\dA^{\ptr}$ is DG-equivalent to the DG category of finitely generated semi-free DG modules $\SFf\dA.$
We will not make a difference between the DG categories $\dA^{\ptr}$ and $\SFf\dA.$
\begin{definition} A DG category $\dA$ is called {\em pretriangulated} if the canonical DG functor $\dA\to\dA^{\ptr}$
is a quasi-equivalence.
\end{definition}
\begin{remark}{\rm
It is equivalent to require that the homotopy category $\Ho(\dA)$ is triangulated
as a subcategory of $\Ho(\Mod\dA).$
}
\end{remark}
The DG category $\dA^{\ptr}$ is always pretriangulated,
so $\Ho(\dA^{\ptr})$ is a triangulated category. We denote $\tr(\dA):=\Ho(\dA^{\ptr}).$
With any small DG category $\dA$ we can also associate another DG category $\prfdg\dA$ that is called the DG category of perfect DG modules.
This category is even more important than $\dA^{\ptr}.$
\begin{definition} A DG category of perfect DG modules $\prfdg\dA$
is the full DG subcategory of $\SF\dA$ consisting of all DG modules which are homotopy
equivalent to a direct summand of a finitely generated semi-free DG module.
\end{definition}
Thus, the DG category $\prfdg\dA$ is pretriangulated and contains $\SFf\dA\cong \dA^{\ptr}.$ Denote by $\prf\dA$ the homotopy
category $\Ho(\prfdg\dA).$ The triangulated category $\prf\dA$ can be obtained from the triangulated category
$\tr(\dA)$ as its idempotent completion (Karubian envelope).
\begin{proposition}
For any small DG category $\dA$ the set of representable objects $\{\mh^Y\}_{Y\in \dA}$ forms a set of
compact generators of $\D(\dA)$ and the subcategory of compact objects $\D(\dA)^c$ coincides with the subcategory of perfect
DG modules $\prf\dA.$
\end{proposition}
\begin{remark}\label{perfect}
{\rm
If $\dA$ is a small pretriangulated DG category and $\Ho(\dA)$ is idempotent complete, then
the natural Yoneda DG functor $\mh: \dA\to\prfdg\dA$ is a quasi-equivalence.
}
\end{remark}
It is well-known that the categories $\D(\dA)$ and $\prf\dA$ are invariant under quasi-equivalences of DG categories.
\begin{proposition}\label{pre-tr_equivalence}
If a DG functor $\mF: \dA\to\dB$ is a quasi-equivalence, then the functors
\[
\mF^{*}: \SFf\dA\lto\SFf\dB,\quad \mF^{*}:\prfdg\dA\lto \prfdg\dB, \quad \mF^*:\SF\dA\lto \SF\dB
\]
are quasi-equivalences too.
\end{proposition}
Furthermore, we have the
following proposition that is essentially equal to
Lemma 4.2 in \cite{Ke} (see also \cite[Prop. 1.15]{LO} and proof there).
\begin{proposition}\cite{Ke}\label{Keller2}
Let $\mF: \dA\hookrightarrow \dB$ be a full embedding of DG categories
and let $\mF^{*}:\SF\dA\to\SF\dB$ (resp. $\mF^*:\prfdg\dA\to\prfdg\dB$) be the extension DG functor. Then the induced
homotopy functor $F^*: \D(\dA)\to \D(\dB)$ (resp. $F^*: \prf\dA\to \prf\dB$) is fully faithful.
If, in addition, the category $\prf\dB$ is classically generated by $\Ob\dA,$ then
$F^*$ is an equivalence.
\end{proposition}
\begin{remark}\label{Keller2_fg}
{\rm
The first statement holds for the functor
$\mF^*:\SFf\dA\to\SFf\dB$ too. The second also holds if we ask that the category
$\tr(\dB)$ is generated by $\Ob\dA$ (not classically).
}
\end{remark}
\begin{definition} Let $\T$ be a triangulated category. An {\em
enhancement} of $\T$ is a pair $(\dA , \varepsilon),$ where $\dA$ is a
pretriangulated DG category and $\varepsilon:\Ho(\dA)\stackrel{\sim}{\to} \T$ is an exact equivalence.
\end{definition}
\subsection{Quasi-functors}
Let $\kk$ be a field. Denote by
$\DGcat_k$ the category of small DG $\kk$\!--linear categories.
It is known that it admits a structure of cofibrantly
generated model category whose weak equivalences are the
quasi-equivalences (see \cite{Ta}).
This implies that the localization $\Hqe$ of
$\DGcat_k$ with respect to the quasi-equivalences has small
$\Hom$\!-sets. This also gives that a morphism from $\dA$ to $\dB$
in the localization can be represented as $\dA\leftarrow \dA_{cof}\to\dB,$ where $\dA\leftarrow \dA_{cof}$ is a cofibrant replacement.
It is not easy to compute the morphism sets in the localization
category $\Hqe$ using a cofibrant replacement. On the other hand, they can be
described in term of quasi-functors.
Consider two small DG categories $\dA$ and $\dB.$
Let $\mT$ be a $\dA\hy\dB$\!--bimodule. It defines a derived tensor functor
\[
(-)\stackrel{\bL}{\otimes}_\dA \mT: \D(\dA) \lto \D(\dB)
\]
between derived categories of DG modules over $\dA$ and $\dB.$
\begin{definition}
An $\dA\hy\dB$\!--bimodule $\mT$ is called a {\em quasi-functor}
from $\dA$ to $\dB$ if the tensor functor
$
(-)\stackrel{\bL}{\otimes}_\dA \mT: \D(\dA) \to \D(\dB)
$
takes every representable $\dA$\!--module to an object which is
isomorphic to a representable $\dB$\!--module.
\end{definition}
Denote by $\Rep(\dA,\; \dB)$ the full subcategory of the derived
category $\D(\dA^{op}\otimes\dB)$ of $\dA\hy\dB$\!--bimodules consisting of
all quasi-functors.
In
other words a quasi-functor is represented by a DG functor $\dA\to \Mod\dB$ whose essential image consists of
quasi-representable DG $\dB$\!--modules (``quasi-representable'' means quasi-isomorphic to a representable DG module).
Since the category of quasi-representable DG
$\dB$\!--modules is equivalent to $\Ho(\dB)$ a quasi-functor
$\mT \in \Rep(\dA,\; \dB)$ defines a functor $\Ho(\mT):\Ho(\dA)\to \Ho(\dB).$
Notice that a quasi-functor $\mF:\dA\to\dB$
defines an exact functor $\tr(\dA)\to \tr(\dB)$ between triangulated categories.
It is now known that quasi-representable functors form morphisms between DG categories in the localization
category
$\Hqe.$
\begin{theorem}\label{quasi-functors}\cite{To} The morphisms from $\dA$ to $\dB$
in the localization $\Hqe$ of $\DGcat_{\kk}$ with respect to
quasi-equivalences are in natural bijection with the isomorphism
classes of $\Rep(\dA,\dB).$
\end{theorem}
Due to this theorem any morphism from $\dA$ to $\dB$ in the localization category $\Hqe$
will be called a quasi-functor.
Let $\mF: \dA\to\dB$ be a quasi-functor. It can be realized as
a roof $\dA\stackrel{\ma}{\stackrel{\sim}{\longleftarrow}}\dA'\stackrel{\mF'}{\lto}\dB,$
where $\ma$ and $\mF'$ are DG functors and $\ma$ is also a quasi-equivalence.
For instance we can take a cofibrant replacement $\dA_{cof}$ as $\dA'.$
The quasi-functor $\mF$ induces functors
\begin{equation}\label{derived_quasi}
\bL F^*= \bL F^{'*}\circ a_*: \D(\dA)\lto \D(\dB)
\quad
\text{and}
\quad
\bR F_*:= F'_* \circ\bL a^{*}: \D(\dB)\lto \D(\dA).
\end{equation}
If now we consider the quasi-functor $F$ as an $\dA\hy\dB$\!--bimodule $\mT,$ then
there are isomorphisms of functors
\[
\bL F^*\cong -\stackrel{\bL}{\otimes}_\dA \mT: \D(\dA) \lto \D(\dB)
\quad
\text{and}
\quad
\bR F_*\cong \bR\Hom_{\dB}(\mT, -): \D(\dB) \lto \D(\dA).
\]
The standard tensor product $\otimes$ on the category $\DGcat_{\kk}$ induces a tensor product
$\stackrel{\bL}{\otimes}$ on the localization $\Hqe.$
It is proved in \cite{To} that the monoidal category $(\Hqe, \stackrel{\bL}{\otimes})$ has internal Hom-functor
$\dR\lHom.$ In particular, there is a quasi-equivalence
\begin{equation}\label{RHom}
\dR\lHom(\dA\otimes\dB,\; \dC)\cong \dR\lHom(\dA, \; \dR\lHom(\dB,\; \dC)).
\end{equation}
\begin{theorem}\cite{To}
For any DG categories $\dA$ and $\dB$ the DG category
$\dR\lHom(\dA, \dB)$ is quasi-equivalent to the full DG subcategory $\dRep(\dA, \dB)\subset \SF(\dA^{\op}\otimes\dB)$
consisting of all objects of $\Rep(\dA, \dB).$
\end{theorem}
Thus, there are equivalences
$
\Ho(\dR\lHom(\dA, \dB))\cong\Ho(\dRep(\dA, \dB))\cong \Rep(\dA, \dB).
$
\section{Commutative and noncommutative schemes}
\subsection{Derived categories of quasi-coherent sheaves and noncommutative schemes}\label{3.1}
In this paper we will consider separated noetherian schemes over an arbitrary field $\kk.$
Let $X$ be such a scheme.
The abelian category $\Qcoh X$ of quasi-coherent sheaves
$\Qcoh X$ is a Grothendieck category and has enough injectives.
Denote by $\dCom X$ the DG category of unbounded complexes of quasi-coherent sheaves on $X.$
This category has enough h-injective complexes (see, e.g. \cite{KSh}). Denote by $\dI(X)$ the full DG subcategory
of h-injective complexes. This DG category gives us a natural DG enhancement for the unbounded derived category
of quasi-coherent sheaves, because $\Ho(\dI(X))\cong \D(\Qcoh X).$ Another natural
enhancement for $\D(\Qcoh X)$ comes from the definition of the derived category and DG version
of Verdier localization \cite{Dr}. Consider the full DG subcategory $\dAc X\subset\dCom X$ of all acyclic complexes.
We can take the quotient DG derived category
$
\dCom X/\dAc X.
$
Of course, $\dI(X)$ and $\dCom X/\dAc X$ are naturally quasi-equivalent enhancements.
There is another enhancement of $\D(\Qcoh X)$ that is very useful when we work with pullback and tensor product functors.
It comes from h-flat complexes. Recall that an (unbounded) complex $\mP^{\cdot}$ of quasi-coherent sheaves on $X$
is called {\em h-flat} if $\Tot^{\oplus}(\mP^{\cdot}\otimes_{\cO_X} \mC^{\cdot})$ is acyclic
for any acyclic $\mC^{\cdot} \in\dAc X.$ Denote by $\dHf X\subset\dCom X$ the full DG subcategory of h-flat complexes.
It was shown in \cite[Prop. 1.1]{AJL} that there are enough h-flat complexes in $\dCom X$ for any separated quasi-compact scheme.
Hence the DG quotient category $\dHf X/\dHAc X,$ where $\dHAc X$ is the DG subcategory of acyclic h-flat complexes, is an enhancement of $\D(\Qcoh X)$ (see \cite[3.10]{KL}).
It is easy to see that for any morphism of schemes $f:X\to Y$ the pullback
$f^*,$ acting componentwise on complexes, sends h-flat complexes to h-flat complexes and h-flat acyclic
complexes to h-flat acyclic complexes. It is also true that the tensor product of an h-flat acyclic complex with
any complex is acyclic (see \cite{Sp}).
Thus, for any morphism of schemes $f: X\to Y$ we obtain a DG functor (not only a quasi-functor)
\[
\mf^*: \dHf Y/\dHAc Y\lto \dHf X/\dHAc X,
\]
which induces the derived inverse image functor $\bL f^*$ on the derived categories of quasi-coherent sheaves.
Similarly, we have a DG tensor functor $(-) \otimes \mP^{\cdot}$ from $\dHf X/\dHAc X$ to itself.
Thus, we have three different DG categories $\dI(X),$ $\dCom X/\dAc X,$ and $\dHf X/\dHAc X,$ which are natural quasi-equivalent
enhancements for $\D(\Qcoh X).$ There is no reason to make difference between them, but sometimes one of them is more favorable
because some quasi-functors can be realized as usual DG functors.
In this paper we work with the DG category $\dHf X/\dHAc X,$ which will be denoted by $\dD(\Qcoh X),$
since pulbacks and tensor products are DG functors on them.
For any morphism of schemes $f: X\to Y$ we also have a DG functor $\mf_*$ from $\dI(X)$ to
$\dCom Y/\dAc Y$ acting componentwise on h-injective complexes. This DG functor induces a quasi-functor that we will denote by the same letter
\[
\mf_*: \dD(\Qcoh X)\stackrel{\sim}{\lto}\dCom X/\dAc X\stackrel{\sim}{\longleftarrow}\dI(X)\stackrel{\mf_*}{\lto}
\dCom Y/\dAc Y
\stackrel{\sim}{\longleftarrow}\dD(\Qcoh X).
\]
Recall now the important notion of a perfect complex on a scheme $X,$ which was introduced in \cite{SGA6}. A
perfect complex is a complex of sheaves which is locally
quasi-isomorphic to a bounded complex of locally free sheaves of finite
type (a good reference is \cite{TT}).
\begin{definition}
Denote by $\prfdg X$ the full DG subcategory of $\dD(\Qcoh X)$ consisting of all perfect complexes.
\end{definition}
The triangulated category $\prf X=\Ho(\prfdg X)$ is a full subcategory of $\D(\Qcoh X).$
Amnon Neeman in \cite{Ne} showed that
the triangulated category $\D(\Qcoh X)$ is compactly generated and $\prf X$ is nothing but the subcategory of compact object in $\D(\Qcoh X).$
In \cite{Ne} this assertion is proved for any quasi-compact and separated scheme, in \cite{BVdB} a generalization of this fact for
$\D_{\Qcoh}(X)$ was established for a quasi-compact and quasi-separated scheme.
For any morphism of schemes $f: X\to Y$ the DG functor $\mf^*$ induces a DG functor
\[
\mf^*:\prfdg Y\to \prfdg X.
\]
Under some conditions on the morphism $f:X\to Y,$ the induced quasi-functor $\mf_*$ from
$\dD(\Qcoh X)$ to $\dD(\Qcoh Y)$ sends $\prfdg X$ to $\prfdg Y$ (see \cite[2.5.4]{TT} and \cite[III]{SGA6}).
Thus, for noetherian schemes $X$ and $Y$ if $f$ is proper and has finite
Tor-dimension we obtain a quasi-functor
\[
\mf_*: \prfdg X\lto \prfdg Y.
\]
Note that it holds for any morphism between smooth and proper schemes.
In \cite{Ne, BVdB} it was proved that the category $\prf X$ admits a classical generator $\mE$ and, hence, $\mE$
is a compact generator of the whole $\D(\Qcoh X).$
Let us take such a generator $\mE\in\prfdg X.$ Denote by $\dE$ its DG algebra of endomorphisms,
i.e. $\dE=\dHom(\mE, \mE).$ Since $\mE$ is perfect, the DG algebra $\dE$ has only finitely many cohomologies.
Proposition \ref{Keller2} implies the following statement.
\begin{statement}\cite[3.1.8]{BVdB}\label{statement}
The DG category $\dD(\Qcoh X)$ is quasi-equivalent to $\SF\dE$ and $\prfdg X$ is quasi-equivalent to
$\prfdg \dE,$ where $\dE$ is a DG algebra with bounded cohomology.
\end{statement}
This fact allows us to suggest a definition of a (derived) noncommutative scheme over $\kk.$
\begin{definition}\label{noncommutative_scheme}
A {\em (derived) noncommutative scheme} over a field $\kk$ is a $\kk$\!--linear DG category of the form $\prfdg\dE,$
where $\dE$ is a cohomologically bounded DG algebra over $\kk.$ The derived category
$\D(\E)$ will be called the derived category of quasi-coherent sheaves on this noncommutative scheme.
\end{definition}
For a noetherian scheme $X$ we consider the abelian category of coherent sheaves
$\coh X.$
Denote by $\D^b(\coh(X))$
the bounded derived category of coherent
sheaves on $X.$
Since $X$ is noetherian the natural functor $\D^b(\coh(X))\to
\D(\Qcoh(X))$ is fully faithful and realizes an equivalence of
$\D^b(\coh(X))$ with the full subcategory
$\D^b(\Qcoh(X))_{\coh}\subset \D(\Qcoh(X))$ consisting of all cohomologically bounded
complexes with coherent cohomology (see \cite[II 2.2.2]{SGA6}).
Because of that, when we consider $\D^b(\coh(X))$ as a subcategory of
$\D(\Qcoh(X))$ we will identify it with the full subcategory
$\D^b(\Qcoh(X))_{\coh},$ adding all isomorphic objects.
The enhancement $\dD(\Qcoh X)$ induces an enhancement of $\D^b(\coh X)$ that we denote
by $\dD^b(\coh X).$
\subsection{Gluing of DG categories}
Let $\dA$ and $\dB$ be two small DG categories and let $\mS$ be a $\dB\hy\dA$\!--bimodule,
i.e. a DG $\dB^{\op}\otimes\dA$\!--module.
We now construct a so called upper triangular DG category corresponding to the data
$(\dA, \dB, \mS).$
\begin{definition}\label{upper_tr}
Let $\dA$ and $\dB$ be two small DG categories and let $\mS$ be a $\dB\hy\dA$\!--bimodule.
The {\em upper triangular} DG category $\dC=\dA\underset{\mS}{\with}\dB$ is defined as follows:
\begin{enumerate}
\item[1)] $\Ob(\dC)=\Ob(\dA)\bigsqcup\Ob(\dB),$
\item[2)]
$
\dHom_{\dC}(X, Y)=
\begin{cases}
\dHom_{\dA}(X, Y), & \text{ when $X,Y\in\dA$}\\
\dHom_{\dB}(X, Y), & \text{ when $X,Y\in\dB$}\\
\mS(Y, X), & \text{ when $X\in\dA, Y\in\dB$}\\
0, & \text{ when $X\in\dB, Y\in\dA$}
\end{cases}
$
\end{enumerate}
with evident composition law coming from DG categories $\dA, \dB$ and the bimodule structure on $\mS.$
\end{definition}
The upper triangular DG category $\dC=\dA\underset{\mS}{\with}\dB$ is not necessary pretriangulated even if the components $\dA$ and $\dB$
are pretriangulated. To make this operation well defined on the class of pretriangulated categories we introduce a so called gluing
of pretriangulated categories.
\begin{definition}\label{gluing_cat}
Let $\dA$ and $\dB$ be two small pretriangulated DG categories and let $\mS$ be a $\dB\hy\dA$\!--bimodule.
A {\em gluing} $\dA\underset{\mS}{\oright}\dB$ of DG categories $\dA$ and $\dB$ via $\mS$ is defined as the pretriangulated hull of $\dA\underset{\mS}{\with}\dB,$ i.e. $\dA\underset{\mS}{\oright}\dB=(\dA\underset{\mS}{\with}\dB)^{\ptr}.$
\end{definition}
\begin{remark}{\rm
The gluing can be defined for any DG categories not only for pretriangulated
(see, for example, \cite{KL}). The resulting DG category is not necessary pretriangulated.
However, we prefer to restrict ourself to the pretriangulated case,
because the definition above is more convenient
for our purposes. Since we use different definition we give different proofs for
Propositions \ref{dg_semiorhtogonal}, \ref{gluing_semi-orthogonal}, and \ref{gluing_quasifunctors} in spite of they were also proved
in \cite{KL}.
}
\end{remark}
Natural fully faithful DG inclusions $\ma: \dA\hookrightarrow \dA\underset{\mS}{\with}\dB$ and
$\mb: \dB\hookrightarrow \dA\underset{\mS}{\with}\dB$ induce fully faithful DG
functors $\ma^*: \dA\hookrightarrow \dA\underset{\mS}{\oright}\dB$ and
$\mb^*: \dB\hookrightarrow \dA\underset{\mS}{\oright}\dB.$
It is easy to see that the restriction functor $\mb_*: \Mod(\dA\underset{\mS}{\with}\dB) \to \Mod\dB$
sends semi-free DG modules to semi-free and we obtain a DG functor
$\SFf(\dA\underset{\mS}{\with}\dB)\to\SFf\dB.$ By assumption $\dB$ is pretriangulated, and
we know that the pretriangulated hull is DG-equivalent to the DG category of finitely generated semi-free DG modules.
Thus we obtain a quasi-functor $\mb_*: \dA\underset{\mS}{\oright}\dB \to \dB$ that is right adjoint to
$\mb^*.$
These quasi-functors induce exact functors
\[
a^*:\Ho(\dA)\lto\Ho(\dA\underset{\mS}{\oright}\dB), \quad b^*:\Ho(\dB)\lto\Ho(\dA\underset{\mS}{\oright}\dB),\quad b_*: \Ho(\dA\underset{\mS}{\oright}\dB)\lto \Ho(\dB)
\]
between triangulate categories such that $a^*, b^*$ are fully faithful, and $b_*$ is right adjoint to $b^*.$
Therefore, there is a semi-orthogonal decomposition
\[
\Ho(\dA\underset{\mS}{\oright}\dB)=\langle \N,\; \Ho(\dB)\rangle
\]
with some triangulated subcategory $\N.$ It is evident that $\Ho(\dA)$ is a full subcategory of $\N$ with respect to the functor $a^*.$
The subcategory $\N$ is left admissible and we have the quotient functor $\Ho(\dA\underset{\mS}{\oright}\dB)\to\N$ that sends
$\Ho(\dB)$ to zero. Since the category $\Ho(\dA\underset{\mS}{\oright}\dB)$ is generated by the union of objects
$a^* \mh^X_{\dA}$ and $b^*\mh^Y_{\dB}$ we obtain that the subcategory $\N$ is generated by the objects $a^* \mh^X_{\dA}.$
Hence $\N$ coincides with the triangulated subcategory $\Ho(\dA)\subset\N,$ because it also contains all these objects.
Thus, we have proved the following proposition.
\begin{proposition}\label{dg_semiorhtogonal} Let the DG category $\dC$ be the gluing $ \dA\underset{\mS}{\oright}\dB.$
Then the DG functors $\ma^*: \dA\to\dC$ and $\mb^*: \dB\to\dC$ induce a semi-orthogonal decomposition
for the triangulated category $\Ho(\dC)$ of the form
$\Ho(\dC)=\langle\Ho(\dA), \Ho(\dB)\rangle.$
\end{proposition}
On the other hand, we can show that any enhancement of a triangulated category with a semi-orthogonal decomposition
can be obtained as a gluing of enhancements of the summands.
\begin{proposition}\label{gluing_semi-orthogonal}
Let $\dC$ be a pretriangulated DG category. Suppose that we have a semi-orthogonal decomposition
$\Ho(\dC)=\langle \A, \B\rangle.$ Then the DG category $\dC$ is quasi-equivalent to the gluing
$\dA\underset{\mS}{\oright}\dB,$ where $\dA, \dB\subset\dC$ are full DG subcategories
with the same objects as $\A$ and $\B,$ respectively, and the $\dB\hy\dA$\!--bimodule is given by the rule
\begin{equation}\label{bimodule}
\mS ( Y, X)=\dHom_{\dC}(X, Y), \quad \text{with}\quad X\in\dA \;\text{and}\; Y\in\dB.
\end{equation}
\end{proposition}
\begin{dok}
Take full DG subcategories $\dA\subset\dC$ and $\dB\subset\dC$ with objects from $\A$ and $\B,$ respectively.
Consider the $\dB\hy\dA$\!--bimodule $\mS$ defined by rule (\ref{bimodule}).
There is a natural inclusion of the upper triangular DG category $\dA\underset{\mS}{\with}\dB$ into $\dC.$
Since $\dC$ is pretriangulated we obtain a quasi-functor from the pretriangulated hull
$\dA\underset{\mS}{\oright}\dB$ to $\dC.$
Since $\A$ and $\B$ are semi-orthogonal, the DG category $\dA\underset{\mS}{\with}\dB$ under the inclusion $\dA\underset{\mS}{\with}\dB \hookrightarrow \dC$ is quasi-equivalent to the full
DG subcategory of $\dC$ on the set of objects $\Ob(\dA)\bigsqcup\Ob(\dB).$
Combining Propositions \ref{pre-tr_equivalence}, \ref{Keller2}, and Remark \ref{Keller2_fg} we obtain that the functor
$\Ho(\dA\underset{\mS}{\oright}\dB)\to\Ho(\dC)$ is fully faithful.
Since the set $\Ob(\dA)\bigsqcup\Ob(\dB)$ generates the category $\Ho(\dC),$ this functor is an equivalence
by Remark \ref{Keller2_fg}.
\end{dok}
\begin{example}\label{ex1}
{\rm Let $X$ be a noetherian scheme and let $\E$ be a vector bundle on $X$ of rank $2.$
Consider the projectivization $\PP(\E^{\vee})$ with projection $p: \PP(\E^{\vee})\to X.$
Denote by $\cO(1)$ the antitautological line bundle on $\PP(\E^{\vee}).$ We know that $\bR p_*\cO(1)\cong \E$
and $p^*$ is fully faithful.
It was shown in \cite{Blow} that there is a semi-orthogonal decomposition of the form
\[
\prf \PP(\E^{\vee})=\langle p^* \prf X,\; p^* \prf X \otimes \cO(1)\rangle.
\]
Furthermore,
the DG category $\prfdg \PP(\E^{\vee})$ is quasi-equivalent to the gluing
$\prfdg X \underset{\mS_{\E}}{\oright}\prfdg X,$ where $\mS_{\E}$ is a DG bimodule of the form
\[
\mS_{\E}(B, A)\cong\dHom_{\prfdg X}(A,\; B\otimes\E),\quad\text{where}\quad A, B\in\prfdg X.
\]
By the same rule the DG category $\dD^b(\coh \PP(\E^{\vee}))$
can be obtain as the gluing of $\dD^b(\coh X)$ with itself via $\mS_{\E}.$}
\end{example}
\begin{example}\label{ex2}
{\rm
Let $\pi: \wt{X}\to X$ be a blowup of a regular scheme $X$ along a closed regular subscheme $Y$ of codimension $2.$
The functor $\bL\pi^*$ is fully faithful. Consider the exceptional divisor $j: E\hookrightarrow \wt{X}.$
The morphism $p: E\to Y$ is the projectivization of the normal bundle to $Y$ in $X.$
The functor $\bR j_* p^*$ is fully faithful as well.
The triangulated category $\prf \wt{X}$ has a semi-orthogonal decomposition of the form
\[
\prf \wt{X}=\langle \bL\pi^* \prf X,\; \bR j_* p^*\prf Y\rangle.
\]
Furthermore,
the DG category $\prfdg \wt{X}$ is quasi-equivalent to the gluing
$\prfdg X \underset{\mS}{\oright}\prfdg Y,$ where $\mS$ is a DG bimodule of the form
\[
\mS(B, A)\cong\dHom_{\prfdg Y}(\mi^* A,\; B),\quad\text{where}\quad A\in\prfdg X,\; B\in\prfdg Y,\quad\text{and}\quad i: Y\hookrightarrow X.
\]
By the same rule the DG category $\dD^b(\coh \wt{X})$
can be obtained as gluing via $\mS.$
}
\end{example}
Let $\ma:\dA\to \dA'$ and $\mb:\dB\to \dB'$ be DG functors between small pretriangulated DG categories. Let $\mS$ and $\mS'$ be bimodules, i.e.
DG modules over $\dB^{\op}\otimes\dA$ and ${\dB'}^{\op}\otimes\dA'$ respectively. Consider the restriction functor on bimodules
\[
(\mb\otimes\ma)_*: \Mod ({\dB'}^{\op}\otimes\dA')\to \Mod (\dB^{\op}\otimes\dA).
\]
Suppose that we have a map of DG modules $\phi:\mS\to (\mb\otimes\ma)_*\mS'.$ Then it is evident from Definition \ref{upper_tr}
that there are DG functors
\[
\ma\underset{\phi}{\with}\mb: \dA\underset{\mS}{\with}\dB \lto \dA'\underset{\mS'}{\with}\dB',
\quad
\text{ and }\quad
\ma\underset{\phi}{\oright}\mb: \dA\underset{\mS}{\oright}\dB \lto \dA'\underset{\mS'}{\oright}\dB'.
\]
Furthermore, assume that $\phi$ is a quasi-isomorphism.
Now if the exact functors $a:\Ho(\dA)\to\Ho(\dA')$ and $b:\Ho(\dB)\to \Ho(\dB')$ are fully faithful, then
\[
a\underset{\phi}{\with}b: \Ho(\dA\underset{\mS}{\with}\dB) \lto \Ho(\dA'\underset{\mS'}{\with}\dB'),
\quad
\text{and}\quad
a\underset{\phi}{\oright}b: \Ho(\dA\underset{\mS}{\oright}\dB) \lto \Ho(\dA'\underset{\mS'}{\oright}\dB')
\]
are fully faithful by Theorem \ref{Keller2} and Remark \ref{Keller2_fg}.
If $\ma$ and $\mb$ are quasi-equivalences and $\phi$ is a quasi-isomorphism, then
$\ma\underset{\phi}{\with}\mb$ is a quasi-equivalence and, by Remark \ref{Keller2_fg}, $\ma\underset{\phi}{\oright}\mb$
is a quasi-equivalence too since the objects of $\dA\underset{\mS}{\with}\dB$ generate $\Ho(\dA'\underset{\mS'}{\oright}\dB')$
in this case.
This statement can be generalized to a class of quasi-functors.
Indeed, quasi-functors $\ma:\dA\to \dA'$ and $\mb:\dB\to \dB'$ induce a quasi-functor
$\mb\otimes\ma: \dB^{\op}\otimes\dA \to {\dB'}^{\op}\otimes\dA'.$ The quasi-functor $\mb\otimes\ma$
induces a derived functor
\[
\bR (\mb\otimes\ma)_*: \D({\dB'}^{\op}\otimes\dA')\lto \D(\dB^{\op}\otimes\dA)
\]
by rule (\ref{derived_quasi}).
Any quasi-functor $\ma:\dC\to \dD$ can be realized as a roof $\dC\stackrel{\sim}{\leftarrow} \dC'\to \dD$
and any morphism of bimodules $\mM\to\mN$ in $\D(\C)$ can be represented as
a roof of the form $\mM\stackrel{\sim}{\leftarrow}\mM'\to \mN.$
Hence, we obtain the following proposition.
\begin{proposition}\label{gluing_quasifunctors}
Let $\ma:\dA\to \dA'$ and $\mb:\dB\to \dB'$ be quasi-functors between small DG categories. Let $\mS$ and $\mS'$ be
DG modules over $\dB^{\op}\otimes\dA$ and ${\dB'}^{\op}\otimes\dA'$ respectively. Assume that there is a morphism
$\phi:\mS\to \bR (\mb\otimes\ma)_*\mS'$ in $\D(\dB^{\op}\otimes\dA).$ Then there are quasi-functors
\[
\ma\underset{\phi}{\with}\mb: \dA\underset{\mS}{\with}\dB \lto \dA'\underset{\mS'}{\with}\dB',
\quad
\text{and}\quad
\ma\underset{\phi}{\oright}\mb:\dA\underset{\mS}{\oright}\dB \lto \dA'\underset{\mS'}{\oright}\dB'.
\]
Moreover, suppose that $\phi$ is a quasi-isomorphism.
If $a:\Ho(\dA)\to\Ho(\dA')$ and $b:\Ho(\dB)\to \Ho(\dB')$ are fully faithful, then
\[
a\underset{\phi}{\with}b: \Ho(\dA\underset{\mS}{\with}\dB) \lto \Ho(\dA'\underset{\mS'}{\with}\dB'),
\quad
\text{and}\quad
a\underset{\phi}{\oright}b:\Ho(\dA\underset{\mS}{\oright}\dB) \lto \Ho(\dA'\underset{\mS'}{\oright}\dB')
\]
are fully faithful.
If $\ma, \mb$ are quasi-equivalences, then both
$\ma\underset{\phi}{\with}\mb$ and $\ma\underset{\phi}{\oright}\mb$ are quasi-equivalences.
\end{proposition}
\subsection{Regular, smooth, and proper noncommutative schemes}\label{reg_proper_noncommutative}
Let $\T$ be a small $\kk$\!--linear triangulated category and let $\dA$ be a small $\kk$\!--linear DG category.
\begin{definition}\label{reg_and_prop} We say that $\T$ is {\em regular} if it has a strong generator,
and we say that $\T$ is proper if
$\bigoplus_{m\in\ZZ}\Hom(X, Y[m])$ is finite dimensional for any two objects $X, Y\in\T.$
\end{definition}
\begin{definition} We call $\dA$ {\em regular} (resp. {\em proper}) if the triangulated category
$\prf\dA$ is regular (resp. proper).
\end{definition}
\begin{remark}
{\rm
Instead of $\prf\dA$ we can consider $\tr(\dA).$ Since $\prf\dA$ is the idempotent completion of $\tr(\dA),$
regularity and properness of these categories hold simultaneously.
}
\end{remark}
\begin{remark}\label{finitness}
{\rm
It is easy to see that $\dA$ is proper if and only if $\bigoplus_i H^i(\dHom(X, Y))$ are finite dimensional
for all $X, Y\in\dA.$ It is evidently necessary due to Yoneda embedding $\dA\subset\prfdg\dA.$
Since $\Ob\dA$ classically generate $\prf\dA$ it is also sufficient.
}
\end{remark}
The following theorem is due to A.~Bondal and M.~Van den Bergh.
\begin{theorem}{\rm \cite[Th. 1.3]{BVdB}}\label{saturated}
Let $\T$ be a regular and proper triangulated category which is idempotent complete (Karoubian).
Then any exact functor from $\T^{\op}$ to the bounded derived category of finite dimensional
vector spaces $\prf\kk$ is representable, i.e. it is of the form $h^{Y}=\Hom(-, Y).$
\end{theorem}
Such a triangulated category is called {\em right saturated} in \cite{BK, BVdB}.
It is proved in \cite[2.6]{BK} that if $\T$ is a right saturated triangulated category and it is a full subcategory
in a proper triangulated category, then it is right admissible there.
By Theorem \ref{saturated} a regular and proper
idempotent complete triangulated category is right saturated. Since the opposite category is also regular and proper,
it is left saturated as well. Thus, we obtain the following proposition.
\begin{proposition}\label{admissible}
Let $\T\subset\T'$ be a full subcategory in a proper triangulated category
$\T'.$ Assume that $\T$ is regular and idempotent complete. Then $\T$ is admissible in $\T'.$
\end{proposition}
The proof of Theorem \ref{saturated} works for DG categories without any changes (see \cite{BVdB}).
Moreover, the DG version can be deduced from Theorem \ref{saturated}.
\begin{theorem}\label{dg_saturated} Let $\dA$ be a small DG category that is regular and proper.
Then a DG module $\mM$ is perfect if and only if $\dim\bigoplus_i H^i(\mM(X))<\infty$
for all $X\in\dA.$
\end{theorem}
\begin{dok}
If $\mM$ is perfect, then $\dim\bigoplus_i H^i(\mM(X))<\infty,$ because $\prf\dA$ is proper.
Assume now that $\dim\bigoplus_i H^i(\mM(X))<\infty.$ This implies that $\dim\bigoplus_i H^i(\dHom(\mP, \mM))<\infty$
for any $\mP\in \prfdg\dA.$
Therefore, the module $\mM$ gives the DG functor
$\dHom(-, \mM)$ from $\prfdg\dA$ to $\prfdg\kk.$ By Theorem \ref{saturated} the induced functor
$\Hom(-, \mM): \prf\dA\to\prf\kk$ is represented by an object $\mN\in\prf\dA$ and there is a canonical map
$\mN\to\mM.$ The cone $C$ of this map in $\D(\dA)$ is an object such that $\Hom(X, C)=0$ for any $X\in\dA.$
This implies that $C=0$ because $\Ob\dA$ is a set of compact generators in $\D(\dA).$
Thus, $\mM$ is a perfect complex.
\end{dok}
\begin{corollary}\label{representable}
Let $\dA$ be a regular and proper pretriangulated DG category for which $\Ho(\dA)$ is idempotent complete. Let $\mM$ be a DG $\dA$\!--module such that
$\dim\oplus_i H^i(\mM(X))<\infty$
for all $X\in\dA.$
Then $\mM$ is quasi-isomorphic to a representable module $\mh^Y=\dHom(-, Y)$ for some $Y\in\dA.$
\end{corollary}
\begin{proof}
It directly follows from the previous theorem and Remark \ref{perfect}.
\end{proof}
The properties of regularity and properness behave well under taking semi-orthogonal summands and gluing.
\begin{proposition}\label{reg_prop}
Let $\T$ be a $\kk$\!--linear triangulated category with a semi-orthogonal decomposition
$\T= \langle\T_1, \T_{2}\rangle.$ The following properties hold
\begin{enumerate}
\item[1)] if $\T$ is proper, then $\T_i$ are proper;
\item[2)] if $\T$ is regular, then $\T_i$ are regular;
\item[3)] if $\T_i,\; i=1,2$ are regular, then $\T$ is regular too.
\end{enumerate}
\end{proposition}
\begin{dok}
1) is evident, since any subcategory of a proper category is proper.
To prove 2) we should note that there are quotient functors from $\T$ to $\T_i.$ Now it is evident that the images of a strong generator
under these functors are strong generators in $\T_i.$
Let $E_i$ be strong generators of $\T_i$ such that $\langle E_i\rangle_{n_i}=\T_i.$ We can take $E=E_1\oplus E_2.$
There are embeddings $\langle E\rangle_{n_i}\supset \langle E_i\rangle_{n_i}=\T_i, i=1,2.$ By definition of
a semi-orthogonal decomposition, for any object $X\in\T$ there is an exact triangle of the form $X_2\to X\to X_1$
with $X_i\in\T_i.$ This implies that $X\in\langle E\rangle_{n_2}\diamond\langle E\rangle_{n_1}=\langle E\rangle_{n_1+n_2}.$
Hence $\T=\langle E\rangle_{n_1+n_2}.$ This proves 3).
\end{dok}
\begin{remark}
{\rm
The proof implies inequality $\dim\T\le \dim \T_1+\dim\T_2 +1.$
}
\end{remark}
\begin{proposition}
Let $\dA$ and $\dB$ be two small pretriangulated DG categories and let $\mS$ be a $\dB\hy\dA$\!--bimodule.
Then the following conditions are equivalent:
\begin{enumerate}
\item the gluing $\dA\underset{\mS}{\oright}\dB$ is regular and proper,
\item $\dA$ and $\dB$ are regular and proper and $\dim\bigoplus_i H^i(\mS(Y, X))<\infty$
for all $X\in\dA, Y\in\dB.$
\end{enumerate}
\end{proposition}
\begin{dok}
(1)$\Rightarrow$(2). Since $\Ho(\dA\underset{\mS}{\oright}\dB)=\langle \Ho(\dA), \Ho(\dB)\rangle$
regularity and properness of $\dA$ and $\dB$ directly follow from Proposition \ref{reg_prop} 1) and 2).
Properness of $\dA\underset{\mS}{\oright}\dB$ implies that $\dim\bigoplus_i H^i(\mS(Y, X))<\infty$ as well.
(2)$\Rightarrow$(1). Regularity of the gluing follows from 3) of Proposition \ref{reg_prop} and Proposition \ref{dg_semiorhtogonal}.
In view of Remark \ref{finitness} properness of $\dA\underset{\mS}{\oright}\dB$ directly follows from the properness of $\dA$ and $\dB$ and the finiteness of $\mS.$
\end{dok}
There is another important property of DG categories that is called smoothness.
\begin{definition}
A small $\kk$\!--linear DG category $\dA$ is called $\kk$\!-smooth if it is perfect as the module over $\dA^{\op}\otimes\dA.$
\end{definition}
This property depends on the base field $\kk.$ For example, a finite inseparable
extension $F\supset\kk$ is not smooth over $\kk$ and it is smooth over itself.
The following statement is proved in \cite{Lu} see Lemmas 3.5. and 3.6.
\begin{proposition}\label{smooth_regular}
If a small DG category $\dA$ is smooth, then it is regular.
\end{proposition}
Smoothness is invariant under Morita equivalence \cite{Lu, LS}. This means that if $\D(\dA)$ and $\D(\dB)$
are equivalent through a functor of the form $(-)\stackrel{\bL}{\otimes_{\dA}}\mT,$ where
$\mT$ is an $\dA\hy\dB$\!--bimodule, then $\dA$ is smooth if and only if $\dB$ is smooth.
Since $\dA\underset{\mS}{\with}\dB$ and $\dA\underset{\mS}{\oright}\dB$ are Morita equivalent we obtain that
smoothness of $\dA\underset{\mS}{\with}\dB$ and $\dA\underset{\mS}{\oright}\dB$ hold simultaneously.
Further, we can compare smoothness of a gluing with smoothness of summands. We get the following.
\begin{theorem}\cite[3.24]{LS}\label{smooth_glue}
Let $\dA$ and $\dB$ be two small pretriangulated DG categories over a field $\kk$ and let $\mS$ be a $\dB^{\op}\otimes\dA$\!--module.
Then the following conditions are equivalent:
\begin{enumerate}
\item the gluing $\dA\underset{\mS}{\oright}\dB$ is smooth;
\item $\dA$ and $\dB$ are smooth and $\mS$ is a perfect $\dB^{\op}\otimes\dA$\!--module.
\end{enumerate}
\end{theorem}
\subsection{Regularity, smoothness, and properness in commutative geometry}
Let us now discuss all these properties of DG categories in context of the usual geometry of schemes.
\begin{proposition}\label{proper}
Let $X$ be a proper scheme. Then the category $\prf\!X$ is proper.
\end{proposition}
\begin{dok}
Let $\E^{\cdot}$ be a perfect complex. Consider the functor $\bR\lHom(\E^{\cdot}, -)$ from $\D(\Qcoh X)$ to itself.
Since a perfect complex is locally quasi-isomorphic to a finite complex of vector bundles we obtain that
any object $\bR\lHom(\E^{\cdot}, \F^{\cdot})$ is perfect when $\E^{\cdot}$ and $\F^{\cdot}$ are perfect.
Let $\pi$ be the canonical morphism from $X$ to $\Spec\kk.$
By \cite[III 4.8.1]{SGA6} (see also \cite[2.5.4]{TT}) since $X$ is proper the object $\bR \pi_* \E^{\cdot}$ is perfect over $\kk$
when $\E^{\cdot}$ is perfect. Hence, the complex
\[
\bR\Hom(\E^{\cdot}, \F^{\cdot})\cong \bR\pi_*\bR\lHom(\E^{\cdot}, \F^{\cdot})
\]
is a perfect complex of $\kk$\!-vector spaces, i.e. $\bigoplus_{k}\Hom(\E^{\cdot}, \F^{\cdot}[k])$ is finite dimensional.
\end{dok}
\begin{theorem}\label{regular}
Let $X$ be a separated noetherian scheme of finite Krull dimension over an arbitrary filed $\kk.$
Assume that the square $X\times X$ is noetherian too. Then the following conditions are equivalent:
\begin{enumerate}
\item $X$ is regular;
\item $\prf X$ is regular, i.e. it has a strong generator.
\end{enumerate}
\end{theorem}
\begin{dok}
At first, note that the affine case $X=\Spec A$ was treated in Corollary 8.4 of \cite{Chr} and the remark immediately following (see also \cite[7.25]{Ro}).
We will use it.
(2)$\Rightarrow$(1)
Take an affine open subset $U\subset X.$ Any perfect complex on $U$ is a direct summand of a perfect
complex restricted from $X$ (\cite[Lemma 2.6]{Ne}). Hence, the category $\prf U$ is strongly generated too.
Thus we have reduced to the affine case.
If $\prf A$ is strongly generated, then the algebra $A$ has finite global dimension, i.e. it is regular.
(1)$\Rightarrow$(2) By \cite[II 2.2.7.1]{SGA6} any regular separated noetherian scheme has an ample family of line bundles, i.e.
there is a family of line bundles $\{\L_{\alpha}\}$ on $X$ such that for any quasi-coherent sheaf $\F,$ the evaluation map
\[
\bigoplus_{\alpha;\; n\ge 1}\Gamma(X, \F\otimes\L_{\alpha}^{\otimes n})\otimes \L_{\alpha}^{\otimes -n}\twoheadrightarrow
\F
\]
is an epimorphism.
In particular, for any coherent sheaf $\F$ there are an algebraic vector bundle $\E$ (i.e. locally free sheaf of finite type)
and an epimorphism $\E\twoheadrightarrow \F.$
Consider an affine covering $X=\bigcup_{i=1}^{m} V_i,$ where $V_i=\Spec A_i.$
Since $X$ is regular, all $A_i$ are regular noetherian algebras of finite dimension and, hence, they have finite global dimension.
This implies that for sufficiently large $n\in \ZZ$ (greater than maximum of global dimensions of $A_i$) for any quasi-coherent sheaf $\F$ there is a global locally free resolution
\begin{equation}\label{locally_free}
0\lto \E^{-n}\lto\cdots\lto\E^0\lto\F\lto 0.
\end{equation}
By \cite[1.4.12]{EGA3} (see also \cite[App. B]{TT}) there exists an integer $k\in \ZZ$ such that for all $p\ge k$
and for all quasi-coherent sheaves $\G,$ one has $\Ext^p(\E, \G)=H^p(X, \E^{\vee}\otimes \G)=0,$ where $\E$ is locally free.
Using a locally free resolution of type (\ref{locally_free}) for a quasi-coherent sheaf $\F,$ one has that
for sufficiently large $N\in \ZZ$ for all $p\ge N$ and all quasi-coherent sheaves $\F, \G,$ we have
$\Ext^p(\F, \G)=0.$ Thus, the abelian category $\Qcoh(X)$ has a finite global dimension. Let us denote it by
$k=\mathrm{gl}.\dim\Qcoh(X).$
Consider the product $X\times_{\kk} X.$ It is known that the family $\{\L_{\alpha}^r\boxtimes\L_{\beta}^s| r,s\ge 1\}$
forms an ample family on $X\times X$ (see \cite[2.1.2.f]{TT}), the scheme $X\times X$ not necessary being regular.
Take the structure sheaf $\cO_{\Delta}$ of the diagonal $\Delta\subset X\times X.$ Since $X$ is separated, $\Delta$ is closed. As $X\times X$ is noetherian, $\cO_{\Delta} $ is a coherent sheaf. Fix an infinite locally free resolution $\E^{\cdot}$ of $\cO_{\Delta}$
\[
\cdots \lto \E^{-n}\lto\cdots\lto\E^0\lto\cO_{\Delta}\lto 0,
\]
where each $\E^{-i}$ is a finite direct sum of sheaves of the form $\L_{\alpha}^{\otimes r}\boxtimes\L_{\beta}^{\otimes s}.$
Take a brutal truncation $\sigma^{\ge -l}\E^{\cdot}$ for a sufficiently large $l\gg 0.$ It has only two cohomology sheaves
$H^{-l}(\sigma^{\ge -l}\E^{\cdot})$ and $H^0(\sigma^{\ge -l}\E^{\cdot})=\cO_{\Delta}.$
Take all $\L_{\alpha}^{\otimes n}$ that appear in $\E^{-i}$ for all $0\le i\le l$ and consider their direct sum. Denote it by
$\cS.$ We have that $\cS$ is an algebraic vector bundle and $\sigma^{\ge -l}\E^{\cdot}\in \langle \cS\boxtimes \cS\rangle_{l+1}.$
For any quasi-coherent sheaf $\F,$ the object $C=\bR\pr_{2*}(\pr_1^*(\F)\otimes \sigma^{\ge -l}\E^{\cdot})$ is a complex
on $X,$ all cohomology $H^j(C)$ of which are trivial when $j> -l+k$ except $H^0(C)$ that is isomorphic to $\F.$
Since $l$ is large enough, we obtain that $\F$ is a direct summand of $C.$ But $C$ belongs to $\widebar{\langle\cS\rangle}_{l+1}.$
Therefore, $\F\in \widebar{\langle\cS\rangle}_{l+1}$ too.
Thus, we obtain that $\widebar{\langle\cS\rangle}_{l+1}$ contains all quasi-coherent sheaves.
Now we can apply the following proposition from \cite{Ro}.
\begin{proposition}\cite[Prop. 7.22]{Ro}
Let $\A$ be an abelian category of finite global dimension $k.$
Let $C$ be a complex of objects from $\A.$
Then, there is a distinguished
triangle in $\D(\A)$
\[
\bigoplus_{i} D_i\lto C\lto \bigoplus_{i} E_i,
\]
where $D_i=\sigma^{\ge ki+1}(\tau^{\le k(i+1)-1} C)$ is a complex with zero terms outside
$[ki+1,\dots k(i+1)-1]$ and $E_i$ is a complex concentrated in degree $ki.$
\end{proposition}
Using this proposition we obtain that any object of $\D(\Qcoh X)$ belongs to $\widebar{\langle\cS\rangle}_{k(l+1)}$
where $k=\mathrm{gl}.\dim\Qcoh(X).$ Indeed $E_i\in \widebar{\langle\cS\rangle}_{l+1}$ and $D_i$ as complexes
of length $k-1$ belong to $\widebar{\langle\cS\rangle}_{(k-1)(l+1)}.$
Finally, by Proposition \ref{compact_gen} we have that $\prf X=\D(\Qcoh(X))^c=\langle\cS\rangle_{k(l+1)}.$
\end{dok}
\begin{remark}
{\rm Recently, Amnon Neeman obtained a more general result in this direction.
In particular, the property to be noetherian for the square is not needed.
}
\end{remark}
\begin{proposition}\label{prop_scheme} Let $X$ be a separated scheme of finite type
over a field $\kk.$ Then $X$ is proper if and only if the category
of perfect complexes $\prf X$ is proper.
\end{proposition}
\begin{proof}
If $X$ is proper, then by Proposition \ref{proper}
the category $\prf X$ is proper.
Suppose that $\prf X$ is proper.
Let us show that $X$ is proper. We will prove by contradiction. Assume that $X$ is not proper.
By Chow's Lemma for any separated scheme of finite type $X$ there is a quasi-projective $X'$ with a proper map
$f: X'\to X.$ If $X$ is not proper $X'$ is not projective. Consider its closure $\widebar{X}'\in \PP^N.$
Take the complement $Y$ to $X'$ in $\widebar{X}'$ and choose a closed point $p\in Y.$
There is an irreducible and reduced projective curve $C\subset \widebar{X}'$ that contains the point $p$ and is not contained in $Y.$
Denote by $C_0\subset C$ the intersection of $C$ with $X'$ and by $\wt{C}$ and $\wt{C}_0$ the normalizations of
$C$ and $C_0$ respectively. Since $p\not\in C_0$ the complement $D$ to $\wt{C}_0$ in $\wt{C}$ is not empty.
The curve $C$ is regular, hence $D$ is a Cartier divisor on $C.$ Since $D$ is effective it is ample (see, e.g. \cite[7.5.5]{Liu}).
This implies that $\wt{C}_0$ is an affine curve.
Now consider the composition map $g:\wt{C}_0\to X$ and take the complex $\bR g_* \cO_{\wt{C}_0}.$
Since $g$ is proper being a composition of proper morphisms, the complex $\bR g_* \cO_{\wt{C}_0}$ is a cohomologically bounded complex with coherent cohomology.
From Theorem 4.1 of \cite{LN} we know that, given any integer $m,$ we may find a perfect complex $P^{\cdot}$ and a morphism $u: P^{\cdot}\to
\bR g_* \cO_{\wt{C}_0}$ so that the induced morphisms $\H^i(u)$ on cohomology sheaves are isomorphisms for all
$i>m.$ Let $m=-k-2$ where $X$ can be covered by $k$ affine open sets. Choose $u: P^{\cdot}\to
\bR g_* \cO_{\wt{C}_0}$ as above and complete to an exact triangle
\[
P^{\cdot}\stackrel{u}{\lto}\bR g_* \cO_{\wt{C}_0}\lto Q\lto P[1].
\]
Since all cohomology sheaves $\H^i(Q)$ are trivial when $i>-N-2$ the map between $\Hom(\cO_X, P^{\cdot})$ and $\Hom(\cO_X, \bR g_* \cO_{\wt{C}_0})$
is an isomorphism. Thus we obtain
\[
\Hom_X(\cO_X, P^{\cdot})\cong \Hom_X(\cO_X, \bR g_* \cO_{\wt{C}_0})\cong \Hom_{C_0}(\cO_{\wt{C}_0}, \cO_{\wt{C}_0})\cong H^0(\wt{C}_0, \cO_{\wt{C}_0})=A,
\]
where $\Spec A=\wt{C}_0.$
The $\kk$\!-space $A$ is infinite dimensional over $\kk,$ while $\cO_X$ an $P^{\cdot}$ are both perfect.
Hence, $\prf X$ can not be proper. This proves the proposition.
\end{proof}
We recall that a scheme of finite type over a field $\kk$ is called {\em smooth} if the scheme
$\widebar{X}=X\otimes_{\kk} \widebar{\kk}$ is regular, where $\widebar{\kk}$ is an algebraic closure
of $\kk.$
\begin{proposition}\label{smooth_proper} Let $X$ be a separated scheme of finite type
over an arbitrary field. Then $X$ is smooth and proper if and only if the DG category
$\prfdg X$ is smooth and proper.
\end{proposition}
\begin{proof}
The statement for smoothness (without properness) is proved in \cite[3.13]{Lu} for a separated scheme of finite type over a perfect field.
On the other hand, the definition of a smooth scheme and a smooth DG category is invariant under a base field change.
Indeed, if $\prfdg X$ is smooth, then $\prfdg \widebar{X}$ is smooth and, hence, by \cite[3.13]{Lu} the scheme $\widebar{X}$ is smooth (regular).
But this is exactly smoothness of $X$ by definition.
The properness of $X$ is proved in Proposition \ref{prop_scheme}.
Now if $X$ is smooth and proper, then properness of the DG category
$\prfdg X$ follows from Proposition \ref{proper}.
Since $\widebar{X}$ is regular we obtain that the DG category $\prfdg \widebar{X}$ is smooth by \cite[3.13]{Lu}.
Finally, we should argue that smoothness of $\prfdg\widebar{X}$ implies
smoothness of $\prfdg X.$ Since $\kk\subset\widebar{\kk}$ is faithfully flat, the following property holds:
for any DG category $\dA$ and any $\dA$\!--module
$\mM$ if $\mM\otimes_{\kk}\widebar{\kk}$ is perfect as $\dA\otimes_{\kk}\widebar{\kk}$\!--module, then
$\mM$ is also perfect.
\end{proof}
\section{Gluing of smooth projective schemes and geometric noncommutative schemes}
\subsection{Geometric noncommutative schemes}
Let $X$ and $Y$ be two smooth projective schemes over a field $\kk.$
Consider DG categories of perfect complexes $\prfdg X$ and $\prfdg Y.$ Since $X$ and $Y$
are smooth these categories are quasi-equivalent to DG categories $\dD^b(\coh X)$ and $\dD^b(\coh Y),$
respectively. Theorem \ref{dg_saturated} tells us that for a regular and proper $X$ there is a quasi-equivalence
\[
\dR\lHom(\prfdg X^{\op},\; \prfdg\kk)\cong \prfdg X.
\]
Therefore, applying the canonical quasi-equivalence (\ref{RHom}) we obtain that
\begin{multline}
\dR\lHom(\prfdg Y\otimes_{\kk}\prfdg X^{\op},\; \prfdg\kk)\cong \dR\lHom(\prfdg Y, \dR\lHom(\prfdg X^{\op},\; \prfdg\kk))
\cong\\
\dR\lHom(\prfdg Y,\; \prfdg X).
\end{multline}
Moreover, there is the following theorem due to B.~To\"en.
\begin{theorem}\cite{To}\label{toen}
Let $X$ and $Y$ be smooth projective schemes over a field $\kk.$
Then there is a canonical isomorphism in $\Hqe$
\[
\dR\lHom(\prfdg Y,\; \prfdg X)\cong\prfdg(X\times_k Y).
\]
In particular, the DG category $\prfdg(X\times_k Y)$ is quasi-equivalent to the DG category of perfect
DG modules over $\prfdg Y^{\op}\otimes_{\kk}\prfdg X.$
\end{theorem}
This quasi-equivalence can be described explicitly.
As was explained in Section \ref{3.1} there are DG functors
\[
\mpr_1^*: \prfdg X\lto \prfdg(X\times Y),\quad \text{and}\quad \mpr_2^*: \prfdg Y\lto \prfdg(X\times Y)
\]
For any perfect complex $\mE^{\cdot}$ on the product $X\times_{\kk} Y$ we can define a bimodule
$\mS_{\mE^{\cdot}}$ by the rule
\[
\mS_{\mE^{\cdot}}(B, A)\cong \dHom_{\prfdg(X\times Y)}(\mpr_1^* A,\; \mpr_2^* B\otimes \mE),
\quad\text{where}
\quad
A\in \prfdg X,\; B\in\prfdg Y.
\]
This is exactly the quasi-equivalence between the DG category $\prfdg(X\times Y)$ and
the DG category of perfect $\prfdg Y\hy\prfdg X$\!--bimodules, i.e. perfect
DG modules over $\prfdg Y^{\op}\otimes_{\kk}\prfdg X.$
Let $\M\subset\prf X$ and $\N\subset\prf Y$ be admissible subcategories, where $X$ and $Y$ are smooth projective schemes
over $\kk.$ Consider the induced DG subcategories $\dM\subset\prfdg X$ and $\dN\subset\prfdg Y$
and the induced DG functor $\mF: \dN^{\op}\otimes\dM\to \prfdg Y^{\op}\otimes\prfdg X$ that is fully faithful.
This DG functor gives the extension quasi-functor
\[
\mF^*:\prfdg(\dN^{\op}\otimes\dM)\lto \prfdg (X\times Y)
\]
that is fully faithful on the homotopy categories by Proposition \ref{Keller2}.
In more detail, for any pair of admissible subcategories $\M\subset\prf X$ and $\N\subset\prf Y$ we can define
a full triangulated subcategory $\M\prd\N$ of the category
$\prf(X\times Y)$ as the minimal triangulated subcategory
of $\prf(X\times Y)$ closed under taking direct summands and containing
all objects of the form $\pr_1^* M\otimes \pr_2^* N$ with $M\in \M$ and $N\in \N.$
Denote by $\dM\prd\dN\subset \prfdg (X\times Y)$ the induced enhancement of $\M\prd\N.$
It is easy to see that $\prfdg(\dN^{\op}\otimes\dM)$ is quasi-equivalent to $\dM\prd\dN$ because
$\prf(\dN^{\op}\otimes\dM)$ and $\M\prd\N$ are classically generated by $\Ob(\dN^{\op}\otimes\dM).$
Being admissible subcategories in DG categories of perfect complexes on smooth and proper schemes,
the DG categories $\dM$ and $\dN$
are smooth and proper (see Theorem \ref{smooth_glue} for smoothness and Proposition \ref{reg_prop} for properness). By Theorem \ref{dg_saturated} there is a quasi-equivalence
\[
\dR\lHom(\dM^{\op},\; \prfdg\kk)\cong \dM.
\]
Therefore, applying the canonical quasi-equivalence (\ref{RHom}) we obtain that
\[
\dR\lHom(\dN\otimes_{\kk}\dM^{\op},\; \prfdg\kk)\cong \dR\lHom(\dN, \dR\lHom(\dM^{\op},\; \prfdg\kk))
\cong\dR\lHom(\dN,\; \dM).
\]
Let us summarize what we have.
\begin{proposition}\label{geometric}
Let $X$ and $Y$ be two smooth projective schemes and $\dM\subset\prfdg X$ and $\dN\subset\prfdg Y$ be full DG subcategories such that
the subcategories $\M=\Ho(\dM)$ and $\N=\Ho(\dN)$ are admissible in $\prf X$ and $\prf Y,$ respectively.
In this case there are quasi-equivalences of DG categories
\[
\dR\lHom(\dN,\; \dM)\cong\prfdg(\dN^{\op}\otimes_{\kk}\dM)\cong \dM\prd\dN\subset\prfdg(X\times_{\kk}Y),
\]
where $\dM\prd\dN$ is a full DG subcategory of $\prfdg(X\times Y)$ that is classically generated by objects of the form
$\pr_1^* M\otimes \pr_2^* N$ with $M\in \M$ and $N\in \N.$
\end{proposition}
We are interested in smooth (or regular) and proper noncommutative scheme $\prfdg\E.$
Smooth and proper geometric noncommutative schemes naturally appear as induced enhancements of
admissible subcategories $\N\subset\prf X$ for some smooth and projective scheme $X.$
\begin{definition}
A noncommutative scheme $\prfdg\E$ (see Definition \ref{noncommutative_scheme})
will be called a {\em geometric noncommutative scheme} if
there are a smooth and projective scheme $X$ and an admissible subcategory $\N\subset \prf X$
such that $\prfdg \E$ is quasi-equivalent to the corresponding enhancement $\dN\subset \prfdg X$ of $\N.$
\end{definition}
We can consider a 2-category of smooth and proper noncommutative schemes
$\RPNS$ over a field $\kk.$
Objects of $\RPNS$ are DG categories $\dA$ of the form $\prfdg\dE,$ where
$\dE$ is a smooth and proper DG algebra; 1-morphisms are
quasi-functors $\mT;$ 2-morphisms are morphisms of quasi-functors, i.e. morphisms in
$\D(\dA^{op}\otimes\dB).$
The 2-category $\RPNS$ has a natural full 2-subcategory of geometric noncommutative schemes $\GNS.$
Evidently, $\GNS$ contains all smooth and proper commutative schemes with $\prf(X\times Y)$ as category of morphisms
between $X$ and $Y.$
The natural question that arises is following.
\begin{question}
Is there a smooth and proper noncommutative scheme that is not geometric?
\end{question}
The first attempt to find such a noncommutative scheme is to glue geometric noncommutative schemes via a bimodule.
Another way is to consider a finite dimensional $\kk$\!--algebra $\Lambda$ of finite global dimension and take
the DG category $\prfdg\Lambda.$
The main goal of this paper is to show that these two approaches do not lead us to new noncommutative schemes. We show that the world of geometric noncommutative schemes is closed under gluing via any perfect bimodule.
More precisely, consider smooth and proper geometric noncommutative schemes $\prfdg\dE_1$ and $\prfdg\dE_2$ such that
$\prfdg\dE_i$ is quasi-equivalent to $\dN_i\subset\prfdg X_i,$ where $X_i$ are smooth and projective and
$\N_i=\Ho(\dN_i)$ are admissible in $\prf X_i,$ respectively.
After that we take a gluing $\prfdg\dE_1\underset{\mS}{\oright}\prfdg\dE_2$ via a perfect bimodule $\mS$ and show that
the resulting noncommutative scheme is geometric too (see Theorem \ref{main2}).
\begin{remark}\label{Hironaka}
{\rm
We work with smooth and projective schemes. On the other hand, over a field of characteristic 0
the category of perfect complexes on any smooth and proper scheme can be realized as an admissible subcategory
in a smooth and projective scheme. Indeed, by Chow's Lemma for a proper scheme $X$ there is a proper birational morphism
$f: Y\to X$ from a projective scheme $Y.$ Now applying Hironaka hut for resolution of the birational map $X\dashrightarrow Y,$
we can find a proper scheme $Z$ with birational maps to $X$ and $Y$ such that the morphism $\pi: Z \to X$
is a sequence of blowups with regular centers. This implies that $Z$ is smooth and also projective because there is a proper
birational morphism to the projective scheme $Y.$ Finally, the inverse image functor $\bL\pi^*$ gives a full embedding
of $\prf X$ into $\prf Z.$
Note that over the complex numbers $\CC$ we can apply a result of Moishezon asserting that any smooth and proper algebraic space over $\CC$
becomes a projective variety after some blowups along smooth centers.
}
\end{remark}
We also show that for any finite dimensional $\kk$\!--algebra $\Lambda$ such that its semisimple part
$S=\Lambda/\rd$ is separable over $\kk$ there are a smooth and projective scheme $X$ and a perfect complex
$\E^{\cdot}$ such that $\bR\Hom(\E^{\cdot}, \E^{\cdot})\cong \Lambda.$
This implies that for the finite dimensional algebra $\Lambda$ the DG category
$\prfdg\Lambda$ is quasi-equivalent to a full DG subcategory of $\prfdg X$ and in the case of finite global dimension
the smooth and proper noncommutative scheme $\prfdg\Lambda$ is geometric (Theorem \ref{algebra}).
\subsection{Perfect complexes as direct images of line bundles}
Let $X$ be a scheme and $\E^{\cdot}$ be a strict perfect complex, i.e. a bounded complex of algebraic vector bundles
(locally free sheaves of finite type). In this section we show that
any such a strict perfect complex $\E^{\cdot}$ can be realized as a direct image of a line bundle with respect to a smooth morphism
$Z\to X.$
\begin{proposition}\label{line} Let $X$ be a scheme. Let $\E^{\cdot}=\{\E^0\to\cdots\to\E^k\}$
be a bounded complex of algebraic vector bundles on $X.$
Then there are a scheme $Z$ with a morphism $f: Z\to X$ and a line bundle $\L$ on $Z$
such that
\begin{enumerate}
\item[1)]
$\bR f_{*}\L\cong \E^{\cdot}$ in the derived category $\D(\Qcoh X);$
\item[2)] $\bR f_{*}\L^{-1}\cong 0;$
\item[3)] the morphism $f: Z\to X$ is a composition of maps $Z=X_n\to X_{n-1}\to\cdots\to X_0=X,$
where each $X_{p+1}$ is the projectivization of a vector bundle $\F_p$ over $X_p.$
\end{enumerate}
\end{proposition}
\begin{dok}
First, we should note that the following construction does not work for the $0$\!--complex and for
a complex $\E^{\cdot}$ that is a line bundle. However, it can be easily improved. In such cases we
change $\E^{\cdot}$ to a quasi-isomorphic complex by adding an acyclic complex of the form $\F\stackrel{\id}{\to}\F.$
Now we will prove the proposition by induction on the length of the complex $\E^{\cdot}.$
If $\E^{\cdot}$ has only one term and it is a vector bundle $\E$ (of rank $>1$), then we take the line bundle $\L=\cO(1)$ on the
projective bundle $f: \PP(\E^{\vee})\to X.$ As result we obtain that $\bR f_* \L= f_*\L\cong\E$
and $\bR f_* \L^{-1}=0.$
Assume that $\E^{\cdot}=\{\E^0\stackrel{d}{\lto}\E^1\}$ is a complex of vector bundles that has only two nontrivial terms in degree $0$ and $1.$
Taking $f_1: X_1=\PP(\E^{1\vee})\to X$ and $\L_1=\cO(1)$ we obtain that $\bR f_{1*}\L\cong\E^1$ as described above.
Now let $X_2=X_1\times\PP^1$ and $f_2$ is the projection on $X.$ Take $\L_2=\L_1\boxtimes\cO(-2).$
It is easy to see that $\bR \pr_{1*}\L_2$ has only one nontrivial term $\bR^1 \pr_{1*}\L_2$ that is isomorphic to $\L_1.$
Hence $\bR f_{2*}\L_2\cong \E^1[-1].$
Thus we obtain a sequence of isomorphisms
\begin{multline*}
\Ext^1_{X_2}(f_2^*\E^0, \L_2)\cong \Ext^1_{X_1}(f_1^*\E^0, \bR \pr_{1*}\L_2)\cong \Hom_{X_1}(f_1^*\E^0, \L_1)\cong\Hom_{X}(\E^0, \bR f_{1*}\L_1)\cong\\
\Hom_{X}(\E^0, \E^1).
\end{multline*}
Under this isomorphism the differential $d$ induces an element $e\in \Ext^1(f_2^*\E^0, \L_2).$
Let us consider the extension
\[
0\lto\L_2\lto\F\lto f_2^*\E^0\lto 0
\]
given by the element $e.$ Applying the functor $\bR f_{2*}$ to this short exact sequence we obtain an exact triangle of the form
\[
\bR f_{2*} \F\lto \E^0 \stackrel{\alpha}{\lto} \bR f_{2*}\L_2[1].
\]
By construction, $\bR f_{2*}\L_2[1]\cong \E^1$ and $\alpha=d.$ Therefore, $\bR f_{2*} \F$ is isomorphic to the complex $\E^{\cdot}.$
Finally, we consider $Z=\PP(\F^{\vee})$ with the natural morphism $f$ to $X$ and $\L=\cO_{\PP(\F^{\vee})}(1).$ We get that $\bR f_{*}\L\cong\E^{\cdot}.$
Since the rank of $\F$ is bigger than one, it is also evident that $\bR f_* \L^{-1}=0.$
The same trick works for any complex of vector bundles
\[
\E=\{\E^0\stackrel{d}{\lto}\E^1\lto\cdots\lto\E_k\}.
\]
Indeed, consider the stupid truncation $\sigma^{\ge 1}\E^{\cdot}.$
By induction we can assume that there is $f_{n-1}: X_{n-1}\to X$ and $\L_{n-1}$ on $X_{n-1}$ such that
$\bR f_{(n-1)*}\cong \sigma^{\ge 1}\E^{\cdot}[1].$
Now repeat the procedure described above.
Let $X_{n}=X_{n-1}\times\PP^1$ and $f_{n}$ be the projection on $X.$
Take $\L_{n}=\pr_1^*\L_{n-1}\boxtimes\cO(-2).$
We have $\bR f_{n*}\L_{n}\cong \sigma^{\ge 1}\E^{\cdot}.$
There is an isomorphisms
\[
\Ext^1(f_{n}^*\E^0, \L_{n})\cong \Ext^1(\E^0, \bR f_{n*}\L_n)\cong
\Hom(\E^0, \sigma^{\ge 1}\E^1[1])
\]
Under this isomorphism the differential $d:\E_0\to \sigma^{\ge 1}\E^1[1]$ induces an element $e\in \Ext^1(f_{n}^*\E^0, \L_{n}).$
Let us consider the extension
\[
0\lto\L_n\lto\F\lto f_n^*\E^0\lto 0
\]
given by the element $e.$ Applying the functor $\bR f_{n*}$ to this short sequence we obtain an exact triangle of the form
\[
\bR f_{n*} \F\lto \E^0 \stackrel{\alpha}{\lto} \bR f_{n*}\L_n[1].
\]
By construction, $\bR f_{n*}\L_2[1]\cong \sigma^{\ge 1}\E^1[1]$ and $\alpha=d.$
Therefore, $\bR f_{n*} \F$ is isomorphic to the complex $\E^{\cdot}.$
Finally, we consider $Z=\PP(\F^{\vee})$ with the natural morphism $f$ to $X$ and $\L=\cO(1).$
We get that $\bR f_{*}\L\cong\E^{\cdot}.$ By construction, the scheme $f:Z\to X$ is a sequence of
projective bundles. Moreover, we have $\bR f_*\L^{-1}=0,$ because the rank of $\F$ is bigger than one and $\bR p_*\L^{-1}=0,$
where $p$ is the projection of $Z=\PP(\F^{\vee})$ to $X_n.$
\end{dok}
\begin{remark}
{\rm
Assume that a quasi-compact and separated scheme $X$ has enough locally free sheaves, i.e. for any quasi-coherent sheaf of finite type
$\F$ there is an algebraic vector bundle $\E$ on $X$ and an epimorphism $\E\twoheadrightarrow\F.$
In this case, any perfect complex is quasi-isomorphic to
a strict perfect complex (see \cite[2.3.1]{TT}) and, hence, Proposition \ref{line} can be applied to any perfect complex up to a shift
in the triangulated category.
Note that any quasi-projective scheme and any separated regular noetherian scheme have enough locally free sheaves.
}
\end{remark}
\subsection{Blowups and gluing of smooth projective schemes}
Let $X_1$ and $X_2$ be two smooth irreducible projective schemes. Let $\E^{\cdot}$ be a perfect complex
on the product $X_1\times X_2.$ Since $X_i$ are projective any perfect complex on $X_1\times X_2$ is globally (not only locally) quasi-isomorphic to
a strictly perfect complex, i.e. a bounded complex of locally free sheaves of finite type (see, e.g. \cite[2.3.1]{TT}).
A strictly perfect complex will be also called a bounded complex of vector bundles.
Applying a shift in the triangulated category we can assume that $\E^{\cdot}\in\prf (X_1\times X_2)$ is
a complex of the form $\{\E^0{\to}\E^1\to\cdots\to\E_k\}.$
By Proposition \ref{line} there is a scheme $Z$ with a morphism $f: Z\to X_1\times X_2$
and a line bundle $\L$ on $Z$ such that $\bR f_* \L\cong \E^{\cdot}.$
Let us fix such $Z$ and $f.$ By the construction of $Z,$ since $X_1$ and $X_2$ are smooth projective the scheme $Z$ is also
smooth and projective and the morphism $f$ is smooth.
Denote by $q_1$ and $q_2$ the canonical morphisms from $Z$ to $X_1$ and $X_2$ respectively.
Fix very ample line bundles $\M_1$ and $\M_2$ on $X_1$ and $X_2$ respectively.
Using Serre's theorem we can find a very ample line bundle $\L'$ on $Z$ such that
the three line bundles $\L_1=q_1^*\M_1^{-1}\otimes\L',$ $\L_2=q_2^*\M_2^{-1}\otimes\L'$ and $
\L_3=\L^{-1}\otimes \L'$ are very ample as well.
Denote by $s_1, s_2, s_3$ the closed immersions of $Z$ to projective spaces
$\PP^{n_1}, \PP^{n_2}, \PP^{n_3}$ induced by $\L_1, \L_2,$ and $\L_3$ respectively.
The product map $i_1=(q_1, s_1, s_3)$ gives a closed immersion of $Z$ to the projective scheme
$P_1=X_1\times \PP^{n_1}\times\PP^{n_3}.$ Similarly, we obtain a closed immersion $i_2=(q_2, s_2, s_3)$ of $Z$ to
$P_2=X_2\times \PP^{n_2}\times\PP^{n_3}.$ It directly follows from construction
that there are isomorphisms
\begin{equation}\label{line_bundles}
\begin{aligned}
& i_1^* (\M_1\boxtimes\cO(1)\boxtimes\cO(-1)) \cong i_2^* (\M_2\boxtimes\cO(1)\boxtimes\cO(-1)) \cong \L,\\
& i_1^* (\M_1\boxtimes\cO(1)\boxtimes\cO(1)) \cong i_2^* (\M_2\boxtimes\cO(1)\boxtimes\cO(1)) \cong \L'\otimes\L_3
\end{aligned}
\end{equation}
Consider these two closed immersions $i_1: Z\to P_1$ and $i_2: Z\to P_2.$ It is known that the gluing $P_1\bigsqcup_{Z} P_2$
of $P_1$ and $P_2$ along $Z$ is a scheme (\cite[3.9]{Schwede} or \cite[5.4]{F}). It has two irreducible components that meet
along $Z.$ Denote this gluing by $T.$ The scheme $T$ is a pushout (fibred coproduct) in the category
of schemes. In our case we also can argue that the scheme $T$ is projective.
\begin{lemma} The scheme $T=P_1\bigsqcup_{Z} P_2$ is projective.
\end{lemma}
\begin{dok} Consider very ample line bundles $\M_1\boxtimes\cO(1)\boxtimes\cO(1)$ and $\M_2\boxtimes\cO(1)\boxtimes\cO(1)$
on $P_1$ and $P_2.$ Since their restrictions on $Z$ are isomorphic to $\L'\otimes\L_3$ we can glue them into a line bundle
$\N$ on $T.$
It follows from \cite[2.6.2]{EGA3} that $\N$ is ample on $T$ (see also \cite[6.3]{F}).
\end{dok}
Consider a closed immersion $j: T\hookrightarrow \PP^N$ to a projective space $\PP^N.$
Denote by $j_1, j_2$ the induced closed immersions of $P_1, P_2$ to $\PP^N.$
Consider the blowup $V'$ of $\PP^N$ along $P_1.$ Take the strict transform $\wt{P}_2$ of $P_2$ in $V'.$
It is the blowup of $P_2$ along the subvariety $Z.$
Denote by $V$ the blowup of $V'$ along $\wt{P}_2.$
This construction can be illustrated by the following diagram (\ref{diagram})
\begin{align}\label{diagram}
\xymatrix{
& \wt{E}_1 \ar[d]_{\rho} \ar@{^{(}->}[r]^{\wt{e}_1} & V \ar[d]_{\pi} & E_2
\ar@{_{(}->}[l]_{e_2} \ar[d]^{h_2}
\\
D \ar[d]_{g} \ar@{^{(}->}[r]^{d_1}& E_1 \ar[d]_{h_1} \ar@{^{(}->}[r]^{e_1} &
V' \ar[d]_{\pi'} & \wt{P}_2 \ar[d]^{\tau} \ar@{_{(}->}[l]_{\wt{j}_2}
& D \ar[d]^{g} \ar@{_{(}->}[l]_{d_2}
\\
Z \ar@{^{(}->}[r]^{i_1} \ar[dr]_{q_1} & P_1 \ar[d]^{p_1} \ar@{^{(}->}[r]^{j_1} & \PP^N & P_2 \ar[d]_{p_2} \ar@{_{(}->}[l]_{j_2} &
Z \ar@{_{(}->}[l]_{i_2} \ar[dl]^{q_2}
\\
& X_1 && X_2
}
\end{align}
where $h_1: E_1\to P_1$ is the exceptional divisor of the first blowup, $g: D\to Z$ is the exceptional divisor
of the induced blowup $\tau: \wt{P}_2\to P_2,$ $\wt{E}_1$ is the strict transformation of the divisor $E_1$ under the second blowup,
and $h_2: E_2\to \wt{P}_2$ is the exceptional divisor
of the second blowup.
Since we started from smooth projective schemes $X_1$ and $X_2,$ we obtain smooth and projective schemes $Z,$ $P_1$ and $P_2.$
A blowup of a smooth projective scheme along a smooth closed subscheme brings to a smooth and projective scheme (see, e.g.
\cite[Th.8.1.19]{Liu}). Thus, we obtain
\begin{lemma}
All schemes in diagram (\ref{diagram}) are projective and smooth.
\end{lemma}
Now let us analyze morphisms in our diagram (\ref{diagram}) and functors induced by them.
\begin{proposition}\label{functors}
In the diagram (\ref{diagram}) the following properties of morphisms hold:
\begin{enumerate}
\item[1)] the morphisms $g, h_1, h_2, p_1, p_2$ are projectivizations of vector bundles, and the functors $g^*, h_1^*, h_2^*, p_1^*, p_2^*$
are fully faithful;
\item[2)] the morphisms $\pi', \pi, \tau, \rho$ are blowups along smooth centers, and the exact functors
$\bL {\pi'}^{*},$ $\bL\pi^*,$ $\bL\tau^*,$ and $\bL\rho^*$ are fully faithful;
\item[3)] functors of the form $\bR d_{2*}(\K\otimes g^*(-)),\; \bR e_{1*}(\K\otimes h_1^*(-)),\; \bR e_{2*}(\K\otimes h_2^*(-)),$
where $\K$ is a line bundle on $D, E_1, E_2,$ respectively, are fully faithful;
\item[4)] the functors $\bR d_{2*}, \bR e_{1*}, \bR e_{2*}$ have right adjoints
$d_2^{\flat}, e_1^{\flat}, e_2^{\flat}$ and there are isomorphisms
\[\hspace*{1cm}
d_2^{\flat}\cong \bL d_2^*(\cO(D)\otimes (-))[-1],\quad
e_1^{\flat}\cong \bL e_1^*(\cO(E_1)\otimes (-))[-1],\quad
e_2^{\flat}\cong \bL e_2^*(\cO(E_2)\otimes (-))[-1].
\]
\end{enumerate}
\end{proposition}
\begin{dok}
1) and 2) follow from the construction and the projection formula, because the derived direct image of the structure sheaf
under each of these morphisms is isomorphic to the structure sheaf of a target.
3) is proved in \cite[4.2]{Blow} or \cite[2.2.7]{O2}.
4) follows from the fact that for any closed immersion $i$ of locally a complete intersection $Z$ to $Y$
the right adjoint $i^{\flat}$ to $\bR i_{*}$ has
the form $\bL i^*(\cdot) \otimes \omega_{Z/Y}[-r],$ where
$\omega_{Z/Y}\cong \Lambda^{r} N_{Z/Y}$ and $r$ is the codimension.
\end{dok}
\begin{theorem}\label{main}
Let $X_1$ and $X_2$ be smooth irreducible projective schemes and let $\E^{\cdot}$ be a perfect complex
on the product $X_1\times X_2.$ Let $V$ be a smooth projective scheme constructed above.
Then the DG category $\prfdg X_1 \underset{\E^{\cdot}}{\oright}\prfdg X_2$ is quasi-equivalent to a full DG subcategory
of $\prfdg V$ and, hence,
the triangulated category $\Ho(\prfdg X_1 \underset{\E^{\cdot}}{\oright}\prfdg X_2)$ is admissible in
$\prf V.$
\end{theorem}
Consider the DG categories $\prfdg X_1$ and $\prfdg X_2$ and the following composition quasi-functors
\begin{equation}
\mPhi:=\mpi^* \me_{1*}( \cO_{E_1}(E_1)\otimes \mh_1^* \mp_1^*(-)), \quad\text{and}\quad
\mPsi:=\me_{2*} \mh_2^* \mtau^* (\mp_2^*(-)\otimes \R)[1]
\end{equation}
from $\prfdg X_1$ and $\prfdg X_2$ to $\prfdg V$ respectively, where $\cO_{E_1}(E_1)$ is the restriction
of the line bundle $\cO(E_1)$ from $V'$ to $E_1,$ and $\R\cong \M_2\boxtimes\cO(1)\boxtimes\cO(-1)$ is a line bundle on $P_2.$
These quasi-functors induce exact composition functors
\begin{equation}\label{funct}
\Phi:=\bL\pi^* \bR e_{1*}( \cO_{E_1}(E_1)\otimes h_1^* p_1^*(-)), \quad\text{and}\quad
\Psi:=\bR e_{2*} h_2^* \bL\tau^* (p_2^*(-)\otimes \R)[1]
\end{equation}
from the triangulated categories $\prf X_1$ and $\prf X_2$ to the triangulated category $\prf V.$
\begin{lemma}\label{lemma_orthogonal}
The functors $\Phi$ and $\Psi$ are fully faithful and the subcategories
$\Phi(\prf X_1)$ and $\Psi(\prf X_2)$ are semi-orthogonal so that
$\Phi(\prf X_1)$ is in the right orthogonal $\Psi(\prf X_2)^{\perp}.$
\end{lemma}
\begin{dok} The functors $\Phi$ and $\Psi$ are fully faithful as compositions
of fully faithful functors. It follows from 1)-3) of Proposition \ref{functors}.
Semi-orthogonality is a consequence of the fact that $\bL\pi^*(\prf V')$ is in
the right orthogonal $\bR e_{2*} h_2^*(\prf \wt{P}_2)^{\perp}.$
The last statement follows from the chain of isomorphisms
\begin{multline*}
\Hom(\bR e_{2*} h_2^* B,\; \bL\pi^* A)\cong
\Hom(h_2^* B,\; e_2^{\flat}\bL\pi^* A)\cong
\Hom(h_2^* B,\; \bL e_2^*\bL\pi^* A\otimes \cO_{E_2}(E_2))\cong\\
\Hom(h_2^* B,\; h_2^*\bL \wt{j}_2^* A\otimes \cO_{E_2}(E_2))\cong
\Hom(B,\; \bL \wt{j}_2^* A\otimes \bR h_{2*}\cO_{E_2}(E_2))=
0
\end{multline*}
where $A\in\prf V', B\in\prf \wt{P}_2.$
The last equality holds because $\cO_{E_2}(E_2)\cong\cO_{E_2}(-1)$ under consideration of $E_2$ as the projectivization of the normal bundle
of $\wt{P}_2$ in $V',$ i.e. we have $\bR h_{2*} \cO_{E_2}(E_2)=0.$
\end{dok}
\begin{proposition}\label{main_prop} Let $X_1$ and $X_2$ be smooth projective schemes and let $\E^{\cdot}$ be a perfect complex
on the product $X_1\times X_2.$ Let $V$ be the smooth projective scheme constructed above and let $\Phi, \Psi$ be exact functors
defined by formula (\ref{funct}).
Let $\F^{\cdot}$ and $\G^{\cdot}$ be perfect complexes on $X_1$ and $X_2,$ respectively.
Then there is an isomorphism
\[
\Hom_{V}(\Phi(\F^{\cdot}),\; \Psi(\G^{\cdot}))\cong \Hom_{X_1\times X_2}(\pr_1^* \F^{\cdot}, \; \pr_2^*\G^{\cdot}\otimes \E^{\cdot}),
\]
where $\pr_i$ denote the projections of $X_1\times X_2$ on $X_i.$
\end{proposition}
\begin{dok}
Firstly, consider objects $A\in\prf V'$ and $B\in\prf \wt{P}_2.$ There is a sequence of isomorphisms
\begin{multline}\label{1_sequence}
\Hom_V(\bL\pi^* A, \; \bR e_{2*} h_2^* B)
\cong
\Hom_{V'}(A, \; \bR\pi_*\bR e_{2*} h_2^* B)
\cong\\
\Hom_{V'}(A, \; \bR\wt{j}_{2*}\bR h_{2*} h_2^* B)
\cong
\Hom_{V'}(A, \; \bR\wt{j}_{2*} B).
\end{multline}
Secondly, take $A\in\prf P_1$ and $B\in\prf P_2.$ Consider the commutative square
\begin{equation}\label{square}
\xymatrix{
D \ar@{^{(}->}[r]^{d_2} \ar[d]_{d_1} & \wt{P}_2 \ar[d]^{\wt{j}_2}\\
E_1 \ar@{^{(}->}[r]^{e_1} & V'
}
\end{equation}
that is a part of our main diagram (\ref{diagram}). The commutative square (\ref{square}) is cartesian.
Moreover, it is Tor-independent. This means that $\Tor^p_{\cO_{V'}}(\cO_{E_1}, \cO_{\wt{P}_2})=0$ for all $p>0.$
Therefore, by \cite[IV 3.1]{SGA6} or \cite[2.5.6]{TT} there is a canonical base change isomorphism of functors
\[
\bL e_1^{*}\bR \wt{j}_2\stackrel{\sim}{\lto}\bR d_{1*}\bL d_2^*.
\]
Using this isomorphism of functors we obtain the following sequence of isomorphisms
\begin{equation}
\begin{split}
&\Hom_{V'}(\bR e_{1*} (\cO_{E_1}(E_1)\otimes h_1^* A), \; \bR \wt{j}_{2*} \bL\tau^* B)
\cong
\Hom_{E_1}(\cO_{E_1}(E_1)\otimes h_1^* A, \; e_1^{\flat}\bR \wt{j}_{2*} \bL\tau^* B)
\cong\\
\label{2_sequence}
&\Hom_{E_1}(h_1^* A, \; \bL e_1^{*}\bR \wt{j}_{2*} \bL\tau^* B[-1])
\cong
\Hom_{E_1}(h_1^* A, \; \bR d_{1*}\bL d_2^* \bL\tau^* B[-1])
\cong\\
&
\Hom_{D}(\bL d_1^* h_1^* A, \; \bL d_2^* \bL\tau^* B[-1])
\cong
\Hom_{D}(g^* \bL i_1^* A, \; g^* \bL i_2^* B[-1])
\cong
\Hom_Z(\bL i_1^* A, \; \bL i_2^* B[-1])
\end{split}
\end{equation}
Now combining (\ref{1_sequence}) and (\ref{2_sequence}), we obtain
\begin{equation}\label{3_sequence}
\begin{split}
&\Hom_{V}(\Phi(\F^{\cdot}),\; \Psi(\G^{\cdot}))
\cong
\Hom_{V'}(\bR e_{1*} (\cO_{E_1}(E_1)\otimes h_1^* p_1^* \F^{\cdot}), \; \bR \wt{j}_{2*} \bL\tau^* (p_2^* \G^{\cdot}\otimes \R)[1])
\cong\\
&\Hom_{Z}(\bL i_1^* p_1^* \F^{\cdot}, \; \bL i_2^* (p_2^* \G^{\cdot}\otimes \R) )
\cong
\Hom_{Z}(q_1^* \F^{\cdot}, \; q_2^* \G^{\cdot}\otimes \L)).
\end{split}
\end{equation}
The last isomorphism is a consequence of the construction of $P_2$ and the line bundle $\R$ on $P_2.$
By (\ref{line_bundles}) the restriction of $\R$ on $Z$ coincides with the line bundle $\L.$
Finally, we have $q_i=\pr_i\cdot f$ for $i=1,2$ and we know that $\bR f_*\L\cong\E^{\cdot}$ by construction
from Proposition \ref{line}. This implies
\begin{multline}\label{4_sequence}
\Hom_{Z}(q_1^* \F^{\cdot},\; q_2^* \G^{\cdot}\otimes \L))\cong
\Hom_{X_1\times X_2}(\pr_1^* \F^{\cdot},\; \pr_2^* \G^{\cdot}\otimes \bR f_*\L))
\cong\\
\Hom_{X_1\times X_2}(\pr_1^* \F^{\cdot},\; \pr_2^* \G^{\cdot}\otimes \E^{\cdot})).
\end{multline}
The isomorphisms (\ref{3_sequence}) and (\ref{4_sequence}) finish the proof of the proposition.
\end{dok}
\bigskip
\noindent{\bf Proof of Theorem \ref{main}.}\;
Let us consider the DG functors
\[
\mPhi: \prfdg X_1\lto \prfdg V,\quad \text{and}\quad \mPsi: \prfdg X_2\lto \prfdg V.
\]
They induce a bimodule
$\mS$ determined by the following rule
\begin{equation}\label{s_bimodule}
\mS(B, A)\cong \dHom_{\prfdg V}(\mPhi A,\; \mPsi B),
\quad\text{where}
\quad
A\in \prfdg X_1,\; B\in\prfdg X_2.
\end{equation}
On the other hand, the calculations from Proposition \ref{main_prop} gives us that the bimodule $\mS$ is quasi-isomorphic
to a bimodules $\mS_{\E^{\cdot}}$ given by the rule
\[
\mS_{\E^{\cdot}}(B, A)\cong \dHom_{\prfdg(X\times Y)}(\mpr_1^* A,\; \mpr_2^* B\otimes \E^{\cdot}),
\quad\text{where}
\quad
A\in \prfdg X_1,\; B\in\prfdg X_2.
\]
Take the pretriangulated DG subcategory $\dC\subset\prfdg V$ that is generated by the DG subcategories
$\mPhi(\prfdg X_1)$ and $\mPsi(\prfdg X_2).$ Lemma \ref{lemma_orthogonal} implies that there is a semi-orthogonal decomposition
\[
\Ho(\dC)\cong \langle \Phi(\prf X_1),\; \Psi(\prf X_2)\rangle.
\]
Since $\Phi$ and $\Psi$ are fully faithful, Propositions \ref{gluing_semi-orthogonal} and \ref{gluing_quasifunctors}
give us that there are quasi-equivalences
\[
\dC\cong \mPhi(\prfdg X_1)\underset{\mS}{\oright}\mPsi(\prfdg X_2)
\cong
\prfdg X_1 \underset{\E^{\cdot}}{\oright}\prfdg X_2,
\]
where $\mS$ is the bimodule given by rule (\ref{s_bimodule}). By Theorem \ref{smooth_glue} the full DG subcategory $\dC\subset\prfdg V$
is smooth and proper as a gluing of smooth and proper DG categories via the perfect bimodule $\mS.$
Hence $\Ho(\dC)\cong \Ho(\prfdg X_1 \underset{\E^{\cdot}}{\oright}\prfdg X_2)$ is admissible in $\prf V.$
\hfill$\Box$
\begin{remark}
{\rm
It is useful to take in account that the category $\prf V$ from Theorem \ref{main} has a semi-orthogonal decomposition of the form
$
\prf V=\langle \T_1,\dots \T_k\rangle
$
such that each $\T_i$ is equivalent to one of the four categories, namely
$\prf\kk,\; \prf X_1,\; \prf X_2,$ and $\prf (X_1\times X_2).$
It follows from the construction of $V$ as a two-step blowup of a projective space $\PP^N$ along
$P_1$ and $\wt{P}_2.$ By definition, $P_1$ and $P_2$ are projective bundles over $X_1$ and $X_2$ respectively and
$\wt{P}_2$ is the blowup of $P_2$ along $Z,$ where $Z$ is a sequence of projective bundles over $X_1\times X_2.$
}
\end{remark}
\subsection{Gluing of geometric noncommutative schemes}
In this section we extend results from the previous section to the case of geometric noncommutative schemes.
Actually all these statements are direct consequences of corresponding assertions for smooth and projective schemes.
Let $X_i,\; i=1,\dots,n$ be smooth and projective schemes. Let $\dN_i,\; i=1,\dots, n$ be small pretriangulated DG categories.
Denote by $\N_i=\Ho(\dN_i)$ the homotopy triangulated categories.
Suppose that for all $i$ there are quasi-functors $\mF_i: \dN_i\to\prfdg X_i$
such that the induced exact functors $F_i: \N_i\to \prf X_i$ are fully faithful and have right and left adjoint functors.
This means that $\N_i$ are admissible subcategories in $\prf X_i$ with respect to the full embeddings given by $F_i.$
This conditions imply that $\dN_i$ are geometric noncommutative schemes and, moreover, the DG categories $\dN_i$
are smooth and proper by Theorem \ref{smooth_glue}.
\begin{theorem}\label{main2}
Let DG categories $\dN_i,\; i=1,\dots, n$ and smooth projective schemes $X_i,\; i=1,\dots, n$ be as above.
Let $\dC$ be a proper pretriangulated DG category with full embeddings
of DG categories $\dN_i\subset \dC$ such that $\C=\Ho(\dC)$ has a semi-orthogonal decomposition of the form
$
\C\cong\langle\N_1, \N_2,\dots, \N_n\rangle,
$
where $\N_i=\Ho(\dN_i).$ Then there are a smooth and projective scheme $V$ and a quasi-functor $\mF: \dC\to\prfdg V$ such that
the induced functor $F: \C\to \prf V$ is fully faithful and has right and left adjoint functors, i.e. $\dC$ is
a geometric noncommutative scheme.
\end{theorem}
\begin{proof}
The case $n=1$ is evident.
Consider the main case $n=2.$
By Proposition \ref{gluing_semi-orthogonal} the DG category $\dC$ is quasi-equivalent
to a gluing of $\dN_1$ and $\dN_2$ via
a $\dN_2\hy\dN_1$\!--bimodule $\mS$ that is defined by the rule
\[
\mS (B, A)=\dHom_{\dC}(A, B), \quad \text{with}\quad A\in\dN_1 \;\text{and}\; B\in\dN_2.
\]
Since $\dC$ is proper the bimodule $\mS$ is a DG functor from $\dN_2\otimes\dN_1^{\op}$ to $\prfdg\kk$
and by Theorem \ref{dg_saturated} it is perfect, because
$\dN_i$ are smooth and proper.
By Proposition \ref{gluing_semi-orthogonal} the DG category $\dC$ is quasi-equivalent to the gluing
$\dN_1\underset{\mS}{\oright}\dN_2.$ By Theorem \ref{smooth_glue} we obtain that $\dC$ is smooth.
Consider quasi-functors $\mF_i: \dN_i\to \prfdg X_i.$ We know that $\mF_i$ establish quasi-equivalences with
enhancements of admissible subcategories in $\prf X_i.$
By Theorem \ref{toen}
the DG category of perfect
DG modules over $\prfdg X_2^{\op}\otimes_{\kk}\prfdg X_1$ is equivalent to $\prfdg(X_1\times X_2).$
Thus the quasi-functors $\mF_i$ induce the extension and induction quasi-functors
\[
(\mF_1\otimes\mF_2)^*:\prfdg (\dN_2^{\op}\otimes\dN_1)\to \prfdg(X_1\times X_2),
\quad
(\mF_1\otimes\mF_2)_*:\prfdg(X_1\times X_2)\to \prfdg (\dN_2^{\op}\otimes\dN_1)
\]
and by Proposition \ref{geometric} the extension functor induces a fully faithful functor
between homotopy categories.
This implies that the bimodule $\mS$ is quasi-isomorphic to
a bimodule of the form $(\mF_1\otimes\mF_2)_* \E^{\cdot}$ for some perfect complex $\E^{\cdot}$ on
$X_1\times X_2.$
By Proposition \ref{gluing_quasifunctors} there is a quasi-functor
\[
\mF_1\underset{\phi}{\oright}\mF_2: \dN_1\underset{\mS}{\oright}\dN_2 \lto \prfdg X_1\underset{\E^{\cdot}}{\oright}\prfdg X_2,
\]
where $\phi$ is a quasi-isomorphism between $\mS$ and $(\mF_1\otimes\mF_2)_* \E^{\cdot}.$
Since the functors $F_i: \N_i\to \prf X_i$ are fully faithful, the induced functor
\[
F_1\underset{\phi}{\oright}F_2: \Ho(\dN_1\underset{\mS}{\oright}\dN_2) \lto \Ho(\prfdg X_1\underset{\E^{\cdot}}{\oright}\prfdg X_2)
\]
is fully faithful too by Proposition \ref{gluing_quasifunctors}.
By Theorem \ref{main} the DG category $\prfdg X_1 \underset{\E^{\cdot}}{\oright}\prfdg X_2$ is quasi-equivalent to a full DG subcategory
of $\prfdg V$ for some smooth and projective scheme $V.$ Consider the composition quasi-functor
\[
\mF: \dC\stackrel{\sim}{\lto} \dN_1\underset{\mS}{\oright}\dN_2 \stackrel{\mF_1 \underset{\phi}{\oright}\mF_2}{\llto} \prfdg X_1\underset{\E^{\cdot}}{\oright}\prfdg X_2\lto \prfdg V.
\]
It induces an exact functor $\C\to\prf V$ that is fully faithful as a composition of fully faithful functors.
The DG category $\dC$ is proper and it is smooth as a gluing of smooth DG categories via a perfect DG bimodule.
Smoothness implies regularity of $\dC$ (see Proposition \ref{smooth_regular}).
Moreover, the category $\C$ is idempotent complete, because $\N_i$ are idempotent complete as admissible subcategories
of $\prf X_i.$ Now, by Proposition \ref{admissible} regularity and properness of $\C$
give that the image of the fully faithful functor $F$ is an admissible
subcategory of $\prf V.$ Hence, $F$ admits right and left adjoint functors.
The general case $n$ is done by induction. Denote by $\N'_2\subset \C$ the left orthogonal to $\N_1$ and denote by
$\dN'_2\subset \dC$ the full DG subcategory consisting of all objects from $\N'_2.$
We have semi-orthogonal decompositions $\N'_2=\langle \N_2,\dots N_n\rangle$ and $\C=\langle \N_1,\N_2\rangle.$
By the induction hypothesis, there are a smooth and projective scheme $V'$ and a quasi-functor $\mF: \dN'_2\to\prfdg V'$ such that
the induced functor $F: \N'_2\to \prf V'$ is fully faithful and has right and left adjoint functors.
Now applying the proof for $n=2$ and the DG subcategories $\dN_1$ and $\dN'_2$ in $\dC,$ we obtain the statement of the theorem for
$\dC.$
\end{proof}
\begin{corollary}\label{emb} Let $Y$ be a proper scheme over a field of characteristic $0.$ Then there are
a smooth projective scheme $V$ and a quasi-functor $\mF: \prfdg Y\to \prfdg V$ such that
the induced functor $F:\prf Y\to \prf V$ is fully faithful.
\end{corollary}
\begin{dok}
It follows from the main theorem of \cite[Th.1.4]{KL} that $Y$ has a so-called categorical resolution.
By the construction of this categorical resolution there is a quasi-functor from $\mG: \prfdg Y\to\dD,$
where $\dD$ is a gluing of DG categories of perfect complexes on smooth proper schemes, and
the induced functor $G:\prf Y\to \D$ is fully faithful.
Now as in Remark \ref{Hironaka} over a field of characteristic $0$ for any smooth proper scheme
there is a sequence of blowups with smooth centers such that the resulting smooth scheme is projective.
Hence $\dD$ is a gluing of geometric noncommutative schemes. By Theorem \ref{main2} there are a smooth projective $V$
and a quasi-functor
from $\dD$ to $\prfdg V$ which is fully faithful on homotopy categories. The composition of these quasi-functors
gives us a quasi-functor $\mF: \prfdg Y\to \prfdg V$ that is also fully faithful on homotopy categories.
\end{dok}
\section{Application to finite algebras and exceptional collections}\label{applications}
\subsection{Finite dimensional algebras}
Let $\Lambda$ be a finite dimensional algebra over a base field $\kk.$
Denote by $\rd$ the (Jacobson) radical of $\Lambda.$ We know that $\rd^n=0$ for some $n.$
Define the index of nilpotency $i(\Lambda)$ of $\Lambda$ as the smallest integer $n$
such that $\rd^n=0.$
Let $S$ be the quotient algebra $\Lambda/\rd.$ It is semisimple and has only a finite number of simple non-isomorphic modules.
Denote by $\Md\Lambda$ and $\md\Lambda$ the abelian categories of all right modules and finite right modules over $\Lambda,$ respectively.
The following amazing result was proved by M.~Auslander.
\begin{theorem}\cite{Au}
Let $\Lambda$ be a finite dimensional algebra of index $n.$ Then the finite dimensional algebra
$\Gamma=\End(\bigoplus_{p=1}^{n} \Lambda/\rd^p)$ has the following properties:
\begin{enumerate}
\item[1)] $\gldim\Gamma\le n+1;$
\item[2)] there is a finite projective $\Gamma$\!--module $P$ such that $\End_{\Gamma}(P)\cong \Lambda.$
\end{enumerate}
\end{theorem}
Let us consider the bounded derived category of finite $\Gamma$\!--modules $\D^b(\md\Gamma).$
Since $\Gamma$ has finite global dimension, $\D^b(\md\Gamma)$ is equivalent to the category
of perfect complexes $\prf\Gamma.$
Some variants of the following theorem are known (see. e.g. \cite{KL}).
\begin{theorem}\label{algebra_exceptional}
Let $\Lambda$ be a finite dimensional algebra of index $n$ and let
$\Gamma=\End(\bigoplus_{p=1}^{n} \Lambda/\rd^p).$
The derived category $\prf \Gamma\cong \D^b(\md\Gamma)$ has a semi-orthogonal decomposition of the form
\[
\prf\Gamma=\langle\N_1,\dots,\N_n\rangle
\]
such that each subcategory $\N_i$ is semisimple, i.e. $\N_i\cong \langle K_i\rangle,$
where $K_i$ is semi-exceptional and for all $i$ the algebras $\End_{\Gamma}(K_i)$ are quotients of the semisimple algebra
$S=\Lambda/\rd.$
\end{theorem}
\begin{dok}
Denote by $M$ the $\Lambda$\!--module $\bigoplus_{p=1}^{n} \Lambda/\rd^p$
and by $M_s$ the $\Lambda$\!--modules $\Lambda/\rd^s,\; s=1,\dots,n.$
Consider the functor $\Hom_{\Lambda}(M, -)$ from the abelian category
$\md\Lambda$ to the abelian category $\md\Gamma.$
Denote by $P_s$ the $\Gamma$\!--modules $\Hom_{\Lambda}(M, M_s).$
They are projective $\Gamma$\!--modules and $\Gamma=\bigoplus_{s=1}^n P_s.$
By Gabriel-Popescu theorem, since $M$ is a generator for $\Md\Lambda$ the functor $\Hom_{\Lambda}(M, -)$ from
$\Md\Lambda$ to $\Md\Gamma$ is fully faithful. Thus, there are isomorphisms
\[
\Hom_{\Gamma}(P_i, P_j)\cong \Hom_{\Lambda}(M_i, M_j)=\Hom_{\Lambda}(\Lambda/\rd^i, \Lambda/\rd^j)
\quad\text{for all}
\quad
1\le i, j\le n.
\]
Moreover, we have $\Hom_{\Lambda}(\Lambda/\rd^i, \Lambda/\rd^j)\cong \Lambda/\rd^j$ when $i\ge j.$
The canonical quotient morphisms $\Lambda/\rd^i\to \Lambda/\rd^j,$ when $i\ge j,$ induce morphisms
$\phi_{i,j}: P_i\to P_j.$
Let us consider $\phi_{i, i-1}$ and the induced exact triangles
\begin{equation}\label{semi_exceptional}
\xymatrix{
K_i\ar[r] & P_i \ar[r]^{\phi_{i, i-1}} & P_{i-1}\ar[r] & K_i[1], & i=2,\dots n
}
\end{equation}
in $\prf \Gamma.$ These triangles define objects $K_i$ for $i=2,\dots, n.$
We also set $K_1=P_1.$
Now, since $P_i$ are projective and $\Hom_{\Gamma}(P_i, P_j)\cong \Lambda/\rd^{j}$ when $i\ge j,$ we have vanishing
\begin{equation}\label{vanish}
\Hom_{\Gamma}(K_i, P_j[l])=0,\quad\text{for all}\quad l \quad\text{when}\quad i>j.
\end{equation}
Using definition (\ref{semi_exceptional}) of $K_i$ we immediately obtain semi-orthogonality conditions
\[
\Hom_{\Gamma}(K_i, K_j[l])=0, \quad\text{for all}\quad l \quad\text{when}\quad i>j.
\]
Finally, we have to compute $\bR\Hom_{\Gamma}(K_i, K_i)$ for all $i.$ Exact triangles (\ref{semi_exceptional}) give us
that the vector spaces $\Hom_{\Gamma}(K_i, K_i[l])$ are cohomology of the complexes
\begin{equation}\label{cohomology}
\Hom_{\Gamma}(P_{i-1}, P_i)\lto \Hom_{\Gamma}(P_{i}, P_i)\bigoplus \Hom_{\Gamma}(P_{i-1}, P_{i-1})
\lto \Hom_{\Gamma}(P_{i}, P_{i-1})
\end{equation}
that coincide with the complexes
\[
\Hom(\Lambda/\rd^{i-1}, \Lambda/\rd^i)\lto \Hom(\Lambda/\rd^{i}, \Lambda/\rd^i)\bigoplus \Hom(\Lambda/\rd^{i-1}, \Lambda/\rd^{i-1})
\lto \Hom(\Lambda/\rd^{i}, \Lambda/\rd^{i-1}).
\]
The morphism $\Hom(\Lambda/\rd^{i-1}, \Lambda/\rd^{i-1})
\to \Hom(\Lambda/\rd^{i}, \Lambda/\rd^{i-1})$ is an isomorphism and the morphism
$\Hom(\Lambda/\rd^{i-1}, \Lambda/\rd^i)\lto \Hom(\Lambda/\rd^{i}, \Lambda/\rd^i)$ is an injection.
This implies that the complex (\ref{cohomology}) has only zero cohomology.
Therefore,
\[
\Hom_{\Gamma}(K_i, K_i[l])=0, \quad\text{for all}\quad l\ne 0 \quad \text{and all}\quad i=1,\dots, n.
\]
Denote by $S_i$ the algebra of endomorphisms $\End_{\Gamma}(K_i),$ where $i=1,\dots, n.$
We know that $S_1=\End_{\Gamma}{P_1}\cong\End_{\Lambda}(\Lambda/\rd)=S$ is semisimple.
Let $a\in S_i$ be an element. It can be presented by a pair of morphisms $(a_{i}, a_{i-1})$ included in commutative diagram
\[
\begin{CD}
P_i & @>\phi_{i, i-1}>> & P_{i-1}\\
@Va_iVV && @VV a_{i-1}V\\
P_i & @>\phi_{i, i-1}>> & P_{i-1}
\end{CD}
\]
The vanishing conditions (\ref{vanish}) implies that the morphism $a_{i-1}$ is uniquely determined by $a_i.$
Thus, the element $a_i\in\Hom_{\Gamma}(P_i, P_i)\cong\Lambda/\rd^i$
induces an endomorphism of $K_i$ and we see that there is a homomorphism of algebras
$\Lambda/\rd^{i}\to \End_{\Gamma}(K_i)$ that is surjective.
If now $a_i\in\End(P_i)=\Lambda/\rd^i$ belongs to $\rd,$ then as an endomorphism
of $\Lambda/\rd^i$ it sends $\rd^{i-1}$ to zero.
This implies that it is induced by a morphism of $\Lambda/\rd^{i-1}$ to $\Lambda/\rd^{i}.$
Thus, we obtain that the pair of morphisms $(a_i, a_{i-1})$ is induced by a morphism from $P_{i-1}$ to $P_i$
if $a_i\in \rd.$
This means that the algebra of endomorphisms $\End_{\Gamma}(K_i)$ is a quotient of the semisimple algebra $S=\Lambda/\rd.$
Therefore, the algebras $S_i=\End_{\Gamma}(K_i)$ are semisimple for all $i=1,\dots, n$ too.
As $P_i,\;i=1,\dots,n$ generate the category $\prf\Gamma$ the objects $K_i,\;i=1,\dots,n$ generate
$\prf\Gamma$ as well, and we obtain a semi-orthogonal decomposition
\[
\prf\Gamma=\langle\langle K_1\rangle,\cdots,\langle K_n\rangle\rangle,
\]
where all $K_i$ are semi-exceptional and $S_i=\End_{\Gamma}(K_i)$ are quotients of the algebra $S=\Lambda/\rd.$
\end{dok}
Consider now the $\Gamma$\!--module $P_n=\Hom_{\Lambda}(M, \Lambda),$ where
$M=\bigoplus_{p=1}^{n} \Lambda/\rd^p.$ It is projective and $\End_{\Gamma}(P_n)\cong\Lambda.$ This object gives us two functors
\[
(-)\otimes_{\Lambda} P_n: \prf\Lambda\to\prf\Gamma\quad\text{and}\quad
\Hom_{\Gamma}(P_n, -):\D^b(\md\Gamma)\to\D^b(\md\Lambda)
\]
The first functor is fully faithful while the second functor is a quotient. Since $\Gamma$ has finite global dimension
there is an equivalence $\prf\Gamma\cong\D^b(\md\Gamma).$
Now if $\Lambda$ also has finite global dimension, then the second functor is right adjoint to the first one and the
category $\prf\Lambda$ is right admissible in $\prf\Gamma$ with respect to the full embedding $(-)\otimes_{\Lambda} P_n.$
Recall that a semisimple algebra $S$ over a field $\kk$ is called separable over $\kk$ if it is a projective
$S^{\op}\otimes_{\kk} S$\!--module. It is well-known that a semisimple algebra $S$ is separable if
it is a direct sum of simple algebras, the centers of which are separable extensions of $\kk.$
\begin{theorem}\label{algebra}
Let $\Lambda$ be a finite dimensional algebra over $\kk.$ Assume that $S=\Lambda/\rd$ is a separable $\kk$\!--algebra.
Then there are a smooth projective scheme
$V$ and a perfect complex $\E^{\cdot}$ such that $\End(\E^{\cdot})\cong\Lambda$ and $\Hom(\E^{\cdot}, \E^{\cdot}[l])=0$
for all $l\ne 0.$
\end{theorem}
\begin{dok}
As above, let $\Gamma=\End(\bigoplus_{p=1}^{n} \Lambda/\rd^p).$
Consider the DG category $\prfdg\Gamma.$ By Theorem
\ref{algebra_exceptional} the triangulated category $\prf\Gamma$ has
a semi-exceptional collection $(K_1,\dots, K_n)$ and
\[
\prf\Gamma=\langle\N_1,\dots,\N_n\rangle
\]
where $\N_i=\langle K_i\rangle$ is semisimple. Thus, for any $i$
the object $K_i$ is a direct sum of the form $\bigoplus_{j=1}^{m_i} K_{ij},$ where $K_{ij}$ are completely orthogonal to each other
for fixed $i$ and different $j.$
Moreover, each $\End_{\Gamma}(K_{ij})$ is a simple algebra, i.e it is a matrix algebra over a division $\kk$\!--algebra $D_{ij}.$
By assumption $S$ is separable. Hence, all $\End_{\Gamma}(K_{ij})$ are separable as quotients of $S.$
Thus we obtain that the centers $\kk_{ij}$ of all $D_{ij}$ are separable extensions of $\kk.$
Now as in Example \ref{Severi-Brauer} we can consider a Severi-Brauer variety $SB(D_{ij})$ that is a smooth projective scheme over $\kk_{ij}$
and over $\kk$ too, because $\kk_{ij}\supset\kk$ is a finite separable extension. It was mentioned in Example \ref{Severi-Brauer}
that there is a vector bundle $E_{ij}$ on $SB(D_{ij})$ such that it is w-exceptional and $\End(E_{ij})\cong D_{ij}.$
This implies that each DG category $\prfdg D_{ij}$ is a full DG subcategory of the DG category $\prfdg SB(D_{ij}),$
and $SB(D_{ij})$ is smooth and projective over $\kk.$
All categories $\N_i$ have complete orthogonal decompositions of the form
$\N_i=\N_{i1}\oplus\cdots\oplus \N_{i m_i},$ where $\N_{ij}=\langle K_{ij}\rangle$ are equivalent to
$\prf D_{ij}.$ These decompositions induce a semi-orthogonal decomposition for $\prf\Gamma$ of the form
\[
\prf\Gamma=\langle\N_{11}, \N_{12},\dots,\N_{1 m_1},\N_{21},\dots, \N_{n m_n}\rangle.
\]
Applying Theorem \ref{main2} we obtain that there are a smooth projective scheme $V$ and a quasi-functor
from $\mF: \prfdg\Gamma\to \prfdg V$ such that the homotopy functor
$F:\prf\Gamma\to\prf V$ is fully faithful and establishes an equivalence with an admissible subcategory
in $\prf V.$ Denote by $\E^{\cdot}$ the perfect complex $F(P_n),$ where $P_n=\Hom_{\Lambda}(M, \Lambda)$
is a projective $\Gamma$\!--module. Since $F$ is fully faithful we have isomorphisms
\[
\Hom_V(\E^{\cdot}, \E^{\cdot}[l])\cong\Hom_{\Gamma}(P_n, P_n[l]).
\]
When $l\ne 0$ it is $0,$ and it is isomorphic to the algebra $\Lambda$ for $l=0.$
\end{dok}
\begin{corollary}\label{algebra_inclusion}
Let $\Lambda$ be a finite dimensional algebra over $\kk$ for which $S=\Lambda/\rd$ is separable $\kk$\!--algebra.
Then there is a smooth projective scheme $V$ such that
the DG category $\prfdg\Lambda$ is quasi-equivalent to a full DG subcategory of $\prfdg V.$ Moreover, if $\Lambda$ has finite global dimension, then
$\prf\Lambda$ is admissible in $\prf V.$
\end{corollary}
\begin{dok}
By Theorem \ref{algebra} there is a smooth projective scheme $V$ and
a perfect complex $\E^{\cdot}$ such that
$\End(\E^{\cdot})\cong\Lambda$ and $\Hom(\E^{\cdot},
\E^{\cdot}[l])=0$ for all $l\ne 0.$ Hence, the DG algebra
$\dHom_{\prfdg V}(\E^{\cdot}, \E^{\cdot})$ is quasi-isomorphic to
the algebra $\Lambda.$ Thus, by Proposition \ref{Keller2} there is a
quasi-functor $\mF:\prfdg\Lambda\to \prfdg V$ induced by the
embedding of $\dHom_{\prfdg V}(\E^{\cdot}, \E^{\cdot})$ into $\prfdg
V$ such that the homotopy functor $F: \prf\Lambda\to \prf V$ is
fully faithful. If $\Lambda$ has finite global dimension, then the
category $\prf\Lambda$ is regular and proper. Hence it is admissible
in $\prf V$ by Proposition \ref{admissible}.
\end{dok}
\begin{remark}
{\rm
Note that over a perfect field all semisimple algebras are separable.
Thus, if $\kk$ is perfect, then results of this section apply to all finite dimensional algebras.
}
\end{remark}
\begin{remark}
{\rm Theorem \ref{algebra} tells us, in particular, that for any
finite dimensional algebra $\Lambda$ of finite global dimension the
category $\prf\Lambda$ can be embedded to a triangulated category
with a full semi-exceptional collection (actually, with
w-exceptional collection).
On the other hand, as was pointed out to me by Theo Raedschelders
there are finite dimensional algebras of finite global dimension for which
the category of perfect complexes does not have a full exceptional collection
(actually, it does not have any exceptional object).
Such examples were discussed by Dieter Happel in \cite{Hap}.
This gives a counterexample to Jordan-H\"older property for triangulated categories of perfect complexes on smooth projective schemes.
More precisely, there are admissible subcategories $\T$ in
$\prf X$ on smooth projective variety $X$ such that $\prf X$ has a full exceptional
collection but $\T$ does not have an exceptional object at all.
Another example coming from a quiver was constructed by Alexei Bondal
and were discussed by Alexander Kuznetsov in \cite{Kuz} from geometric point of view.
}
\end{remark}
\subsection{Exceptional collections}\label{exceptional_coll}
In this section we describe a more useful procedure of constructing a scheme that admits a full exceptional collection and contains
as a subcollection an exceptional collection given in advance.
Let $\dA$ be a small smooth and proper pretriangulated DG category over a field $\kk$ such that the homotopy category
$\Ho(\dA)$ has a semi-orthogonal decomposition of the form
\[
\Ho(\dA)=\langle \N,\; \langle E\rangle\rangle,
\]
where $\N$ is a full admissible subcategory and $E$ is an exceptional object, i.e.
$\langle E\rangle\cong \prf\kk.$
Assume that the enhancement of $\N$ induced from $\dA$ is
quasi-equivalent to a full DG subcategory $\dN\subset \prfdg X$
for a smooth and projective irreducible scheme $X$ such that $H^i(X,
\cO_X)=0$ for all $i>0,$ i.e. the structure sheaf $\cO_X$ is
exceptional.
\begin{remark}
{\rm
The assumption $H^i(X, \cO_X)=0$ for all $i>0$ is not restrictive. Indeed, for any smooth and projective scheme
we can consider a closed immersion into a projective space $\PP^N$ for some large $N.$
Take the blowup $Z$ of $\PP^N$ along $X.$ Then $\prfdg X$ is quasi-equivalent to a full DG subcategory
in $\prfdg Z$ and $H^i(Z, \cO_{Z})=0$ for all $i>0.$ Now we can take $Z$ instead of $X.$
}
\end{remark}
We have that $\N\cong\Ho(\dN)$ is an admissible subcategory in $\prf X.$
By Propositions \ref{gluing_semi-orthogonal} and \ref{gluing_quasifunctors} the DG category $\dA$ is quasi-equivalent to a gluing of $\dN$ and $\prfdg\kk$
via some $\dN$\!--module $\mS.$
Since $\prfdg X$ and $\dN$ are saturated by Theorem \ref{dg_saturated}, the DG module $\mS$ can be represented by a perfect
complex $\cS^{\cdot}$ on $X,$ i.e the DG $(\prfdg X)$\!--module $\dHom(-,\; \cS^{\cdot})$ after restriction of $\dN$
is quasi-isomorphic to the DG $\dN$\!--module $\mS.$
Shifting the complex $\cS^{\cdot}$ by $[m]$ for an appropriate $m\in\ZZ$ we can suppose that
\[
\cS^{\cdot}\cong\{\cS^0\to \cS^1\to\cdots\to \cS^k\},
\]
where all $\cS^i$ are vector bundles on $X.$
By Proposition \ref{line} there is a smooth morphism $f: Z\to X$ and a line bundle $\L$
on $Z$ such that $\bR f_* \L\cong \cS^{\cdot}$ and $\bR f_*\L^{-1}=0.$
Moreover, the morphism $f$ is a sequence of projective bundles. Hence $\bR f_*\cO_Z\cong \cO_X$
and the inverse image functor $f^*:\prf X \to\prf Z$ is fully faithful.
Since
$\bR f_*\L^{-1}=0$ we have
\[
\Hom_Z(\L, \; f^* A)\cong \Hom_Z(\cO_Z, \; f^* A\otimes \L^{-1})\cong \Hom_X(\cO_X, \; A\otimes\bR f_*\L^{-1})=0
\]
for any $A\in\prf X.$ Therefore $f^*(\prf X)$ is in the right orthogonal $\langle \L\rangle ^{\perp}.$
Since $\cO_X$ is exceptional the structure sheaf
$\cO_Z$ and any line bundle on $Z$ are also exceptional. Therefore, the admissible subcategory
$\T\subset \prf Z$ which is generated by $f^*(\N)$ and $\L$ has a semi-orthogonal decomposition of the form
$
\T\cong\langle \N, \prf\kk\rangle,
$
Denote by $\dT$ enhancement of $\T$ the induced from $\prfdg Z.$ By Propositions \ref{gluing_semi-orthogonal} and \ref{gluing_quasifunctors}
the DG category $\dT$ is quasi-equivalent to $\dA$ because both of them
are quasi-equivalent to the gluing $\dN\underset{\mS}{\oright}\prfdg\kk$ via $\mS.$
The procedure described above can be considered as an induction step in the proof of the following theorem
while the base case is the point $\Spec\kk.$ Thus, we obtain.
\begin{theorem}\label{exc_col}
Let $\dA$ be a small DG category over $\kk$ such that the homotopy category
$\Ho(\dA)$ has a full exceptional collection
\[
\Ho(\dA)=\langle E_1,\dots, E_n\rangle.
\]
Then there are a smooth projective scheme $X$ and an exceptional collection of line bundles
$\sigma=(\L_1,\dots, \L_n)$ on $X$ such that the DG subcategory of $\prfdg X,$
generated by $\sigma,$ is quasi-equivalent to $\dA.$
Moreover, $X$ is a sequence of projective bundles and has a full exceptional collection.
\end{theorem}
\begin{remark}
{\rm
The scheme $X$ has a full exceptional collection as a sequence of projective bundles
(see \cite{Blow}). Furthermore, it follows from construction that a full exceptional collection on $X$ can be chosen in a way that
it contains the collection $\sigma=(\L_1,\dots, \L_n)$ as a subcollection.
}
\end{remark}
\subsection{Noncommutative projective planes}
In this section we consider a particular case of noncommutative projective planes, in sense of noncommutative deformations of the
usual projective plane, and
present explicit embeddings of categories of perfect complexes on them to categories of perfect complexes on smooth projective commutative schemes.
Noncommutative deformations of the projective plane have been described in \cite{ATV}.
The category $\prf \PP^2$ has a full exceptional collection $(\cO, \cO(1), \cO(2)).$ Note also that mirror symmetry relations for
noncommutative planes is described in \cite{AKO}.
Any deformation of the category $\prf \PP^2$ is a category with three ordered objects
$F_0, F_1, F_2$
and with three-dimensional spaces of homomorphisms from $F_i$ to $F_j$ when $j-i=1$ and a six-dimensional
vector space as Hom from $F_0$ to $F_2.$
Any such category is determined by
the composition tensor $\mu: V\otimes U\to W,$ where $\dim V=\dim U=3$ and
$\dim W=6.$ This map should be surjective. Denote by $T$ the kernel of $\mu$ and by $\nu: T\to V\otimes U.$
We will consider only the nondegenerate (geometric)
case, where the restrictions $\nu_{u^*}: T\to V$ and
$\nu_{v*}: T\to U$ have rank at least two for all nonzero elements
$u^*\in U^{\vee}$ and $v^*\in V^{\vee}.$
The equations $\det \nu_{u^*}=0$ and $\det\nu_{v^*}=0$ define closed subschemes
$\Gamma_U\subset\PP(U^{\vee})$ and $\Gamma_V\subset\PP(V^{\vee}).$
Namely, up to projectivization the set of points of $\Gamma_U$
(resp.\ $\Gamma_V$) consists of all
$u^*\in U^{\vee}$ (resp.\ $v^*\in V^{\vee}$) for which the rank of
$\nu_{u^*}$ (resp.\ $\nu_{v*}$) is equal to $2.$
It is easy to see that the correspondence which associates the kernel of the map
$\nu_{v^*}^{\vee}: U^{\vee}\to T^{\vee}$ to
a vector $v^*\in V^{\vee}$ defines
an isomorphism between $\Gamma_V$ and $\Gamma_U$.
Moreover, under these circumstances $\Gamma_V$ is either the entire
projective plane $\PP(V^{\vee})$ or a cubic
in $\PP(V^{\vee}).$ If $\Gamma_V=\PP(V^{\vee}),$
then $\mu$ is isomorphic to the tensor $V\otimes V\to S^2 V,$
i.e. we get the usual projective plane $\PP^2.$
Thus, the non-trivial case is the situation, where $\Gamma_V$ is a cubic, which we will denote by $E$.
This curve comes equipped with two embeddings into the
projective planes $\PP(U^{\vee})$ and $\PP(V^{\vee})$ respectively; by restriction of
$\cO(1)$ these embeddings
determine two line bundles $\L_1$ and $\L_2$ of degree $3$ on $E$,
and it can be checked that $\L_1\ne \L_2.$ This construction has an inverse:
\begin{construction}\label{constr:mu}
{\rm
The tensor $\mu$ can be reconstructed from the triple
$(E, \L_1, \L_2).$
Namely, the spaces $U, V$ are isomorphic to
$H^0(E, \L_1)$ and $H^0(E, \L_2)$ respectively, and the tensor
$\mu: V\otimes U\to W$ is nothing but the canonical map
$
H^0(E, \L_2) \otimes H^0(E, \L_1)\lto H^0(E, \L_2\otimes\L_1).
$
}
\end{construction}
\begin{remark}\label{rmk:comm}
{\rm
Note that we can also consider a triple $(E, \L_1, \L_2)$ such that
$\L_1\cong \L_2.$ Then the procedure described above produces
a tensor with $\Gamma_V\cong \PP(V^{\vee}),$ which defines the usual
commutative projective plane. In this case the tensor $\mu$ does not depend on the curve $E.$
The details of these constructions and statements can be found in \cite{ATV}.
}
\end{remark}
Now let us see what our construction gives in the case of noncommutative planes.
In some sense we repeat the construction from the proof of Proposition \ref{line} in this case.
The subcategory generated by $(F_0, F_1)$ is a subcategory
of $(\cO, \cO(1))$ on the usual $\PP^2=\PP(U^{\vee}).$ Now we should glue to this category
the object $F_2.$ The projection of $F_2$ on the subcategory generated by
$(F_0, F_1)$ can be represented by the complex
\begin{equation}
\label{resol}
T\otimes\cO\stackrel{\nu}{\lto} V\otimes\cO(1)
\end{equation}
on $\PP^2.$ This complex is a resolution of the cokernel of this map. It is isomorphic
to the sheaf $\cO_{E}(\L_1\otimes \L_2),$
where $E$ is a curve of degree 3 on $\PP^2,$ $\L_1$ is the restriction of $\cO(1)$
on $E,$ and $\L_2$ is another line bundle of degree 3 on $E.$
At first, we take the projectivization of $V\otimes\cO(1).$ We obtain $\PP(U^{\vee})\times\PP(V^{\vee})$
and the line bundle $\cO(1,1)$ on it.
The direct image of this bundle on the first component is isomorphic to $V\otimes\cO(1)$ on $\PP(U^{\vee}).$
After that we consider $Y=\PP(U^{\vee})\times\PP(V^{\vee})\times \PP^1$ and the line bundle
$\cO(1, 1, -2)$ on it. The morphism $\nu$ induces an element
$\epsilon\in \Ext^1_Y(T\otimes\cO_Y,\; \cO(1,1,-2)).$
Now we take a vector bundle $\F$ on $Y$ that is extension
\begin{equation}\label{extension}
0\lto \cO(1,1,-2)\lto\F\lto T\otimes\cO_Y\lto 0.
\end{equation}
Finally, we take $Z=\PP(\F)$ and the line bundle $\L=\cO_Z(1)$.
The direct image of $\L$ with respect to the projection on
$\PP(U^{\vee})$ is isomorphic to the complex (\ref{resol}). Now, if we consider
three line bundles $\cO_Z,$ the pull back of $\cO(1)$ from $\PP(U^{\vee}),$
and $\L=\cO_Z(1)$ on $Z,$
then it is an exceptional collection on $Z$
and the corresponding subcategory in $\prf Z,$ generated by them, is equivalent
to the category of perfect complexes on the noncommutative projective plane.
Different noncommutative projective planes correspond to the different vector bundles
$\F$ that depend on the element $\epsilon.$
\begin{proposition}
For any noncommutative deformation of the projective plane $\PP^2_{\mu}$ the DG category
$\prfdg \PP^2_{\mu}$ is quasi-equivalent to a full DG subcategory
of $\prfdg Z,$ where $Z$ is the projectivization of a 4-dimensional vector bundle $\F,$
defined as extension (\ref{extension}), over $Y=\PP^2\times\PP^2\times \PP^1.$
\end{proposition}
New results on realizations of the categories $\prf \PP^2_{\mu}$ on noncommutative projective planes
$\PP^2_{\mu}$ as admissible subcategories of the categories on smooth projective varieties can be found in \cite{O3}.
| 139,546
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TITLE: If $E$ is a connected space, then so is $\overline{E}$.
QUESTION [2 upvotes]: I have to prove that if $E$ is a connected space, then so is $\overline{E}$ (the closure of $E$) a connected space. I tried to prove the contrapositive. So suppose that $\overline{E}$ is not connected in a metric space $X$. This implies there are $A\subset X$ and $B\subset X$ such that $A\cap B=\emptyset$, $A\cap \overline{E}\neq \emptyset$, $B\cap \overline{E}\neq \emptyset$ and $\overline{E}\subset A\cup B$. I will try to prove that this $A$ and $B$ will also work for $E$. Suppose that $A\cap E=\emptyset$ and $B\cap E=\emptyset$, this implies that $A\cup B$ is a subset of the limit points of $E$, because $A\cap \overline{E}\neq \emptyset$ and $B\cap \overline{E}\neq \emptyset$. This contradicts the fact that $\overline{E}\subset A\cup B$, so we have to conclude that $A\cap E\neq \emptyset\neq B\cap E$. Since $E\subset\overline{E}\subset A\cup B$, we have that $E$ is also not connected.
My question is whether my proof is correct, because I have a little bit doubt. If it's not correct, what is the best I can do? Thanks in advance!
Edit1: The definition we use for a connected space $E$ in a metric space $X$ is that if $E$ is connected, then there are no open $A,B\subset X$ such that $A\cap B=\emptyset$, $A\cap E\neq \emptyset\neq B\cap E$ and $E\subset A\cup B$.
Edit2: $A$ and $B$ must be open.
REPLY [3 votes]: A subset $E⊆X$ is connected iff the only continuous functions $f:E→\{0,1\}$ are the constant ones. A continuous function $g:X→Y$ where $Y$ is Hausdorff is uniquely determined by a dense subset of $X$, and $\{0,1\}$ with the discrete metric is evidently Hausdorff. $E$ is dense in $\overline E$ and continuous functions $f:E→\{0,1\}$ are constant by assumption.
Another proof my proceed by contraposition. Suppose that $\overline E$ is disconencted by $U,V$. Then $E$ being dense has nonempty intersection with both $U,V$, so it is disconnected by $U'=E\cap U\;,\; V'=V\cap E$, for $U',V'$ are disjoint nonempty and relatively open in $E$, and $E=V'\cup U'$ by $\bar E=V\cup U$.
| 85,871
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<<
Our team is what sets us apart. Our experienced team provides us the advantage in delivering the best solution to our Client.
We challenge. We innovate. We collaborate.
We believe that challenging each other’s ideas is the best way to reach our full potential. We stay out of the ordinary to provide an unparalleled yet simplified procurement solution. Our team welcomes collaboration, communication and exchanging ideas.
Meet the SimPPLY leadership & management team
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Partner
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| 95,801
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The Rainey Store was built in the 1880s in Ona, FL, one of Hardee County’s oldest settlements. In 1911 a railroad was built through Hardee County, and Ona became a major posting, shipping, and receiving point for western and central Florida. A general store and post office, the Rainey store thrived during Ona’s boom, and the Rainey family raised four children in the living space on the second story. When US-17 was built through Wauchula in the 1930s, however, Ona faded into rural obscurity. The building served as a general store until the 1960s; by the 1980s it was hollow and empty, but still standing. It was donated to Cracker Country in 1988 by Reid and Gussie Rainey, and lives on as Cracker Country’s General Store.
Built 1800s, moved to Cracker Country in 1987, first Florida State Fair 1988
| 269,193
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TITLE: Why does the electric field and not the magnetic field remains in the same direction after reflection from a medium?
QUESTION [1 upvotes]: I was reading up about reflection and transmission when an electromagnetic wave is normally incident on a surface.
I came across this figure :
My question is why is the direction of electric field same while the magnetic field's direction is inverted , can't it be vice versa?
REPLY [0 votes]: Yes it can. It just depends on what convention or meaning you adopt for the reflection coefficient.
Assuming $n_2 > n_1$, then the arrangement as drawn results in a negative reflection coefficient. i.e. The reflected electric field is actually in the opposite direction to that drawn. However, the E-field and B-field directions are not independent; their vector product should be in the (fixed) propagation direction for the reflected ray. So if the E-field direction reverses, then so does the B-field.
If $n_2 > n_1$ and you start with a diagram that has both the reflected E-field and B-field reversed to that shown in your picture (both must be reversed to be consistent with the wave propagation direction), then all that happens is you get a positive reflection coefficient.
| 96,695
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TITLE: compositum of galois extensions
QUESTION [3 upvotes]: Let $E/K$ and $F/K$ be Galois extensions and $E\cap F=K$. Show that $EF/K$ is Galois and $Gal(EF/K) \cong Gal(E/K)\times Gal(F/K)$. Here $EF$ denotes the smallest field containing both $E$ and $F$.
I know how to prove that $EF/F$ is Galois. And we can define a map from $Gal(EF/K)$ to $Gal(E/K)\times Gal(F/K)$ by $\sigma \rightarrow (\sigma|_{E}, \sigma|_{F})$.
In order to show that this is an injection, we have to show that any automorphism in $Gal(EF/F)$ fixing $E$ and $F$ must be identity. For surjectivity, we have to show that for any $\sigma|_{1}\in Gal(E/K)$ and $\sigma|_{2}\in Gal(F/K)$, we can find a $\sigma$ s.t. $\sigma|_{E}=\sigma|_{1}$ and $\sigma|_{F}=\sigma_{2}$.I'm stuck in these two parts.
REPLY [4 votes]: With the basic Galois correspondence in place this goes according to the following plan. Leaving the details to you.
Let $N/K$ be a normal closure of $EF/K$. We then know that $N/K$ is Galois. This is immediate in char zero, and follows from separability of $E/K$ and $F/K$ otherwise (that is a bit non-trivial, but may also be a non-issue). Let $G=Gal(N/K)$, $H_1=Gal(N/E)\le G$, $H_2=Gal(N/F)\le G$.
Because $EF$ is the smallest subfield of $N$ containing both $E$ and $F$ we have $Gal(N/EF)=H_1\cap H_2$.
Because $E/K$ and $F/K$ are Galois, $H_1\unlhd G$ and $H_2\unlhd G$. Therefore $H_1\cap H_2\unlhd G$. By Galois correspondence $EF/K$ is Galois. Therefore $N=EF$ and $H_1\cap H_2=\{1_G\}$.
Because $H_1\unlhd G$, $H_2\unlhd G$ and $H_1\cap H_2=\{1_G\}$ a standard exercise in group theory shows that $h_1h_2=h_2h_1$ for all $h_1\in H_1$ and $h_2\in H_2$.
The subgroup $H_1H_2$ is the smallest subgroup of $G$ containing both $H_1$ and $H_2$. Therefore it corresponds with the intermediate field $E\cap F$, and $Gal(N/(E\cap F))=H_1H_2$.
Because $E\cap F=K$ we have $Gal(N/(E\cap F))=G$.
Therefore $G$ is the (inner) direct product $H_1\times H_2$.
All the above implies that every automorphism $\sigma\in Gal(EF/F)$ can be writtern uniquely in the form $\sigma=\tau_1\tau_2=\tau_2\tau_1$, where
$\tau_1\in Gal(EF/E)$ and $\tau_2\in Gal(EF/F)$. The rest should be easy.
Recall that $Gal(E/K)\simeq Gal(EF/K)/Gal(EF/E)$ and similarly
$Gal(F/K)\simeq Gal(EF/K)/Gal(EF/F)$.
| 155,924
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From: Ethical Alliance Daily News - 8 Dec 2016
China: China to help 1MDB settle multi-billion-dollar legal dispute with Abu Dhabi: Financial TimesChina: China to help 1MDB settle multi-billion-dollar legal dispute with Abu Dhabi: Financial Times
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To celebrate International Anti-Corruption Day on December 9, ethiXbase looks back at some of the key anti-corruption developments of 2016. Read this post and access our 2016 SlideShare for an overview of what has been a tumultuous year in anti-corruption developments.
Romania: Teva investigating allegations by tipster of bribery in Romania
Dec 08, 2016 07:00 pm
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Teva Pharmaceutical Industries is investigating claims by an anonymous tipster that the company bribed state healthcare workers in Romania to get them to prescribe its medication, a Teva spokeswoman said. The internal probe, which has not previously been reported, comes...
Philippines: Gaming tycoon Jack Lam flees Philippines, has casinos shut down amid sabotage and bribery allegations
Dec 08, 2016 06:30 pm
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Switzerland: Switzerland makes progress in money laundering fight
Dec 08, 2016 06:00 pm
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Switzerland has boosted anti-money laundering measures over the last decade but could still do more to prevent financial crime, the inter-governmental Financial Action Task Force (FATF) said on Wednesday. Once a haven for untaxed and illicit assets, the small but...
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Jan. 8: Public trap shoot, every Sunday, 10 a.m.-2 p.m., Juneau Gun Club.
Jan. 11: Public trap shoot, every Wednesday, 6-9 p.m., Juneau Gun Club.
Jan. 12: Winter Trap League Shoot, 5 p.m., Juneau Gun Club.
Jan. 12: Open house to discuss possible Forest Service cabin closures, 5-8 p.m., Juneau District Office, 8465 Old Dairy Road. Details: 586-8800.
Jan. 12: "Journey to the White Continent" lecture, 7:30-8:30 p.m., Egan Auditorium. Sponsored by Juneau Audubon Society.
Jan. 14: Amateur Trapshooting Association/Pacific International Trapshooting Association registered shoot, 9 a.m., Juneau Gun Club.
Jan. 15: Cowboy Action Shoot, 9 a.m., Hank Harmon Rifle Range.
Jan. 19: Winter Trap League shooting, 5 p.m., Juneau Gun Club.
Jan. 21: Southeast Road Runners potluck and award ceremony, 5 p.m., DIPAC Macaulay Salmon Hatchery. Details: Deborah, 789-4260.
Jan. 26: Winter Trap League shooting, 5 p.m., Juneau Gun Club.
Feb. 11: Sweethearts' Relay run, 10 a.m., Douglas Firehall. Details: John or Jamie, 789-5997.
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Ok, so finally getting around to editing this and posting some of the changes that are currently up on PTS along with some that I have floating around in my head. Keep in mind that this list is a work in progress and will be updated regularly. If you don't see something on the list right now, that doesn't mean that I'm not thinking about it or working on it.
Warlord:
- Warlord’s Soul Gift – Each point spent in Warlord increases damage by 1%.
Void Knight:
Talents:Talents:
- Accords are no longer affected by pacts. They give buffs equal to what they would have with 10 pacts now.
- Spark: No longer generates an Attack Point
- Ragestorm: Changed to Air damage from Physical. Cooldown lowered to global. Consumes up to 3 pacts on use.
- Fusion of Flesh: No longer consumes pacts.
- Devouring Shield: No longer a finisher. Absorbs a flat amount. Off GCD. 2m cooldown. Stacks with Rift Shield and Fusion of Flesh.
Tier 1:
(Repurposed) Energy Retention: Increases weapon damage and attack power by 2/4/6/8/10%.
Tier 2:
Tier 4:Tier 4:
- Accord of Resilience swaps places with Void Propulsion. 4 point root ability to Tier 2 in tree.
- Destructive Forces and Empowered armor are now swapped in the talent tree.
- (Adjusted) Accord of Resilience: Does not stack with Shield of the Hero (paladin). Same stats as Shield of the Hero (3% armor, 9% resist).
- Empowered Armor is now Increases Armor by 2/4/6/8/10%.
- Efficient Conversion: Added “Increases duration of pacts by 5/10/15s” to make up for Energy Retention going away.
Paladin:Paladin:
- Destructive Forces is now 0.5/1.0/1.5%
- Fixed an error in Destructive Forces that made it deal less damage that it was supposed to.
Tier 1:Tier 1:
- Interfere: No longer generates an Attack Point
- Light’s Benediction: Lowered damage
- Reverent Protection: Now an ranged life damage AoE. Same 30s cooldown. 20m range. Needs to be renamed.
A Good Defense swaps places with Vengeful Wrath. Tier 1 to Tier 2 in tree.
(Repurposed) Vengeful Wrath: Increases weapon damage and attack power by 2/4/6/8/10%.
Tier 2:
(Adjusted) A Good Defense: Reworded to “Increases damage and healing by abilities that generate attack points by 2/4/6/9/10%”.
Lot of good ideas already in this thread and I've still got plenty of notes from the last thread. Keep them ideas coming! As always, keep the thread constructive.
Bookmarks
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TITLE: If there are y blocks in each street $v_i$ with $0≤i≤x$, why the sum of the blocks isn't y(1+x)?
QUESTION [0 upvotes]: In the solution, I don't understand the part $y_0+y_1+y_2+\cdots +y_x=y$. Since there are y+1 parallel lines, there are y blocks in each vertical street $v_0, v_1, ..., v_x$. So $y_0+y_1+y_2+\cdots +y_x$ equals not y, but yx.
Source: A Path to Combinatorics for Undergraduates: Counting Strategies by Titu Andreescu, Zuming Feng.
REPLY [2 votes]: You write:
In the solution, I don't understand the part $y_0+y_1+y_2+\cdots +y_x=y$.
Since there are $y+1$ parallel lines, there are $y$ blocks in each vertical street $v_0, v_1, ..., v_x$. So $y_0+y_1+y_2+\cdots +y_x$ equals not $y$, but $yx$.
You’re counting all of the blocks on all of the vertical streets (though that is actually $(x+1)y$, as in the red addition to the page, not $xy$); ). That is, you’re assuming that $y_k=y$ for $k=0,\ldots,x$, as if $y_k$ were the whole number of blocks on the street $v_k$. It’s not: $y_k$ is the number of blocks on street $v_k$ that are part of the path that we’re considering at the moment. The point is that it doesn’t matter which path from $A$ to $B$ we’re considering. All movement upward is along one of the $x+1$ streets $v_0,v_1,\ldots,v_x$. If $y_k$ is the number of blocks that we move up while on street $v_k$, then $$y_0+y_1+\ldots+y_x$$ is the total number of blocks that we move upward. We start at $A$ at height $0$ and finish at $B$ at height $y$, so we must move upward a total of $y$ blocks, and therefore
$$y_0+y_1+\ldots+y_x=y\;.$$
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Last in the package last night, and if you’re looking for some creative ways to get rid of your clutter, or tips on selling on eBay, Amazon or Craigslist, his e-books have some great tips and tricks for you to use. The videos are definitely worth it as well. Highly recommended. Sell Your Crap E-book – Clutter Crusher Edition)
In any event, in one of the videos I was watching Adam interviews J.D. Roth of Get Rich Slowly fame, and asks him about falling prey to consumerism, and if things have changed since he has gotten his finances under control, and started earning a better income. Basically - is it harder to not fall prey to consumerism when you can afford to buy things more?
J.D.’s answer might be surprising to some in that he didn’t think it was harder for him to avoid buying things he might want now. He talked about how in the past he did fall prey to a voracious appetite for buying comic books, and all the shiny new gadgets that came out. The problem was, that all of the things he would buy would just turn to clutter. He would buy all the new comic books that came out, but he would only read certain ones. The rest would just sit in a box in the corner. Essentially it was a waste of money.
After J.D. got his finances under control, and was better able to afford things, he realized that while he still wanted to buy certain things that he enjoyed, he didn’t want all the clutter and baggage that went along with buying everything, and buying just out of habit. Instead he developed a laser-like focus that honed in on a few things that he would like to buy, and he would buy those. Instead of buying 10 new comics, he would buy only 1 or 2 that he knew he would read, and forgo the rest.
So for J.D., the keys to avoiding the temptations of consumerism, even though he had the money to spend included:
- Realizing that buying things comes with baggage like debt and clutter
- It’s OK to buy things that you enjoy, especially if you have the money.
- Have laser like focus, and don’t buy everything that you think you might want. Figure out what you like best, and buy that only
Is It Harder Not To Spend, When You Have The Money?
While J.D.’s story really got me thinking, it’s still an interesting question to ask. For a majority of people, is it harder not to buy the things that you want, and fall prey to a high consumption lifestyle if you have the money to spend? Or does having more money to spend go hand in hand with having better financial discipline, and thus not falling prey to consumerism as easily?
I know for me, our family income has gone up substantially in the last couple years, and we can afford to buy more things than we used to. To be honest, I have probably bought more things that I wanted during that time than I did in the past, but I haven’t necessarily gone out of control – cluttering up the house or creating debt. I’ve only paid cash for everything, and the things that I’ve bought tend to be high quality, things that will last for years.
I think for me it was harder to not fall prey to consumerism when I didn’t have any money. I would use credit cards to pay for spring break trips, and buy the newest game console on my plastic. In a way, buying things was a coping mechanism, and an escape towards a better life I was dreaming about. The problem is, the things you buy can far too often take over your life, and the dreams of a more prosperous life they came with, can often turn into a curse of debt and despair.
So what do you think – would it be harder for you to not fall prey to consumerism when you can afford to buy things more? Or have you overspent more often on things you don’t need when you didn’t have the money to spare? Tell us your thoughts in the comments.
Last Edited: 26th September 2010
{ 11 comments… read them below or add one }
From my experience, after we paid off our house and could afford most things we wanted – we found it easier to avoid consumerism. For me, it was being able to avoid focusing on how I could get (save, find deals, etc) what I wanted. Not that I stopped saving or looking for deals, but we stopped the relentless focus on finding the savings and deals and at the same time, God has helped me to put in His perspective on buying things that I wanted. God’s perspective has helped me to realize that my money could be used better giving to the poor, sick, and in prison (Matthew 25) than on buying more stuff.
I can relate to buying things as a coping mechanism. I used to buy things while depressed. I did manage to stay more or less out of debt, only amassing debt on a motorcycle during the worst of those times. Looking back I am so glad it was never easy for me to get a credit card. With shopping as a coping mechanism and a soft spot for the latest gadgets and games I would likely still be trying to recover!
Now that my walk with God has greatly improved and I have a family I do not fight depression nearly as much. If only I had read the bible, or worked a profitable hobby as a coping mechanism back then I bet I’d be even more well off today.
If you chart my earnings growth vs my savings..I do spend more, but my savings is increasing much faster than my spending. I also think the reason I spend more is because I have 2 kids now. I think if you took the kids out of the equation, we’d be at about the same level of spending.
For me, it helps having a smaller house because when we think of buying something the immediate thought that enters my head is “where would I put that?”
I actually had more desire to accumulate things when I was poor than I do now. Perhaps it’s an age thing too..you accumulate when you’re in your 20′s and then you edit your belongings to what you really need in your 30s.
I don’t know if it’s consumerism so much as convenience. Rather than making time to buy groceries or getting up earlier to make a lunch, if I have the money to spend I’ll just get a salad at a local restaurant. If I’m cold at work I’ll duck around the corner and buy a cardigan. If I have a long night of events, I’ll take a cab rather than wait for the bus. As my future husband works to launch a business we hope will be wildly successful, I look at these bad habits and want to break them, because I know if I don’t do this now they have the chance of getting really out of control as our income increases. “Long day at work. I think I’ll just have Jeeves bring the yacht around.”
I think our spending will definitely increase with our income — although I’ve already told my husband that, once he starts earning income, I still want to live on my main one and try to bank everything else… minus the occasional indulgence.
But I think the one loophole JD and folks aren’t thinking about is how everything is turning digital. That means plenty of spending without clutter. Video games, comics, and, of course, iTunes/apps are all available for download. Oh, and let’s not forget e-books.
Consumerism can be a part of any income level. It is an attitude that has been promoted by the media, shopping centers, and even the government. People are led to believe that consumerism is the only necessary thing to keep the economy moving and without it our society would collaspe. Americans are doing a tremendous job of paying down debt and saving that will ultimately create a stronger base to move the economy forward but these acheivements are not congratulated or applauded.
My answer would be a resounding “yes”.
If you don’t have money, you can’t spend it, can you?
That eliminates a big part of the equation.
On the surface it seems the answer to this question would be a resounding YES. But after careful consideration it makes sense that people with means are better able to control financial urges rather than financial urges controlling them. We can clearly see these truths evidenced by their ability to work through the disciplined process of building a sound financial foundation
I’ve been without money and with money.
In both cases, it’s been hard not to spend, but I found that ironically with more money, I find it easier not to spend because I’d rather see the money in my bank account (but I don’t deprive myself).
Without money, I found it harder not to spend because I didn’t see any money in my bank account. Yes, my debt was dropping, but there was no real physical benefit to seeing my money being “saved” for my debt.
Similar to JD, I found minimalism and now I just buy what I want/need, when I want, and only if I actually need/want it. It’s cut down on a lot of spending surprisingly. I hover around the $1000/month range in spending, so I feel pretty good.
I think it is harder to avoid buying when you have the money to just get what you want. It takes discipline, and I’m afraid many people are missing that these days. For me, the “baggage” of having more stuff to store, clean around, display, protect, carry insurance on, etc, just makes me want to save my cash. There’s nothing quite like having some money in the bank and knowing that you already have everything that you need to be happy.
Larry
When all is well, retirement not far away, college funded, mortgage almost paid, it’s tough to ever tell the misses “we really can’t afford that.” Because we can.
There’s an age/income/asset level that prompts the “you can’t take it with you” conversation.
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- 5
Something that a GM needs to remember when designing foes for the players - especially foes created using the same rules the players use (humanoid enemies with class levels), is the difference in role between a PC and an enemy.
A PC sticks around. He faces, (in theory) on average, four encounters a day, every day. Generally, the player who makes him is planning on playing him for the duration of the campaign. The PC is created with the intent to live.
An enemy doesn't last all that long. One direct encounter and he's gone. Caput. So long, sirs, and thanks for all the fish. An enemy is created with the intent to die. He's supposed to challenge the PCs, push them hard, but in the end, if all goes according to plan, he should be laying down and…Read more > …Read more >
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TITLE: Well ordering principle question
QUESTION [0 upvotes]: Is every non empty subset of the integers well ordered and does this mean that every subset contains a least element?
Are the positive rationals well ordered? i believe not.
Is this because of the fact that this set has no minimal element? Does every subset strictly smaller than the set of positive rationals have a least element? i believe it does but not sure
thanks
REPLY [2 votes]: When a linear order is non-well-ordered, lots of its proper subsets are also non-well-ordered. For example, consider the set of negative even integers or the set of reciprocals of integer powers of $2$: these are proper subsets of the integers and the positive rationals respectively, and neither has a least element.
The definition of well-orderedness - "Every subset has a least element" - can feel weird at first. It's helpful to think instead in terms of descending sequences: a linear order $(L,<)$ is well-ordered iff it has no infinite descending sequence $a_1>a_2>a_3>...$. Thinking this way it should be clear e.g. that the positive rationals aren't well-ordered, because we can "count down" $${1\over 1}>{1\over 2}>{1\over 3}>{1\over 4}>...$$ An important thing to keep in mind is that a counterexample to well-orderedness - that is, an infinite descending sequence in the linear order in question - will not be unique: we can't speak of "the" descending sequence, but rather "a" descending sequence.
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Stop by with your questions about the program. No registration required. Current students may register for classes during advising hours.
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For more information about or academic advising for the MBA Program please contact:
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TITLE: Finding every $(m,n)\ (m\gt n)\in \mathbb N$ such that $m\pm n$ are square numbers and $mn$ is a cubic number.
QUESTION [0 upvotes]: Question : Find every $(m,n)\ (m\gt n)\in \mathbb N$ such that $m\pm n$ are square numbers and $mn$ is a cubic number.
Example : I've already found the followings:
$$m=12500a^6, n=10000a^6$$
where $a$ is a natural number.
We can easily show that these satisfy the conditions :
$$m-n=(50a^3)^2, m+n=(150a^3)^2, mn=(500a^4)^3.$$
However, I can neither find the other solutions nor prove that there is no other solution.
Motivation : I've been interested in the following question :
Do there exist $(m,n)\ (m\gt n)\in \mathbb N$ such that each of $m-n, m+n, mn$ is a square number?
The answer is No.
Proof : Letting $m+n=a^2, m-n=b^2, mn=c^2$, we get
$$a^4-b^4=(2c)^2$$
because of $(m+n)^2-(m-n)^2=4mn$.
On the other hand, if $x^4-y^4=z^2$ has a solution, letting $m=2(x^2+y-2), n=2(x^2-y^2)$, we get
$$m+n=(2x)^2, m-n=(2y)^2, mn=(2z)^2.$$
By the way, since it is well known that $x^4-y^4=z^2$ doesn't have any solution, then we now know that the answer for the above question is no.
REPLY [3 votes]: $$ r = 2 \cdot 3^2 \cdot 13^2 \cdot t^6 $$
$$ m = 13 r^2, \; \; n = 12 r^2, \; \; m+n = 25 r^2 , \; \; m - n = r^2 $$
$$ r^4 = 2^4 \cdot 3^8 \cdot 13^8 \cdot t^{24} $$
$$ mn = 2^2 \cdot 3 \cdot 13 \cdot r^4 = 2^6 \cdot 3^9 \cdot 13^9 \cdot t^{24} $$
There are lots of these, and strongly related to Pythagorean triples. Begin with the fact that $m$ must be the sum of two squares, as $2m$ is. If there is just one reasonable expression for $2m = a^2 + b^2,$ which will be the case for $m$ prime and $m \equiv 1 \pmod 4,$ then $2n = a^2 - b^2$ in just one interesting way. Then just multiply both by $r^2$ and decide what $r$ needs to be to end up with a cube.
So: $m=17, 2m = 34 = 5^2 + 3^2, 2n = 5^2 - 3^2 = 16, n=8. $ Then switch to $m=17 r^2, n = 8 r^2.$ As 8 is already a cube, we just need $17 r^4$ a cube, minimum is $r=17^2.$ Finally $r = 17^2 t^6.$ So $$ r = 17^2 t^6, \; \; m = 17 r^2, \; \; n = 8 r^2, \; \; r^4 = 17^8 t^{24}, \; \; mn = 17^9 2^3 t^{24} . $$
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\begin{document}
\title{Calculus of Cost Functions}
\author{
Andr\'e Nies}
\keywords{computability, randomness, lowness, cost functions}
\subjclass{Primary: 03F60; Secondary: 03D30}
\thanks{
Research partially supported by the Marsden Fund of New
Zealand, grant no.\ 08-UOA-187, and by the Hausdorff Institute of Mathematics, Bonn.}
\begin{abstract} Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive strength. We show that the class of additive cost functions describes the $K$-trivial sets. We prove a cost function basis theorem, and give a general construction for building computably enumerable sets that are close to being Turing complete.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
In the time period from 1986 to 2003, several constructions of computably enumerable (c.e.) sets appeared. They turned out to be closely related.
\bi \item[(a)] Given a \ML\ random (ML-random for short) $\DII$ set $Y$,
\n \Kuc~\cite{Kucera:86} built a c.e.\ incomputable set $A\leT Y$. His construction is interesting because in the case that $Y <_T \Halt$, it provides a c.e.\ set~$A$ such that $\ES <_T A <_T \Halt$, without using injury to requirements as in the traditional proofs. ($\Halt$ denotes the halting problem.)
\item[(b)] \Kuc\ and Terwijn~\cite{Kucera.Terwijn:99} built a c.e.\ incomputable set $A$ that is low for ML-randomness: every ML-random set is already ML-random relative to $A$.
\item[(c)] $A$ is called $K$-trivial if $K(A\uhr n) \le K(n)+ O(1)$, where $K$ denotes prefix-free descriptive string complexity. This means that the initial segment complexity of $A$ grows as slowly as that of a computable set. Downey et al.\ \cite{Downey.Hirschfeldt.ea:03} gave a very short construction (almost a ``definition'') of a c.e., but incomputable $K$-trivial set.
\ei
The sets in (a) and (b) enjoy a so-called lowness property, which says that the set is very close to computable. Such properties can be classified according to various paradigms introduced in \cite{Nies:ICM, Greenberg.Hirschfeldt.ea:12}. The set in (a) obeys the \emph{Turing-below-many} paradigm which says that $A$ is close to being computable because it is easy for an oracle set to compute it.
A frequent alternative is the \emph{weak-as-an-oracle} paradigm: $A$ is weak in a specific sense when used as an oracle set in a Turing machine computation. An example is the oracle set in (b), which is so weak that it useless as an extra computational device when testing for ML-randomness. On the other hand, $K$-triviality in (c) is a property stating that the set is far from random: by the Schnorr-Levin Theorem, for a random set $Z$ the initial segment complexity grows fast in that $K(Z \uhr n) \ge n - O(1)$. For background on the properties in (a)-(c) see \cite{Downey.Hirschfeldt:book} and \cite[Ch.\ 5]{Nies:book}.\footnote{We note that the result (c) has a complicated history. Solovay \cite{Solovay:75} built a $\DII$ incomputable set $A$ that is $K$-trivial. Constructing a c.e.\ example of such a set was attempted in various sources such as \cite{Calude.Grozea:96}, and unpublished work of Kummer.}
A central point for starting our investigations is the fact that the constructions in (a)--(c) look very similar. In hindsight this is not surprising: the classes of sets implicit in (a)-(c) coincide! Let us discuss why.
\vsps
\n {\it (b) coincides with (c):} Nies \cite{Nies:AM}, with some assistance by Hirschfeldt, showed that lowness for ML-randomness is the same as $K$-triviality. For this he introduced a method now known as the ``golden run''.
\vsps
\n {\it (a) coincides with (b):} The construction in (a) is only interesting if $Y \not \ge_T \Halt$. Hirschfeldt, Nies and Stephan~\cite{Hirschfeldt.Nies.ea:07} proved that if $A$ is a c.e.\ set such that $A\leT Y$ for some ML-random set $Y \not \ge_T \Halt$, then $A$ is $K$-trivial, confirming the intuition sets of the type built by \Kuc\ are close to computable. They asked whether, conversely, for every $K$-trivial set $A$ there is a ML-random set $Y\ge_T A$ with $Y \not \ge_T \Halt$. This question became known as the ML-covering problem. Recently the question was solved in the affirmative by combining the work of seven authors in two separate papers. In fact, there is a single ML-random $\DII$ set $Y \not \ge_T \Halt$ that is Turing above all the $K$-trivials. A summary is given in~\cite{Bienvenu.Day.ea:14}.
\vsps
The common idea for these constructions is to ensure lowness of $A$ dynamically, by restricting the overall manner in which numbers can be enumerated into $A$. This third lowness paradigm has been called \emph{inertness} in \cite{Nies:ICM}: a set $A$ is close to computable because it is computably
approximable with a small number of changes.
The idea is implemented as follows. The enumeration of a number $x$ into $A$ at a stage~$s$ bears a cost $\cc(x,s)$, a non-negative rational that can be computed from $x$ and~$s$. We have to enumerate $A$ in such a way that the sum of all costs is finite. A construction of this type will be called a {\em cost function construction}.
If we enumerate at a stage more than one number into $A$, only the cost for enumerating the least number is charged. So, we can reduce cost by enumerating $A$ in ``chunks''.
\subsection{Background on cost functions} The general theory of cost functions began in~\cite[Section 5.3]{Nies:book}. It was further developed in \cite{Greenberg.Nies:11,Greenberg.Hirschfeldt.ea:12,Diamondstone.Greenberg.ea:nd}. We use the language of~\cite[Section 5.3]{Nies:book} which already allows for the constructions of $\DII$ sets. The language is enriched by some notation from~\cite{Diamondstone.Greenberg.ea:nd}. We will see that most examples of cost functions are based on randomness-related concepts.
\begin{deff} \label{def:c.f.} {\rm A \emph{cost function} is a computable function \bc $\cc:
\NN \times \NN \ria \{x \in \QQ \colon \, x \ge 0\}$. \ec }
\end{deff}
\n Recall that a \emph{computable approximation} is a computable sequence of finite sets $\seq{ A_s}_{s\in \NN}$ such that $\lim_s A_s(x)$ exists for each $x$.
\begin{deff} \label{df:obey cf}
{\rm
\n (i). Given a computable approximation $\seq{ A_s}_{s\in \NN}$
and a cost function~$\cc$, for $s>0$ we let \bc $\cc_s(A_s) = \cc(x,s) $ where $ x< s \lland x \mmbox{\rm is
least s.t.} A_{s-1} ( x) \neq A_{s} (x) $; \ec
if there is no such $x$ we let $\cc_s(A_s) =0$. This is the cost of changing $A_{s-1}$ to~$A_s$.
We let
\begin{equation*} \cc \seq{A_s}\sN s = \sum_{s>0} \cc_s(A_s) \end{equation*}
be the total cost of all the $A$-changes. We will often write $\cc\seq {A_s}$ as a shorthand for $\cc \seq{A_s}\sN s$.
\vsps
\n (ii) We say that $\seq{A_s}_{s\in \NN}$ \emph{obeys} $\cc$ if $\cc \seq{A_s}$ is finite. We denote this by
\bc $\seq{A_s} \models \cc$. \ec
\vsps
\n (iii) We say that a $\DII$ set~$A$ \emph{obeys} $\cc$, and write $A \models \cc$, if some computable approximation of $A$ obeys $\cc$. }
\end{deff}
A cost function $\cc$ acts like a global restraint, which is successful if the condition $ \cc {\seq{A_s}} < \infty$ holds. \Kuc's construction mentioned in (a) above needs to be recast in order to be viewed as a cost-function construction \cite{Greenberg.Nies:11,Nies:book}. In contrast, (b) and (c) can be directly seen as cost function constructions.
In each of (a)--(c) above, one defines a cost function $\cc$ such that any set $A$ obeying $\cc$ has the lowness property in question. For, if $A \models \cc$, then one can enumerate an auxiliary object that has in some sense a bounded weight.
In (a), this object is a Solovay test that accumulates the errors in an attempted computation of~$A$ with oracle $Y$. Since $Y$ passes this test, $Y$ computes $A$.
In (b), one is given a $\Sigma^0_1(A)$ class $\+ V \sub \cantor$ such that the uniform measure $\leb \+ V$ is less than~$1$, and the complement of $\+ V$ consists only of ML-randoms. Using that $A$ obeys $\cc$, one builds a $\Sigma^0_1$ class $\+ S \sub \cantor$ containing $\+ V$ such that still $\leb \+ S <1$. This implies that $A$ is low for ML-randomness.
In (c) one builds a bounded request set (i.e., Kraft-Chaitin set) which shows that $A$ is $K$-trivial.
The cost function in (b) is adaptive in the sense that $\cc(x,s) $ depends on $A_{s-1}$. In contrast, the cost functions in (a) and (c) can be defined in advance, independently of the computable approximation of the set $A$ that is built.
The main existence theorem, which we recall as Theorem~\ref{thm:cfconstr} below, states that for any cost function $\cc$ with the limit condition $\lim_x \liminf_s \cc(x,s) = 0$, there is an incomputable c.e.\ set $A$ obeying $\cc$. The cost functions in (a)-(c) all have the limit condition. Thus, by the existence theorem, there is an incomputable c.e.\ set $A$ with the required lowness property.
Besides providing a unifying picture of these constructions, cost functions have many other applications. We discuss some of them.
Weak 2-randomness is a notion stronger than ML-randomness: a set $Z$ is weakly 2-random if $Z$ is in no $\PI 2$ null class. In 2006, Hirschfeldt and Miller gave a characterization of this notion: a ML-random is weakly 2-random if and only if it forms a minimal pair with $\Halt$. The implication from left to right is straightforward. The converse direction relies on a cost function related to the one for \Kuc's result (a) above. (For detail see e.g.\ \cite[Thm.\ 5.3.6]{Nies:book}.) Their result can be seen as an instance of the randomness enhancement principle~\cite{Nies:ICM}: the ML-random sets get more random as they lose computational complexity.
The author \cite{Nies:AM} proved that the single cost function $\cost$ introduced in \cite{Downey.Hirschfeldt.ea:03} (see Subsection~\ref{ss:standard_costfunction} below) characterises the $K$-trivials. As a corollary, he showed that every $K$-trivial set $A$ is truth-table below a c.e.\ $K$-trivial $D$. The proof of this corollary uses the general framework of change sets spelled out in Proposition~\ref{prop:c.e. ub cf} below. While this is still the only known proof yielding $A\ltt D$, Bienvenu et al.\
\cite{Bienvenu.Downey.ea:15} have recently given an alternative proof using Solovay functions in order to obtain the weaker reduction $A \leT D$.
In model theory, one asks whether a class of structures can be described by a first order theory. Analogously, we ask whether an ideal of the Turing degrees below $\mathbf 0'$ is given by obedience to all cost functions of an appropriate type. For instance, the $K$-trivials are axiomatized by $\cost$.
Call a cost function $\cc$ {\em benign} if from $n$ one can compute a bound on the number of disjoint intervals $[x,s)$ such that $\cc(x,s) \ge \tp{-n}$. Figueira et al.~\cite{Figueira.ea:08} introduced the property of being strongly jump traceable (s.j.t.), which is an extreme lowness property of an oracle $A$, even stronger than being low for $K$. Roughly speaking, $A$ is s.j.t.\ if the jump $J^A(x)$ is in $T_x$ whenever it is defined, where $\seq{T_x}$ is a uniformly c.e.\ sequence of sets such that any given order function bounds the size of almost all the $T_x$. Greenberg and Nies~\cite{Greenberg.Nies:11} showed that the class of benign cost functions axiomatizes the c.e.\ strongly jump traceable sets.
Greenberg et al.\ \cite{Greenberg.Hirschfeldt.ea:12} used cost functions to show that each strongly jump-traceable c.e.\ set is Turing below each $\omega$-c.e. ML-random set. As a main result, they also obtained the converse. In fact they showed that any set that is below each superlow ML-random set is s.j.t.
The question remained whether a general s.j.t.\ set is Turing below each $\omega$-c.e.\ ML-random set. Diamondstone et al.\ \cite{Diamondstone.Greenberg.ea:nd} showed that each s.j.t.\ set $A$ is Turing below a c.e., s.j.t.\ set $D$. To do so, as a main technical result they provided a benign cost function $\cc$ such that each set $A$ obeying $\cc$ is Turing below a c.e.\ set $D$ which obeys every cost function that $A$ obeys. In particular, if $A$ is s.j.t., then $A \models \cc$, so the c.e.\ cover $D$ exists and is also s.j.t.\ by the above-mentioned result of Greenberg and Nies~\cite{Greenberg.Nies:11}. This gives an affirmative answer to the question.
Note that this answer is analogous to the result~\cite{Bienvenu.Day.ea:14} that every $K$-trivial is below an incomplete random.
\subsection{Overview of our results}
The main purpose of the paper is a systematic study of cost functions and the sets obeying them. We are guided by the above-mentioned analogy from first-order model theory: cost functions are like sentences, sets are like models, and obedience is like satisfaction. So far this analogy has been developed only for cost functions that are monotonic (that is, non-increasing in the first component, non-decreasing in the stage component). In Section~\ref{s:look_ahead} we show that the conjunction of two monotonic cost functions is given by their sum, and implication $\cc \to \dd$ is equivalent to $\ul \dd = O(\ul \cc)$ where $\ul \cc(x) = \sup_s \cc(x,s)$ is the limit function.
In Section~\ref{s:additive cf} we show that a natural class of cost functions introduced in Nies~\cite{Nies:ICM} characterizes the $K$-trivial sets: a cost function $\cc$ is additive if $\cc(x,y)+ \cc(y,z) = \cc(x,z)$ for all $x< y< z$. We show that such a cost function is given by an enumeration of a left-c.e.\ real, and that implication corresponds to Solovay reducibility on left-c.e.\ reals. Additive cost functions have been used prominently in the solution of the ML-covering problem~\cite{Bienvenu.Day.ea:14}. The fact that a given $K$-trivial $A$ obeys every additive cost function is used to show that $A \leT Y$ for the Turing incomplete ML-random set constructed by Day and Miller~\cite{Day.Miller:14}.
Section~\ref{s:random_low_K-triv} contains some more applications of cost functions to the study of computational lowness and $K$-triviality. For instance, strengthening the result in \cite{Greenberg.Hirschfeldt.ea:12} mentioned above, we show that each c.e., s.j.t.\ set is below any complex $\omega$-c.e.\ set $Y$, namely, a set $Y$ such that there is an order function $g$ with $g(n ) \lep K(Y \uhr n)$ for each $n$. In addition, the use of the reduction is bounded by the identity. Thus, the full ML-randomness assumed in \cite{Greenberg.Hirschfeldt.ea:12} was too strong a hypothesis. We also discuss the relationship of cost functions and a weakening of $K$-triviality.
In the remaining part of the paper we obtain two existence theorems.
Section~\ref{s:costf_basis_theorem}
shows that given an arbitrary monotonic cost function $\cc$, any nonempty $\PPI$ class contains a $\DII$ set $Y$ that is so low that each c.e.\ set $A \leT Y$ obeys~$\cc$. In Section~\ref{ss:dual} we relativize a cost function $\cc$ to an oracle set $Z$, and show that there is a c.e.\ set $D$ such that $\Halt$ obeys $\cc^D$ relative to $D$. This much harder ``dual'' cost function construction can be used to build incomplete c.e.\ sets that are very close to computing $\Halt$. For instance, if $\cc$ is the cost function $\cost$ for $K$-triviality, then $D$ is LR-complete.
\section{Basics}
We provide formal background, basic facts and examples relating to the discussion above. We introduce classes of cost functions: monotonic, and proper cost functions. We formally define the limit condition, and give a proof of the existence theorem.
\subsection{Some easy facts on cost functions}
\mbox{}
\begin{deff} \label{def:c.f.-2}
{\rm We say that a cost function~$\cc$
is \emph{nonincreasing in the main argument} if \bc $\fa x, s \, [ \cc(x+1, s) \le \cc(x, s)]$. \ec We say that~$\cc$
is \emph{nondecreasing in the stage} if $\cc(x, s) = 0 $ for $x > s$ and \bc $\fa x, s \, [ \cc(x, s) \le \cc(x, s+1)] $. \ec
If $\cc$ has both properties we say that $\cc$ is \emph{monotonic}. This means that the cost $ \cc(x,s)$ does not decrease when we enlarge the interval $[x,s]$. }
\end{deff}
\begin{fact} \label{easy1} Suppose $A\models \cc$. Then for each $\epsilon >0$ there is a computable approximation $\seq{A_s}\sN s$ of $A$ such that $\cc {\seq{A_s}\sN s} < \epsilon$. \eop \end{fact}
\begin{proof} Suppose $\seq{\widehat A_s}\sN s \models \cc$. Given $x_0$ consider the modified computable approximation $\seq{ A_s}\sN s$ of $A$ that always outputs the final value $A(x)$ for each $x\le x_0$. That is, $A_s(x) =A(x)$ for $x \le x_0$, and $A_s(x) =\widehat A_s(x)$ for $x > x_0$.
Choosing $x_0$ sufficiently large, we can ensure $\cc \seq A_s < \epsilon$.
\end{proof}
\begin{definition} \label{def:proper cf} {\rm Suppose that a cost function $ \cc(x,t)$ is non-increasing in the main argument~$x$. We say that $\cc$ is {\it proper} if $\fao x \ex t \, \cc(x,t) > 0$.} \end{definition}
If a cost function that is non-increasing in the main argument is not proper, then every $\DII$ set obeys~$\cc$. Usually we will henceforth assume that a cost function $\cc$ is proper. Here is an example how being proper helps.
\begin{fact} Suppose that $\cc$ is a proper cost function and $S= \cc {\seq{A_s} } < \infty $ is a computable real. Then $A$ is computable. \end{fact}
\begin{proof} Given an input $x$, compute a stage $t$ such that $\delta = \cc(x,t)>0$ and $S- \cc {\seq{A_s}_{s \le t} } < \delta$. Then $A(x) = A_t(x)$.\end{proof}
A \emph{computable enumeration} is a computable approximation $\seq{ B_s}_{s\in \NN}$
such that $B_s \sub B_{s+1}$ for each $s$.
\begin{fact} \label{fact:fukd} Suppose $\cc$ is a monotonic cost function and $A \models \cc$ for a c.e.\ set~$A$. Then there is a computable \emph{enumeration} $\seq{\wt A_s}$ that obeys $\cc$. \end{fact}
\begin{proof} Suppose $\seq{A_s} \models \cc$ for a computable approximation $\seq{A_s}$ of $A$. Let $\seq{B_t}$ be a computable enumeration of $A$. Define $\seq{\wt A_s} $ as follows. Let $\wt A_0(x)= 0$; for $s> 0$ let $\wt A_s (x)
= \wt A_{s-1}(x)$ if $ \wt A_{s-1}(x)=1$; otherwise let $\wt A_{s}(x)= A_t(x)$ where $t \ge s$ is least such that $A_t(x)= B_t(x)$.
Clearly $\seq{\wt A_s}$ is a computable enumeration of $A$. If $ \wt A_s(x) \neq \wt A_{s-1}(x)$ then $A_{s-1}(x)= 0 $ and $A_s(x)=1$. Therefore $\cc {\seq{\wt A_s}} \le \cc {\seq{A_s}} < \infty $. \end{proof}
\subsection{The limit condition and the existence theorem}
\mbox{}
\n For a cost function $\cc$, let
\begin{equation} \ul \cc (x) = \liminf_s \cc(x,s). \end{equation}
\begin{deff} \label{df:lc} We say that a cost function~$\cc$ satisfies the {limit condition} if $\lim_x
\, \ul \cc(x) = 0$. That is, for each~$e$, for almost every~$x$ we have \bc $\ex^\infty s \, [ \cc(x,s) \le \tp{-e}]$. \ec
\end{deff}
In previous works such as \cite{Nies:book}, the limit condition was defined in terms of $\sup_s \cc(x,s)$, rather than $\liminf_s \cc(x,s)$. The cost functions previously considered were usually nondecreasing in the stage component, in which case $ \sup_s \cc(x,s) = \liminf_s \cc(x,s)$ and hence the two versions of the limit condition are equivalent. Note that the limit condition is a $\PI 3$ condition on cost functions that are nondecreasing in the stage, and $\PI 4$ in general.
The basic existence theorem says that a cost function with the limit condition has a c.e., incomputable model. This was proved by various authors for particular cost functions. The following version of the proof appeared in \cite{Downey.Hirschfeldt.ea:03} for the particular cost function~$\cost$ defined in Subsection~\ref{ss:standard_costfunction} below, and then in full generality in
\cite[Thm 5.3.10]{Nies:book}.
\begin{thm} \label{thm:cfconstr} Let~$\cc$ be a cost function
with the limit condition.
\bi \item[(i)] There is a simple set~$A$ such that~$A\models \cc$. Moreover, $A$ can be obtained uniformly in (a computable index for)~$\cc$.
\item[(ii)] If $\cc$ is nondecreasing in the stage component, then we can make $A$ promptly simple. \ei
\end{thm}
\begin{proof} (i) We meet the usual simplicity requirements
\bc $S_e$: \ $\# W_e =\infty \RRA W_e \cap A \neq \ES. $ \ec
To do so, we define a
computable enumeration $\seq{A_s}\sN{s}$ as follows.
{Let $A_0 = \ES$. At stage $s>0$, for each $e < s$, if
$S_e$ has not been met so far and there is $x \ge 2e$ such that $x
\in W_{e, s} $ and $ \cc(x,s) \le 2^{-e}$,
put~$x$ into $A_s$. Declare $S_e$ met.}
\vsps
To see that $\seq{A_s}\sN{s} $ obeys~$ \cc$, note that at most one number is
put into~$A$ for the sake of each requirement. Thus $\cc {\seq{A_s}} \le \sum_e \tp{-e} =2$.
If $W_e$ is infinite, then there is an $x\ge 2e$ and $s>x$ such that $x \in W_{e,s}$
and $ \cc(x,s) \le 2^{-e}$, because~$\cc$ satisfies the limit condition. So we meet $S_e$.
Clearly the construction of~$A$ is uniform in an index for the computable function~$\cc$.
\vsps
\n (ii) Now we meet the prompt simplicity requirements \bc $PS_e$: \ $\# W_e =\infty \RRA \exo s \exo x [ x \in W_{e,s } - W_{e,s-1} \lland x \in A_{s}]. $ \ec
{Let $A_0 = \ES$. At stage $s>0$, for each $e < s$, if
$PS_e$ has not been met so far and there is $x \ge 2e$ such that $x
\in W_{e, s} - W_{e,s-1} $ and $ \cc(x,s) \le 2^{-e}$,
put~$x$ into $A_s$. Declare $PS_e$ met.}
If $W_e$ is infinite, there is an $x\ge 2e$ in $W_e$
such that $ \cc(x,s) \le 2^{-e}$ for all $s>x$, because~$\cc$ satisfies the limit condition and is nondecreasing in the stage component. We enumerate such an
$x$ into~$A$ at the stage $s>x$ when~$x$ appears in $W_e$, if
$PS_e$ has not been met yet by stage~$s$. Thus~$A$ is promptly simple.
\end{proof}
Theorem~\ref{thm:cfconstr}(i) was strengthened in \cite[Thm 5.3.22]{Nies:book}. As before let $\cc$ be a cost function
with the limit condition. Then for each low c.e.\ set~$B$,
there is a c.e.\ set~$A$ obeying~$\cc$ such that $A\not \leT B$. The proof of \cite[Thm 5.3.22]{Nies:book} is for the case of the stronger version of the limit condition $\lim_x \sup_s \cc(x,s)=0$, but in fact works for the version given above.
The assumption that $B$ be c.e.\ is necessary: there is a low set Turing above all the $K$-trivial sets by~\cite{Kucera.Slaman:09}, and the $K$-trivial sets can be characterized as the sets obeying the cost function $\cost$ of Subsection~\ref{ss:standard_costfunction} below.
The following fact implies the converse of Theorem~\ref{thm:cfconstr} in the monotonic case.
\begin{fact} \label{ex:limit cond necc}
\n Let $\cc$ be a monotonic cost function. If a computable approximation $\seq{A_s}_{s\in
\NN}$ of an incomputable set~$A$ obeys~$\cc$, then~$\cc$ satisfies the limit condition. \end{fact}
\begin{proof} Suppose the limit condition fails for~$e$. There is $s_0$ such
that \bc $ \sum_{s\ge s_0} \sum_{x< s} \cc_s(A_s) \le \tp{-e}$. \ec To compute~$A$, on input~$n$ compute $s > \max (s_0, n)$ such that $ \cc(n,s) > \tp{-e}$. Then $A_s(n) = A(n)$. \end{proof}
\begin{convention} \label{rmk:all finite} {\rm For a \emph{monotonic} cost function $\cc$, we may forthwith assume that $\ul \cc (x)< \infty$ for each $x$. For, firstly, if $\fao x [ \ul \cc (x) = \infty]$, then $A \models \cc$ implies that $A$ is computable. Thus, we may assume there is $x_0$ such that $ \ul \cc (x)$ is finite for all $x\ge x_0$ since $ \ul \cc (x)$ is nonincreasing. Secondly, changing values $ \cc(x,s)$ for the finitely many $x< x_0$ does not alter the class of sets $A$ obeying~$\cc$. So fix some rational $q> \cc(x_0)$ and, for $x< x_0$ redefine $ \cc(x,s) = q$ for all~$s$.} \end{convention}
\subsection{The cost function for $K$-triviality} \label{ss:standard_costfunction} \
\n Let $K_s(x) = \min \{ \sssl \colon \, \UM_s(\sss) = x\}$ be the value of prefix-free descriptive string complexity of $x$ at stage~$s$. We use the conventions $K_s(x) = \infty $ for $x \ge s$ and $\tp{-\infty} =0$. Let \begin{equation} \label{eqn:stcf} \cost(x,s) = \sum_{ w =x+1 }^s 2^{-K_s(w)}. \end{equation}
Sometimes $\cost$ is called the \emph{standard cost function}, mainly because it was the first example of a cost function that received attention. Clearly, $\cost$ is monotonic.
Note that
$\ul \cost(x) = \sum_{w>x} \tp{-K(w)}$. Hence $\cost$ satisfies the limit condition: given $e\in \NN$, since $\sum _w \tp{-K(w)} \le 1$,
there is an $x_0$ such that
\bc $\sum_{w\ge x_0} \tp{-K(w)} \le \tp{-e}$. \ec
Therefore $\ulcost(x) \le \tp{-e}$ for all $x \ge x_0$.
The following example illustrates that in Definition~\ref{df:obey cf}, obeying $\cost$, say, strongly depends on the chosen enumeration. Clearly, if we enumerate $A=\NN$ by putting in $x$ at stage $x$, then the total cost of changes is zero.
\begin{prop}\label{ex:slow N} There is a computable enumeration $\seq{A_s}_{s\in
\NN}$ of $\NN$ in the order $0,1,2,\ldots$ (i.e., each $A_s$ is an initial segment of
$\NN$) such that $\seq{A_s}_{s\in
\NN}$ does not obey~$\cost$.
\end{prop}
\begin{proof} Since $K(2^j) \lep 2 \log j$, there is an increasing computable function~$f$ and a
number $j_0$ such that $\fa j \ge j_0 \, K_{f(j)}(2^j) \le j-1$.
Enumerate the set $A=\NN$ in order, but so slowly that for each $j \ge j_0$ the elements of $(2^{j-1}, 2^j]$ are enumerated only after stage $f(j)$, one by one. Each such
enumeration costs at least $\tp{-(j-1)}$, so the cost for each interval $(2^{j-1}, 2^j]$ is 1. \end{proof}
Intuitively speaking, an infinite c.e.\ set $A$ can obey the cost function $\cost$ only because during an enumeration of $x$ at stage $s$ one merely pays the current cost $\cost(x,s)$, not the limit cost $\ulcost (x)$.
\begin{fact} \label{fa: enumlate} If a c.e.\ set $A$ is infinite, then $\sum_{x \in A} \ulcost(x) = \infty$. \end{fact}
\begin{proof} Let $f$ be a 1-1 computable function with range $A$. Let $L$ be the bounded request set $\{\la r, \max_{i \le \tp{r+1}} f(i)\ra \colon \, r \in \NN \}$. Let $M$ be a machine for $L$ according to the Machine Existence Theorem, also known as the Kraft-Chaitin Theorem. See e.g.\ \cite[Ch.\ 2]{Nies:book} for background. \end{proof}
In \cite{Nies:AM} (also see \cite[Ch.\ 5]{Nies:book}) it is shown that $A$ is $K$-trivial iff $A\models \cost$. So far, the class of $K$-trivial sets has been the only known natural class that is characterized by a single cost function. However, recent work with Greenberg and Miller suggests that for a c.e.\ set $A$, being below both halves $Z_0, Z_1$ of some \ML-random $Z = Z_0 \oplus Z_1$ is equivalent to obeying the cost function $\cc(x,s) = \sqrt{\Om_s - \Om_x}$.
\subsection{Basic properties of the class of sets obeying a cost function}
In this subsection, unless otherwise stated, cost functions will be monotonic.
Recall from Definition~\ref{def:proper cf} that a cost function~$\cc$ is called \emph{proper} if $\fao x \exo t \cc(x,t) > 0$.
We investigate the class of models of a proper cost function~$\cc$. We also assume Convention~\ref{rmk:all finite} that $\ul \cc (x)< \infty$ for each $x$.
The first two results together show that $A\models \cc$ implies that $A$ is weak truth-table below a c.e.\ set $C$ such that $C \models \cc$.
Recall that a $\DII$ set $A$ is called $\omega$-c.e.\ if there is a computable approximation $\seq{A_s}$ such that the number of changes $\# \{s\colon \, A_s(x) \neq A_{s-1}(x)\}$ is computably bounded in $x$; equivalently, $A \lwtt \Halt$ (see \cite[1.4.3]{Nies:book}).
\begin{fact} Suppose that $\cc$ is a proper monotonic cost function. Let $A \models \cc$. Then $A$ is $\omega$-c.e. \end{fact}
\begin{proof} Suppose $\seq{A_s} \models \cc$. Let $g$ be the computable function given by $g(x) = \mu t. \, \cc(x,t) > 0$. Let $\hat A_s(x) = A_{g(x)} (x)$ for $s< g(x)$, and $\hat A_s(x) = A_s(x) $ otherwise. Then the number of times $\hat A_s(x)$ can change is bounded by $\cc {\seq{A_s}} / \cc(x,g(x))$. \end{proof}
Let $V_e$ denote the $e$-th $\omega$-c.e.\ set (see \cite[pg.\ 20]{Nies:book}).
\begin{fact} \label{ex:wtt index cost} For each cost function $\cc$, the index set $\{e\colon \, V_e \models \cc\}$ is $\SI{3}$.
\end{fact}
\begin{proof} Let $D_n$ denote the $n$-th finite set of numbers. We may view the $i$-th partial computable function $\Phi_i$ as a (possibly partial) computable approximation $\seq {A_t}$ by letting $A_t \simeq D_{\Phi_i(t)}$ (the symbol $\simeq$ indicates that 'undefined' is a possible value). Saying that $\Phi_i$ is total and a computable approximation of $V_e$ is a ~$\PI{2}$ condition of~$i$ and~$e$. Given that~$\Phi_i$ is total, the condition that $\seq{A_t}\models \cc$ is~$\SI{2}$.
\end{proof}
The {\it change set} (see \cite[1.4.2]{Nies:book}) for a {computable approximation} $\seq{A_s}_{s\in \NN}$ of a $\DII$ set~$A$ is a c.e.\ set $C\ge_T A$ defined as follows: if $s>0$ and $A_{s-1}(x)\neq A_{s}(x)$ we put $\la x, i \ra$ into $C_{s}$, where $i $ is least such that $\la x, i \ra \not \in C_{s-1}$. If~$A$ is $\omega$-c.e.\ via this approximation then $C \ge_{tt} A$. The change set can be used to prove the implication of the Shoenfield Limit Lemma that $A \in \DII$ implies $A\leT \Halt$; moreover, if $A$ is $\omega$-c.e., then $A \lwtt \Halt$.
\begin{prop}[\cite{Nies:book}, Section 5.3] \label{prop:c.e. ub cf} Let the cost function $\cc$ be non-increasing in the first component.
If a computable approximation $\seq{A_s}_{s\in \NN}$ of a set~$A$ obeys~$\cc$, then its change set $C$ obeys~$\cc$ as well. \end{prop}
\begin{proof} Since $x < \la x,i\ra $ for each $x,i$, we have
\bc $C_{s-1}(x)
\neq C_s(x) \ria A_{s-1}\uhr x \neq A_s \uhr x$ \ec for each $x,s$.
Then, since $ \cc(x,s)$ is non-increasing in~$x$, we have $\cc \seq{C_s} \le \cc {\seq{A_s}} < \infty$. \end{proof}
This yields a limitation on the expressiveness of cost functions. Recall that $A$ is superlow if $A' \ltt \Halt$.
\begin{cor} \label{ex:superlow cost} There is no cost function $\cc$ monotonic in the first component such that $A \models \cc$ iff $A$ is superlow. \end{cor}
\begin{proof} Otherwise, for each superlow set~$A$ there is a c.e.\ superlow set $C\geT A$.
This is clearly not the case: for instance~$A$ could be ML-random, and hence of diagonally non-computable degree, so that any c.e.\ set $C \ge_T A$ is Turing complete. \end{proof}
For $X \sub \NN$ let $2X$ denote $\{2x\colon \, x \in X\}$. Recall that $A \oplus B = 2A \cup (2B +1)$.
We now show that the class of sets obeying $\cc$ is closed under $\oplus$ and closed downward under a restricted form of weak truth-table reducibility.
Clearly, $E \models \cc \lland F \models \cc$ implies $E \cup F \models \cc$.
\begin{prop} Let the cost function $\cc$ be monotonic in the first component. Then $A \models \cc \lland B \models \cc$ implies $A \oplus B \models \cc$. \end{prop}
\begin{proof} Let $\seq{A_s} $ by a computable appoximation of $A$.
By the monotonicity of $\cc$ we have $ \cc {\seq{A_s}} \ge \cc {(2A_s)} $. Hence $2A\models \cc$. Similarly, $2B +1 \models \cc$. Thus $A\oplus B \models \cc$. \end{proof}
Recall that there are superlow c.e.\ sets $A_0,A_1$ such that $A_0 \oplus A_1$ is Turing complete (see \cite[6.1.4]{Nies:book}). Thus the foregoing result yields a a stronger form of Cor.\ \ref{ex:superlow cost}: no cost function characterizes superlowness within the c.e.\ sets.
\section{Look-ahead arguments} \label{s:look_ahead}
This core section of the paper introduces an important type of argument. Suppose we want to construct a computable approximation of a set $A$ that obeys a given monotonic cost function. If we can anticipate that $A(x)$ needs to be changed in the future, we try to change it as early as possible, because earlier changes are cheaper. Such an argument will be called a \emph{look-ahead argument}. (Also see the remark before Fact~\ref{fa: enumlate}.) The main application of this method is to characterize logical properties of cost functions algebraically.
\subsection{Downward closure under $\le_{ibT}$}
Recall that $A \le_{ibT} B$ if $A\lwtt B$ with use function bounded by the identity. We now show that the class of models of $\cc$ is downward closed under $\le_{ibT}$.
\begin{prop} \label{prop:downward_closure} Let $\cc$ be a monotonic cost function. Suppose that $B\models \cc$ and $A = \Gamma^B$ via a Turing reduction $\Gamma$ such that each oracle query on an input $x$ is at most $x$. Then $A \models \cc$. \end{prop}
\begin{proof}
Suppose $B \models \cc$ via a computable approximation $\seq {B_s} \sN s$. We define a computable increasing sequence of stages $\seq {s(i)} \sN i$ by $s(0) = 0$
and \bc $s(i+1) = \mu s > s(i) \, [ \Gamma^B \uhr {s(i)}[s] \DA ]$. \ec
In other words, $s(i+1)$ is the least stage $s$ greater than $s(i)$ such that at stage~$s$, $\Gamma^B(n)$ is defined for each $n < s(i)$.
We will define $A_{s(k)}(x) $ for each $k\in \NN$. Thereafter we let $A_s(x) = A_{s(k)}(x)$ where $k$ is maximal such that $s(k) \le s$.
Suppose $s(i) \le x < s(i+1)$. For $k<i$ let $A_{s(k)}(x)=v$, where $v =\Gamma^B(x)[s(i+2)]$. For $k \ge i$, let $A_{s(k)}(x)= \Gamma^B(x)[s(k+2)]$. (Note that these values are defined. Taking the $\Gamma^B(x)$ value at the large stage $s(k+2)$ represents the look-ahead.)
Clearly $\lim_s A_s(x) = A(x)$. We show that $\cc {\seq{A_s}} \le \cc {\seq{B_t}}$.
Suppose that $x$ is least such that $A_{s(k)}(x) \neq A_{s(k)-1}(x) $. By the use bound on the reduction procedure $\Gamma$, there is $y \le x$ such that $ B_t(y ) \neq B_{t-1}(y)$ for some~$t$, $ s(k+1)< t \le s(k+2)$. Then $ \cc(x, s(k)) \le \cc(y,t) $ by monotonicity of~$\cc$. Therefore $\seq { A_s} \models \cc$. \end{proof}
\subsection{Conjunction of cost functions} In the remainder of this section we characterize conjunction and implication of monotonic cost functions algebraically. Firstly,
we show that a set $A$ is a model of $\cc$ and $\dd$ if and only if $A$ is a model of $\cc + \dd$. Then we show that $\cc$ implies $\dd$ if and only if $\ul \dd = O(\ul \cc)$.
\begin{thm} \label{thm:conj} Let $\cc,\dd $ be monotonic cost functions. Then
\bc $A \models \cc \lland A \models \dd \LLR A \models \cc+\dd$. \ec \end{thm}
\begin{proof} \lapf This implication is trivial.
\n \rapf We carry out a look-ahead argument of the type introduced in the proof of Proposition~\ref{prop:downward_closure}. Suppose that $\seq {E_s}\sN s$ and $\seq {F_s} \sN s$ are computable approximations of a set $A$ such that $\seq {E_s} \models \cc$ and $\seq {F_s} \models \dd$. We may assume that $E_s(x) = F_s(x) =0$ for $s<x$ because changing $E(x)$, say, to $1$ at stage $x$ will not increase the cost as $\cc(x,s) = 0$ for $x>s$. We define a computable increasing sequence of stages $\seq {s(i)} \sN i$ by letting $s(0) = 0$
and \bc $s(i+1) = \mu s > s(i) \, [ E_s \uhr {s(i)} = F_s \uhr {s(i)} ]$. \ec
We define $A_{s(k)}(x) $ for each $k\in \NN$. Thereafter we let $A_s(x) = A_{s(k)}(x)$ where $k$ is maximal such that $s(k) \le s$.
Suppose $s(i) \le x < s(i+1)$. Let $A_{s(k)}(x)=0$ for $k<i$. To define $A_{s(k)}(x)$ for $k \ge i$, let $j(x)$ be the least $j \ge i$ such that $v= E_{s(j+1)}(x)= F_{s(j+1)}(x)$.
\bc $A_{s(k)}(x) = \begin{cases} v & \text{if} \ i \le k \le j(x) \\
E_{s(k+1)}(x) = F_{s(k+1)}(x) & \text{if} \, k > j(x). \end{cases}$ \ec
Clearly $\lim_s A_s(x) = A(x)$. To show $(\cc+\dd) \seq{A_s} < \infty$, suppose that $A_{s(k)}(x) \neq A_{s(k)-1}(x) $. The only possible cost in the case $ i \le k \le j(x) $
is at stage $s(i)$ when $v=1$. Such a cost is bounded by $\tp{-x}$. XX Now consider a cost in the case $k > j(x)$. There is a least $y $ such that $ E_t(y ) \neq E_{t-1}(y)$ for some $t$, $ s(k)< t \le s(k+1)$. Then $y \le x$, whence $ \cc(x, s(k)) \le \cc(y,t) $ by the monotonicity of $\cc$. Similarly, using $\seq{F_s}$ one can bound the cost of changes due to $\dd$. Therefore
$ (\cc+\dd){\seq{A_s}} \le 4+ \cc \seq {E_s} + \dd {\seq {F_s}} < \infty$.
\end{proof}
\subsection{Implication between cost functions}
\begin{definition} \label{def:cf_implication} {\rm For cost functions $\cc$ and $\dd $, we write $\cc \ria \dd$ if $A \models \cc$ implies $A \models \dd$ for each ($\DII$) set $A$.} \end{definition}
If a cost function $\mathbf \cc$ is monotonic in the stage component, then $\ul \cc(x) = \sup_s \cc(x,s)$. By Remark~\ref{rmk:all finite} we may assume $\ul \cc(x)$ is finite for each $x$.
We will show $\cc \ria \dd$ is equivalent to $ \ul \dd(x) = O( \ul \cc(x))$. In particular, whether or not $A \models \cc$ only depends on the limit function $ \ul \cc$.
\begin{thm} \label{thm:imply} Let $\cc,\dd$ be cost functions that are monotonic in the stage component. Suppose $\cc$ satisfies the limit condition in Definition~\ref{df:lc}. Then
\bc $\cc \ria \dd \LLR \exo N \fao x \big [ N \cc(x) > \dd(x) \big ]$. \ec \end{thm}
\begin{proof} \lapf
We carry out yet another look-ahead argument. We define a computable increasing sequence of stages $s(0) < s(1) < \ldots$ by $s(0) = 0$
and \bc $s(i+1) = \mu s > s(i) . \fao x < s(i) \, \big [N \cc(x,s) > \dd(x,s) \big ]$. \ec
Suppose $A$ is a $\DII$ set with a computable approximation $\seq{A_s} \models \cc$. We show that $\seq {\wt A_t} \models \dd$ for some computable approximation $\seq { \wt A_t}$ of $A$. As usual, we define $\wt A_{s(k)}(x) $ for each $k\in \NN$. We then let $\wt A_s(x) = \wt A_{s(k)}(x)$ where $k$ is maximal such that $s(k) \le s$.
Suppose $s(i) \le x < s(i+1)$. If $k < i+1$ let $\wt A_{s(k)}(x)=A_{s(i+2)}(x)$. If $k\ge i+1$ let $\wt A_{s(k)}(x)=A_{s(k+1)}(x)$.
Given $k$, suppose that $x$ is least such that $\wt A_{s(k)}(x) \neq \wt A_{s(k)-1}(x)$. Let $i$ be the number such that $s(i) \le x < s(i+1)$. Then $k \ge i+1$. We have $A_t(x) \neq A_{t-1}(x)$ for some $t$ such that $s(k) < t \le s(k+1)$. Since $x< s(i+1) \le s(k)$, by the monotonicity hypothesis this implies $N \cc(x,t) \ge N \cc(x,s(k) )> \dd(x, s(k))$. So $\dd {\seq{\wt A_s}} \le N \cdot \cc {\seq{A_s}} < \infty$. Hence $A \models \dd$.
\vsp
\n \rapf Recall from the proof of Fact~\ref{ex:wtt index cost} that we view the $e$-th partial computable function $\Phi_e$ as a (possibly partial) computable approximation $\seq{B_t}$, where $B_t \simeq D_{\Phi_e(t)}$.
Suppose that $ \exo N \fao x \, \big [ N \ul \cc(x) > \ul \dd(x) \big ]$ fails. We build a set $A \models \cc$ such that for no computable approximation $\Phi_e$ of $A$ we have $\dd \, \Phi_e \le 1$. This suffices for the theorem by Fact~\ref{easy1}.
We meet the requirements
\bc $R_e\colon \, \Phi_e \ttext{is total and approximates} A \RRA \Phi_e \not \models \dd.$ \ec
The idea is to change $A(x)$ for some fixed $x$ at sufficiently many stages $s$ with $N \cc(x,s) < \dd(x,s)$, where $N$ is an appropriate large constant. After each change we wait for recovery from the side of $\Phi_e$. In this way our $\cc$-cost of changes to $A$ remains bounded, while the opponent's $\dd$-cost of changes to $\Phi_e$ exceeds~$1$.
For a stage $s$, we let $ \init_s(e) \le s$ be the largest stage such that $R_e$ has been initialized at that stage (or $0$ if there is no such stage). Waiting for recovery is implemented as follows. We say that $s$ is \emph{$e$-expansionary} if $s=\init_s(e)$, or $s> \init_s(e) $ and, where $u$ is the greatest $e$-expansionary stage less than $s$,
\bc $\ex t \in [u,s) \, [ \Phi_{e,s}(t) \DA \lland \Phi_{e,s}(t) \uhr u = A_u \uhr u]$. \ec
The strategy for $R_e$ can only change $A(x)$ at an $e$-expansionary stage $u$ such that $x< u$. In this case it preserves $A_u \uhr u$ until the next $e$-expansionary stage. Then, $\Phi_e $ also has to change its mind on~$x$: we have \bc $x\in \Phi_e(u-1) \leftrightarrow x \not \in \Phi_e(t)$ for some $t \in [u,s)$. \ec
We measure the progress of $R_e$ at stages $s$ via a quantity $ \aaa_s(e)$. When $R_e$ is initialized at stage $s$, we set $ \aaa_s(e)$ to~$0$.
If $R_e$ changes $A(x)$ at stage $s$, we increase $ \aaa_s(e)$ by $\cc(x,s)$. $R_e$ is declared satisfied when $ \aaa_s(e)$ exceeds $\tp{-b-e}$, where $b$ is the number of times $R_e$ has been initialized.
\vsp
\n {\it Construction of $\seq{A_s}$ and $\seq { \aaa_s}$.}
Let $A_0= \ES$. Let $\aaa_{0}(e)=0$ for each $e$.
\n {\it Stage $s> 0$.} Let $e$ be least such that $s$ is $e$-expansionary and $\aaa_{ s-1}(e) \le \tp{-b-e}$ where $b$ is the number of times $R_e$ has been initialized so far. If $e$ exists do the following.
Let $x$ be least such that $\init_s(e) \le x < s $, $ \cc(x,s) < \tp{-b-e}$ and
\bc $\tp{b+e} \cc(x,s) < \dd(x,s)$. \ec
If $x$ exists let $A_s(x)= 1-A_{s-1}(x)$. Also let $A_s(y) = 0$ for $x< y < s$. Let $\aaa_s(e)= \aaa_{s-1}(e)+ \cc(x,s)$. Initialize the requirements $R_i$ for $i>e$ and let $\aaa_s(i)=0$. (This preserves $A_s\uhr s$ unless $R_e$ itself is later initialized.) We say that $R_e$ \emph{acts}.
\vsps
\n \verif If $s$ is a stage such that $R_e$ has been initialized for~$b$ times, then $\aaa_s(e) \le \tp{-b-e+1}$. Hence the total cost of changes of $A$ due to $R_e$ is at most $\sum_b \tp{-b-e+1}= \tp{-e+2}$. Therefore $\seq{A_s} \models \cc$.
We show that {\it each $R_e$ only acts finitely often, and is met.} Inductively, $\init_s(e)$ assumes a final value $s_0$. Let $b$ be the number of times $R_e$ has been initialized by stage $s_0$.
Since the condition $ \exo N \fao x [ N \ul \cc(x) > \ul \dd(x)]$ fails, there is $x \ge s_0$ such that for some $s_1 \ge x$, we have $\fa s \ge s_1 \, [\tp{b+e} \cc(x,s) < \dd(x,s)]$. Furthermore, since $\cc$ satisfies the limit condition, we may suppose that $ \ul \cc(x) < \tp{-b-e}$. Choose $x$ least.
If~$\Phi_e$ is a computable approximation of $A$, there are infinitely many $e$-expansionary stages $s \ge s_1$. For each such $s$, we can choose this $ x$ at stage $s$ in the construction. So we can add at least $ \cc(x,s_1)$ to $\aaa(e)$. Therefore $\aaa_t(e)$ exceeds the bound $\tp{-b-e}$ for some stage $t\ge s_1$, whence $R_e$ stops acting at $t$. Furthermore, since $\dd$ is monotonic in the second component and by the initialization due to $R_e$, between stages $s_0$ and $t$ we have caused $\dd \, \Phi_e $ to increase by at least $\tp{b+e} \aaa_t(e) > 1$. Hence $R_e$ is met.
\end{proof}
The foregoing proof uses in an essential way the ability to change $A(x)$, for the same $x$, for a multiple number of times. If we restrict implication to c.e.\ sets, the implication from left to right in Theorem~\ref{thm:imply} fails. For a trivial example, let $ \cc(x,s)= 4^{-x}$ and $ \dd(x,s) = \tp{-x}$. Then each c.e.\ set obeys $\dd$, so $\cc \to \dd$ for c.e.\ sets. However, we do not have $ \dd(x)= O( \cc(x))$.
We mention that Melnikov and Nies (unpublished, 2010) have obtained a sufficient algebraic condition for the non-implication of cost functions via a c.e.\ set. Informally speaking, the condition $ \dd(x)= O( \cc(x))$ fails ``badly''.
\begin{proposition}
Let $\cc$ and $\dd$ be monotonic cost functions satisfying the limit condition such that
$\sum_{x \in \NN} \ul \dd(x) = \infty$ and, for each $N>0$,
\[ \sum \ul \dd (x) \Cyl{ N\ul \cc (x)> \ul \dd (x)} < \infty. \]
Then there exists a c.e. set $A$ that obeys $\cc$, but not $\dd$.
\end{proposition}
The hope is that some variant of this will yield an algebraic criterion for cost function implication restricted to the c.e.\ sets.
\section{Additive cost functions} \label{s:additive cf}
We discuss a class of very simple cost functions introduced in \cite{Nies:ICM}. We show that a $\DII$ set obeys all of them if and only if it is~$K$-trivial. There is a universal cost function of this kind, namely $ \cc(x,s) = \Om_s - \Om_x$. Recall Convention~\ref{rmk:all finite} that $\ul \cc(x)< \infty$ for each cost function $\cc$.
\begin{deff}[\cite{Nies:ICM}] We say that a cost function $\cc$ is \emph{additive} if $\cc (x,s) =0$ for $x >s $, and for each $x< y<z$ we have
\bc $ \cc(x,y) + \cc(y,z)= \cc(x,z)$. \ec \end{deff}
Additive cost functions correspond to nondecreasing effective sequences $\seq {\beta_s}\sN s$ of non-negative rationals, that is, to effective approximations of left-c.e.\ reals $\beta$. Given such an approximation $\langle \beta \rangle = \seq {\beta_s}\sN s$, let for $x \le s$
\bc $\cc_{\langle \beta \rangle} (x,s) = \beta_s - \beta_x$. \ec
Conversely, given an additive cost function $\cc$, let $\beta_s= \cc(0,s)$. Clearly the two effective transformations are inverses of each other.
\subsection{$K$-triviality and the cost function $\cc_{\seq \Om}$}
The standard cost function $\cost$ introduced in (\ref{eqn:stcf}) is \emph{not} additive. We certainly have $\cost(x,y) + \cost(y,z)\le \cost(x,z)$, but by stage $z$ there could be a shorter description of, say, $x+1$ than at stage $y$, so that the inequality may be proper. On the other hand, let $g$ be a computable function such that $\sum_w \tp{-g(w)} < \infty$; this implies that $K(x) \lep g(x)$. The ``analog'' of $\cost $ when we write $g(x) $ instead of $K_s(x)$, namely $\cc_g(x,s)= \sum_{w=x+1}^s \tp{-g(w)}$ is an additive cost function.
Also, $\cost$ is dominated by an additive cost function $\cc_{\langle \Om \rangle}$ we introduce next. Let $\UM$ be the standard universal prefix-free machine (see e.g.\ \cite[Ch.\ 2]{Nies:book}). Let $\langle \Om \rangle$ denote the computable approximation of $\Om$ given by $\Om_s = \leb \, \dom (\UM_s)$. (That is, $\Om_s$ is the Lebesgue measure of the domain of the universal \PF machine at stage~$s$.)
\begin{fact} \label{fa:Omfact} For each $x\le s$, we have $\cost(x,s) \le \cc_{\seq \Om}(x,s) = \Om_s - \Om_x$. \end{fact}
\begin{proof} Fix $x$. We prove the statement by induction on $s\ge x$. For $s=x$ we have $\cost(x,s) = 0$. Now \bc $\cost(x,s+1) - \cost(x,s) = \sum_{ w =x+1 }^{s+1} 2^{-K_{s+1}(w)}- \sum_{ w =x+1 }^s 2^{-K_s(w)} \le \Om_{s+1} - \Om_s$, \ec because the difference
is due to convergence at stage $s$ of new $\UM$-computations.
\end{proof}
\begin{thm} Let $A$ be $\DII$. Then the following are equivalent.
\bi \itone $A$ is $K$-trivial.
\ittwo $A$ obeys each additive cost function.
\itthree $A$ obeys $\cc_{\seq \Om}$, where $\Om_s = \leb \dom ( \UM_s)$. \ei \end{thm}
\begin{proof}
(ii) $\ria$ (iii) is immediate, and (iii) $\ria$ (i) follows from Fact~\ref{fa:Omfact}. It remains to show (i)$\ria$(ii).
Fix some computable approximation $\seq{A_s}\sN{s}$ of~$A$. Let $\cc$ be an additive cost function. We may suppose that $ \ul \cc(0) \le 1$.
For $w>0$ let $r_w \in \NN \cup {\infty} $ be least such that $\tp{-r_w} \le c(w-1,w)$ (where $\tp{-\infty } =0$). Then $\sum_w \tp{-r_w} \le 1$. Hence by the Machine Existence Theorem we have $K(w)\lep r_w$ for each $w$. This implies $\tp{-r_w} = O(\tp{-K(w)})$, so $\sum_{w> x} \tp{-r_w} = O(\ul \cost(x))$ and hence $\ul \cc(x) = \sum_{w>x} c(w-1,w) = O(\ul \cost(x))$. Thus $\cost \to \cc$ by Theorem~\ref{thm:imply}, whence the $K$-trivial set $A$ obeys $\cc$. (See \cite{Bienvenu.Greenberg.ea:16} for a proof not relying on Theorem~\ref{thm:imply}.)
\end{proof}
Because of Theorem~\ref{thm:imply}, we have $\cc_{\seq \Om} \leftrightarrow \cost$. That is,
\bc $\Om - \Om_x \sim \sum_{w= x+1}^\infty \tp{-K(w)}$. \ec
This can easily be seen directly: for instance, $\cost \le \cc_{\seq \Om}$ by Fact~\ref{fa:Omfact}.
\subsection{Solovay reducibility}
Let $\Qbinary $ denote the dyadic rationals, and let the variable $ q$ range over $\Qbinary$. Recall Solovay reducibility on left-c.e.\ reals: $\beta \le_S \aaa$ iff there is a partial computable $\phi \colon \, \Qbinary \cap [0,\aaa) \ria \Qbinary \cap [0,\beta) $ and $N \in \NN$ such that \bc $\fao q < \aaa \big [ \beta - \phi(q) < N(\aaa - q)\big ]$. \ec Informally, it is easier to approximate $\beta$ from the left, than $\aaa$. See e.g.\ \cite[3.2.8]{Nies:book} for background.
We will show that reverse implication of additive cost functions corresponds to Solovay reducibility on the corresponding left-c.e.\ reals. Given a left-c.e.\ real $\gamma$, we let the variable $\seq \gamma$ range over the nondecreasing effective sequences of rationals converging to $\gamma$.
\begin{prop} \label{prop:heyheyey} Let $\aaa, \beta$ be left-c.e.\ reals. The following are equivalent.
\bi \itone $\beta \le_S \aaa$
\ittwo $ \fa \seq \aaa \ex \seq \beta \, [c_{\seq \aaa} \ria c_{\seq \beta}]$
\itthree $ \ex \seq \aaa \ex \seq \beta \, [ c_{\seq \aaa} \ria c_{\seq \beta}] $. \ei
\end{prop}
\begin{proof}
\n (i) $\ria$ (ii). Given an effective sequence $\seq \aaa$, by the definition of $\le_S$ there is an effective sequence~$\seq \beta$ such that $\beta - \beta_x = O(\aaa - \aaa_x)$ for each $x$. Thus $\ul \cc_{\seq \beta} = O( \ul \cc_{\seq \aaa} )$. Hence $\cc_{\seq \aaa } \ria \cc_{\seq \beta}$ by Theorem~\ref{thm:imply}.
\vsps
\n (iii) $\ria$ (i). Suppose we are given $\seq \aaa$ and $\seq \beta$ such that $\ul \cc_{\seq \beta} = O( \ul \cc_{\seq \aaa} )$.
Define a partial computable function $\phi$ by $\phi(q) = \beta_x$ if $\aaa_{x-1} \le q < \aaa_x$. Then $\beta \le_S \aaa$ via $\phi$.
\end{proof}
\subsection{The strength of an additive cost function}
Firstly, we make some remarks related to Proposition~\ref{prop:heyheyey}. For instance, it implies that an additive cost function can be weaker than $\cc_{\la \Om \ra}$ without being obeyed by all the~$\DII$ sets.
\begin{prop} There are additive cost functions $\cc,\dd$ such that $\cc_{\la \Om \ra} \ria \cc$, $\cc_{\la \Om \ra} \ria \dd$ and $\cc,\dd$ are incomparable under the implication of cost functions. \end{prop}
\begin{proof} Let $\cc,\dd$ be cost functions corresponding to enumerations of Turing (and hence Solovay) incomparable left-c.e.\ reals. Now apply Prop.~\ref{prop:heyheyey}. \end{proof}
Clearly, if $\beta$ is a computable real then any c.e.\ set obeys $\cc_{\la \beta \ra}$. The intuition we garner from Prop.\ \ref{prop:heyheyey} is that a more complex left-c.e.\ real $\beta$ means that the sets $A \models \cc_{\la \beta \ra}$ become less complex, and conversely. We give a little more evidence for this principle: if $\beta$ is non-computable, we show that a set $A \models \cc_{\la \beta \ra}$ cannot be weak truth-table complete. However, we also build a non-computable $\beta$ and a c.e.\ Turing complete set that obeys $\cc_{\la \beta \ra}$
\begin{prop}
Suppose $\beta$ is a non-computable left-c.e.\ real and $A \models \cc_\seq \beta$.
Then $A$ is not weak truth-table complete.
\end{prop}
\begin{proof} Assume for a contradiction that $A$ is weak truth-table complete. We can fix a computable approximation $\seq {A_s}$ of $A$ such that $ \cc_\seq \beta \seq {A_s} \le 1$. We build a c.e.\ set $B$. By the recursion theorem we can suppose we have a weak truth-table reduction $\Gamma$ with computable use bound $g$ such that $B = \Gamma^A$. We build $B$ so that $\beta - \beta_{g(\tp{e+1})} \le \tp{-e}$, which implies that $\beta$ is computable.
Let $I_e = [2^e, 2^{e+1})$. If ever a stage $s$ appears such that $\beta_s - \beta_{g(\tp{e+1})} \le \tp{-e}$, then we start enumerating into $B \cap I_e$ sufficiently slowly so that $A \uhr {g(2^{e+1})}$ must change $2^e$ times. To do so, each time we enumerate into $B$, we wait for a recovery of $B = \Gamma^A$ up to $2^{(e+1)}$. The $A$-changes we enforce yield a total cost $>1$ for a contradiction.
\end{proof}
\begin{prop} There is a non-computable left-c.e.\ real $\beta$ and a c.e.\ set $A \models \cc_\seq \beta$
such that $A$ is Turing complete.
\end{prop}
\begin{proof} We build a Turing reduction $\Gamma$ such that $\Halt = \Gamma(A)$. Let $\gamma_{k,s}+1$ be the use of the computation $\Gamma^\Halt(k)[s]$. We view $\gamma_{k}$ as a movable marker as usual. The initial value is $\gamma_{k,0} = k$. Throughout the construction we maintain the invariant
\bc $\beta_s - \beta_{\gamma_{k,s}} \le \tp{-k}$. \ec
Let $\seq {\phi_e}$ be the usual effective list of partial computable functions. By convention, at each stage at most one computation $\phi_e(k)$ converges newly.
To make $\beta$ non-computable, it suffices to meet the requirements
\bc $R_k\colon \, \phi_k(k) \DA \RRA \beta - \beta_{\phi_k(k)} \ge \tp{-k}$. \ec
\n {\it Strategy for $R_k$.} If $\phi_k(k) $ converges newly at stage $s$, do the following.
\bi \item[1.] Enumerate $\gamma_{k,s}$ into $A$. (This incurs a cost of at most $\tp{-k}$.)
\item[2.] Let $\beta_s = \beta_{s-1} + \tp{-k}$.
\item[3.] Redefine $\gamma_i$ ($i\ge k$) to large values in an increasing fashion.
\ei
In the construction, we run the strategies for the $R_k$. If $k$ enters $\Halt$ at stage $s$, we enumerate $\gamma_{k,s}$ into $A$.
Clearly each $R_k$ acts at most once, and is met. Therefore $\beta$ is non-computable. The markers $\gamma_k$ reach a limit. Therefore $\Halt = \Gamma(A)$. Finally, we maintain the stage invariant, which implies that the total cost of enumerating $A$ is at most $4$.
\end{proof}
As pointed out by Turetsky, it can be verified that $\beta$ is in fact Turing complete.
Next, we note that if we have two computable approximations from the left of the same real, we obtain additive cost functions with very similar classes of models.
\begin{prop} Let $\seq {\aaa}, \seq \beta$ be left-c.e.\ approximations of the same real. Suppose that $A \models \cc_{\seq \aaa}$. Then there is $B \equiv_m A$ such that $ B \models \cc_{\seq \beta}$. If $A$ is c.e., then $B$ can be chosen c.e.\ as well. \end{prop}
\begin{proof} Firstly, suppose that $A$ is c.e. By Fact~\ref{fact:fukd} choose a computable enumeration $\seq{A_s} \models \cc_{\seq \aaa}$.
By the hypothesis on the sequences $\seq {\aaa}$ and $ \seq \beta$, there is a computable sequence of stages $s_0 < s_1 < \ldots$ such that $|\aaa_{s_i} - \beta_{s_i}| \le \tp{-i}$. Let $f$ be a strictly increasing computable function such that $\aaa_x \le \beta_{f(x)}$ for each~$x$.
To define $B$, if $x $ enters $A$ at stage $s$, let $i$ be greatest such that $s_i \le s$. If $f(x) \le s_i$ put $f(x)$ into $B$ at stage $s_i$.
Clearly \bc $\aaa_s - \aaa_x \ge \aaa_{s_i} - \aaa_x \ge \aaa_{s_i} - \beta_{f(x)} \ge \beta_{s_i} - \beta_{f(x) } - \tp{-i}$. \ec So $ \cc_{\seq \beta} \seq {B_s} \le \cc_{\seq \aaa} \seq {A_s}+ \sum_i \tp{-i}$.
Let $R$ be the computable subset of $A$ consisting of those $x$ that are enumerated early, namely $x $ enters $A$ at a stage $s$ and $ f(x) > s_i$ where $i$ is greatest such that $s_i \le s$. Clearly $B = f(A-R)$. Hence $B \equiv_m A$.
The argument can be adapted to the case that $A$ is $\DII$. Given a computable approximation $\seq{A_s}$ obeying $\cc_{\seq \aaa }$, let $t$ be the least $s_i$ such that $s_i \ge f(x)$. For $s \le t$ let $B_s(f(x)) = A_t(x)$. For $s > t$ let $B_s(f(x)) = A_{s_i}(x)$ where $s_i \le s < s_{i+1}$. \end{proof}
\section{Randomness, lowness, and $K$-triviality}
\label{s:random_low_K-triv}
Benign cost functions were briefly discussed in the introduction.
\begin{deff}[\cite{Greenberg.Nies:11}] \label{df:benign} {\rm A monotonic cost function $\cc$ is called {\it benign} if there is a computable function~$g$ such that for all $k$,
\bc $x_0< x_1 < \ldots < x_k \lland \fa i < k \, [ \cc(x_i, x_{i+1}) \ge \tp{-n}]$ implies $k \le g(n)$. \ec } \end{deff}
Clearly such a cost function satisfies the limit condition. Indeed, $\cc$ satisfies the limit condition if and only if the above holds for some $g \leT \Halt$. For example, the cost function $\cost $ is benign via $g(n) = 2^n$.
Each additive cost function is benign where $g(n) = O(2^n)$. For more detail see \cite{Greenberg.Nies:11} or~\cite[Section~8.5]{Nies:book}.
For definitions and background on the extreme lowness property called strong jump traceability, see \cite{Greenberg.Nies:11,Greenberg.Hirschfeldt.ea:12} or \cite[Ch.\ 8 ]{Nies:book}.
We will use the main result in \cite{Greenberg.Nies:11} already quoted in the introduction: a c.e.\ set $A$ is strongly jump traceable iff $A$ obeys each benign cost function.
\subsection{A cost function implying strong jump traceability}
The following type of cost functions first appeared in \cite{Greenberg.Nies:11} and \cite[Section 5.3]{Nies:book}. Let $Z \in \DII$ be ML-random. Fix a computable approximation $\seq{Z_s}$ of $Z$ and let $\cc_Z$ (or, more accurately, $\cc_\seq {Z_s}$) be the cost function defined as follows. Let
$\cc_Z(x,s) = \tp{-x} $ for each $x\ge s$; if $x<s$, and~$e< x$ is
least such that
$Z_{s-1} ( e) \neq Z_s ( e)$, we let
\begin{equation} \label{eqn: cZ} c_Z(x,s) =
\max(c_Z(x,s-1), \tp{-e} ). \end{equation} Then $A \models \cc_Z$ implies $A \leT Z$ by the aforementioned result
from \cite{Greenberg.Nies:11}, which is proved like its variant above.
A {Demuth test} is a sequence of c.e.\ open sets $(S_m)\sN{m}$ such that
\bi \item $\fao m \leb S_m \le \tp{-m}$, and there is a function $f$ such that $S_m $ is the $\SI 1$ class $\Opcl{W_{f(m)}}$; \item $f(m) = \lim_s g(m,s)$ for a computable function $g$ such that the size of the set $\{s\colon \, g(m,s) \neq g(m,s-1)\}$ is bounded by a computable function $h(m)$. \ei
A set~$Z$ {passes} the test if $ Z\not \in S_m$ for almost every~$m$.
We say that~$Z$ is {Demuth random} if~$Z$ passes each Demuth test.
For background on Demuth randomness see \cite[pg.\ 141]{Nies:book}.
\begin{prop} Suppose $Y$ is a Demuth random $\DII$ set and $A \models c_Y$. Then $A \leT Z$ for each $\omega$-c.e.\ ML-random set $Z$. \end{prop}
In particular, $A$ is strongly jump traceable by \cite{Greenberg.Hirschfeldt.ea:12}.
\begin{proof} Let $G^s_e = [ Y_t \uhr e]$ where $t\le s $ is greatest such that $Z_t(e) \neq Z_{t-1}(e)$. Let $G_e = \lim_s G^s_e$. (Thus, we only update $G_e$ when $Z(e)$ changes.) Then $(G_e) \sN e$ is a Demuth test. Since $Y$ passes this test, there is $e_0$ such that
\bc $\fa e \ge e_0 \, \fao t [ Z_t(e) \neq Z_{t-1}(e) \ria \ex s > t \ Y_{s-1} \uhr e \neq Y_s \uhr e] $. \ec
We use this fact to define a computable approximation $(\hat Z_u)$ of $Z$ as follows: let $\hat Z_u(e) = Z(e)$ for $e \le e_0$; for $e > e_0$ let $\hat Z_u(e) = Z_s(e)$ where $s\le u$ is greatest such that $Y_{s-1} \uhr e \neq Y_s \uhr e $.
Note that $\cc_{\hat Z}(x,s) \le c_Y(x,s)$ for all $x, s$. Hence $A \models \cc_{\hat Z}$ and therefore $A \leT Z$. \end{proof}
Recall that some Demuth random set is $\DII$. \Kuc\ and Nies \cite{Kucera.Nies:11} in their main result strengthened the foregoing proposition in the case of a c.e.\ sets $A$: if $A \leT Y$ for some Demuth random set $Y$, then $A$ is strongly jump traceable. Greenberg and Turetsky \cite{Greenberg.Turetsky:14} obtained the converse of this result: every c.e.\ strongly jump traceable is below a Demuth random.
\begin{remark} \label{spoon} {\rm For each $\DII$ set $Y$ we have $\cc_Y(x)= \tp{-F(x)}$ where $F$ is the $\DII$ function such that \bc $F(x) = \min \{e\colon \, \ex s >x \, Y_s(e) \neq Y_{s-1}(e)\}$. \ec Thus $F$ can be viewed as a modulus function in the sense of~\cite{Soare:87}. }
\end{remark}
For a computable approximation $\Phi$ define the cost function $\cc_\Phi $ as in (\ref{eqn: cZ}). The following (together with Rmk.\ \ref{spoon}) implies that any computable approximation $\Phi$ of a ML-random Turing incomplete set changes late at small numbers, because the convergence of $\Om_s$ to $\Om$ is slow.
\begin{cor}
Let $Y <_T \Halt $ be a ML-random set. Let $\Phi$ be any computable approximation of $Y$. Then $\cc_{\Phi} \ria \cost$ and therefore $O(c_\Phi(x)) = \cc_{\seq \Om}(x)$. \end{cor}
\begin{proof} If $A \models \cc_\Phi$ then $C \models \cc_\Phi$ where $C\geT A$ is the change set of the given approximation of $A$ as in Prop.\ \ref{prop:c.e. ub cf}. By \cite{Hirschfeldt.Nies.ea:07} (also see \cite[5.1.23]{Nies:book}), $C$ and therefore $A$ is $K$-trivial. Hence $A \models \cc_{\seq \Om}$. \end{proof}
\subsection{Strongly jump traceable sets and d.n.c.\ functions}
Recall that we write $X \le_{ibT} Y$ if $X \leT Y$ with use function bounded by the identity. When building prefix-free machines, we use the terminology of \cite[Section 2.3]{Nies:book} such as Machine Existence Theorem (also called the Kraft-Chaitin Theorem), bounded request set etc.
\begin{thm} \label{thm:Kuc-g} Suppose an $\omega$-c.e.\ set $Y$ is diagonally noncomputable via a function that is weak truth-table below $ Y$. Let $A$ be a strongly jump traceable c.e.\ set. Then $A \le_{ibT} Y$. \end{thm}
\begin{proof} By \cite{Kjos.ea:2005} (also see \cite[4.1.10]{Nies:book}) there is an order function $h$ such that $2h(n) \lep K(Y\uhr n)$ for each $n$.
The argument of the present proof goes back to \Kuc's injury free solution to Post's problem (see \cite[Section 4.2]{Nies:book}). Our proof is phrased in the language of cost functions, extending the similar result in \cite{Greenberg.Nies:11} where $Y$ is ML-random (equivalently, the condition above holds with $h(n)= \lfloor n/2 \rfloor +1$.
Let $\seq {Y_s}$ be a computable approximation via which $Y$ is $\omega$-c.e. To help with building a reduction procedure for $A \le_{ibT} Y$, via the Machine Existence Theorem we give prefix-free descriptions of initial segments $Y_s\uhr e$. On input $x$, if at a stage $s>x$, $e$ is least such that $Y(e)$ has changed between stages $x$ and $s$, then we still hope that $Y_s \uhr e$ is the final version of $Y \uhr e$. So whenever $A(x)$ changes at such a stage $s$, we give a description of $Y_s\uhr e$ of length $h(e)$. By hypothesis $A$ is strongly jump traceable, and hence obeys each benign cost function. We define an appropriate benign cost function $\cc$ so that a set $A $ that obeys~$\cc$ changes little enough that we can provide all the descriptions needed.
To ensure that $A \le_{ibT} Y$, we define a computation $ \Gamma(Y \uhr x) $ with output $A(x)$ at the least stage $t\ge x $ such that $Y_t\uhr x$ has the final value. If $Y$ satisfies the hypotheses of the theorem, $A(x) $ cannot change at any stage $s> t$ (for almost all $x$), for otherwise $Y \uhr e$ would receive a description of length $h(e)+O(1)$, where $e$ is least such that
$Y(e)$ has changed between $x$ and $s$.
We give the details. Firstly we give a definition of a cost
function $\cc$ which generalizes the definition in (\ref{eqn: cZ}). Let $ \cc(x,s) = 0 $ for each $x\ge s$. If $x<s$, and~$e< x$ is
least such that
$Y_{s-1} ( e) \neq Y_s ( e)$, let
\begin{equation} \label{eqn:defn of cY} \cc(x,s) =
\max( \cc(x,s-1), \tp{-h(e)} ). \end{equation}
Since $Y$ is $\omega$-c.e., $\cc$ is benign. Thus each strongly jump traceable c.e.\ set obeys $\cc$ by the main result in \cite{Greenberg.Nies:11}. So it suffices to show that $A \models \cc$ implies $A \le_{ibT} Y$ for any set $A$. Suppose that $\cc{\seq{A_s}} \le 2^u$. Enumerate a bounded request set~$L$ as follows. When $A_{s-1}(x)\neq A_s(x)$ and $e$ is least such that $e=x$ or $Y_{t-1} ( e) \neq Y_t ( e)$ for some $t \in [x,s)$, put the request $\la u+ h(e), Y_s \uhr e \ra$ into $L$. Then $L$ is indeed a bounded request set.
Let $d$ be a coding constant for $L$ (see \cite[Section 2.3]{Nies:book}). Choose $e_0$ such that $h(e) + u+ d < 2h(e)$ for each $e \ge e_0$. Choose $s_0\ge e_0$ such that $Y \uhr{e_0}$ is stable from stage $s_0$ on.
To show $A \le_{ibT} Y$, given an input $x\ge s_0$, using~$Y$ as an oracle, compute $t >x$ such that $Y_t \uhr x =Y \uhr x$.
We claim that $A(x) = A_t(x)$. Otherwise $A_{s}(x) \neq A_{s-1}(x)$ for some $s > t$. Let $e \le x$ be the largest number such that $Y_r\uhr e = Y _t \uhr e$ for all~$r$ with $t < r \le s$. If $e < x$ then $Y(e)$ changes in the interval $(t,s]$ of stages. Hence, by the choice of $t\ge s_0$, we cause $K(y) < 2h(e)$ where $y= Y_t \uhr {e} = Y\uhr {e}$, contradiction. \end{proof}
\begin{example} {\rm For each order function $h$ and constant $d$, the class
\bc $P_{h, d }= \{Y \colon \, \fao n 2h(n) \le K(Y\uhr n)+d \}$ \ec
is $\PPI$. Thus, by the foregoing proof, each strongly jump traceable c.e.\ set is \emph{ibT} below each $\omega$-c.e.\ member of $P_{h,d}$. } \end{example}
We discuss the foregoing Theorem~\ref{thm:Kuc-g}, and relate it to results in \cite{Greenberg.Hirschfeldt.ea:12,Greenberg.Nies:11}.
\vsp
\n 1. In \cite[Thm 2.9]{Greenberg.Hirschfeldt.ea:12} it is shown that given a non-empty $\PPI$ class $P$, each jump traceable set $A$ Turing below each superlow member of $P$ is already strongly jump traceable. In particular this applies to superlow c.e.\ sets $A$, since such sets are jump traceable \cite{Muenster}. For many non-empty $\PPI$ classes such a set is in fact computable. For instance, it could be a class where any two distinct members form a minimal pair. In contrast, the nonempty among the $\PPI$ classes $P=P_{h,d}$ are examples where being below each superlow (or $\omega$-c.e.) member characterizes strong jump traceability for c.e.\ sets.
\vsp
\n 2. Each superlow set $A$ is weak truth-table below {\it some} superlow set $Y$ as in the hypothesis of Theorem~\ref{thm:Kuc-g}. For let $P$ be the class of $\{0,1\}$-valued d.n.c.\ functions. By \cite[1.8.41]{Nies:book} there is a set $Z \in P$ such that $(Z \oplus A)' \ltt A'$. Now let $Y = Z \oplus A$. This contrasts with the case of ML-random covers: if a c.e.\ set $A$ is not $K$-trivial, then each ML-random set Turing above $A$ is already Turing above $\Halt$ by \cite{Hirschfeldt.Nies.ea:07}. Thus, in the case of {\it ibT} reductions, Theorem~\ref{thm:Kuc-g} applies to more oracle sets $Y$ than \cite[Prop.\ 5.2]{Greenberg.Nies:11}.
\vsp
\n 3.
Greenberg and Nies \cite[Prop.\ 5.2]{Greenberg.Nies:11} have shown that for each order function~$p$, each strongly jump traceable c.e.\ set is Turing below below each $\omega$-c.e.\ ML-random set, via a reduction with use bounded by $p$.
We could also strengthen Theorem~\ref{thm:Kuc-g} to yield such a ``$p$-bounded'' Turing reduction.
\iffalse
\subsection{More on benign cost functions}
In the following we strengthen a result of \cite{Greenberg.Nies:11} that a c.e.\ set obeying all benign cost functions is strongly jump traceable: we remove the hypothesis that the given set be c.e.\
\begin{thm}
Suppose a $\DII$ set $A$ obeys all benign cost functions. Then~$A$ is strongly jump traceable. \end{thm}
\begin{proof} One can derive the theorem from results in \cite{Greenberg.Hirschfeldt.ea:12}. For each $\omega$-c.e.\ set $Y$, the set $A$ obeys the benign cost function $\cc_Y$ defined in (\ref{eqn: cZ}). Hence $A \leT Y$. By \cite[Thm. 2.1]{Greenberg.Hirschfeldt.ea:12} this implies that $A$ is strongly jump traceable.
It is instructive to give a direct proof of the theorem, relying only on technical result from
\cite{Greenberg.Hirschfeldt.ea:12} about so-called restrained approximations. Since $A \models \cost$, $A$ is $K$-trivial. Hence $A$ is jump traceable and superlow.
Given an order function~$h$, we will define a jump trace $(T_n) \sN n$ with bound $h$ for~$A$. Define a cost function~$\cc$ similar to (\ref{eqn: cZ}): let $ \cc(x,s) =0 $ for each $x\ge s$. If $x<s$ and~$n<x$ is
least such that
$J^A(n) \DA [s]$ but $J^A(n) \UA [s-1]$, then let $ \cc(x,s) = \max( \cc(x,s-1), 1/h(n) )$.
Since $(A_s, \wt J_s)\sN s$ is a restrained approximation it is easy to see that~$\cc$ is benign. So choose a computable enumeration
$(\hat A_r)\sN{r}$ of~$A$
obeying~$\cc$.
\end{proof}
\fi
\subsection{A proper implication between cost functions}
In this subsection we study a weakening of $K$-triviality using the monotonic cost function
\[ \cc_{\max } (x, s) = \max \{\tp{-K_s(w)} \colon \, x < w \le s \}.\]
Note that $\cc_{\max }$ satisfies the limit condition, because \bc $\ul \cc_{\max }(x) =\max \{\tp{-K(w)} \colon \, x < w \} $. \ec
Clearly $\cc_{\max } (x,s) \le \cost(x,s) $, whence $\cost \ria \cc_{\max }$. We will show that this implication of cost functions is proper. Thus, some set obeys $\cc_{\max } $ that is not $K$-trivial.
Firstly, we investigate sets obeying $\cc_{\max }$. For a string $\aaa$, let $g(\aaa)$ be the longest prefix of $\aaa$ that ends in $1$, and $g(\aaa)= \estring$ if there is no such prefix.
\begin{definition} {\rm We say that a set $A$ is \emph{weakly $K$-trivial} if \bc $\fao n [ K(g(A\uhr n))\lep K(n)]$. \ec } \end{definition}
Clearly, every $K$-trivial set is weakly $K$-trivial. By the following, every \emph{c.e.}\ weakly $K$-trivial set is already $K$-trivial.
\begin{fact} If $A$ is weakly $K$-trivial and not h-immune, then $A$ is $K$-trivial. \end{fact}
\begin{proof} By the second hypothesis, there is an increasing computable function~$p$ such that $ [p(n), p(n+1)) \cap A \neq \ES $ for each~$n$. Then
\bc $K(A \uhr {p(n)})\lep K(g(A \uhr {p(n+1)}))\lep K(p(n+1))\lep K(p(n))$. \ec
This implies that $A$ is $K$-trivial by \cite[Ex.\ 5.2.9]{Nies:book}. \end{proof}
We say that a computable approximation $\seq {A_s} \sN s$ is \emph{erasing} if for each~$x$ and each $s>0$, $A_s(x) \neq A_{s-1}(x)$ implies $A_s(y)= 0 $ for each $y$ such that $x< y \le s$. For instance, the computable approximation built in the proof of the implication ``$\RA$'' of Theorem~\ref{thm:imply} is erasing by the construction.
\begin{prop} \label{prop:CMax_implies_weakly_K-trivial} Suppose $\seq {A_s} \sN s$ is an erasing computable approximation of a set $A$, and $\seq {A_s} \models \cc_{\max }$. Then $A$ is weakly $K$-trivial. \end{prop}
\begin{proof} This is a modification of the usual proof that every set $A$ obeying $ \cost$ is $K$-trivial (see, for instance, \cite[Thm.\ 5.3.10]{Nies:book}).
To show that $A$ is weakly $K$-trivial, one builds a bounded request set $W$. When at stage $s>0$ we have $r = K_s(n) < K_{s-1}(n)$, we put the request $\la r+1, g(A\uhr n)\ra$ into $W$. When $A_s(x) \neq A_{s-1}(x)$, let $r$ be the number such that $\cc_{\max }(x,s) = \tp{-r}$, and put the request $\la r+1, g(A\uhr {x+1})\ra$ into $W$.
Since the computable approximation $\seq {A_s} \sN s$ obeys $\cc_{\max }$, the set $W$ is indeed a bounded request set; since $\seq {A_s} \sN s$ is erasing, this bounded request set shows that $A$ is weakly $K$-trivial.
\end{proof}
We now prove that $\cc_{\max } \not \ria \cost$. We do so via proving a reformulation that is of interest by itself.
\begin{thm} \label{thm:cmax_cK_separation} For every $b \in \NN$ there is an $x$ such that $\ulcost(x) \ge \tp b \ul \cc_{\max }(x)$. In other words, \bc $ \sum \{\tp{-K(w)} \colon \, x < w \} \ge \tp b \max \{\tp{-K(w)} \colon \, x < w \} $. \ec \end{thm}
By Theorem~\ref{thm:imply}, the statement of the foregoing Theorem
is equivalent to $\cc_{\max } \not \rightarrow \cost$. Thus, as remarked above, some set $A$ obeys $\cc_{\max }$ via an erasing computable approximation, and does not obey $\cost$. By Proposition~\ref{prop:CMax_implies_weakly_K-trivial} we obtain a separation.
\begin{cor} Some weakly $K$-trivial set fails to be $K$-trivial. \end{cor}
Melnikov and Nies~\cite[Prop.\ 3.7]{Melnikov.Nies:12} have given an alternative proof of the preceding result by constructing a weakly $K$-trivial set that is Turing complete.
\begin{proof}[Proof of Theorem~\ref{thm:cmax_cK_separation}] Assume that there is $b \in \NN$ such that \bc $\fao x [\ulcost(x) < \tp b \ul \cc_{\max }(x)]$. \ec
To obtain a contradiction, the idea is that $\cost(x, s)$, which is defined as a sum, can be made large in many small bits; in contrast, $\cc_{\max }(x,s)$, which depends on the value $\tp{-K_s(w)}$ for a single $w$, cannot.
We will define a sequence $0=x_0 < x_1 < \ldots < x_N$ for a certain number $N$. When $x_v$ has been defined for $v<N$, for a certain stage $t> x_v$ we cause $\cost (x_v,t)$ to exceed a fixed quantity proportional to $1/N$. We wait until the opponent responds at a stage $s > t$ with some $w> x_v $ such that $\tp{-K_s(w)}$ corresponding to that quantity. Only then, we define $x_{v+1}=s$. For us, the cost $\cost(x_i,x_j)$ will accumulate for $i<j$, while the opponent has to provide a new~$w$ each time. This means that eventually he will run out of space in the domain of the prefix-free machine giving short descriptions of such $w$'s.
In the formal construction, we will build a bounded request set $L$ with the purpose to cause $\cost(x,s)$ to be large when it is convenient to us. We may assume by the recursion theorem that the coding constant for $L$ is given in advance (see \cite[Remark 2.2.21]{Nies:book} for this standard argument). Thus, if we put a request $\la n, y+1 \ra$ into $L$ at a stage $y$, there will be a stage $t>y$ such that $K_t(y+1 ) \le n+d$, and hence $\cost(x,t) \ge \cost (x,y) + \tp{-n-d}$.
Let $\k= \tp{b+d+1}$. Let $N= \tp \k$.
\vsp
\n \emph{Construction of $L$ and a sequence $0=x_0 < x_1 < \ldots < x_N$ of numbers.}
\vsps
\n Suppose $v< N$ and $x_v$ has already been defined. Put $\la \k, x_v+1 \ra$ into $L$. As remarked above, we may wait for a stage $t>x_v$ such that $\cost(x_v, t)\ge \tp{-\k-d}$. Now, by our assumption, we have $\ulcost(x_i) < \tp b \ul \cc_{\max }(x_i)$ for each $i \le v$. Hence we can wait for a stage $s> t$ such that
\begin{equation} \label{eqn:wait_cost_bounded}
\fa i \le v \, \exo w \big [ x_i < w \le s \lland \cost(x_i, s ) \le \tp{b -K_s(w)}]. \end{equation}
Let $x_{v+1}=s$. This ends the construction.
\vsp
\verif Note that $L$ is indeed a bounded request set. Clearly we have $\cost(x_i, x_{i+1}) \ge \tp{-\k-d}$ for each $i<N$.
\begin{claim} \label{claim:Dali} Let $r\le \k$. Write $R = \tp r$. Suppose $p+R \le N$. Let $s = x_{p+R}$. Then we have
\begin{equation} \label{eqn:claim_ineq} \sum_{w = x_p+1}^{x_{(p+R)}}\min (\tp{ - K_s(w)}, \tp{-\k-b-d+r}) \ge (r+1) \tp{-\k-b-d+r-1}.
\end{equation}
\end{claim}
\n For $r= \k$, the right hand side equals $(\k+1) \tp{-(b+d+1)}>1$, which is a contradiction because the left hand side is at most $\Om \le 1$.
\vsp
We prove the claim by induction on $r$.
To verify the case $r=0$, note that by (\ref{eqn:wait_cost_bounded}) there is $w \in (x_p, x_{p+1}]$ such that $\cost(x_p, x_{p+1}) \le \tp {b - K_s(w)}$. Since $\tp{-\k-d} \le \cost(x_p, x_{p+1})$, we obtain \bc $\tp{-\k - b - d} \le \tp{-K_s(w)}$ (where $s= x_{p+1}$). \ec Thus the left hand side in the inequality (\ref{eqn:claim_ineq}) is at least $\tp{-\k - b - d}$, while the right hand side equals $\tp{-\k - b - d-1}$, and the claim holds for $r=0$.
\vsps
In the following, for $i< j \le N$, we will write $\SS(x_i,x_j)$ for a sum of the type occurring in (\ref{eqn:claim_ineq}) where $w$ ranges from $x_i+1$ to $x_j$.
Suppose inductively the claim has been established for $r<\k$. To verify the claim for $r+1$, suppose that $p + 2R \le N$ where $R = \tp r$ as before. Let $s= x_{p+2R}$. Since $\cost(x_i, x_{i+1}) \ge \tp{-\k-d}$, we have
\bc $\cost(x_p,s) \ge 2R \tp{-k-d} = \tp{-\k-d+r+1}$. \ec
By (\ref{eqn:wait_cost_bounded}) this implies that there is $w$, $x_p< w \le s$, such that
\begin{equation} \label{eqn:hurray_w}\tp{-\k - b - d+r+1} \le \tp{-K_s(w)}. \end{equation}
Now, in sums of the form $\SS (x_q,x_{q+R})$, because of taking the minimum, the ``cut-off'' for how much $w$ can contribute is at $\tp{-\k-b-d+r}$. Hence we have
\bc $\SS(x_p, x_{p+2R}) \ge \tp{-\k-b-d+r} + \SS(x_p,x_{p+R}) + \SS(x_{p+R}, x_{p+2R}) $. \ec
The additional term $\tp{-\k-b-d+r}$ is due to the fact that $w$ contributes at most $\tp{-\k-b-d+r}$ to $ \SS(x_p,x_{p+R}) + \SS(x_{p+R}, x_{p+2R})$, but by (\ref{eqn:hurray_w}), $w$ contributes $\tp{-\k-b-d-r+1}$ to $\SS(x_p, x_{p+2R})$. By the inductive hypothesis, the right hand side is at least
\bc $\tp{-\k-b-d+r} + 2 \cdot (r+1) \tp{-\k-b-d+r-1}= (r+2) \tp{-\k-b-d+r}$, \ec as required.
\end{proof}
\section{A cost function-related basis theorem for $\PPI$ classes} \label{s:costf_basis_theorem}
The following strengthens \cite[Thm 2.6]{Greenberg.Hirschfeldt.ea:12}, which relied on the extra assumption that the $\PPI$ class is contained in the ML-randoms.
\begin{thm} Let $\P$ be a nonempty $\PPI$ class, and let~$\cc$ be a monotonic cost function with the limit condition. Then there is a $\DII$ set $Y \in \P$ such that each c.e.\ set $A \leT Y$ obeys $\cc$.
\end{thm}
\begin{proof} We may assume that $ \cc(x, s) \ge \tp{-x}$ for each $x\le s$, because any c.e.\ set that obeys $\cc$ also obeys the cost function $ \cc(x,s)+ \tp{-x}$.
Let $\seq{A_e, \Psi_e}\sN e$ be an effective listing of all pairs consisting of a c.e.\ set and a Turing functional. We will define a $\DII$ set $Y \in \P$ via a computable approximation ${Y_s} \sN s$, where~$Y_s$ is a binary string of length $s$. We meet the requirements
\bc $N_e\colon \, A_e = \Psi_e(Y) \RRA A_e $ obeys $ \cc$. \ec
We use a standard tree construction at the $\ES''$ level. Nodes on the tree $\strcantor$ represent the strategies. Each node $\aaa$ of length~$e$ is a strategy for $N_e$. At stage $s$ we define an approximation $\delta_s$ to the true path.
We say that $s$ is an \emph{$\aaa$-stage} if $\aaa \prec \delta_s$.
Suppose that a strategy $\aaa$ is on the true path. If $\aaa 0 $ is on the true path, then strategy~$\aaa$ is able to build a computable enumeration of $A_e$ via which $A_e$ obeys~$\cc$. If $\aaa 1$ is on the true path, the strategy shows that $A_e \neq \Psi_e(Y)$.
Let $\P^\estring $ be the given class $\P$. A strategy $\aaa$ has as an environment a $\PPI$ class $\P^\aaa$. It defines $\P^{\aaa 0} = \P^\aaa$, but usually let $\P^{\aaa 1}$ be a proper refinement of $\P^{\aaa}$.
Let $\aaal =e$. The length of agreement for $e$ at a stage $t$ is $\min\{ y \colon \, A_{e,t} (y) \neq \Psi_{e,t} (Y_t)\}$. We say that an $\aaa$-stage $s$ is $\aaa$-\emph{expansionary} if the length of agreement for $e$ at stage~$s$ is larger than at $u$ for all previous $\aaa$-stages $u $.
Let $w^n_0 = n$, and
\begin{equation} \label{eqn:wi} w^n_{i+1} \simeq \mu v > w^n_i . \, \cc(w^n_i, v) \ge 4^{-n}. \end{equation}
Since $\cc$ satisfies the limit condition, for each $n$ this sequence breaks off.
Let $a= w^n_i$ be such a value. The basic idea is to {\it certify} $A_{e,s} \uhr w$, which means to ensure that all $X \succ Y_s\uhr {n+d}$ on $\PP^\aaa$ compute $A_{e,s} \uhr w$. If $A \uhr w$ changes later then also $Y\uhr {n+d}$ has to change. Since $Y\uhr {n+d}$ can only move to the right (as long as $\aaa$ is not initialized), this type of change for $n$ can only contribute a cost of $4^{-n+1} \tp{n+d} = \tp{-n+d+2}$.
By \cite[p.\ 55]{Nies:book}, from an index $\Q$ for a $\PPI$ class in $\cantor$ we can obtain a computable sequence $(\Q_s) \sN s$ of clopen classes such that $\Q_s \supseteq \Q_{s+1}$ and $\Q = \bigcap_s \Q_s$. In the construction below we will have several indices for $\QI$ classes $\Q$ that change over time. At stage $s$, as usual by $\Q[s]$ we denote the value of the index at stage $s$. Thus $(\Q[s])_s$ is the clopen approximation of $\Q[s]$ at stage $s$.
\vsps
\n {\em Construction of} $Y$.
\n \emph{Stage $0$. Let $\delta_0 = \estring$ and $\P^\estring = \P$. Let $Y_0 = \estring$.}
\n \emph{Stage $s>0$.} Let $\P^\estring = \P$.
\n For each $\beta$ such that $\delta_{s-1} <_L \beta$ we initialize strategy~$\beta$. We let $Y_s$ be the leftmost path on the current approximation to $\P^{\delta_{s-1}}$, i.e., the leftmost string $y$ of length $s-1$ such that $[y ] \cap (\P^{\delta_{s-1}}[s-1])_{s} \neq \ES$. For each $\aaa, n$, if $Y_s \uhr {n+d} \neq Y_{s-1} \uhr {n+d} $ where $d= \init_s(\aaa) $, then we declare each existing value $w^n_i$ to be $(\aaa, n)$-\emph{unsatisfied}.
\n \emph{Substage $k$}, $0\le k < s$. Suppose we have already defined $\aaa = \delta_s \uhr k$. Run strategy~$\aaa$ (defined below) at stage $s$, which defines an outcome $r \in \twoset$ and a $\PPI$ class $\P^{\aaa r}$. Let $\delta_s(k) = r$.
We now describe the strategies $\aaa$ and procedures $\SS^\aaa_n$ they call. To initialize a strategy $\aaa$ means to cancel the run of this procedure. Let
\bc $d= \init_s(\aaa) = \aaal + $the last stage when $\aaa$ was initialized. \ec
\n \emph{Strategy $\aaa$ at an $\aaa$-stage $s$.}
\n \bi \item[(a)] If no procedure for $\aaa$ is running, call procedure $\SS^\aaa_n$ with parameter~$w$, where~$n$ is least, and $i$ is chosen least for $n$, such that $w= w^n_i\le s$ is not $(\aaa, n)$-satisfied. Note that $n$ exists because $w^s_0 =s$ and this value is not $(\aaa, n)$-satisfied at the beginning of stage $s$.
By calling this procedure, we attempt to certify $A_{e,s} \uhr w$ as discussed above.
\item[(b)] While such a procedure $\SS^\aaa_n$ is running, give outcome $ 1$.
\n (This procedure will define the current class $\P^{\aaa 1}$.)
\item[(c)] If a procedure $\SS^\aaa_n$ returns at this stage, goto (d).
\item[(d)] If $s$ is $\aaa$-expansionary, give outcome $0$, let $\P^{\aaa 0}= \P^\aaa$, and continue at (a) at the next $\aaa$-stage.
Otherwise, give outcome $ 1$, let $\P^{\aaa 1}= \P^\aaa$, and stay at (d). \ei
\n \emph{Procedure $\SS^\aaa_n$ with parameter~$w$ at a stage $s$.}
\n If $n+d \ge s-1$ let $\P^{\aaa 1} = \P^\aaa$. Otherwise, let
\begin{equation} \label{eqn:Q} \Q = \P^\aaa \cap \{X \succ z \colon \, \Psi^X_e \not \succ A_{e,s}\uhr w\},
\end{equation}
where $z = Y_{s}\uhr {n + d}$. (Note that each time $Y\uhr {n + d} $ or $A_{e}\uhr w$ has changed, we update this definition of $\Q$.)
\bi \item[(e)] If $\Q_s \neq \ES$ let $\P^{\aaa 1} = \Q$. If the definition of $\P^{\aaa 1}$ has changed since the last $\aaa$-stage, then
each $\beta$ such that $\aaa 1 \preceq \beta$ is initialized.
\item[(f)] If $\Q_s= \ES$, declare $w$ to be \emph{$(\aaa, n)$-satisfied} and return. ($A_{e,s} \uhr w$ is certified as every $X \in \P^\aaa$ extending $z$ computes $A_{e,s} \uhr w$ via $\Psi_e$. If $A_e \uhr w$ changes later, the necessarily $z \not \preceq Y$.)
\ei
\vsps
\begin{claim} \label{claim:1}
Suppose a strategy $\aaa$ is no longer initialized after stage $s_0$. Then for each $n$, a procedure $\SS^\aaa_n$ is only called finitely many times after $s_0$.
\end{claim}
There are only finitely many values $w=w^n_i$ because $\cc$ satisfies the limit condition. Since $\aaa$ is not initialized after $s_0$, $\P^\aaa$ and $d= \init_s(\aaa) $ do not change. When a run of $\SS^\aaa_n$ is called at a stage~$s$, the strategies $\beta \succeq \aaa1 $ are initialized, hence $\init_t(\beta) \ge s > n+d$ for all $t \ge s$.
So the string $Y_s\uhr {n+d}$ is the leftmost string of length $n+d$ on $\P^\aaa$ at stage $s$. This string has to move to the right between the stages when $\SS^\aaa_n$ is called with the same parameter $w$, because $w$ is declared $(\aaa, n)$-unsatisfied before $\SS^\aaa_n$ is called again with parameter $w$. Thus, procedure $\SS^\aaa_n$ can only be called $\tp{n+d}$ times with parameter $w$.
\begin{claim} \label{claim:2} $\seq {Y_s} \sN s$ is a computable approximation of a $\DII$ set $Y \in \P$. \end{claim}
Fix $k\in \NN$. For a stage $s$, if $Y_s\uhr k $ is to the left of $ Y_{s-1} \uhr k$ then there are $\aaa, n$ with $n + \init_s(\aaa) \le k$ such that $\P^\aaa[s] \neq \P^\aaa [{s-1}]$ because of the action of a procedure $\SS^\aaa_n$ at (e) or (f).
There are only finitely many pairs $\aaa, s$ such that $ \init_s(\aaa) \le k$. Thus by Claim~\ref{claim:1} there is stage $s_0$ such that at all stages $s\ge s_0$, for no $\aaa $ and $n$ with $ n+ \init_s(\aaa) \le k$, a procedure $\SS^\aaa_n$ is called.
While a procedure $\SS^\aaa_n$ is running with a parameter $w$, it changes the definition of $\P^{\aaa 1}$ only if $A_e \uhr w$ changes ($e = \aaal$), so at most $w$ times. Thus there are only finitely many $s$ such that $Y_s\uhr k \neq Y_{s-1} \uhr k$.
By the definition of the computable approximation $\seq {Y_s} \sN s$ we have $Y \in \P$. This completes Claim~\ref{claim:2}.
As usual, we define the true path $f$ by $f(k) = \liminf_s \delta_s(k)$. By Claim~\ref{claim:1} each $\aaa \prec f$ is only initialized finitely often, because each $\beta$ such that $\beta 1 \prec \aaa$ eventually is stuck with a single run of a procedure $\SS^\beta_m$.
\begin{claim} If $e = \aaal $ and $\aaa 1 \prec f$, then $A_e \neq \Psi_e^Y$. \end{claim}
Some procedure $\SS^\aaa_n$ was called with parameter $w$, and is eventually stuck at (e) with the final value
$A_e\uhr w$. Hence the definition $\Q= \PP^{\aaa1}$ eventually stabilizes at $\aaa$-stages $s$. Since $Y \in \Q$, this implies $A_e \neq \Psi_e^Y$.
\begin{claim} \label{claim:4}
If $e = \aaal $ and $\aaa 0 \prec f$, then $ A_e$ obeys $\cc$.
\end{claim}
Let $A= A_e$. We define a computable enumeration $(\hat A_p) \sN p$ of $A$ via which $A$ obeys $\cc$.
Since $\aaa 0 \prec f$, each procedure $\SS^\aaa_n$ returns. In particular, since $\cc$ has the limit condition and by Claims~\ref{claim:1} and~\ref{claim:2}, each value $w= w^n_i$ becomes permanently $(\aaa, n)$-satisfied. Let $d= \init_s(\aaa)$. Let $s_0$ be the least $\aaa0$-stage such that $s_0 \ge d$, and let
\vsps
$s_{p+1} = \mu s \ge s_p+2 \, [ s \ttext{is $\aaa0$-stage} \lland $
\bc $\forall n, i \, ( w = w^n_i < s_p \ria \, w \ttext{is $(\aaa, n)$-satisfied at} s ) ]$. \ec
As in similar constructions such as~\cite{Nies:book}, for $p \in \NN$ we let \bc $\hat A_p = A_{s_{p+2}} \cap [0,p)$. \ec
Consider the situtation that $p > 0$ and $x\le p$ is least such that $\hat A_p(x) \neq \hat A_{p-1}(x)$. We call this situation an \emph{$n$-change} if $n$ is least such that $x< w^n_i < s_p$ for some $i$. (Note that $n\le p+1$ because $w_0^{p+1} = p+1$.) Thus $(x,s_p)$ contains no value of the form $w^{n-1}_j$, whence $ \cc(x, p) \le \cc(x,s_p) \le 4^{-n+1}$. We are done if we can show there are at most $\tp{n+d}$ many $n$-changes, for in that case the total cost $\cc \seq {\hat A_p } $ is bounded by $\sum_n 4^{-n+1} \tp{n+d} =O(2^d)$.
Recall that $\P^\aaa$ is stable by stage $s_0$. Note that $Y\uhr{n+d}$ can only move to the right after the first run of $\SS^\aaa_n$, as observed in the proof of Claim~\ref{claim:1}.
Consider $n$-changes at stages $p<q$ via parameters $w = w^n_i$ and $w'= w^n_k$ (where possibly $k<i$). Suppose the last run of $\SS^\aaa_n$ with parameter $w$ that was started before $s_{p+1}$ has returned at stage $t \le s_{p+2}$, and similarly, the last run of $\SS^\aaa_n$ with parameter $w'$ that was started before $s_{q+1}$ has returned at stage $t'$. Let $z= Y_t \uhr {n+d}$ and $z'= Y_{t'} \uhr {n+d}$. We show $z <_L z'$; this implies that there are at most $\tp{n+d}$ many $n$-changes.
At stage $t$, by definition of returning at (f) in the run of $\SS^\aaa_n$, we have $\Q = \ES$. Therefore $ \Psi^X_{e,t} \succ A_{e,t}\uhr w$ for each $X$ on $ \P^\aaa_t$ such that $ X \succ z $. Now \bc $\hat A_p(x) \neq \hat A_{p-1}(x)$, $x<w$ and $t \le s_{p+1}$, \ec so $A_{s_{p+2}} \uhr w \neq A_t \uhr w$, The stage $s_{p+2}$ is $\aaa0$-expansionary, and $Y_{s_{p+2}}$ is on $\P^\aaa_t$. Therefore
\bc $Y_{r-1} \uhr {n+d}\, <_L Y_{r} \uhr {n+d}$ \ec
for some stage $r$ such that $t < r \le s_{p+2}$. Thus, at stage $r$, the value $w'$ was declared $(\aaa, n)$-unsatisfied. Hence a new run of $\SS^\aaa_n$ with parameter $w'$ is started after $r$, which has returned by stage $s_{q+1} \ge s_{p+2}$. Thus $r< t'$. So $z \le_L Y_{r-1} \uhr {n+d} <_L Y_{r} \uhr {n+d} \le_L z'$, whence $z <_L z'$ as required.
This concludes Claim~\ref{claim:4} and the proof.
\end{proof}
\section{A dual cost function construction} \label{ss:dual}
Given a relativizable cost function $\cc$, let $D \rightarrow W^D$ be the c.e.\ operator given by the cost function construction in Theorem~\ref{thm:cfconstr} relative to the oracle~$D$. By pseudo-jump inversion there is a c.e.\ set $D$ such that $W^D \oplus D \equiv_T \Halt$, which implies $D <_T \Halt$.
Here, we give a direct construction of a c.e.\ set $D <_T \Halt$ so that the total cost of $\Halt$-changes as measured by $\cc^D$ is finite. More precisely, there is a $D$-computable enumeration of $\Halt $ obeying~$\cc^D$.
If $\cc$ is sufficiently strong, then
the usual cost function construction builds an incomputable c.e.\ set $A$ that is close to being computable. The dual cost function construction then builds a c.e.\ set $D$ that is close to being Turing complete.
\subsection{Preliminaries on cost functionals} Firstly we clarify how to relativize cost functions, and the notion of obedience to a cost function. Secondly we provide some technical details needed for the main construction.
\begin{deff} {\rm (i) A \emph{cost functional} is a Turing functional $\cc^Z(x,t)$ such that for each oracle $Z$, $\cc^Z$ either is partial, or is a cost function relative to $Z$. We say that $\cc$ is non-increasing in main argument if this holds for each oracle $Z$ such that $\cc^Z $ is total. Similarly, $\cc$ is non-decreasing in the stage argument if this holds for each oracle $Z$ such that $\cc^Z $ is total. If both properties hold we say that $\cc$ is monotonic.
\n (ii) Suppose $A \leT Z'$. Let $\seq{A_s}$ be a $Z$-computable approximation of $A$.
\n We write $\seq{A_s} \models^Z \cc^Z$ if
\vsps
$ {\cc^Z} {\seq{A_s}} = \sum_{x,s} \cc^Z(x,s) $
\hfill $
\Cyl{ x< s \lland
\cc^Z(x,s) \DA \lland x \ttext{ is
least s.t.} A_{s-1} ( x) \neq A_{s} (x) } $
\vsps
\n is finite.
We write $A \models^Z c^Z$ if $\seq{A_s} \models^Z c^Z$ for some $Z$-computable approximation $\seq{A_s}$ of $A$. } \end{deff}
\n For example, $\cost^Z(x,s) = \sum_{x < w \le s} 2^{-K^Z_s(w)}$ is a total monotonic cost functional. We have $A\models^Z \cost^Z$ iff $A $ is $K$-trivial relative to $Z$.
\vsp
We may convert a cost functional $\cc$ into a total cost functional $\wt \cc$ such that $\wt \cc^Z(x)= \cc^Z(x)$ for each $x$ with $\fao t \cc^Z(x,t)\DA$, and, for each $Z,x,t$, the computation $\wt \cc^Z(x,t)$ converges in $t$ steps. Let
\bc $\wt \cc^Z(x,s) = \cc^Z(x,t)$ where $t\le s$ is largest such that $\cc^Z(x,t)[s] \DA$. \ec Clearly, if $\cc$ is monotonic in the main/stage argument then so is $\wt \cc$.
Suppose that $D$ is c.e.\ and we compute $\cc^D(x,t)$ via hat computations \cite[p.\ 131]{Soare:87}: the use of a computation $\cc^D(x,t)[s]\DA$ is no larger than the least number entering $DÄ$ at stage $s$. Let $N_D $ be the set of non-deficiency stages; that is, $s \in N_D$ iff there is $x\in D_s- D_{s-1}$ such that $D_s\uhr x = D\uhr x$. Any hat computation existing at a non-deficiency stage is final. We have
\begin{equation} \label{hat cost} \cc^D(x) = \sup_{s \in N_D} \wt \cc^{D_s}(x,s). \end{equation}
For, if $\cc^D(x,t)[s_0] \DA$ with $D$ stable below the use, then $\cc^D(x,t) \le \wt \cc^{D_s}(x,s)$ for each $s \in N_D$. Therefore $ \cc^D(x) \le \sup_{s \in N_D} \wt \cc^{D_s}(x,s)$. For the converse inequality, note that for $s \in N_D$ we have $\wt \cc^{D_s}(x,s) = \cc^D(x,t)$ for some $t \le s$ with $D$ stable below the use.
\subsection{The dual existence theorem}
\begin{thm} Let $\cc$ be a total cost functional that is nondecreasing in the stage component and satisfies the limit condition for each oracle. Then there is a Turing incomplete c.e.\ set $D$ such that $\Halt \models^D \cc^D$. \end{thm}
\begin{proof} We define a cost functional $\Gamma^Z(x,s)$ that is nondecreasing in the stage. We will have $\ul \Gamma^D(x) = \cc^D(x)$ for each $x$, where $\ul \Gamma^D(x) = \lim_t \Gamma^D(x,t)$, and $\Halt$ with its given computable enumeration obeys $\Gamma^D$. Then $\Halt \models^D \cc^D$ by the easy direction `$\LA$' of Theorem~\ref{thm:imply} relativized to $D$.
Towards $ \Gamma^D(x) \ge \cc^D(x)$, when we see a computation $\wt \cc^{D_s}(x,s) = \aaa$ we attempt to ensure that $\Gamma^D(x,s) \ge \aaa$.
To do so we enumerate relative to $D$ a set $G$ of ``wishes'' of the form
\bc $\rho = \la x, \aaa \ra^u$, \ec
where $x\in \NN$, $\aaa$ is a nonnegative rational, and $u+1$ is the use. We say that $\rho$ is a {\it wish about $x$}. If such a wish is enumerated at a stage $t$ and $D_t \uhr u$ is stable, then the wish is granted, namely, $\Gamma^D(x,t) \ge \aaa$. The converse inequality $ \Gamma^D(x) \le \cc^D(x)$ will hold automatically.
To ensure $D <_T \Halt$, we enumerate a set $F$, and meet the requirements
\bc $N_e \colon \, F \neq \Phi_e^D$. \ec
Suppose we have put a wish $\rho = \la x, \aaa \ra^u$ into $G^D$. To keep the total $\Gamma^D$-cost of the given computable enumeration of $\Halt$ down, when $x$ enters $\Halt$ we want to remove $\rho$ from $G^D$ by putting $u$ into $D$. However, sometimes $D$ is preserved by some $N_e$. This will generate a {\it preservation cost}. $N_e$ starts a run at a stage $s$ via some parameter $v$, and ``hopes'' that $\Halt_s \uhr v$ is stable. If $\Halt \uhr v$ changes after stage $s$, then this run of $N_e$ is cancelled. On the other hand, if $x\ge v$ and $x $ enters $\Halt$, then the ensuing preservation cost can be afforded. This is so because we choose $v$ such that $\wt c_s^{D_s}(v,s) $ is small. Since $\wt \cc^D$ has the limit condition, eventually there is a run $N_e(v)$ with such a low-cost $v$ where $\Halt \uhr v$ is stable. Then the diagonalization of $N_e$ will succeed.
\vsp
\n {\it Construction of c.e.\ sets $F,D$ and a $D$-c.e.\ set $G$ of wishes.}
\n {\it Stage $s > 0$. } We may suppose that there is a unique $n \in \ES'_s - \ES'_{s-1}$.
\vsps
\n {\it 1.\ Canceling $N_e$'s.} Cancel all currently active $N_e(v)$ with $v > n$.
\n {\it 2. Removing wishes.} For each $\rho= \la x, \aaa \ra^u\in G^D[s-1]$ put in at a stage $t<s$, if $ \Halt_s \uhr {x+1} \neq \Halt_{t} \uhr {x+1}$ and $\rho$ is not held by any $N_e(v) $, then put $u-1$ into $D_s$, thereby removing $\rho $ from $G^D$.
\n {\it 3. Adding wishes.} For each $x< s$ pick a large $u$ (in particular, $u \not \in D_s$) and put a wish $\la x, \aaa \ra ^u$ into $G$ where $\aaa = \wt \cc^{D_s}(x,s)$. The set of queries to the oracle $D$ for this enumeration into $G$ is contained in $[0,r) \cup \{ u\}$, where $r $ is the use of $ \wt \cc^{D_s}(x,s)$ (which may be much smaller than $s$). Then, from now on this wish is kept in $G^D$
unless (a) $D\uhr r$ changes , or (b) $u$ enters $D$.
\n {\it 4. Activating $N_e(v)$.} For each $e < s$ such that $N_e$ is not currently active, see if there is $v$, $e \le v \le n$ such that
\bi \item[--] $\wt \cc^{D_s}(v,s) \le 3^{-e}/2$,
\item[--] $v> w$ for each $w$ such that $N_i(w)$ is active for some $i< e$, and
\item[--] $\Phi_e^D \uhr{x+1} =F \uhr {x+1} $ where $x = \la e, v, |\Halt \cap [0,v)|\ra$,
\ei
If so, choose $v$ least and activate $N_e(v)$. Put $x $ into $F$. Let $N_e$ {\it hold} all wishes for some $y \ge v$ that are currently in $G^D$. Declare that such a wish is no longer held by any $N_i(w)$ for $i\neq e$. (We also say that $N_e$ {\it takes over} the wish.)
Go to stage $s'$ where $s'$ is larger than any number mentioned so far.
\vsp
\n {\bf Claim 1.} {\it Each requirement $N_e$ is activated only finitely often, and met. Hence $F \not \leT D$.}
\n Inductively suppose that $N_i$ for $i<e$ is no longer activated after stage $t_0$. Assume for a contradiction that $F= \Phi_e^D$. Since $\cc^D$ satisfies the limit condition, by (\ref{hat cost}) there is a least $v$ such that $\wt \cc^{D_s}(v,s)\le 3^{-e}/2$ for infinitely many $s> t_0$. Furthermore, $v> w$ for any $w$ such that some $N_i(w)$, $i< e$, is active at $t_0$. Once $N_e(v) $ is activated, it can only be canceled by a change of $\Halt \uhr v$. Then there is a stage $s> t_0$, $\wt \cc^{D_s}(v,s)\le 3^{-e}/2$, such that $\Halt \uhr v$ is stable at $s$ and $\Phi_e^D \uhr{x+1} =F \uhr {x+1} $ where $x = \la e, v, |\Halt \cap [0,v)|\ra$. If some $N_e(v')$ for $v' \le v$ is active after (1.) of stage $s$ then it remains active, and $N_e$ is met. Now suppose otherwise.
Since we do not activate $N_e(v)$ in (4) of stage $s$, some $N_e(w)$ is active for $w> v$. Say it was activated last at a stage $t< s$ via $x= \la e,w, | \ES'_t \cap [0,w]|$. Then $x' = \la e, v, |\Halt_t \cap [0,v)|\ra$ was available to activate $N_e(v)$ as $x' \le x$ and hence $\Phi_e^D \uhr {x'+1} = F \uhr {x'+1} [t]$. Since $w$ was chosen minimal for $e$ at stage $t$, we had
$\wt \cc^{D_t}(v,t) > 3^{-e}/2$. On the other hand, $\wt \cc^{D_s}(v,s)\le 3^{-e}/2$, hence $D_t \uhr t \neq D_s\uhr t$. When $N_e(w)$ became active at $t$ it tried to preserve $D\uhr t$ by holding all wishes about some $y \ge w$ that were in $G^D[t]$. Since $N_e(w)$ did not succeed, it was cancelled by a change $\Halt_t \uhr w \neq \Halt_s \uhr w$. Hence $N_e(w)$ is not active at stage $s$, contradiction. \hfill $\Diamond$
\vsps
We now define $\Gamma^Z(x,t)$ for an oracle $Z$ (we are interested only in the case that $Z=D$). Let $s$ be least such that $D_s \uhr t = Z\uhr t$. Output the maximum $\aaa $ such that some wish $\la x, \aaa \ra^u $ for $u \le t$ is in $G^D [s]$.
\vsp
\n {\bf Claim 2.} {\it (i) $\Gamma^D(x,t)$ is nondecreasing in $t$. (ii) $\fao x \ul \Gamma^D(x) = \ul \cc^D(x)$.}
\n (i). Suppose $t' \ge t$. As above let $s$ be least such that $D_{s} \uhr {t}$ is stable. Let $s'$ be least such that $D_{s'} \uhr {t'}$ is stable. Then $s' \ge s$, so a wish as in the definition of $\Gamma^D(x,t)$ above is also in $G^D[s']$. Hence $\Gamma^D(x,t') \ge \Gamma^D(x,t)$.
\n (ii). Given $x$, to show that $\Gamma^D(x) \ge \cc^D(x)$ pick $t_0$ such that $ \Halt \uhr {x+1}$ is stable at $t_0$. Let $s \in N_D$ and $s > t_0$. At stage $s$ we put a wish $\la x, \aaa\ra^u$ into $G_D$ where $\aaa= \wt \cc^{D_s}(x,s)$. This wish is not removed later, so $\Gamma^D(x) \ge \aaa$.
For $\Gamma^D(x) \le \cc^D(x)$, note that for each $s \in N_D$ we have
$\wt \cc^{D_s}(x,s) \ge \Gamma^{D_s}(x,s)$ by the removal of a wish in 3(a) of the construction when the reason the wish was there disappears.
\hfill $\Diamond$
\vsps
\n {\bf Claim 3.} {\it The given computable enumeration of $\Halt $ obeys $\Gamma^D$. }
\n First we show by induction on stages $s$ that {$N_e$ holds in total at most $3^{-e}$ at the end of stage $s$, namely,
\begin{equation} \label{heyyy} 3^{-e} \ge \sum_x \max \{ \aaa \colon \, N_e \ttext{holds a wish} \la x, \aaa \ra^u\} \end{equation}
Note that once $N_e(v)$ is activated and holds some wishes, it will not hold any further wishes later, unless it is cancelled by a change of $\Halt \uhr v$ (in which case the wishes it holds are removed).
We may assume that $N_e(v)$ is activated at (3.) of stage $s$. Wishes held at stage $s$ by some $N_i(w)$ where $ i < e$ will not be taken over by $N_e(v)$ because $w<v$. Now consider wishes held by a $N_i(w)$ where $ i > e$. By inductive hypothesis the total of such wishes is at most $\sum_{i > e} 3^{-i} = 3^{-e}/2 $ at the beginning of stage $s$. The activation of $N_e(v)$ adds at most another $3^{-e}/2 $ to the sum in (\ref{heyyy}).
To show $\Gamma^D \seq {\Halt_s} < \infty$, note that any contribution to this quantity due to $n$ entering $\Halt $ at stage $s$ is because a wish $\la n, \delta \ra^u $ is eventually held by some $N_e(v) $. The total is at most $\sum_e 3^{-e}$. }
\end{proof}
The study of non-monotonic cost function is left to the future. For instance, we conjecture that there are cost functions $\cc, \dd$ with the limit condition such that for any $\DII$ sets $A,B$,
\bc $A \models \cc$ and $B \models \dd$ $\RA$ $A,B$ form a minimal pair. \ec
It is not hard to build cost functions $\cc,\dd$ such that only computable sets obey both of them. This provides some evidence for the conjecture.
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When the system is approaching the loadability boundary, e.g., nose point of a $2$-bus system, the point A and point B will come closer. Therefore, we need to algebraically characterize the operating points,
especially for those on or close to the boundary of the feasible power flow region. Commonly, these types of analysis are done through the power flow Jacobian, and in particular, the singularity of the Jacobian matrix has long been used to characterize the solvability and stability of power flow solutions~\cite{KundurEtAl1994a}. In this section, we show that the Jacobian is \emph{not always sufficient to identify the boundary of the power flow region}, and we propose a different method of identifying whether a solution is on the boundary by using a linear program in the rectangular coordinates.
\subsection{The Limitation of Singularity Analysis of Jacobian Matrix}
The power flow Jacobian matrix, $\bl J$, is normally defined by the first order partial derivatives of active and reactive powers with respect to the state variables. In our analysis, these partial derivatives are taken with respect to the real and imaginary parts of bus voltages:
\begin{equation*}
\bl J= \begin{bmatrix} \pd{\bl p}{\bl v_{r}} & \pd{\bl p}{\bl v_i} \\
\pd{\bl q}{\bl v_r} & \pd{\bl q}{\bl v_i} \end{bmatrix},
\end{equation*}
where the elements are given by
\begin{subequations} \label{eqn:partial_rec}
\begin{equation}
\frac{\partial p_d}{\partial v_{k,r}} = \left\{
\begin{array}{ll}
2t_{d,1}v_{d,r}+t_{d,2},~~& \text{if } d = k,\\
g_{kd}v_{d,r}+b_{kd}v_{d,i},~~& \text{if } d \neq k.
\end{array} \right.
\end{equation}
\begin{equation}
\frac{\partial p_d}{\partial v_{k,i}} = \left\{
\begin{array}{ll}
2t_{d,1}v_{d,i}+t_{d,3}, & \text{if } d = k,\\
-b_{kd}v_{d,r}+g_{kd}v_{d,i}, & \text{if } d \neq k.
\end{array} \right.
\end{equation}
\begin{equation}
\frac{\partial q_d}{\partial v_{k,r}} = \left\{
\begin{array}{ll}
2t_{d,4} v_{d,r}-t_{d,3}, & \text{if } d = k,\\
-b_{kd}v_{d,r}+g_{kd}v_{d,i}, & \text{if } d \neq k.
\end{array} \right.
\end{equation}
\begin{equation}
\frac{\partial q_d}{\partial v_{k,i}} = \left\{
\begin{array}{ll}
2t_{d,4} v_{d,i}+t_{d,2}, & \text{if } d = k,\\
-g_{kd}v_{d,r}-b_{kd}v_{d,i}, & \text{if } d \neq k.
\end{array} \right.
\end{equation}
\end{subequations}
For the partial derivatives above, $t_{d,1}$ and $t_{d,4}$ are both constant given network parameters, and $t_{d,2}$ and $t_{d,3}$ are \emph{linear} in the variables. Therefore, each of the partial derivatives in \eqref{eqn:partial_rec} is linear in the state variables $v_{d,r}$ and $v_{d,i}$.
The Jacobian is normally used via the inverse function theorem, which states that the power flow equations stabilize around an operating voltage if the Jacobian is non-singular. This condition is necessary since every stable point must have a nonsingular Jacobian. However, the singularity of the Jacobian is insufficient~\cite{Bijwe03}, especially for the loadability boundary. As the next example will show, a singular Jacobian does not imply that the operating voltage is at the boundary of the power flow feasibility region.
Again, consider the $3$-bus network in Fig.~\ref{fig:tri_all}. For simplicity, we assume the lines are purely resistive (all line admittances are $1$ per unit) and only consider active powers. Let bus $1$ be the slack bus and buses $2$ and $3$ be load buses consuming positive amount of active powers.
In this case, the Jacobian becomes:
\begin{equation}\label{eqn:Jacobian_tri}
\bl J=\begin{bmatrix} 1-4v_2+v_3 & v_2 \\ v_3 & 1-4v_3+v_2 \end{bmatrix},
\end{equation}
where $v_2$ and $v_3$ are the voltages at bus $2$ and bus $3$, respectively. Fig. \ref{fig:jacobian} shows the feasible power flow region of power consumptions at buses $2$ and $3$. The red lines show the points where the Jacobian is singular.
Here, we focus on two particular points in Fig.~\ref{fig:jacobian}, points $F$ and $H$.
At these points, $v_1=v_2=v$ due to the symmetry of the network. Then, finding the determinant of $J$ and equating it to $0$, we obtain $v=0.25$ (point $H$) or $v=0.5$ (point $F$). We emphasize that these two points are qualitatively different. Point $F$ is on the boundary of the feasible region, and therefore is a loadability point. However, point $H$ is well within the strict interior of the ``feasible region", therefore it is not on the loadability boundary.
Points like $H$ are sometimes called cusp bifurcation points in stability analysis \cite{Vournas99,Harlim07}. Therefore, if we are interested in finding whether a point is on the boundary or characterizing its loadability margin, just looking at the determinant of the Jacobian is insufficient.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.4]{Region3.eps}
\caption{Feasible power flow region for the $3$-bus network. The red points denote operating points when the Jacobian is singular. They separate into two parts: the boundary of the region (e.g., point F) and points in the strict interior (e.g., point H).}
\label{fig:jacobian}
\end{figure}
In addition to the purely resistive network to generate Fig.~\ref{fig:jacobian}, we also plot the feasibility region and the points where the Jacobian is singular for a network with complex impedances as in~Fig.~\ref{fig:Jacobian2}. Other cases have results similar to Fig.~\ref{fig:jacobian} and Fig.~\ref{fig:Jacobian2}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.45]{region_reactive.eps}
\caption{The active power injections where the Jacobian is singular for a $3$-bus fully connected network, with reactance $j$ per unit on each of the lines. There are two distinct types of points (marked in red), the boundary and the interior, that both have a singular Jacobian matrix.}
\label{fig:Jacobian2}
\end{figure}
\subsection{Verify Loadability Points via Pareto-Front Method}
To isolate just the points on the boundary, we need to look deeper into the power flow equations than just the determinant of the Jacobian. Here, we focus on a network where the buses are loads. Geometrically, a point is on the \emph{loadability boundary} (or simply boundary) if there does not exist another point that can consume more power:
\begin{mydef}\label{def:boundary}
Let $\bl v$ be the complex voltages and $\bl p=(p_1,\cdots,p_n)$ be the corresponding bus active powers. We say that the operating point is on the loadability boundary if there does not exist another operating point $\hat{\bl p}=(\hat{p}_1,\cdots,\hat{p}_n)$ such that $\hat{p}_k \geq p_k$ for all bus index $k$ (nonnegative change in load) and $\hat{p}_d > p_d$ for at least one $d\neq k$ (positive change in at least one load).
\end{mydef}
This definition coincides with the definition of the Pareto-Front since we are modeling each bus (except the slack) as a load bus. It can be easily extended to a network where some buses are generators by changing the direction of inequalities in the definition above. Instead of looking at the determinant of the Jacobian, the next theorem gives a linear programming condition for the points on the boundary:
\begin{theorem}~\label{thm:boundary}
Checking whether an operating point is on the boundary of the feasible power flow region is equivalent to solving a linear programming problem, e.g., \eqref{opt:verify}.
\end{theorem}
Let
\begin{align}
\bl h_d=[\pd{p_d}{v_{1,r}}, \pd{p_d}{v_{1,i}}, \pd{p_d}{v_{2,r}}, \cdots , \pd{p_d}{v_{n,i}}]^T \in R^{2n}
\end{align}
be the gradient of $p_d$ with respect to all the state variables. Therefore, $\bl h_d$ is the transpose of the $d^{th}$ row of the Jacobian matrix. Let $\bl z \in R^{2n}$ be a direction, towards which we move the real and imaginary part of the voltages. Then, by Definition~\ref{def:boundary}, a point is on the boundary if there does not exist a direction to move where the consumption of one bus is increased without decreasing the consumption at other buses.
Therefore, we can check if there is a direction $\bl y$ that makes the following problem feasible. Suppose that $\bl h_1,\cdots,\bl h_n$ are given.
\begin{subequations} \label{opt:feasible}
\begin{align}
& \bl y^T \bl h_d \geq 0, \mbox{ for all } d=1,\cdots,n ,\label{eqn:verify_all} \\
& \sum_{d=1}^n \bl y^T \bl h_d =1. \label{eqn:verify_1}
\end{align}
\end{subequations}
The constraint \eqref{eqn:verify_all} specifies that moving in the direction $\bl y$ cannot decrease any of the active powers. The constraint \eqref{eqn:verify_1} is equivalent to stating that at least one bus' active power must strictly increase. This comes from the fact that $\bl y$ is not a constraint. Therefore, as long as $\bl y^T\bl h_d >0$ for some $d$, the sum $\sum_{d=1}^n \bl y^T \bl h_d$ can be scaled to be $1$. If the problem (\ref{opt:feasible}) is feasible, the corresponding power pair is not on the boundary. If the problem (\ref{opt:feasible}) is infeasible, this means that the point is on the Pareto-Front. So, it is on the loadability boundary.
By adding a constant objective, we can encode this condition in a linear programming (LP) feasibility problem. This is because, in a constraint optimization problem, a solver usually tries to firstly find a feasible set by using the constraints. Then, it will use searching methods, e.g., gradient descent method, for the objective in the feasible region. Therefore, by converting the feasibility problem (\ref{opt:feasible}) into an optimization form, we can use the state-of-the-art solver in convex optimization tool set, which is quite efficient.
Then, we solve the following:
\begin{subequations} \label{opt:verify}
\begin{align}
\min_{\bl y} \;\; & 1 \\
\mbox{s.t. } & \bl y^T \bl h_d \geq 0, \mbox{ for all } d=1,\cdots,n ,\label{eqn:verify_all_2} \\
& \sum_{d=1}^n \bl y^T \bl h_d =1. \label{eqn:verify_1_2}
\end{align}
\end{subequations}
In this optimization problem, the objective is irrelevant since we are only interested in whether the problem is feasible. Finally, an operating point is on the boundary if and only if the problem in \eqref{opt:verify} is infeasible.
A system operating on the boundary limit will lose stability before our conditions are checked. So, we provide an alarm when a system is approaching this boundary for practical interest. In the following, we change the optimization (\ref{opt:verify}) slightly to provide an alarm by setting up an $\epsilon$ value for earlier alarming.
\begin{subequations} \label{opt:verify2}
\begin{align}
\min_{\bl y} \;\; & 1 \\
\mbox{s.t. } & \bl y^T \bl h_d \geq \epsilon, \mbox{ for all } d=1,\cdots,n ,\label{eqn:verify_all_3} \\
& \sum_{d=1}^n \bl y^T \bl h_d =1. \label{eqn:verify_1_3}
\end{align}
\end{subequations}
\begin{figure*}[ht]
\centering
\subfloat[Margin according to Thevenin method.]{ \label{fig:margin_th}
\includegraphics[width=0.45\textwidth]{margin_Thevenin.eps}}
\subfloat[Margin using the proposed method.]{ \label{fig:margin}
\includegraphics[width=0.45\textwidth]{margin.eps}}
\caption{Comparison of the margin computed by (a) the Thevenin equivalent method and (b) our proposed method.}
\label{fig:3-bus}
\end{figure*}
\begin{remark}
In the system operating on or near the loadability boundary, computation speed is important, otherwise the system may lose stability before we can compute anything. The algorithm we propose involves solving a simple linear program that is on the order of the system size, e.g., bus numbers, making the computational speed very short even for large systems. In contrast, some of other nonlinear calculators require iterative methods that solve successive nonlinear problems, resulting in a much slower process.
\end{remark}
For the example in Fig.~\ref{fig:jacobian}, point $A$ is given by $v_2=v_3=0.5$ and point $B$ is given by $v_2=v_3=0.25$. Using \eqref{eqn:partial_rec}, we have $\bl h_2=[-0.5, 0.5]^T$ and $\bl h_3=[0.5, -0.5]^T$ at point $A$, and $\bl h_2=\bl h_3=[0.25, 0.25]^T$ at point $B$. It is easy to check that a $\bl y$ feasible for \eqref{opt:verify} exists at point $B$. However, it is impossible to find such a $\bl y$ at point $A$. Therefore, we can conclude that $A$ is on the boundary whereas $B$ is not, even though both have singular Jacobians.
For how to add more constraint, please refer to Appendix \ref{sec:add}.
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It's hard to guess how important spring training is to a team's success during the season. Come November, when the last manager is thrown in the air after the Konami Cup, someone will say the pennant-winning effort really began on the first day of camp, when the manager said this or the player's rep said that.
That's news on the order of saying this wonderful year began on Jan. 1, imagine that! January? Who would have thought it? Nearly every player's season starts in camp, and every manager, coach, trainer and player has a list of spring training dos and don'ts.
One don't is don't have bad weather. This February has been one of the worst in memory. On Sunday afternoon, Orix international director Takashi Miyata said his club had spent three-quarters of its first nine practice days indoors.
For managers and coaches, who need as much time as possible to evaluate before the season starts, bad weather is a handicap and the main reason why clubs have gradually moved south for the spring. It was a big reason why the Chiba Lotte Marines trained in Australia the last two years.
Unfortunately another don't is don't do things players will refuse to deal with, and Miyata, a former Lotte official, criticized moving the Marines boot camp on that basis.
"Sure it's cold in Kagoshima, but the food is great," said Miyata, who believes the players prefer familiar cuisine to sunshine.
Miyata ran off a list of Kagoshima delicacies that made it sound like anyone who chose to train elsewhere simply lacked taste buds.
And while the Marines got lots of work done--a number of players were less than eager to go abroad.
When it rains, as it has a lot this year, it forces teams indoors and then the facilities play a factor in how much can get done. For the Hiroshima Carp in the city of Okinawa, this is particularly tough. Nowhere can fans and players interact more than in the Carp's camp, but trying to get your swings in when the main stadium is unusable is Hiroshima headache No. 1.
On Saturday afternoon, while hitters swung in the three dimly lit batting cages, the catchers ran through defensive drills in the main stadium's bullpen.
If the batting cages are dark, the bullpen is worse. The cramped niche under the dilapidated stands is closed off from the outside by a steel screen, giving the space the look of a dank Edo-period prison. Yet, instead of inmates' screams, the sounds from the sweating Carp closeted inside were those of professionals going about their work.
And--if one puts aside thoughts of tasty shabu-shabu--work is what players come for.
After the Hokkaido Nippon Ham Fighters finished fourth in 2005, then-manager Trey Hillman said he had underestimated the need for hard drilling in fundamentals. His practices the following February went very much back to basics. Although always a believer in quality vs quantity, Hillman's 2006 camp was two hours longer per day, giving his staff the time to attack fundamental flaws head on.
Hillman went into that year highly critical of a 2005 fielding unit that did little right but catch fly balls and turn double plays. His revised camp made a difference in the Fighters' defensive efficiency that was key in the club's back-to-back Pacific League pennant-winning runs.
While effort is necessary, there is always a risk in overdoing it too early. There is no bigger don't than don't get hurt in practice. This must be the reason why the Tigers' situational infield drill on Saturday was a parody. Runners, ostensibly there to force speed from fielders, moved so slowly that the Tigers mascot Toraki could have made the plays. OK, it was done indoors in the cold, but it was hard to see much benefit from it.
Although Chunichi's camp gets poor marks for fan and media access, the Dragons' energy the next morning was impressive. On a cold day, their situational practice was at regular-season speed. Dragons players running the bases hard and sliding to beat throws was a stark contrast to the Tigers' play acting the day before.
Although no team has ever won a game in practice, the purposeful energy is a good sign that the champs are very focused on winning again this season.
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\begin{document}
\title{From Matrix to Operator Inequalities}
\author{Terry A. Loring}
\address{Department of Mathematics and Statistics, University of New Mexico,
Albuquerque, NM 87131, USA.}
\keywords{$C*$-algebras, matrices, bounded operators, relations, operator
norm, order, commutator, exponential, residually finite dimensional.}
\subjclass{46L05, 47B99}
\urladdr{http://www.math.unm.edu/\textasciitilde{}loring/}
\maketitle
\markright {$C^*$-Algebra Relations (preprint version)}\markright {$C^*$-Algebra Relations (preprint version)}
\begin{center}
\ifthenelse{\boolean{Details}}{{\Large Fully Detailed Version}}{}
\par\end{center}
\begin{abstract}
We generalize L\"{o}wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition for $C^{*}$-relations of
being residually finite dimensional.
Our main result is a meta-theorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved, and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators and ``positive'' noncommutative $*$-polynomials.
\end{abstract}
\section{Introduction}
Our topic is bounded operators that satisfy relations that involve
algebraic relations, the operator norm, functional calculus and positivity.
The word positive, when applied to matrices, shall mean positive semidefinite.
The $*$-strong topology can bridge the gap between representations
of relations by bounded operators on Hilbert space and representations
by matrices. This feature of the $*$-strong topology has been noted
before, for example by L\"{o}wner in \cite{BendatSherman}, or in
the context of residually finite dimensional $C^{*}$-algebras in
\cite{ExelLoringRFDfreeProd}.
This article is essentially independent of our previous paper \cite{LoringCstarRelations}
on $C^{*}$-relations. We minimize the role of universal $C^{*}$-algebras.
Perhaps someone will see how to strip out the $C^{*}$-algebras and
get a result that works for norms other than the operator norm.
Theorem~\ref{thm:RFDimpliesClosedReduction}, our main result, a
meta-theorem. We first define $C^{*}$-relation, and define on $C^{*}$-relations
the concepts of closed and residually finite dimensional (RFD). The
meta-theorem is that, given a theorem about matrices which states
that an RFD $C^{*}$-relation implies a closed $C^{*}$-relation,
we may conclude the same implication holds for all bounded operators.
The author is grateful for inspiring conversations with R.\ Bhatia,
T.\ Shulman, V.\ Shulman and S.\ Silvestrov.
\section{$C^{*}$-Relations}
\begin{defn}
Suppose $\mathcal{X}$ is a set. A statement $R$ about functions
$f:\mathcal{X}\rightarrow A$ into various $C^{*}$-algebras is a
\emph{$C^{*}$-relation} if the following four axioms hold:
\begin{description}
\item [{R1}] the unique function $\mathcal{X}\rightarrow\{0\}$ satisfies
$R;$
\item [{R2}] if $\varphi:A\hookrightarrow B$ is an injective $*$-homomorphism
and $\varphi\circ f:\mathcal{X}\rightarrow B$ satisfies $R$ for
some $f$ then $f:\mathcal{X}\rightarrow A$ also satisfies $R;$
\item [{R3}] if $\varphi:A\rightarrow B$ is a $*$-homomorphism and $f:\mathcal{X}\rightarrow A$
satisfies $R$ then $\varphi\circ f:\mathcal{X}\rightarrow B$ also
satisfies $R;$
\item [{R4f}] if $f_{j}:\mathcal{X}\rightarrow A_{j}$ satisfy $R$ for
$j=1,\ldots n$ the so does $f=\prod f_{j}$ where\[
f:\mathcal{X}\rightarrow\prod_{j=1}^{n}A_{j}\]
sends $x$ to $\left\langle f_{1}(x),\ldots,f_{n}(x)\right\rangle .$
\end{description}
\end{defn}
Examples of $C^{*}$-relations include the zero-sets of $*$-polynomials
in noncommuting variables, henceforth called NC $*$-polynomials.
When $\mathcal{X}=\{1,2,\ldots,m\},$ the case we really care about,
we use $a_{1},\ldots,a_{m}$ in place of the function notation $j(q)=a_{q}.$
Given a NC $*$-polynomial $p(x_{1},\ldots,x_{m})$ with constant
term zero, its zero-set is the $C^{*}$-relation\[
p(a_{1},\ldots,a_{m})=0.\]
Other $C^{*}$-relations associated to $p$ include\[
p(a_{1},\ldots,a_{m})\geq0\]
and\[
\left\Vert p(a_{1},\ldots,a_{m})\right\Vert \leq C\]
for a constant $C>0,$ as well as\[
\left\Vert p(a_{1},\ldots,a_{m})\right\Vert <C.\]
Given a set $\mathcal{R}$ of $C^{*}$-relations, a function $f:\mathcal{X}\rightarrow A$
to a $C^{*}$-algebra $A$ is called a \emph{representation} \emph{of}
$\mathcal{R}$ \emph{in} $A$ if every statement in $\mathcal{R}$
is true for $f.$ A function $\iota:\mathcal{X}\rightarrow U$ into
a $C^{*}$-algebra is \emph{universal} for $\mathcal{R}$ if $\iota$
is a representation of $\mathcal{R}$ and for every representation
$f:\mathcal{X}\rightarrow A$ of $\mathcal{R}$ there is a unique
$*$- homomorphism $\varphi:U\rightarrow A$ so that $\varphi\circ\iota=f.$
It is important to note that often there is no universal $C^{*}$-algebra
and no universal representation. See \cite{LoringCstarRelations}.
We use the notation $C^{*}\left\langle \mathcal{X}\left|\mathcal{R}\right.\right\rangle $
for $U$ and call it the universal $C^{*}$-algebra. Notice that universal
representation $\iota$ is usually what we should be talking about.
Notice also that $\iota$ need not be injective, but still we often
say that the representation $f:\mathcal{X}\rightarrow A$ of $\mathcal{R}$
extends to a unique $*$- homomorphism $\varphi:U\rightarrow A$ with
the requirement $\varphi(\iota(x))=f(x).$ A good exercise is to show
that $\iota(\mathcal{X})$ must generate $C^{*}\left\langle \mathcal{X}\left|\mathcal{R}\right.\right\rangle $
as a $C^{*}$-algebra. Alternately, this is clear from the proof of
Theorem 2.6 in \cite{LoringCstarRelations}.
Given a set $\mathcal{R}$ of $C^{*}$-relations on a set $\mathcal{X},$
we let $\textnormal{rep}_{\mathcal{R}}(\mathcal{X},A)$ denote the
set of all representations of $\mathcal{X}$ in $A.$ If $\mathbb{H}$
is a Hilbert space then we set \[
\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})=\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{B}(\mathbb{H})).\]
The notation\[
\prod_{\lambda\in\Lambda}A_{\lambda}\]
shall denote the $C^{*}$-algebra product consisting of all bounded
sequences or families $\left\langle a_{\lambda}\right\rangle _{\lambda\in\Lambda}$
that have $a_{\lambda}$ in $A_{\lambda}.$ Given a family of functions
$f_{\lambda}:\mathcal{X}\rightarrow A_{\lambda}$ we say it is \emph{bounded}
if $\sup_{\lambda}\left\Vert f_{\lambda}(x)\right\Vert $ is finite
for all $x$ in $\mathcal{X}.$ For such a bounded family we define
their \emph{product} to be the function \[
\prod_{\lambda\in\Lambda}f_{\lambda}:\mathcal{X}\rightarrow\prod_{\lambda\in\Lambda}A_{\lambda}\]
that sends $x$ to the family $\left\langle f_{\lambda}(x)\right\rangle .$
It could be argued that the above be called the sum of the representations,
c.f.\ Section II.6.1 in \cite{BlackadarOperatorAlgebras}. Notice
that given Hilbert spaces $\mathbb{H}_{\lambda}$ we have the inclusion
(block diagonal) \[
\prod_{\lambda}\mathbb{B}\left(\mathbb{H}_{\lambda}\right)\subseteq\mathbb{B}\left(\bigoplus_{\lambda}\mathbb{H}_{\lambda}\right).\]
A potential source of confusion is that we are talking about representations
of relations in $C^{*}$-algebras, but then often want to represent
various $C^{*}$-algebras on Hilbert space. Sometime we cut out the
middle man and talk of representations of relations on Hilbert space.
However, what we allow to be called $C^{*}$-relations on a set of
operators are properties that can be determined by how they sit in
the $C^{*}$-algebra they generate.
\begin{defn}
Suppose $\mathcal{R}$ is a set of $C^{*}$-relations on a set $\mathcal{X}.$
We say $\mathcal{R}$ is \emph{closed} if
\begin{description}
\item [{R4b}] for every bounded family $f_{\lambda}:\mathcal{X}\rightarrow A_{\lambda}$
of representations of $\mathcal{R}$ the product function $\prod f_{\lambda}$
is also a representation of $\mathcal{R}.$
\end{description}
We say $\mathcal{R}$ is \emph{compact} if it satisfies the stronger
axiom
\begin{description}
\item [{R4}] every family $f_{\lambda}:\mathcal{X}\rightarrow A_{\lambda}$
of representations of $\mathcal{R}$ is bounded and the associated
product function $\prod f_{\lambda}$ is a representation of $\mathcal{R}.$
\end{description}
Following Hadwin, Kaonga, Mathes (\cite{Hadwin-Kaonga-Mathes}) Phillips
(\cite{PhillipsProCstarAlg}) and others, we showed in \cite{LoringCstarRelations}
that $\mathcal{R}$ is compact if and only if there is a universal
$C^{*}$-algebra for $\mathcal{R}.$
\end{defn}
For example, \[
\left\{ x^{2}-x=0,\ x^{*}-x=0\right\} \]
is compact, and has universal $C^{*}$-algebra isomorphic to $\mathbb{C}.$
Also compact is \[
\left\{ \left\Vert x^{2}-x\right\Vert \leq\frac{1}{8},\ x^{*}-x=0\right\} \]
and this also has a universal $C^{*}$-algebra that is commutative.
On the other hand\[
\left\{ x^{2}-x=0\right\} \]
is closed but not compact, while\[
\left\{ \left\Vert x^{2}-x\right\Vert <\frac{1}{8},\ x^{*}-x=0\right\} \]
is not even closed.
It is sometimes easier to look at unital relations and unital $C^{*}$-algebras.
Everything here carries over. Notice we are not putting $1$ in $\mathcal{X},$
but use it symbolically in relations to stand for the unit in $A$
when considering a function $f:\mathcal{X}\rightarrow A.$ Rather
than introduce notation for this, we limit our discussion of unital
relations to quoting other papers.
The relations associated to NC $*$-polynomials are not the only interesting
$C^{*}$-relations. The relation\[
0\leq x\leq1\]
is compact, with universal $C^{*}$-algebra $C_{0}(0,1].$ Another
relation on $\{ x,y,z\}$ is\[
0\leq\left[\begin{array}{cc}
y & x^{*}\\
x & z\end{array}\right],\]
which is to be interpreted so that $X,$ $Y$ and $Z$ in a $C^{*}$-algebra
$A$ form a representation if and only if the matrix\[
\left[\begin{array}{cc}
Y & X^{*}\\
X & Z\end{array}\right]\]
is a positive element in $\mathbf{M}_{2}(A).$
Many results about operators relative to the operator norm, or positivity,
can be stated in the form where one set of $C^{*}$-relations implies
another. For example\[
x^{*}x=xx^{*}\implies\left[\begin{array}{cc}
\left|x\right| & x^{*}\\
x & \left|x\right|\end{array}\right]\geq0\]
or\[
0\leq k\leq1,\left\Vert h\right\Vert \leq1,\left\Vert hk-kh\right\Vert \leq\epsilon\implies\left\Vert hk^{\frac{1}{2}}-k^{\frac{1}{2}}h\right\Vert \leq\frac{5}{4}\epsilon\]
or\[
x^{*}x=xx^{*},\ yx=xy\implies yx^{*}=x^{*}y.\]
In many cases, such a theorem will follow from the special case of
that theorem restricted to the matrix case. Whether this leads to
a new result, a shorter proof of a known result, or a new but harder
proof of a known result, depends on the example. What seems interesting
is just how many theorems in the literature involve $C^{*}$-relations
that are residually finite dimensional.
\section{Residually Finite Dimensional $C^{*}$-Relations}
We certainly want to look at relations that are compact and have universal
$C^{*}$-algebra that is residually finite dimensional (RFD). For
example, for some $0<\epsilon\leq2$ the relations\[
u^{*}u=uu^{*}=v^{*}v=vv^{*}=1,\ \left\Vert uv-vu\right\Vert \leq\epsilon\]
have universal $C^{*}$-algebra that has been dubbed ``the soft torus''
by Exel, and this $C^{*}$-algebra has been shown to be residually
finite dimensional by Eilers and Exel in \cite{EilersExelSoftTorusRFD}.
A $C^{*}$-algebra is \emph{residually finite dimensional} if there
is a separating family of representations of $A$ on finite dimensional
Hilbert spaces.
The restriction to compact relations is artificial in operator theory.
A very important example is the relation $\left\Vert xy+yx\right\Vert \leq\epsilon$
on $\{ x,y\}.$ (The obvious ``and'' operation turns a set of $C^{*}$-relations
into a single relation, so we tend to use relation and set of relations
interchangeably.) We could discuss RFD $\sigma$-$C^{*}$-algebras,
but prefer to take as our starting point an alternate characterization
of RFD $C^{*}$-algebras in \cite{ExelLoringRFDfreeProd}.
On $\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})$ we will
consider the pointwise $*$-strong topology, whenever $\mathbb{H}$
is a Hilbert space and $\mathcal{R}$ is a $C^{*}$-relation on a
set $\mathcal{X}.$
Compare this to \[
\textnormal{rep}(A,\mathbb{H})=\left\{ \pi:A\rightarrow\mathbb{B}(\mathbb{H})\left|\ \pi\mbox{ is a }*\mbox{-homomorhpism}\right.\right\} \]
for a $C^{*}$-algebra $A$ with the pointwise $*$-strong topology.
Equivalently, consider this with the pointwise strong topology. A
representation $\pi$ is said to be \emph{finite dimensional} if its
essential subspace is finite dimensional. The relevant result from
\cite{ExelLoringRFDfreeProd} is that $A$ is RFD if and only if for
all $\mathbb{H}$ the finite dimensional representations are dense
in $\textnormal{rep}(A,\mathbb{H}).$ For more characterizations of
a $C^{*}$-algebra being RFD see \cite{ArchboldRFD}.
We need a definition of finite dimensional for $f\in\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H}).$
We define the \emph{essential subspace} of $f$ to be\[
\left\{ \xi\in\mathbb{H}\left|f(x)\xi=\left(f(x)\right)^{*}\xi=0,\ \forall x\in\mathcal{X}\right.\right\} ^{\perp}.\]
We say $f$ is \emph{finite dimensional} if its essential subspace
is finite dimensional. Notice this property has nothing to do with
$\mathcal{R}.$
If $\mathcal{R}$ is a compact set of $C^{*}$-relations then the
essential space of $f$ is the same as the essential space of the
associated representation $\pi$ of $C^{*}\left\langle \mathcal{X}\left|\mathcal{R}\right.\right\rangle .$
Thus $f$ is finite dimensional if and only if $\pi$ is finite dimensional.
\begin{defn}
A set of $C^{*}$-relations $\mathcal{R}$ on $\mathcal{X}$ is \emph{residually
finite dimensional (RFD)} if there are finite constants \[
C(x,r)\quad(x\in X,\ r\in[0,\infty))\]
so that, for every $\mathbb{H}$ and any choice of nonnegative constants
$r_{x},$ every $f\in\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})$
satisfying\[
\| f(x)\|\leq r_{x}\quad(\forall x\in\mathcal{X})\]
is in the $*$-strong closure of \[
\left\{ g\in\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})\left|\ g\mbox{ is finite dimensional and }g(x)\leq C(x,r_{x})\ \forall x\in\mathcal{X}\right.\right\} .\]
\end{defn}
We say $f$ in $\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})$
is \emph{cyclic} if there is a vector $\xi$ so that $A\xi$ is dense
in $\mathbb{H},$ where $A=C^{*}(f(\mathcal{X})).$ Even if $A\xi$
is not dense, we call its closure a \emph{cyclic subspace} for $f.$
We say $f$ is \emph{unitarily equivalent} to $g$ in $\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{K})$
if there is a unitary $U:\mathbb{K}\rightarrow\mathbb{H}$ so that
$g(x)=U^{-1}f(x)U.$ If $\mathbb{K}$ is a reducing subspace for all
operators in $f(\mathcal{X})$ then let $U$ be the inclusion of $\mathbb{K}$
in $\mathbb{H}$ and let $g(x)=U^{*}f(x)U.$ By \textbf{R3} this is
also a representation and we call $g$ a \emph{subrepresentation of}
$f.$
\begin{lem}
Suppose $\mathcal{R}$ is a set of $C^{*}$-relations on $\mathcal{X}.$
Every Hilbert space representation of $\mathcal{R}$ is unitarily
equivalent to a product of cyclic representations.
\end{lem}
\begin{proof}
\ifthenelse{\boolean{Details}}{ Suppose $f$ is a representation
of $\mathcal{R}$ on $\mathbb{H}.$ Consider the set of all sets of
pairwise orthogonal cyclic subspaces. Inclusion makes this a partial
order that is not empty; the set containing the zero subspace qualifies.
Upper bounds for chains clearly exist, so Zorn's Lemma tells us there
is a maximal set of orthogonal cyclic subspaces. If the union of these
subspaces is not dense, the orthogonal complement is a nontrivial
reducing subspace. Take any nonzero vector there, form the the cyclic
subspace it determines and add it to this set. This is a larger qualified
set, giving us a contradiction.
Since $\mathbb{H}$ is the internal direct sum of cyclic subspaces
for $f,$ each of which is of course reducing, we find $f$ is unitarily
equivalent to the product of cyclic representations.
}{The proof is almost identical to that of the same result for representations
of $C^{*}$-algebras.}
\end{proof}
\begin{lem}
\label{lem:RFDapproxCyclicSuffices} A set $\mathcal{R}$ of $C^{*}$-relations
is RFD if and only if there are finite constants $C(x,r)$ for $x\in X$
and $r\in[0,\infty))$ so that, for every $\mathbb{H},$ every cyclic
$f\in\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})$ is in
the $*$-strong closure of \[
\left\{ g\in\textnormal{rep}_{\mathcal{R}}(\mathcal{X},\mathbb{H})\left|\ g\mbox{ is finite dimensional and }g(x)\leq C(x,\| f(x)\|)\right.\right\} .\]
\end{lem}
\begin{proof}
The forward implication is obvious, so assume the condition on the
cyclic representations holds for some choice of $C(x,r).$
We may as well assume\[
f=\prod_{\gamma\in\Gamma}f_{\gamma}\]
where $f_{\lambda}$ is a cyclic representation on $\mathbb{H}_{\gamma}$
and $\mathbb{H}=\bigoplus\mathbb{H}_{\gamma}.$ Suppose $\epsilon>0$
and $\xi=\left\langle \xi_{\gamma}\right\rangle $ is a unit vector.
There is a finite set $\Gamma_{0}$ so that when we define $\eta=\left\langle \eta_{\gamma}\right\rangle $
by\[
\eta_{\gamma}=\left\{ \begin{array}{cc}
\xi_{\gamma} & \mbox{if }\gamma\in\Gamma_{0}\\
0 & \mbox{if }\gamma\notin\Gamma_{0}\end{array}\right.\]
we have $\left\Vert \xi-\eta\right\Vert \leq\delta$ for \[
\delta=\frac{\epsilon}{2}\left(\| f(x)\|+C(x,\| f(x)\|)\right)^{-1}.\]
(Without loss of generality, $C(x,r)\neq0.$) Suppose $\Gamma_{0}$
has $q$ elements. For each $\gamma$ in $\Gamma_{0}$ there is a
finite dimensional representation $g_{\gamma}:\mathcal{X}\rightarrow\mathbb{H}_{\gamma}$
so that\[
\left\Vert g_{\gamma}(x)\right\Vert \leq C(x,\| f(x)\|)\]
and\[
\left\Vert g_{\gamma}(x)\xi_{\gamma}-f_{\gamma}(x)\xi_{\gamma}\right\Vert ,\left\Vert \left(g_{\gamma}(x)\right)^{*}\xi_{\gamma}-\left(f_{\gamma}(x)\right)^{*}\xi_{\gamma}\right\Vert \leq\frac{\epsilon}{2q}.\]
For $\gamma\notin\Gamma_{0}$ set $g_{\gamma}(x)=0.$ Let $g=\prod_{\gamma}g_{\gamma},$
which is a representation, first in $\prod_{\gamma\in\Gamma_{0}}\mathbb{B}(\mathbb{H}_{\gamma})$
by \textbf{R4f}, and then on $\mathbb{H}$ by \textbf{R3}. It satisfies
the norm condition since\[
\left\Vert g(x)\right\Vert =\sup_{\gamma}\left\Vert g_{\gamma}(x)\right\Vert .\]
The essential space of $g$ is just the sum of the orthogonal essential
spaces of the $g_{\gamma}$ for $\gamma\in\Gamma_{0}$ and so $g$
is also finite dimensional. For each $x$ we have\begin{eqnarray*}
\left\Vert g(x)\xi-f(x)\xi\right\Vert & \leq & \left\Vert g(x)-f(x)\right\Vert \left\Vert \xi-\eta\right\Vert +\left\Vert g(x)\eta-f(x)\eta\right\Vert \\
& \leq & \left(\| f(x)\|+C(x,\| f(x)\|)\right)\delta+\sum_{\gamma\in\Gamma_{0}}\left\Vert g_{\gamma}(x)\xi_{\gamma}-f_{\gamma}(x)\xi_{\gamma}\right\Vert \\
& \leq & \left(\| f(x)\|+C(x,\| f(x)\|)\right)\delta+q\rho\\
& = & \epsilon\end{eqnarray*}
and \ifthenelse{\boolean{Details}}{\begin{eqnarray*}
\left\Vert \left(g(x)\right)^{*}\xi-\left(f(x)\right)^{*}\xi\right\Vert & \leq & \left\Vert g(x)-f(x)\right\Vert \left\Vert \xi-\eta\right\Vert +\left\Vert \left(g(x)\right)^{*}\eta-\left(f(x)\right)^{*}\eta\right\Vert \\
& \leq & \left(\| f(x)\|+C(x,\| f(x)\|)\right)\delta+\sum_{\gamma\in\Gamma_{0}}\left\Vert \left(g_{\gamma}(x)\right)^{*}\xi_{\gamma}-\left(f_{\gamma}(x)\right)^{*}\xi_{\gamma}\right\Vert \\
& \leq & \left(\| f(x)\|+C(x,\| f(x)\|)\right)\delta+q\rho\\
& = & \epsilon.\end{eqnarray*}
}{ similarly\[
\left\Vert \left(g(x)\right)^{*}\xi-\left(f(x)\right)^{*}\xi\right\Vert \leq\epsilon.\]
}
\end{proof}
\begin{prop}
If $\mathcal{R}$ is a compact set of $C^{*}$-relations on a set
$\mathcal{R}$ then $\mathcal{R}$ is RFD if and only if $C^{*}\left\langle \mathcal{X}\left|\mathcal{R}\right.\right\rangle $
is RFD.
\end{prop}
\begin{proof}
By the discussion above, this follows directly from Theorem 2.4 in
\cite{ExelLoringRFDfreeProd}.
\end{proof}
Every $C^{*}$-algebra is isomorphic to the universal $C^{*}$-algebra
of some $C^{*}$-relations, c.f.\ Section 2 in \cite{LoringCstarRelations}.
There is an abundant supply of RFD $C^{*}$-algebras and so an abundant
supply of RFD $C^{*}$-relations. Examples include the subhomogeneous
$C^{*}$-algebras.
Given a specific $C^{*}$-algebra it can be difficult to find a nice
universal set of generator and relations. Conversely, given a set
of $C^{*}$-relations, it can be difficult to get a description of
its universal $C^{*}$-algebra that is more useful than the given
universal property. For present purposes it is best to work directly
with representations of $C^{*}$-relations.
\begin{lem}
Suppose $\mathcal{\mathcal{R}}$ is a set of $C^{*}$-relations $\mathcal{R}$
on $\mathcal{X}.$ If $\mathcal{R}$ is closed and $\left\langle f_{\lambda}\right\rangle _{\lambda\in\Lambda}$
is a bounded net in $\textnormal{rep}(A,\mathbb{H})$ that converges
to the function $f:\mathcal{X}\rightarrow\mathbb{B}(\mathbb{H})$
then $f\in\textnormal{rep}(A,\mathbb{H}).$
\end{lem}
\begin{proof}
The key is noticing inside\[
\prod_{\lambda\in\Lambda}\mathbb{B}(\mathbb{H})\]
the $C^{*}$-algebra $A$ of all bounded nets $\left\langle a_{\lambda}\right\rangle $
indexed by $\Lambda$ that have $*$-strong limits\[
L\left(\left\langle a_{\lambda}\right\rangle \right)=\lim_{\lambda}a_{\lambda}\quad(*\mbox{-strong}).\]
Recall, say from \cite[I.3.2.1]{BlackadarOperatorAlgebras}, that
we need boundedness to gain joint continuity of multiplication in
the $*$-strong topology. Here we let $\lambda$ range over the directed
set $\Lambda$ that indexes the net $f_{\lambda}.$
The $f_{\lambda}$ form a bounded family of representations, so $\prod f_{\lambda}$
determines a representation of $\mathcal{R}$ in $A.$ Now we use
the naturality property for $C^{*}$-relations and conclude $f=L\circ\prod f_{\lambda}$
is a representation.
\end{proof}
Now the main theorem.
\begin{thm}
\label{thm:RFDimpliesClosedReduction} Suppose $\mathcal{R}$ and
$\mathcal{S}$ are $C^{*}$-relations on $\mathcal{X}.$ If $\mathcal{R}$
is residually finite dimensional and $\mathcal{S}$ is closed, and
if every finite-dimensional representation of $\mathcal{R}$ is a
representation of $\mathcal{S},$ then every representation of $\mathcal{R}$
is a representation of $\mathcal{S}.$
\end{thm}
\begin{proof}
Given a representation $f:\mathcal{X}\rightarrow\mathbb{B}(\mathbb{H})$
of $\mathcal{R},$ the fact that $\mathcal{R}$ is RFD tells us that
there are functions $f_{\lambda}:\mathcal{X}\rightarrow\mathbb{B}(\mathbb{H})$
with finite dimensional essential spaces that are representations
of $\mathcal{R}$ and so that $f_{\lambda}(x)$ converges $*$-strongly
to $f(x).$ By assumption, the $f_{\lambda}$ are also representation
of $\mathcal{S}$ and, since $\mathcal{S}$ is closed, we conclude
that $f$ is a representation of $\mathcal{S}.$
\end{proof}
\section{Examples}
It is easy to find lots of closed $C^{*}$-relations. Here are enough
to keep us busy while we attack the harder problem of finding RFD
$C^{*}$-relations. Blackadar noted in \cite{Blackadar-shape-theory}
the importance of ``softening'' a relation $p(x_{1},\ldots,x_{n})=0$
to $\| p(x_{1},\ldots,x_{n})\|\leq\epsilon.$
\begin{prop}
\label{pro:NCpolyRelationsAreClosed} Suppose $\epsilon>0$ is real
number. If $p$ is a NC $*$-polynomial in $x_{1},\ldots,x_{n},$
with constant term zero, then each of\[
p\left(x_{1},\ldots,x_{n}\right)=0,\]
\[
p\left(x_{1},\ldots,x_{n}\right)\geq0,\]
\[
\left\Vert p\left(x_{1},\ldots,x_{n}\right)\right\Vert \leq\epsilon\]
is a closed $C^{*}$-relation.
\end{prop}
\begin{proof}
Consider the last relation, for illustration. The other parts of the
proof are similar.
Certainly \[
\left\Vert p\left(0,0,0,\ldots,0\right)\right\Vert =\left\Vert 0\right\Vert =0\leq\epsilon.\]
The evaluation of a NC $*$-polynomial does not depend on the ambient
algebra. Therefore axiom \textbf{R2} holds.
NC $*$-polynomials are natural, so \textbf{R3} holds.
Given families $\left\langle x_{j}^{(\lambda)}\right\rangle $ with
\[
\sup_{\lambda}\left\Vert x_{j}^{(\lambda)}\right\Vert <\infty\]
we immediately get elements in the product $C^{*}$-algebra $\prod A_{\lambda}.$
All NC $*$-polynomials respect products, so\begin{eqnarray*}
& & \left\Vert p\left(\left\langle x_{1}^{(\lambda)}\right\rangle ,\left\langle x_{2}^{(\lambda)}\right\rangle ,\left\langle x_{3}^{(\lambda)}\right\rangle ,\ldots,\left\langle x_{n}^{(\lambda)}\right\rangle \right)\right\Vert \\
& = & \left\Vert \left\langle p\left(x_{1}^{(\lambda)},x_{2}^{(\lambda)},x_{3}^{(\lambda)},\ldots,x_{n}^{(\lambda)}\right)\right\rangle \right\Vert \\
& = & \sup_{\lambda}\left\Vert p\left(x_{1}^{(\lambda)},x_{2}^{(\lambda)},x_{3}^{(\lambda)},\ldots,x_{n}^{(\lambda)}\right)\right\Vert \\
& \leq & \epsilon.\end{eqnarray*}
Therefore \textbf{R4b} holds.
\end{proof}
The relation that inspired this study is $x\leq y,$ which is easily
shown to be RFD using an approximate unit.
\begin{prop}
If $f$ is a continuous, real-valued function on $[0,\infty),$ then\[
\left\{ x^{*}=x,\ y^{*}=y,\ f(x)\leq f(y)\right\} \]
is a closed set of $C^{*}$-relations.
\end{prop}
\begin{proof}
This follows from trivial facts such as $f(0)\leq f(0)$ and well-known
facts about the functional calculus.
\end{proof}
\begin{prop}
The relation $x=x^{*}$ is closed $C^{*}$-relation.
\end{prop}
\begin{proof}
This a special case of Proposition~\ref{pro:NCpolyRelationsAreClosed}
\end{proof}
\begin{prop}
The relation $0\leq x$ is closed $C^{*}$-relation.
\end{prop}
\begin{proof}
A key fact about $C^{*}$-algebras is that positivity ($x=y^{*}y$
for some $y$) does not depend on the ambient $C^{*}$-algebra either.
See, for example, Section 1.6.5 in \cite{DixmierCstarAlgebras}.
\end{proof}
\begin{prop}
If $f$ is a holomorphic function on the complex plane and $\epsilon$
is real number such that $\epsilon\geq|f(0)|$ then $\left\Vert f(x)\right\Vert \leq\epsilon$
is a closed $C^{*}$-relation.
\end{prop}
\begin{proof}
The functional calculus is to be applied in the unitization of the
ambient $C^{*}$-algebra. We know $f(0)=f(0)\mathbb{1}$ for the zero
operator and so $\{0\}$ is a representation. The holomorphic functional
calculus is natural, does not depend on the surrounding $C^{*}$-algebra,
and respects finite products since polynomial do.
\end{proof}
\begin{prop}
The union of two closed sets of $C^{*}$-relations on the same set
is closed. If a set of $C^{*}$-relations on a set $\mathcal{X}$
is closed, the same properties applied to a larger set $\mathcal{Y}\supseteq\mathcal{X}$
form a closed set of $C^{*}$-relations.
\end{prop}
\begin{proof}
These statements should be obvious.
\end{proof}
\begin{prop}
\label{pro:fractionalNCpolysAreClosed} If $p$ is a NC $*$-polynomial
in $x_{1},\ldots,x_{n}$ with constant term zero, and give $t_{1},\ldots,t_{m}$
positive exponents and possibly repeated indices $j_{1},\ldots,j_{m},$
then \[
\left\{ x_{j}^{*}=x_{j},\ p\left(x_{j_{1}}^{t_{1}},\ldots,x_{j_{m}}^{t_{m}}\right)\geq0\right\} \]
is a closed set of $C^{*}$-relations. If only integer powers of $x_{r}$
are used the relation $x_{r}^{*}=x_{r}$ may be dropped and the result
is still a closed set of $C^{*}$-relations.
\end{prop}
\begin{proof}
Pile the functional calculus higher and deeper. Just to illustrate,
we are talking about a relation such as $x^{\frac{1}{3}}yx^{\frac{2}{3}}\geq0$
applied to pairs $(x,y)$ of operators where $x$ is positive.
\end{proof}
Harder is the task of finding RFD relations. We start with the classic
that kicked off this investigation.
\begin{thm}
The set of $C^{*}$-relations\[
\left\{ x^{*}=x,\ y^{*}=y,\ x\leq y\right\} \]
is RFD.
\end{thm}
\begin{proof}
We dress L\"{o}wner's argument (\cite{BendatSherman}) in categorical
clothing.
By Lemma~\ref{lem:RFDapproxCyclicSuffices} we need only consider
representations on a separable Hilbert space $\mathbb{H}$. Suppose
$x$ and $y$ are bounded operators on $\mathbb{H}$ that are self-adjoint
and that $x\leq y.$ Let $p_{n}$ be the projection onto the first
$n$ elements in some fixed orthonormal basis. Define $x_{n}=p_{n}xp_{n}$
and $y_{n}=p_{n}yp_{n}$ so that $x_{n}$ and $y_{n}$ have norms
bounded by $\| x\|$ and $\| y\|$ and $0\leq x_{n}\leq y_{n}.$ The
operators $x$ and $y$ form finite dimensional relations, and they
converge $*$-strongly to $x$ and $y.$
\end{proof}
\begin{thm}
\label{lem:rightBoundOnRealPartIsRFD} Suppose $\beta>0$ is a real
number. The $C^{*}$-relation $\textnormal{Re }x\leq\beta$ is RFD.
\end{thm}
\begin{proof}
This is almost identical to the last proof. \ifthenelse{\boolean{Details}}{We
adopt that notation. Given $x$ on $\mathbb{H}$ with $x+x^{*}\leq2\beta$
let $x_{n}=p_{n}xp_{n}.$ Then \begin{eqnarray*}
x_{n}+x_{n}^{*} & = & p_{n}(x+x^{*})p_{n}\\
& \leq & p_{n}(2\beta)p_{n}\\
& \leq & 2\beta\end{eqnarray*}
and $\| x_{n}\|\leq\left\Vert x\right\Vert .$}{}
\end{proof}
\begin{thm}
Suppose $\beta>1$ is a real number. The $C^{*}$-relation $\left\Vert e^{\textnormal{Re }x}\right\Vert \leq\beta$
is RFD.
\end{thm}
\begin{proof}
Since the real part of $x$ is Hermitian, the exponential of the real
part is hermitian with spectrum contained in the positive real line.
Therefore $\left\Vert e^{\textnormal{Re }x}\right\Vert \leq e^{\beta}$
holds if and only if $e^{\textnormal{Re }x}\leq e^{\beta}$ which
holds if and only if $\textnormal{Re }x\leq\ln(\beta).$ This relation
has the same representations as the relation in Lemma~\ref{lem:rightBoundOnRealPartIsRFD},
so must itself be RFD.
\end{proof}
\begin{thm}
The empty set of relations on any set $\mathcal{X}$ is a closed $C^{*}$-relation.
\end{thm}
\begin{proof}
This is more-or-less contained in \cite{GoodearlMenalRFDandFree}.
Given a Hilbert space $\mathbb{H}$ and operators $a_{j}$ on $\mathbb{H}$
with $j\in\mathcal{X},$ take any net of finite-rank projections $p_{\lambda}$
converging strongly to the identity. Then $p_{\lambda}a_{j}p_{\lambda}$
is bounded in norm by $\| a_{j}\|$ and converges $*$-strongly to
$a_{j}.$
\end{proof}
Many amalgamated products $A\ast_{C}B$ turn out to be RFD when $A$
and $B$ are RFD. The simplest theorem of this sort, proved in \cite{ExelLoringRFDfreeProd},
is that $A$ and $B$ being RFD implies $A\ast B$ is RFD. This generalizes
easily here to something very useful.
\begin{thm}
Suppose $\mathcal{X}$ and $\mathcal{Y}$ are disjoint sets. If $\mathcal{R}$
is an RFD set of $C^{*}$-relations on $\mathcal{X}$ and $\mathcal{S}$
is an RFD closed set of $C^{*}$-relations on $\mathcal{Y}$ then,
regarding both sets as relations on $\mathcal{X}\cup\mathcal{Y},$
the set $\mathcal{R}\cup\mathcal{S}$ is an RFD set of $C^{*}$-relations.
\end{thm}
\begin{proof}
All we need to know is that if $A$ is a set of operators on $\mathbb{H}$
that are zero on the orthogonal complement of the finite dimensional
subspace $\mathbb{H}_{1},$ and if $B$ is a set of operators on $\mathbb{H}$
that are zero on the orthogonal complement of the finite dimensional
subspace $\mathbb{H}_{2},$ then the union is a set of operators that
is zero on the orthogonal complement of the subspace $\mathbb{H}_{1}+\mathbb{H}_{2},$
which is also finite dimensional.
\end{proof}
\begin{prop}
\label{pro:softNCpolyRelationsAreRFD} If $p_{1},\ldots,p_{m}$ are
NC $*$-polynomials in $x_{1},\ldots,x_{n},$ are homogeneous of degrees
that can vary, and $\epsilon_{s}>0$ are real constants and $0\leq n_{1}\leq n_{2}\leq n$
then\begin{eqnarray*}
& & 0\leq x_{j},\quad(j=1,\ldots,n_{1})\\
& & x_{j}^{*}=x_{j},\quad(j=n_{1}+1,\ldots,n_{2})\\
& & \left\Vert p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert \leq\epsilon_{s}\quad(s=1,\ldots,m)\end{eqnarray*}
form a closed set of $C^{*}$-relations.
\end{prop}
\begin{proof}
Assume $x_{1},\ldots,x_{n}$ are in $\mathbb{B}(\mathbb{H})$ where
$\mathbb{H}$ is separable and that these satisfy the above relations.
Let $u_{k}$ be a countable approximate identity for the compact operators,
with $0\leq u_{k}\leq1,$ that is quasi-central for $x_{1},\ldots,x_{n}.$
Such an approximate identity exists by Corollary~3.12.16 in \cite{Pedersen-C*-algebrasBook}.
Applying a decreasing perturbation to each $u_{k}$ we may further
assume each $u_{k}$ is finite-rank.
Let $x_{j,k}=u_{k}x_{j}u_{k}.$ Clearly $\| x_{j,k}\|\leq\| x_{j}\|$
and $0\leq x_{j,k}$ for $j\leq n_{1}$ and $x_{j,k}^{*}=x_{j,k}$
for $n_{1}<j\leq n_{2}.$ Also $x_{j,k}\rightarrow x_{j}$ in the
$*$-strong topology. For fixed $k$ the $x_{j,k}$ all act as zero
on the complement of range of $u_{k},$ which is finite dimensional.
However, we need to modify the $x_{j,k}$ to make the last line of
relations hold.
Suppose $p_{s}$ is homogeneous of degree $d_{s}.$ This means $u_{k}$
appears $2d_{s}$ times in each monomial in $p_{s}.$ Since $u_{k}$
is quasi-central for the $x_{j}$ we have\[
\lim_{k\rightarrow\infty}\left\Vert p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)-u_{k}^{2d_{s}}p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert =0.\]
Therefore\begin{eqnarray*}
\limsup_{k\rightarrow\infty}\left\Vert p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)\right\Vert & = & \limsup_{k\rightarrow\infty}\left\Vert u_{k}^{2d_{s}}p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert \\
& \leq & \left\Vert p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert .\end{eqnarray*}
It is easy to show that if $a_{\lambda}\rightarrow a$ in the strong
topology then\[
\liminf_{\lambda\rightarrow\infty}\left\Vert a_{\lambda}\right\Vert \geq\left\Vert a\right\Vert .\]
The $x_{j,k}$ are bounded sequences converging to $x_{j}$ in the
$*$-strong topology, so\[
\lim_{k}p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)=p_{s}\left(x_{1},\ldots,x_{n}\right)\quad(*\mbox{-strong})\]
which means\[
\limsup_{k\rightarrow\infty}\left\Vert p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)\right\Vert =\left\Vert p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert .\]
If $p_{s}\left(x_{1},\ldots,x_{n}\right)=0$ let $\alpha_{j,k}=1,$
and otherwise let \[
\alpha_{j,k}=\max\left(1,\left(\frac{\left\Vert p_{s}\left(x_{1},\ldots,x_{n}\right)\right\Vert }{\left\Vert p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)\right\Vert }\right)^{\frac{1}{d_{s}}}\right).\]
Let $y_{j,k}=\alpha_{j,k}x_{j,k}.$ The $\alpha_{j,k}$ are are most
$1$ so $\| y_{j,k}\|\leq\| x_{j}\|.$ We are scaling by positive
factors so $0\leq y_{j,k}$ for $j\leq n_{1}$ and $y_{j,k}^{*}=y_{j,k}$
for $n_{1}<j\leq n_{2}.$ Since $\alpha_{j,k}\rightarrow1$ we see
that $y_{j,k}\rightarrow x_{j}$ in the $*$-strong topology. For
fixed $k$ the $y_{j,k}$ still all act as zero on the complement
of range of $u_{k}.$ What we have gained are the final relations,\begin{eqnarray*}
\left\Vert p_{s}\left(y_{1,k},\ldots,y_{n,k}\right)\right\Vert & = & \left\Vert \alpha_{j,k}^{d_{s}}p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)\right\Vert \\
& = & \alpha_{j,k}^{d_{s}}\left\Vert p_{s}\left(x_{1,k},\ldots,x_{n,k}\right)\right\Vert \\
& \leq & \epsilon_{s}.\end{eqnarray*}
\end{proof}
\section{Applications}
We have several corollaries to Theorem\ \ref{thm:RFDimpliesClosedReduction}.
All these results refer to the operator norm or order relations. Of
course, we recover the result of L\"{o}wner that matrix monotone
for all orders implies operator monotone.
\begin{cor}
Let $a$ be a bounded operator. Then\[
\left\Vert e^{a}\right\Vert \leq\left\Vert e^{\textnormal{Re}(a)}\right\Vert .\]
\end{cor}
\begin{proof}
Theorem IX.3.1 of \cite{BhatiaMatrixAnalysis} tells us this result
is true for any matrix, and indeed with any unitarily invariant norm.
Since $\left\Vert e^{x}\right\Vert \geq1$ for any operator $x,$
we can rephrase this to say that for each $\alpha\geq1,$ we have\[
\left\Vert e^{\textnormal{Re}(a)}\right\Vert \leq\alpha\implies\left\Vert e^{a}\right\Vert \leq\alpha.\]
As the first relation is RFD and the second is closed, we are done
by Theorem\ \ref{thm:RFDimpliesClosedReduction}.
\end{proof}
The NC $*$-polynomial version (in the original variables, not their
fractional powers) of the following can be proven by Helton's sum-of-squares
theorem, Theorem\ 1.1 in \cite{McCulloughPutinarNCsumofSquares},
which is essentially in \cite{HeltonPositiveNCpoly}.
\begin{cor}
Suppose that $p$ is a NC $*$-polynomial in $x_{1},\ldots,x_{n}$
with constant term zero, and that $t_{1},\ldots,t_{m}$ are positive
exponents and $j_{1},\ldots,j_{m}$ are between $1$ and $n.$ If
\[
p\left(x_{j_{1}}^{t_{1}},\ldots,x_{j_{m}}^{t_{m}}\right)\geq0\]
for all self-adjoint matrices $x_{1},\ldots,x_{n}$ then the same
hold true for all self-adjoint operators on Hilbert space. If only
integer powers of $x_{r}$ are used the relation $x_{r}^{*}=x_{r}$
may be dropped.
\end{cor}
\begin{proof}
The null set of relations is RFD so Theorem\ \ref{thm:RFDimpliesClosedReduction}
still applies.
\end{proof}
\begin{cor}
Let $a,$ $b$ and $x$ be a bounded operators, with $a\geq0$ and
$b\geq0$. Then, for $0\leq\nu\leq1,$\[
\left\Vert a^{\nu}xb^{1-\nu}+a^{1-\nu}xb^{\nu}\right\Vert \leq\left\Vert ax+xb\right\Vert .\]
\end{cor}
\begin{proof}
This is in \cite{BhatiaMatrixAnalysis} as Corollary IX.4.10, restricted
to matrices but for any unitarily invariant norm. Using the operator
norm version of that result, Proposition\ \ref{pro:softNCpolyRelationsAreRFD}
and Proposition\ \ref{pro:fractionalNCpolysAreClosed} we find again
the the operator result follows from the matrix result.
\end{proof}
\begin{cor}
If $C$ is a constant so that\begin{equation}
\| a\|\leq1\mbox{ and }b\geq0\implies\left\Vert ab^{\frac{1}{2}}-b^{\frac{1}{2}}a\right\Vert \leq C\left\Vert ab-ba\right\Vert ^{\frac{1}{2}}\label{eq:commutatorSquareRoot}\end{equation}
for all matrices $a$ and $b,$ then (\ref{eq:commutatorSquareRoot})
is true for all bounded operators on Hilbert space (or $C^{*}$-algebra
elements).
\end{cor}
\begin{proof}
We all hope that the constant $C=1$ works here, c.f.~\cite{BhatiaKittanehInequalitiesNormsCommutators,PedersenCommutatorInequality}.
Perhaps this reduction to the matrix case will make that easier to
prove.
We can rephrase this as\[
\| a\|\leq1,\ b\geq0,\ \left\Vert ab-ba\right\Vert \leq\delta\implies\left\Vert ab^{\frac{1}{2}}-b^{\frac{1}{2}}a\right\Vert \leq C\delta^{\frac{1}{2}}.\]
The set of relations on the left is RFD by Proposition\ \ref{pro:softNCpolyRelationsAreRFD}
and those on the right are closed by Proposition\ \ref{pro:fractionalNCpolysAreClosed}.
\end{proof}
| 215,764
|
Enlightened – Season 3 Episode 3 – Video Preview – Higher Power
Here’s the video preview of Enlightened Season 3 Episode 3: “Higher Power” on HBO. Airs January 27, 2013.
Spoilers in the synopsis.
Amy gets a letter from Levi in Hawaii, where he refuses to buy into Open Air’s rehabilitation philosophy and decides to accompany a pair of younger, fellow addicts for a night of indulgence at a nearby hotel.
Latest News on Enlightened
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| 116,269
|
Poll Question
FREEPORT, Maine — Independent former Gov. Angus King took an early lead and cruised to victory Tuesday in the race to become Maine’s next U.S. senator, eight months after Olympia Snowe shocked the political establishment by announcing she wouldn’t seek another term.
King had claimed 54 percent of the vote with 15 percent of precincts reporting. Republican Charlie Summers trailed with 30 percent while Democrat Cynthia Dill had the support of 13 percent of votes.
Independents Andrew Ian Dodge, Danny Dalton and Steve Woods each captured about 1 percent of the vote.
In his victory speech at a Freeport hotel at about 9 p.m., where he was greeted by bagpipes, King said that Maine voters sent a clear message that they want less division in politics.
“They’re tired of the false choice that always seems to confront them,” he said. “As a guy said on this campaign, ‘I always wanted a chance to vote for none of the above. And you’re it.’”
King joked that former George W. Bush adviser Karl Rove invigorated his campaign with new energy when his group Crossroads GPS started airing anti-King ads this fall.
“I spent a lot of time promoting Maine and Maine tourism,” said the former two-term governor, “and I’ve never said this, but I hope that man never comes to Maine.”
Summers conceded the race around 10:40 p.m. before family members and supporters at the Portland Regency Hotel.
“We have a very difficult situation that this country faces, and I think it’s critical that we line up behind our representatives,” Summers said. “Sen.-elect King certainly needs our support.”
Thanking his family, Summers alluded to his three previous unsuccessful runs to represent Maine’s 1st District in Congress. “They’ve been with me through a number of elections, a number of these announcements, unfortunately,” he said.
Snowe, who largely sat out the race although Summers served as her state director for nine years, issued a congratulatory statement to King just after 9 p.m., well before Summers had conceded.
“I have known Angus for many years and worked closely with him on issues critical to Maine during his two terms as governor,” she said. “I know he cares deeply about Maine people and the future of our nation.
“We had a very good conversation — I offered anything I could do to assist him with a smooth transition, and we will be meeting toward that end in the near future.”
Sen. Susan Collins, who will be Maine’s senior senator, also congratulated King, saying, “I’m sure that Angus and I will be meeting soon to discuss committee assignments and how we can work together to meet the challenges facing our state and our nation.”
It was an election contest that, at the start of the year, political observers didn’t think would happen and that the national Republican Party didn’t think it would have to worry about losing.
Snowe’s late-February departure had prominent Maine politicians on both sides of the aisle weighing a Senate bid. But on the Democratic side, big names like former Gov. John Baldacci and U.S. Reps. Chellie Pingree and Mike Michaud ultimately backed away when independent former Gov. Angus King threw his hat into the ring.
While King was an early favorite and maintained a lead in every poll taken in the Senate race, Republicans and affiliated groups didn’t make the campaign an easy one for him.
Nearly $7.4 million poured into the state from outside groups, with more than half — $4.24 million — paying for advertising designed to peel support away from King and boost Summers.
The U.S. Chamber of Commerce was the first in a string of right-leaning groups to launch an ad offensive against King in an effort to cut into his favorability and create an opening for Summers, the Chamber’s endorsed candidate. The U.S. Chamber spent $1.35 million on three separate ad campaigns that attacked King for his fiscal record during his two terms as governor.
The National Republican Senatorial Committee also joined the anti-King ad wars in September, with accusations related to King’s wind energy business before King boosters shot back.
While King hasn’t revealed intentions to caucus with either party if elected, national Democrats have largely expected him to caucus with them.
| 89,279
|
\begin{document}
\title[Quantum Heisenberg group algebra]
{${}^*$-representations of a quantum Heisenberg group algebra}
\author{Byung--Jay Kahng}
\date{}
\address{Department of Mathematics\\ University of Kansas\\
Lawrence, KS 66045}
\email{bjkahng@math.ukans.edu}
\subjclass{46L87, 81R50, 22D25}
\begin{abstract}
In our earlier work, we constructed a specific non-compact quantum group
whose quantum group structures have been constructed on a certain twisted
group $C^*$-algebra. In a sense, it may be considered as a ``quantum
Heisenberg group $C^*$-algebra''. In this paper, we will find, up to
equivalence, all of its irreducible ${}^*$-representations. We will point
out the Kirillov type correspondence between the irreducible representations
and the so-called ``dressing orbits''. By taking advantage of its
comultiplication, we will then introduce and study the notion of {\em inner
tensor product representations\/}. We will show that the representation
theory satisfies a ``quasitriangular'' type property, which does not appear
in ordinary group representation theory.
\end{abstract}
\maketitle
{\sc Introduction.}
Recently in \cite{BJKp2}, we constructed a specific non-compact $C^*$-algebraic
quantum group, via deformation quantization of a certain non-linear Poisson--Lie
bracket on an exponential solvable Lie group. The underlying $C^*$-algebra of
this quantum group has been realized as a twisted group $C^*$-algebra of a
nilpotent Lie group.
From its construction, it is reasonable to view it as a ``quantum Heisenberg
group $C^*$-algebra'' (This observation will be made a little clearer after the
first section.). Focusing on its $C^*$-algebra structure, we study in this paper
its irreducible ${}^*$-representations. It is not difficult to see that there
exists a Kirillov type, one-to-one correspondence between the irreducible
${}^*$-representations and the ``dressing orbits'' at the level of its
Poisson--Lie group counterpart.
Since the object of our study is actually a Hopf $C^*$-algebra (i.\,e. quantum
group), we may use its comultiplication to define and study {\em inner tensor
product\/} of representations. This is a generalization to quantum case of the
inner tensor product representations of an ordinary group. We will show that
unlike in the case of an ordinary group and in the cases of many earlier examples
of non-compact quantum groups (e.\,g. \cite{VD}, \cite{SZ},\cite{Rf5}, \cite{Ld}),
the inner tensor product representations of our Hopf $C^*$-algebra satisfy a
certain ``quasitriangularity'' property.
Here, we only study representation theory of the specific example of \cite{BJKp2}.
However, our earlier results (\cite{BJKp1,BJKp2}) imply that this quantum group
is just one example of a larger class of solvable quantum groups having twisted
group $C^*$-algebras or (more general) twisted crossed product algebras as
underlying $C^*$-algebras. One of the main purposes of \cite{BJKp2} and this
paper is to present a case study, so that we are later able to develop a
procedure to construct and study more general class of locally compact quantum
groups. Eventually, we wish to further develop a generalized orbit theory of
Kirillov type, which would then be used to study the harmonic analysis of the
locally compact quantum groups. This will be our forthcoming project.
Since this paper is essentially a continuation of \cite{BJKp2}, we will
keep the same notation as in that paper. Some of the notation is reviewed
in section 1. This section is followed by Appendix, where we discuss the
classical counterparts (Poisson--Lie groups) to our quantum groups. Main
purpose here is to calculate the dressing orbits on these Poisson--Lie groups.
Although the calculations are not difficult, these results could not be found
in the literatures. Results here will be useful in our future study.
In the second section, we discuss the representation theory of our example.
We find all the irreducible ${}^*$-representations up to equivalence. We then
point out the Kirillov type one-to-one correspondence between these representations
and the dressing orbits.
In the last section, we study inner tensor product representations. By using
the quasitriangular quantum $R$-matrix operator obtained in \cite{BJKp2}, we
will show some interesting properties that do not appear in ordinary group
representation theory.
\section{The Hopf $C^*$-algebras}
Our objects of study are the Hopf $C^*$-algebras $(A,\Delta)$ and $(\tilde{A},
\tilde{\Delta})$ constructed in \cite{BJKp2}. As a $C^*$-algebra, $A$ is
isomorphic to a twisted group $C^*$-algebra. That is, $A\cong C^*\bigl(H/Z,
C_{\infty}(\Gg/\Gq),\sigma\bigr)$, where $H$ is the $(2n+1)$ dimensional Heisenberg
Lie group (see Appendix) and $Z$ is the center of $H$. Whereas, $\Gg=\Gh^*$ is
the dual space of the Lie algebra $\Gh$ of $H$ and $\Gq=\Gz^{\bot}$, for $\Gz
\subseteq\Gh$ corresponding to $Z$. We denoted by $\sigma$ the twisting cocycle
for the group $H/Z$. As constructed in \cite{BJKp2}, $\sigma$ is a continuous
field of cocycles $\Gg/\Gq\ni r\mapsto\sigma^r$, where
\begin{equation}\label{(sigma)}
\sigma^r\bigl((x,y),(x',y')\bigr)=\bar{e}\bigl[\eta_{\lambda}(r)\beta(x,y')\bigr].
\end{equation}
Following the notation of the previous paper, we used: $\bar{e}(t)=e^{(-2\pi i)t}$
and $\eta_{\lambda}(r)=\frac{e^{2\lambda r}-1}{2\lambda}$. The elements $(x,y)$,
$(x',y')$ are group elements in $H/Z$.
In \cite{BJKp2}, we showed that the $C^*$-algebra $A$ is a strict deformation
quantization (in the sense of Rieffel) of $C_{\infty}(\Gg)$, the commutative
algebra of continuous functions on $\Gg$ vanishing at infinity. For convenience,
the deformation parameter $\hbar$ has been fixed ($\hbar=1$), which is the reason
why we do not see it in the definition of $A$. When $\hbar=0$ (i.\,e. classical
limit), it turns out that $\sigma\equiv1$. So $A_{\hbar=0}\cong C_{\infty}(G)$.
Throughout this paper (as in \cite{BJKp2}), we write $A=A_{\hbar=1}$.
We could also construct a ``(regular) multiplicative unitary operator'' $U\in
{\mathcal B}({\mathcal H}\otimes{\mathcal H})$ associated with $A$, in the sense
of Baaj and Skandalis. Thus $(A,\Delta)$ becomes a Hopf $C^*$-algebra, whose
comultiplication $\Delta$ is determined by $U$. Similar realization as a twisted
group $C^*$-algebra also exists for the ``extended'' Hopf $C^*$--algebra $(\tilde{A},
\tilde{\Delta})$. Again, its comultiplication $\tilde{\Delta}$ is determined by
a certain regular multiplicative unitary operator $\tilde{U}$.
Actually, $(A,\Delta)$ is an example of a {\em locally compact quantum group\/},
equipped with the counit, antipode, and Haar weight, as constructed in \cite{BJKp2}.
We do not intend to give here the correct definition of a locally compact quantum
group, which is still at a primitive stage (But see \cite{KuV}, \cite{MN}, \cite{Wr7}
for some recent developments.). Since the main goal of the present paper is in the
study of ${}^*$-representations of $A$, it would be rather sufficient to focus on
the Hopf $C^*$-algebra structure of $(A,\Delta)$. For this reason, our preferred
terminology throughout this paper for $(A,\Delta)$ will be the ``Hopf $C^*$-algebra'',
although much stronger notion of the ``locally compact quantum group'' is still
valid. Similar comments holds also for the extended Hopf $C^*$-algebra $(\tilde{A},
\tilde{\Delta})$.
Before we continue our discussion, we insert the following Appendix. It serves
three purposes: First, it briefly reviews the notation introduced in \cite{BJKp2}
and is being used here. Second, it briefly gives a summary of the Poisson--Lie
group theory for those readers who are less familiar with the subject. Third,
we calculate the dressing orbits on the Poisson--Lie groups which are classical
counterparts to our quantum groups.
\bigskip
\begin{center}
{\sc Appendix: Poisson--Lie groups, dressing actions and dressing orbits}
\end{center}
\noindent {\bf A.1. The Poisson--Lie groups.}
Let $H$ be the $(2n+1)$--dimensional Heisenberg Lie group such that the space
for the group is isomorphic to $\mathbb{R}^{2n+1}$ and the multiplication on it
is defined by
$$
(x,y,z)(x',y',z')=\bigl(x+x',y+y',z+z'+\beta(x,y')\bigr),
$$
for $x,y,x',y'\in\mathbb{R}^n$ and $z,z'\in\mathbb{R}$. Here $\beta(\ ,\ )$
is the usual inner product on $\mathbb{R}^n$, following the notation of
\cite{BJKp2}.
Consider also the extended Heisenberg Lie group $\tilde{H}$, with the group law
defined by
$$
(x,y,z,w)(x',y',z',w')=\bigl(x+e^{w}x',y+e^{-w}y',z+z'+(e^{-w})\beta(x,y'),
w+w'\bigr).
$$
The notation is similar as above, with $w,w'\in\mathbb{R}$. This group contains
$H$ as a normal subgroup.
In \cite{BJKp2}, we obtained the ``dual Poisson--Lie group'' $G$ of $H$. It is
determined by the multiplication law:
$$
(p,q,r)(p',q',r')=(e^{\lambda r'}p+p',e^{\lambda r'}q+q',r+r'),
$$
while the dual Poisson--Lie group $\tilde{G}$ of $\tilde{H}$ is determined by the
multiplication law:
$$
(p,q,r,s)(p',q',r',s')=(e^{\lambda r'}p+p',e^{\lambda r'}q+q',r+r',s+s').
$$
Here $\lambda\in\mathbb{R}$ is a fixed constant, which determines a certain
non-linear Poisson structure on $G$ (or $\tilde{G}$) when $\lambda\ne0$.
Although we do not explicitly mention the Poisson brackets here (see instead
\cite{BJKp2}), we can show indeed that $\tilde{H}$ and $\tilde{G}$ (similarly,
$H$ and $G$) are mutually dual Poisson--Lie groups. For definition and some
important results on Poisson--Lie groups, see for example the article by Lu and
Weinstein \cite{LW} or the book by Chari and Pressley \cite{CP}.
In \cite{BJKp2}, using the realization that the Poisson bracket on $G$ is a
non-linear Poisson bracket of the ``cocycle perturbation'' type, we have been
able to construct a quantum version of $G$: The (non-commutative) Hopf
$C^*$--algebra $(A,\Delta)$, whose underlying $C^*$--algebra is a twisted group
$C^*$--algebra. Similarly for $\tilde{G}$, we constructed the Hopf $C^*$--algebra
$(\tilde{A},\tilde{\Delta})$. These are the main objects of study in \cite{BJKp2}
and in this paper.
In this Appendix, we will study a special kind of a Lie group action of a
Poisson--Lie group on its dual Poisson--Lie group, called a ``dressing action''.
The orbits under the dressing action are the ``dressing orbits''. We are
going to compute here the dressing orbits for our examples $\tilde{G}$ and
$\tilde{H}$ (and also $G$ and $H$).
\bigskip
\noindent {\bf A.2. Basic definitions: Dressing actions.}
Let $G$ be a Poisson--Lie group, let $G^*$ be its dual Poisson--Lie group,
and let $\Gg$ and $\Gg^*$ be the corresponding Lie algebras. Together,
$(\Gg,\Gg^*)$ forms a Lie bialgebra. On the vector space $\Gg\oplus\Gg^*$,
we can define a bracket operation by
\begin{align}
&\bigl[(X_1,\mu_1),(X_2,\mu_2)\bigr] \notag\\
&=\bigl([X_1,X_2]_{\Gg}-\operatorname{ad}^*_{\mu_2}X_1
+\operatorname{ad}^*_{\mu_1}X_2,[\mu_1,\mu_2]_{\Gg^*}
+\operatorname{ad}^*_{X_1}\mu_2-\operatorname{ad}^*_{X_2}\mu_1\bigr),
\tag{A.1}
\end{align}
where $\operatorname{ad}^*_{X}\mu$ and $\operatorname{ad}^*_{\mu}X$ are,
respectively, the coadjoint representations of $\Gg$ on $\Gg^*$ and of
$\Gg^*$ on $\Gg=(\Gg^*)^*$. This is a Lie bracket on $\Gg\oplus\Gg^*$,
which restricts to the given Lie brackets on $\Gg$ and $\Gg^*$. We denote
the resulting Lie algebra by $\Gg\bowtie\Gg^*$, the {\em double Lie algebra\/}
\cite{LW}.
Let $D=G\bowtie G^*$ be the connected, simply connected Lie group
corresponding to $\Gg\bowtie\Gg^*$. There are homomorphisms of Lie groups
$$
G\hookrightarrow D\hookleftarrow G^*,
$$
lifting the inclusion maps of $\Gg$ and $\Gg^*$ into $\Gg\bowtie\Gg^*$. Thus
we can define a product map $G\times G^*\to D$. We will assume from now on
that the images of $G$ and $G^*$ are closed subgroups of $D$ and that the map
is a global diffeomorphism of $G\times G^*$ onto $D$. In this case, we will
say that $D$ is a {\em double Lie group\/}. In particular, each element of
$D$ has a unique expression $g\cdot\gamma$, for $g\in G$ and $\gamma\in G^*$.
Suppose we are given a double Lie group $D=G\bowtie G^*$. For $g\in G$ and
$\gamma\in G^*$, regarded naturally as elements in $D$, the product $\gamma\cdot
g$ would be factorized as
$$
\gamma\cdot g=g^{\gamma}\cdot\gamma^g,
$$
for some $g^{\gamma}\in G$ and $\gamma^g\in G^*$. We can see without difficulty
that the map $\lambda:G^*\times G\to G$ defined by
$$
\lambda_{\gamma}(g)=g^{\gamma}
$$
is a left action of $G^*$ on $G$. Hence,
\begin{defna1}\label{dressing}
The map $\rho:G\times G^*\to G$ defined by
$$
\rho_{\gamma}(g)=g^{(\gamma^{-1})}
$$
is a right action, called the {\em dressing action\/} of $G^*$ on $G$.
\end{defna1}
\begin{rem}
Sometimes, the action $\lambda$ is called the left dressing action, while the
action $\rho$ is called the right dressing action. It is customary to call the
right action $\rho$ the dressing action. Semenov--Tian--Shansky \cite{Se} first
proved that the (right) dressing action of $G^*$ on $G$ is a Poisson action
(i.\,e. it preserves the respective Poisson structures). The notion of dressing
action still exists (at least locally), even if the assumption that $G\times G^*
\cong D$ is not satisfied. See \cite{LW}.
\end{rem}
For any Lie algebra $\Gh$, it can always be regarded as a Lie bialgebra by
viewing its dual vector space $\Gg=\Gh^*$ as an abelian Lie algebra. Then
the dressing action of $H$ on $G$ actually coincides with the coadjoint
action of $H$ on $\Gh^*\,(=\Gg\cong G)$. In this sense, we may regard the
dressing action as a generalization of the coadjoint action. This is the
starting point for the attempts to generalize the Kirillov's orbit theory,
and this point of view has been helpful throughout this paper.
We conclude the subsection by stating the following important result by
Semenov--Tian--Shansky, which exhibits the close relationship between dressing
actions and the geometric aspects of Poisson--Lie groups. This result appeared
in \cite{Se}, where he discusses the dressing actions in relation to the study
of complete integrable systems. For the proof of the theorem, see \cite{Se}
or \cite{LW}.
\begin{theorema2}\label{a2}
The dressing action $G^*$ on $G$ is a Poisson action. Moreover, the orbit of
the dressing action through $g\in G$ is exactly the symplectic leaf through
the point $g$ for the Poisson bracket on $G$.
\end{theorema2}
\bigskip
\noindent {\bf A.3. Dressing orbits for the Poisson--Lie groups $\tilde{G}$
and $\tilde{H}$.}
Let us consider our specific Poisson--Lie groups $\tilde{G}$ and $\tilde{H}$,
or equivalently, the Lie bialgebra $(\tilde{\Gg},\tilde{\Gh})$. We will
construct here the double Lie group, dressing action, and dressing orbits.
Along the way, we will obtain the corresponding results for $G$ and $H$.
By equation (A.1) and by using the Lie brackets on $\tilde{\Gg}$ and on
$\tilde{\Gh}$ (see \cite{BJKp2}), we can construct the double Lie algebra
$\tilde{\Gg}\bowtie\tilde{\Gh}$. The space for it is $\tilde{\Gg}\oplus
\tilde{\Gh}$, on which the following Lie bracket is defined:
\small
\begin{align}
&\bigl[(p,q,r,s;x,y,z,w),(p',q',r',s';x',y',z',w')\bigr] \notag \\
&=\bigl(\lambda(r'p-rp')+(w'p-wp')+(r'y-ry'),\lambda(r'q-rq')+
(wq'-w'q)+(rx'-r'x),0,\notag \\
&\qquad(p'\cdot x-p\cdot x')+(q\cdot y'-q'\cdot y);(wx'-w'x)+
\lambda(rx'-r'x),(w'y-wy')+\lambda(ry'-r'y), \notag \\
&\qquad\beta(x,y')-\beta(x',y)+\lambda(p'\cdot x-p\cdot x')+
\lambda(q'\cdot y-q\cdot y'),0\bigr). \notag
\end{align}
\normalsize
We then calculate the corresponding Lie group $\tilde{D}$. Using the notation
$\eta_{\lambda}(r)=\frac{e^{2\lambda r}-1}{2\lambda}$ (the function introduced
in Definition 2.3 of \cite{BJKp2}), the group $\tilde{D}$ is given by the
following multiplication law:
\small
\begin{align}
&(p,q,r,s;x,y,z,w)(p',q',r',s';x',y',z',w') \notag \\
&=\bigl(e^{\lambda r'}p+e^{-w}p'+e^{-\lambda r'}\eta_{\lambda}(r')y,
e^{\lambda r'}q+e^{w}q'-e^{-\lambda r'}\eta_{\lambda}(r')x,r+r',
\notag \\
&\qquad s+s'+(e^{-\lambda r'-w})p'\cdot x-(e^{-\lambda r'+w})q'\cdot y-
\eta_{\lambda}(-r')\beta(x,y);e^{-\lambda r'}x+e^{w}x',e^{-\lambda r'}y
+e^{-w}y', \notag \\
&\qquad z+z'+(e^{-\lambda r'-w})\beta(x,y')+\lambda(e^{-\lambda r'-w})p'
\cdot x+\lambda(e^{-\lambda r'+w})q'\cdot y+\lambda\eta_{\lambda}(-r')
\beta(x,y),w+w'\bigr). \notag
\end{align}
\normalsize
If we identify $(p,q,r,s)\in\tilde{G}$ with $(p,q,r,s;0,0,0,0)$ and
$(x,y,z,w)\in{\tilde{H}}$ with $(0,0,0,0;x,y,z,w)$, it is easy to see that
$\tilde{G}$ and $\tilde{H}$ are (closed) Lie subgroups of $\tilde{D}$.
This defines a global diffeomorphism of $\tilde{G}\times\tilde{H}$ onto
$\tilde{D}$, since any element $(p,q,r,s;x,y,z,w)$ of $\tilde{D}$ can be
written as
$$
(p,q,r,s;x,y,z,w)=(p,q,r,s;0,0,0,0)(0,0,0,0;x,y,z,w).
$$
In other words, $\tilde{D}$ is the {\em double Lie group\/} of $\tilde{G}$
and $\tilde{H}$, which will be denoted by $\tilde{G}\bowtie\tilde{H}$.
Meanwhile, if we consider only the $(p,q,r)$ and the $(x,y,z)$ variables,
we obtain in the same way the double Lie group $D=G\bowtie H$ of $G$ and $H$.
By Definition A.1, the {\em dressing action\/} of $\tilde{H}$ on $\tilde{G}$
is:
\begin{align}
\bigl(\rho(x,y,z,w)\bigr)(p,q,r,s)&=\bigl(e^{w}p-e^{-\lambda r+w}\eta
_{\lambda}(r)y,e^{-w}q+e^{-\lambda r-w}\eta_{\lambda}(r)x,r,
\notag \\
&\qquad s-e^{-\lambda r}p\cdot x+e^{-\lambda r}q\cdot y
+e^{-2\lambda r}\eta_{\lambda}(r)\beta(x,y)\bigr). \notag
\end{align}
The dressing orbits, which by Theorem A.2 are the symplectic leaves
in $\tilde{G}$ for its (non-linear) Poisson bracket, are:
\begin{itemize}
\item $\tilde{\mathcal O}_s=\{(0,0,0,s)\}$, when $(p,q,r)=(0,0,0)$.
\item $\tilde{\mathcal O}_{p,q}=\{(ap,\frac1a q,0,c):a>0,c\in\mathbb{R}\}$,
when $r=0$ but $(p,q)\ne(0,0)$.
\item $\tilde{\mathcal O}_{r,s}=\bigl\{\bigl(a,b,r,s-\frac1{\eta_{\lambda}
(r)}a\cdot b\bigr):(a,b)\in\mathbb{R}^{2n}\bigr\}$, when $r\ne0$.
\end{itemize}
Here $a\cdot b$ denotes the inner product. The $\tilde{\mathcal O}_s$ are
1--point orbits, the $\tilde{\mathcal O}_{p,q}$ are 2--dimensional orbits,
and the $\tilde{\mathcal O}_{r,s}$ are $2n$--dimensional orbits.
Similarly, if we only consider the $(p,q,r)$ and the $(x,y,z)$ variables,
we obtain the expression for the dressing action of $H$ on $G$ as follows:
$$
\bigl(\rho(x,y,z)\bigr)(p,q,r)=\bigl(p-e^{-\lambda r}\eta_{\lambda}(r)y,
q+e^{-\lambda r}\eta_{\lambda}(r)x,r\bigr).
$$
So the dressing orbits in $G$ are:
\begin{itemize}
\item (1--point orbits): ${\mathcal O}_{p,q}=\{(p,q,0)\}$, when
$r=0$.
\item ($2n$--dimensional orbits): ${\mathcal O}_r=\{(a,b,r):(a,b)
\in\mathbb{R}^{2n}\}$, when $r\ne0$.
\end{itemize}
Meanwhile, to calculate the dressing action of $\tilde{G}$ on $\tilde{H}$,
it is convenient to regard $\tilde{D}$ as the double Lie group $\tilde{H}
\bowtie\tilde{G}$ of $\tilde{H}$ and $\tilde{G}$. Indeed, there exists a
global diffeomorphism between $\tilde{H}\times\tilde{G}$ and $\tilde{D}$
defined by
\small
\begin{align}
&\bigl((x,y,z,w),(p,q,r,s)\bigr)\mapsto(0,0,0,0;x,y,z,w)(p,q,r,s,0,0,0,0)
\notag \\
&=\bigl(e^{-w}p+e^{-\lambda r}\eta_{\lambda}(r)y,e^{w}q-e^{-\lambda r}
\eta_{\lambda}(r)x,r,s+(e^{-\lambda r-w})p\cdot x-(e^{-\lambda r+w})q\cdot y
+\eta_{\lambda}(r)\beta(x,y); \notag \\
&\qquad e^{-\lambda r}x,e^{-\lambda r}y,z+\lambda(e^{-\lambda r-w})p\cdot x
+\lambda(e^{-\lambda r+w})q\cdot y-\lambda\eta_{\lambda}(r)\beta(x,y),w\bigr).
\notag
\end{align}
\normalsize
Similarly, we can show that $D$ is the double Lie group of $H$ and $G$.
Using this characterization of the double Lie group $\tilde{H}\bowtie
\tilde{G}$, the dressing action of $\tilde{G}$ on $\tilde{H}$ is obtained
by Definition A.1. That is,
\begin{align}
&\bigl(\rho(p,q,r,s)\bigr)(x,y,z,w) \notag \\
&=\bigl(e^{-\lambda r}x,e^{-\lambda r}y,z+\lambda(e^{-\lambda r})
p\cdot x +\lambda(e^{-\lambda r})q\cdot y-\lambda(e^{-2\lambda r})
\eta_{\lambda}(r)\beta(x,y),w\bigr). \notag
\end{align}
The dressing orbits in $\tilde{H}$ are:
\begin{itemize}
\item (1--point orbits): $\tilde{\mathcal O}_{z,w}=\{(0,0,z,w)\}$, when
$(x,y)=(0,0)$.
\item (2--dimensional orbits): $\tilde{\mathcal O}_{x,y,w}=\{(\alpha x,
\alpha y,\gamma,w):\alpha>0,\gamma\in\mathbb{R}\}$, when $(x,y)\ne(0,0)$.
\end{itemize}
Similarly, the dressing action of $G$ on $H$ is given by
$$
\bigl(\rho(p,q,r)\bigr)(x,y,z)=\bigl(e^{-\lambda r}x,e^{-\lambda r}y,
z+\lambda(e^{-\lambda r})p\cdot x-\lambda(e^{-\lambda r})q\cdot y
-\lambda(e^{-2\lambda r})\eta_{\lambda}(r)\beta(x,y)\bigr),
$$
and the dressing orbits in $H$ are:
\begin{itemize}
\item (1--point orbits): ${\mathcal O}_{z}=\{(0,0,z)\}$, when $(x,y)
=(0,0)$.
\item (2--dimensional orbits): ${\mathcal O}_{x,y}=\{(\alpha x,\alpha y,
\gamma):\alpha>0,\gamma\in\mathbb{R}\}$, when $(x,y)\ne(0,0)$.
\end{itemize}
\begin{rem}
As pointed out earlier, the dressing action is usually regarded as a
generalization of the coadjoint action. In the present case, we can
see easily that the dressing orbits in $\tilde{H}$ are exactly the
coadjoint orbits in $\tilde{\Gh}\cong\tilde{H}$. This illustrates
the point that the Poisson bracket on $\tilde{H}$ is just the linear
Poisson bracket. On the other hand, for $\tilde{G}$, which has a
non-linear Poisson bracket, this is no longer the case. The orbits
$\tilde{\mathcal O}_{r,s}$ are different from the coadjoint orbits
in $\tilde{\Gg}$. Nevertheless, we can still see close resemblance.
\end{rem}
\section{Quantum Heisenberg group algebra representations}
From its construction, we can see that $(A,\Delta)$ is at the same time a
``quantum $C_{\infty}(G)$'' and ``quantum $C^*(H)$''. In the previous paper
\cite{BJKp2}, the first viewpoint has been exploited: As we already mentioned,
$(A,\Delta)$ has been constructed as a deformation quantization of $C_{\infty}(G)$.
The construction of the counit, antipode, and Haar weight for $(A,\Delta)$ all
comes from the corresponding structures on $G$.
In this article, we wish to focus our attention to the second viewpoint. To
see this, recall that $A\cong C^*\bigl(H/Z,C_{\infty}(\Gg/\Gq),\sigma\bigr)$,
where the twisting cocycle $\sigma$ for $H/Z$ is defined as in \eqref{(sigma)}.
Since we may put $\eta_{\lambda}(r)=r$ for $\lambda=0$, it is a simple exercise
using Fourier inversion theorem that $A\cong C^*(H)$ when $\lambda=0$ (The reader
may refer to the article \cite{BJKp1} or \cite{Rf2} for the definition of a
twisted group $C^*$-algebra.). For this reason, we will on occasion call $(A,
\Delta)$ the {\em quantum Heisenberg group $C^*$-algebra\/}.
\begin{rem}
The notion of the ``quantum Heisenberg group ($C^*$-)algebra'' introduced above
is different from the notion of the ``quantum Heisenberg algebra'' used in some
physics literatures \cite{GF}, \cite{Ke}. They are different as algebras.
Another significant distinction is that ours is equipped with a (non-cocommutative)
comultiplication, while the other one does not consider any coalgebra structure.
Hence the slight difference in the choice of the terminologies.
\end{rem}
The study of the ${}^*$-representations of $(A,\Delta)$ (and also of $(\tilde{A},
\tilde{\Delta})$) will be a generalization of the study of the Heisenberg group
representation theory (which, by a standard result, is equivalent to the
${}^*$-representation theory of $C^*(H)$). The topic itself is of interest to us
(providing us with the properties like ``quasitriangularity''). But the success
of the (ordinary) Heisenberg group representation theory in various applications
also suggests that this is a worthwhile topic to develop.
Before we begin our discussion, let us fix our terminology. Suppose $(A,\Delta)$
is a general Hopf $C^*$-algebra (in the sense of \cite{Va}, \cite{BS}). By a
{\em representation\/} of $(A,\Delta)$, we will just mean a non-degenerate
${}^*$-representation of the $C^*$-algebra $A$ on a certain Hilbert space.
On the other hand, by a {\em coaction\/} of $(A,\Delta)$ on a $C^*$-algebra $B$,
we will mean a non-degenerate ${}^*$-homomorphism $\delta_B:B\to M(B,A)$ such that
$$
(\operatorname{id}_B\otimes\Delta)\delta_B=(\delta_B\otimes\operatorname{id}_A)
\delta_B.
$$
Here $M(B,A)$ is the set $\{x\in M(B\otimes A):x(1_{M(B)}\otimes A)+(1_{M(B)}
\otimes A)x\subseteq B\otimes A\}$, which is a $C^*$-subalgebra of the multiplier
algebra $M(B\otimes A)$. Similarly, a {\em (unitary) corepresentation\/} of the
Hopf $C^*$-algebra $(A,\Delta)$ on a Hilbert space ${\mathcal H}$ is a unitary
$\Pi\in M\bigl({\mathcal K}({\mathcal H})\otimes A\bigr)$ such that:
\begin{equation}\label{(corep)}
(\operatorname{id}\otimes\Delta)(\Pi)=\Pi_{12}\Pi_{13}.
\end{equation}
Here $\Pi_{12}$ is understood as an element in $M\bigl({\mathcal K}({\mathcal H})
\otimes A\otimes A\bigr)$ such that it acts as $\Pi$ on the first and the second
variables while the remaining variable is unchanged. The notation $\Pi_{13}$
is understood in the similar manner.
\begin{rem}
Corepresentations of $(A,\Delta)$ are actually the ``representations of the
coalgebra structure on $(A,\Delta)$''. So in many articles on quantum groups,
they are often called ``(unitary) representations of the locally compact quantum
group $(A,\Delta)$''. In particular, representation theory in this sense of
{\em compact quantum groups\/} \cite{Wr2}, which are themselves Hopf $C^*$-algebras,
have been neatly studied by Woronowicz in \cite{Wr3}. However, note that we will
use the terminologies ``representations'' and ``corepresentations'' of a Hopf
$C^*$-algebra in the sense defined above. This would make things a little
simpler. Moreover, this is closer to the spirit of this paper, trying to view
our $(A,\Delta)$ as a quantum Heisenberg group $C^*$-algebra.
\end{rem}
Recall that in our case, the $C^*$-algebra $A$ is isomorphic to the twisted
group $C^*$-algebra $C^*\bigl(H/Z,C_{\infty}(\Gg/\Gq),\sigma\bigr)$. So by
slightly modifying Theorem 3.3 and Proposition 3.4 of \cite{BuS}, we are able
to obtain the representations of $A$ from the so-called ``representing pairs''
$(\mu,Q^{\sigma})$. Such a pair $(\mu,Q^{\sigma})$ consists of a nondegenerate
representation $\mu$ of $C_{\infty}(\Gg/\Gq)$ and a generalized projective
representation $Q^{\sigma}$ of $H/Z$, satisfying the following property:
\begin{equation}\label{(genproj)}
Q^{\sigma}_{(x,y)}Q^{\sigma}_{(x',y')}=\mu\bigl(\sigma\bigl((x,y),(x',y')
\bigr)\bigr)Q^{\sigma}_{(x+x',y+y')},
\end{equation}
for $(x,y),(x',y')\in H/Z$.
Given a representing pair $(\mu,Q^{\sigma})$, we can construct its ``integrated
form''. On the dense subspace of Schwartz functions, it reads:
\begin{equation}\label{(integratedform)}
\pi(f)=\int_{H/Z}\mu\bigl(f(x,y;\cdot)\bigr)Q^{\sigma}_{(x,y)}\,dxdy.
\end{equation}
By natural extension to the $C^*$-algebra level (Due to the amenability of the
group involved, there is no ambiguity. See \cite{BJKp1}.), we obtain in this way
a representation of $A$.
\begin{rem}
Let us from now on denote by ${\mathcal A}$ the dense subspace $S_{3c}(H/Z
\times\Gg/\Gq)$ of $A$, which is the space of Schwartz functions in the
$(x,y;r)$ variables having compact support in the $r\,(\in\Gg/\Gq)$ variable.
This is a dense subalgebra (under the twisted convolution) of our twisted group
$C^*$-algebra $A$, and it has been used throughout \cite{BJKp2} (However,
we should point out that our usage of ${\mathcal A}$ is slightly different
from that of \cite{BJKp2}: There, ${\mathcal A}$ is contained in $S(\Gg)$, while
at present we view it as functions in the $(x,y;r)$ variables. Nevertheless,
they can be regarded as the same if we consider these functions as operators
contained in our $C^*$-algebra.). Similarly for $\tilde{A}$, we will consider
the dense subalgebra $\tilde{\mathcal A}$ of Schwartz functions in the $(x,y,r,w)$
variables having compact support in the $r$ and $w$.
\end{rem}
To find irreducible representations of $A$, let us look for some representing
pairs $(\mu,Q^{\sigma})$ consisting of irreducible $\mu$ and $Q^{\sigma}$.
Irreducible representations of the commutative algebra $C_{\infty}(\Gg/\Gq)$
are just the pointwise evaluations at $r\in\Gg/\Gq$. So let us fix $r\in
\Gg/\Gq$ and the corresponding 1-dimensional representation $\mu$ of $C_{\infty}
(\Gg/\Gq)$, given by $\mu(v)=v(r)$, for $v\in C_{\infty}(\Gg/\Gq)$. Then the
condition for $Q^{\sigma}$ becomes:
\begin{align}
Q^{\sigma}_{(x,y)}Q^{\sigma}_{(x',y')}&=\sigma\bigl((x,y),(x',y');r\bigr)
Q^{\sigma}_{(x+x',y+y')} \notag \\
&=\sigma^r\bigl((x,y),(x',y')\bigr)Q^{\sigma}_{(x+x',y+y')}. \label{(projrepn)}
\end{align}
That is, $Q^{\sigma}$ satisfies the condition for an (ordinary) projective
representation of $H/Z$ with respect to the ordinary $\mathbb{T}$--valued
cocycle $\sigma^r$.
Using $\sigma^r$, we may define an extension group $E$ of $H/Z$. Its
underlying space is $H/Z\times\mathbb{T}$ and its multiplication law is
given by
\begin{align}
(x,y;{\theta})(x',y';{\theta}')&=\bigl(x+x',y+y';{\theta}{\theta}'\sigma^r
\bigl((x,y),(x',y')\bigr)\bigr) \notag\\
&=\bigl(x+x',y+y';{\theta}{\theta}'\bar{e}\bigl[\eta_{\lambda}(r)\beta(x,y')
\bigl]\bigr). \notag
\end{align}
Standard theory tells us that the unitary projective representations of $(H/Z,
\sigma^r)$ come from the unitary group representations of $E$, which is easier
to study: The next lemma gives us all the irreducible unitary representations
of $E$, up to equivalence.
\begin{lem}
Let $E$ be the extension group of $(H/Z,\sigma^r)$ as defined above. Then its
irreducible unitary representations are equivalent to one of the following:
\begin{itemize}
\item For each $(p,q)\in\mathbb{R}^{2n}$, there exists a 1--dimensional
representation $Q_{p,q}$ defined by
$$
Q_{p,q}(x,y;\theta)=\bar{e}(p\cdot x+q\cdot y).
$$
\item There also exists a (unique) infinite dimensional representation $Q_r$
on $L^2(\mathbb{R}^n)$ defined by
$$
\bigl(Q_r(x,y;\theta)\xi\bigr)(u)=\theta\bar{e}\bigl[\eta_{\lambda}(r)
\beta(u,y)\bigr]\xi(u+x).
$$
\end{itemize}
\end{lem}
\begin{proof}
Observe that $E$ is a semi-direct product of two abelian groups $X=\{(x,0,1):
x\in\mathbb{R}^n\}$ and $Y\times\mathbb{T}=\{(0,y,\theta):y\in\mathbb{R}^n,
\theta\in\mathbb{T}\}$. So by using Mackey analysis, every irreducible
representation of $E$ is obtained as an ``induced representation'' \cite{FD},
\cite{Tay}, \cite{Rfm}.
Since $[E,E]=\mathbb{T}$, we have: $E/[E,E]=X\times Y$, which is abelian. So
all the irreducible representations of $E/[E,E]$ are one-dimensional. By
lifting from these 1-dimensional representations, we obtain the (irreducible)
representations $\{Q_{p,q}\}_{(p,q)\in\mathbb{R}^{2n}}$ of $E$ that are trivial
on the commutator $[E,E]$.
The infinite dimensional representation $Q_r$ is the induced representation
$\operatorname{Ind}_{Y\times\mathbb{T},\chi}^E$, where $\chi$ is the
representation of $Y\times\mathbb{T}$ defined by: $\chi(y,\theta)=\theta$.
It turns out (by using standard Mackey theory) that $\{Q_{p,q}\}_{(p,q)\in
\mathbb{R}^{2n}}$ and $Q_r$ exhaust all the irreducible representations of
$E$, up to equivalence.
\end{proof}
We are now able to find the irreducible projective representations of $(H/Z,
\sigma)$. Check equation \eqref{(projrepn)} and we obtain the following
representing pairs consisting of irreducible $\mu$ and $Q^{\sigma}$.
\begin{enumerate}
\item (When $r=0\in\Gg/\Gq$): For each $(p,q)\in\mathbb{R}^{2n}$, there is a
pair $(\mu,Q^{\sigma})$ given by
\begin{itemize}
\item $\mu(v)=v(0)$, $v\in C_{\infty}(\Gg/\Gq)$.
\item $Q^{\sigma}(x,y)=\bar{e}(p\cdot x+q\cdot y),\quad (x,y)\in H/Z$.
\end{itemize}
\item (When $r\ne0\in\Gg/\Gq$): There is a pair $(\mu,Q^{\sigma})$ given by
\begin{itemize}
\item $\mu(v)=v(r)$, $v\in C_{\infty}(\Gg/\Gq)$.
\item On $L^2(\mathbb{R}^n)$,
$$
\bigl(Q^{\sigma}(x,y)\xi\bigr)(u)=\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,y)\bigr]
\xi(u+x),\quad (x,y)\in H/Z.
$$
\end{itemize}
\end{enumerate}
Therefore, we obtain the following proposition. Observe the similarity
between this result and the representation theory of the Heisenberg group
$H$ or the Heisenberg group $C^*$-algebra $C^*(H)$ (This is not surprising,
given the remark at the beginning of this section.).
\begin{prop}\label{repA}
Consider the twisted convolution algebra ${\mathcal A}$. Its irreducible
representations are equivalent to one of the following representations,
which have been obtained by integrating the representing pairs $(\mu,
Q^{\sigma})$ of the preceding paragraph.
\begin{itemize}
\item For $(p,q)\in\mathbb{R}^{2n}$, there is a 1-dimensional
representation ${\pi}_{p,q}$ of ${\mathcal A}$, defined by
$$
\pi_{p,q}(f)=\int f(x,y,0)\bar{e}(p\cdot x+q\cdot y)\,dxdy.
$$
\item For $r\in\mathbb{R}$, there is a representation ${\pi}_r$ of
${\mathcal A}$, acting on the Hilbert space ${\mathcal H}_r=L^2(\mathbb{R}^n)$
and is defined by
$$
\bigl(\pi_r(f)\xi\bigr)(u)=\int f(x,y,r)\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,y)
\bigr]\xi(u+x)\,dxdy.
$$
\end{itemize}
Since ${\mathcal A}$ is a dense subalgebra of our $C^*$-algebra $A$, we thus
obtain all the irreducible representations (up to equivalence) of $A$ by
naturally extending these representations. We will use the same notation,
$\pi_{p,q}$ and $\pi_r$, for the representations of $A$ constructed in this way.
\end{prop}
Let us now consider the representations of the $C^*$-algebra $\tilde{A}$.
They are again obtained from representations of the dense subalgebra $\tilde
{\mathcal A}$, which have been identified with the twisted convolution algebra
of functions in the $(x,y,r,w)$ variables (where $(x,y,w)\in\tilde{H}/Z$ and
$r\in\tilde{\Gg}/{\Gz^{\bot}}$) having compact support in the $r$ and $w$
variables. We may employ the same argument as above to find (up to equivalence)
the irreducible representations of $\tilde{\mathcal A}$. The result is given
in the following proposition:
\begin{prop}\label{repAtilde}
The irreducible representations of $\tilde{A}$ are obtained by naturally
extending the following irreducible representations of the dense subalgebra
$\tilde{\mathcal A}$.
\begin{itemize}
\item For $s\in\mathbb{R}$, there is a 1-dimensional representation
$\tilde{\pi}_s$ defined by
$$
\tilde{\pi}_{s}(f)=\int f(x,y,0,w)\bar{e}(sw)\,dxdydw.
$$
\item For $(p,q)\in\mathbb{R}^{2n}$, there is a representation $\tilde
{\pi}_{p,q}$ acting on the Hilbert space $\tilde{\mathcal H}_{p,q}=L^2
(\mathbb{R})$ defined by
$$
\bigl(\tilde{\pi}_{p,q}(f)\zeta\bigr)(d)=\int f(x,y,0,w)\bar{e}(e^d p\cdot x+e^{-d}
q\cdot y)\zeta(d+w)\,dxdydw.
$$
\item For $(r,s)\in\mathbb{R}^2$, there is a representation $\tilde{\pi}
_{r,s}$ acting on the Hilbert space $\tilde{\mathcal H}_{r,s}=L^2
(\mathbb{R}^n)$ defined by
$$
\bigl(\tilde{\pi}_{r,s}(f)\xi\bigr)(u)=\int f(x,y,r,w)\bar{e}(sw)\bar{e}\bigl
[\eta_{\lambda}(r)\beta(u,y)\bigr](e^{-\frac{w}2})^n\xi(e^{-w}u+e^{-w}x)
\,dxdydw.
$$
\end{itemize}
We will use the same notation, $\tilde{\pi}_s$, $\tilde{\pi}_{p,q}$ and
$\tilde{\pi}_{r,s}$, for the corresponding representations of $\tilde{A}$.
\end{prop}
\begin{proof}
As before, let us first fix $r\in{\Gg}/{\Gz^{\bot}}$. Look for the
irreducible projective representations of $\tilde{H}/Z$, with respect to
the ($\mathbb{T}$-valued) cocycle for $\tilde{H}/Z$ defined by
$$
\tilde{\sigma}^r:\bigl((x,y,w),(x',y',w')\bigr)\mapsto\bar{e}
\bigl[e^{-w}\eta_{\lambda}(r)\beta(x,y')\bigr].
$$
To do this, we consider the extension group $\tilde{E}$ of $\tilde{H}/Z$,
whose underlying space is $\tilde{H}/Z\times\mathbb{T}$ and whose
multiplication law is given by
$$
(x,y,w;\theta)(x',y',w';\theta')=\bigl(x+e^wx',y+e^{-w}y',w+w';
\theta\theta'\bar{e}\bigl[e^{-w}\eta_{\lambda}(r)\beta(x,y')\bigr]\bigr).
$$
Again, all the irreducible representations of $\tilde{E}$ are obtained by
``inducing''. Up to equivalence, they are:
\begin{itemize}
\item For each $s\in\mathbb{R}$, there exists a 1--dimensional representation
$\tilde{Q}_s$ defined by
$$
\tilde{Q}_s(x,y,w;\theta)=\bar{e}(sw).
$$
\item For each $(p,q)\in\mathbb{R}^{2n}$, there exists a representation
$\tilde{Q}_{p,q}$ on $L^2(\mathbb{R})$ defined by
$$
\bigl(\tilde{Q}_{p,q}(x,y,w;\theta)\zeta\bigr)(d)=\bar{e}(e^dp\cdot x+e^{-d}q\cdot y)
\zeta(d+w).
$$
\item For $s\in\mathbb{R}$, there exists an infinite dimensional irreducible
representation $\tilde{Q}_{r,s}$ on $L^2(\mathbb{R}^n)$ defined by
$$
\bigl(\tilde{Q}_{r,s}(x,y,w;\theta)\xi\bigr)(u)=\theta\bar{e}(sw)\bar{e}\bigl[
\eta_{\lambda}(r)\beta(u,y)\bigr](e^{-\frac{w}2})^n\,\xi(e^{-w}u+e^{-w}x).
$$
\end{itemize}
Vary $r\in{\Gg}/{\Gz^{\bot}}$ and check the compatibility condition just
like \eqref{(projrepn)}, to find the appropriate representing pairs. Then
the integrated form of these pairs will give us the irreducible representations
of $\tilde{\mathcal A}$, which are stated in the proposition.
\end{proof}
As in the case of Proposition \ref{repA} (for the $C^*$-algebra $A$),
we can see clearly the similarity between the result of Proposition
\ref{repAtilde} and the representation theory of the ordinary group
$C^*$-algebra $C^*(\tilde{H})$. Indeed, except when we study later the
notion of ``inner tensor product representations'' (taking advantage of the
Hopf structures of $A$ and $\tilde{A}$), the representation theories of $A$
and $\tilde{A}$ are very similar to those of $C^*(H)$ and $C^*(\tilde{H})$.
In this light, it is interesting to observe that the irreducible
representations of $A$ and $\tilde{A}$ are in one-to-one correspondence
with the dressing orbits (calculated in Appendix) in $G$ and $\tilde{G}$,
respectively. To emphasize the correspondence, we used the same subscripts
for the orbits and the related irreducible representations. In this paper,
we will point out this correspondence only. However, it is still true
(as in the ordinary Lie group representation theory) that orbit analysis
sheds some helpful insight into the study of quantum group representations.
In the following, we give a useful result about the irreducible
representations of $\tilde{A}$ and those of $A$. This has been
motivated by the orbit analysis, and it is an analog of a similar
result for the group representations of $\tilde{H}$ and $H$. See
the remark following the proposition.
\begin{defn}\label{restriction}
Suppose we are given a representation $\tilde{\pi}$ of $\tilde{A}$. Since
it is essentially obtained from a representation $\tilde{Q}$ of $\tilde{E}$,
we may consider its restriction $\tilde{Q}|_E$ to $E$. Let us denote by
$\tilde{\pi}|_A$ the representation of $A$ corresponding to the representation
$\tilde{Q}|_E$ of $E$. In this sense, we will call $\tilde{\pi}|_A$ the
{\em restriction\/} to $A$ of the representation $\tilde{\pi}$.
\end{defn}
\begin{prop}\label{decomposition}
Let the notation be as above and consider the restriction to $A$ of the
irreducible representations of $\tilde{A}$. We then have:
$$
\tilde{\pi}_{r,s}|_A=\pi_r\qquad\text{and}\qquad\tilde{\pi}_{p,q}|_A=
\int^{\oplus}_{\mathbb{R}}\pi_{e^wp,e^{-w}q}\,dw.
$$
Here $\int^{\oplus}_{\mathbb{R}}$ denotes the direct integral (\cite{D1})
of representations.
\end{prop}
\begin{proof}
For any $\xi\in\tilde{\mathcal H}_{r,s}=L^2(\mathbb{R}^n)$, we have:
$$
\bigl(\tilde{Q}_{r,s}|_E(x,y;\theta)\xi\bigr)(u)=\theta\bar{e}\bigl[\eta_{\lambda}(r)
\beta(u,y)\bigr]\xi(u+x)=\bigl(Q_r(x,y;\theta)\xi\bigr)(u).
$$
It follows that $\tilde{\pi}_{r,s}|_A=\pi_r$. Next, for any $\zeta\in\tilde
{\mathcal H}_{p,q}=L^2(\mathbb{R})$,
$$
\bigl(\tilde{Q}_{p,q}|_E(x,y;\theta)\zeta\bigr)(w)=\bar{e}(e^wp\cdot x+e^{-w}q\cdot y)
\zeta(w)=\bigl(Q_{e^wp,e^{-w}q}(x,y;\theta)\zeta\bigr)(w).
$$
By definition of the direct integrals, we thus obtain:
$$
\tilde{Q}_{p,q}|_E=\int^{\oplus}_{\mathbb{R}}Q_{e^wp,e^{-w}q}\,dw.
$$
It follows that: $\tilde{\pi}_{p,q}|_E=\int^{\oplus}_{\mathbb{R}}
\pi_{e^wp,e^{-w}q}\,dw$.
\end{proof}
\begin{rem}
This result has to do with the fact that $E$ is a normal subgroup of
$\tilde{E}$ with codimension 1. Compare this result with Theorem 6.1 of
Kirillov's fundamental paper \cite{Ki2} or the discussion in section 2.5
of \cite{CG}, where the analysis of coadjoint orbits was used to obtain
a similar result for the representations of ordinary Lie groups. Although
we proved this proposition directly, this strongly indicates the possibility
of formulating the proposition via generalized (dressing) orbit theory.
\end{rem}
So far we have considered only the representations of $A$ and $\tilde{A}$.
In this article, we are not going to discuss the corepresentations of $(A,
\Delta)$ (or of $(\tilde{A},\tilde{\Delta})$). In fact, it turns out that
the corepresentation theory is equivalent to the representation theory of
the dual group $G$. This is a by-product of the Hopf $C^*$-algebra duality
between $(A,\Delta)$ and $(C^*(G),\hat{\Delta})$, provided by the ``regular''
multiplicative unitary operator $U$ associated with $(A,\Delta)$ (The
definition of $\hat{\Delta}$ depends on $U$.). Since this is the case,
the corepresentation theory of $(A,\Delta)$ is actually simpler.
\section{Inner tensor product of representations}
Given any two representations of a Hopf $C^*$-algebra $(B,\Delta)$, we
can define their ``(inner) tensor product'' \cite[\S10]{Dr}, \cite[\S5]{CP}
as in the below. There is also a corresponding notion for corepresentations.
But in the present article, we will not consider this dual notion.
\begin{defn}\label{tensor-product}
Let $\pi$ and $\rho$ be representations of a Hopf $C^*$-algebra $(B,\Delta)$,
acting on the Hilbert spaces ${\mathcal H}_{\pi}$ and ${\mathcal H}_{\rho}$.
Then their {\em inner tensor product\/}, denoted by $\pi\boxtimes\rho$, is a
representation of $B$ on ${\mathcal H}_{\pi}\otimes {\mathcal H}_{\rho}$
defined by
$$
(\pi\boxtimes\rho)(b)=(\pi\otimes\rho)\bigl(\Delta(b)\bigr),\qquad
b\in B.
$$
Here $\pi\otimes\rho$ denotes the (outer) tensor product of the representations
$\pi$ and $\rho$, which is a representation of $B\otimes B$ naturally extended
to $M(B\otimes B)$.
\end{defn}
As the name suggests, this notion of inner tensor product is a generalization
of the inner tensor product of group representations \cite{FD}. For instance,
in the case of an ordinary group $C^*$-algebra $C^*(G)$ equipped with its
cocommutative (symmetric) comultiplication $\Delta_0$, Definition \ref
{tensor-product} is just the integrated form version of the inner tensor product
group representations. In this case, since $\Delta_0$ is cocommutative, the
flip $\sigma:{\mathcal H}_{\pi}\otimes{\mathcal H}_{\rho}\to{\mathcal H}_{\rho}
\otimes{\mathcal H}_{\pi}$ provides a natural intertwining operator between
$\pi\boxtimes\rho$ and $\rho\boxtimes\pi$.
For general (non-cocommutative) Hopf $C^*$-algebras, however, this is not
necessarily true. In general, $\pi\boxtimes\rho$ need not even be equivalent
to $\rho\boxtimes\pi$. Thus for any Hopf $C^*$-algebra (or quantum group),
it is an interesting question to ask whether two inner tensor product
representations $\pi\boxtimes\rho$ and $\rho\boxtimes\pi$ are equivalent
and if so, what the intertwining unitary operator between them is.
When a Hopf $C^*$-algebra is equipped with a certain ``quantum universal
$R$-matrix'' (\cite{Dr}, \cite{CP}, and Definitions 6.1, 6.2 of \cite{BJKp2}
for the $C^*$-algebra version), we can give positive answers to these questions.
The following result is relatively well known:
\begin{prop}\label{equivitp}
Let $(B,\Delta)$ be a Hopf $C^*$-algebra. Suppose that there exists a
quantum universal $R$-matrix $R\in M(B\otimes B)$ for $B$. Then given any
two representations $\pi$ and $\rho$ of $B$ on Hilbert spaces ${\mathcal H}_{\pi}$
and ${\mathcal H}_{\rho}$, their inner tensor products $\pi\boxtimes\rho$ and
$\rho\boxtimes\pi$ are equivalent. The equivalence is established by the
(unitary) intertwining operator $T_{\pi\rho}:{\mathcal H}_{\pi}\otimes
{\mathcal H}_{\rho}\to{\mathcal H}_{\rho}\otimes{\mathcal H}_{\pi}$, defined by
$$
T_{\pi\rho}=\sigma\circ(\pi\otimes\rho)(R).
$$
Here $\pi\otimes\rho$ is understood as the natural extension to $M(B\otimes B)$
of the tensor product $\pi\otimes\rho:B\otimes B\to{\mathcal B}({\mathcal H}_{\pi}
\otimes{\mathcal H}_{\rho})$ and $\sigma$ is the flip. That is,
$$
T_{\pi\rho}\bigl((\pi\boxtimes\rho)(b)\bigr)=\bigl((\rho\boxtimes\pi)(b)\bigr)
T_{\pi\rho},\qquad b\in B.
$$
Furthermore, if the $R$-matrix is triangular, then we also have:
$$
T_{\rho\pi}T_{\pi\rho}=I_{{\mathcal H}_{\pi}\otimes{\mathcal H}_{\rho}}
\quad\text{and}\quad T_{\pi\rho}T_{\rho\pi}=I_{{\mathcal H}_{\rho}
\otimes{\mathcal H}_{\pi}}.
$$
\end{prop}
\begin{proof}
Let us first calculate how $T_{\pi\rho}$ acts as an operator. If $\zeta
_1\in{\mathcal H}_{\pi}$ and $\zeta_2\in{\mathcal H}_{\rho}$, we have:
$$
T_{\pi\rho}(\zeta_1\otimes\zeta_2)=\sigma\circ\bigl((\pi\otimes\rho)(R)
\bigr)(\zeta_1\otimes\zeta_2)=\bigl((\rho\otimes\pi)(R_{21})\bigr)
(\zeta_2\otimes\zeta_1).
$$
Note that we have $T_{\pi\rho}=\bigl((\rho\otimes\pi)(R_{21})\bigr)
\circ\sigma$, as an operator. To verify that $T_{\pi\rho}$ is an
intertwining operator between $\pi\boxtimes\rho$ and $\rho\boxtimes\pi$,
let us consider an element $b\in B$. Then
\begin{align}
T_{\pi\rho}\bigl((\pi\boxtimes\rho)(b)\bigr)&=T_{\pi\rho}\bigl
((\pi\otimes\rho)(\Delta b)\bigr)=\bigl((\rho\otimes\pi)(R_{21})\bigr)
\bigl((\rho\otimes\pi)(\Delta ^{\text{op}}b)\bigr)\circ\sigma \notag \\
&=\bigl((\rho\otimes\pi)(R_{21}\Delta ^{\text {op}}b)\bigr)\circ\sigma. \notag
\end{align}
From $R\Delta(b)R^{-1}=\Delta^{\text {op}}(b)$, we have: $R_{21}\Delta
^{\text {op}}(b)R_{21}^{-1}=\Delta(b)$. It follows that
\begin{align}
T_{\pi\rho}\bigl((\pi\boxtimes\rho)(b)\bigr)&
=\bigl((\rho\otimes\pi)(\Delta(b)R_{21})\bigr)\circ\sigma
=\bigl((\rho\otimes\pi)(\Delta b)\bigr)\bigl((\rho\otimes\pi)(R_{21})
\bigr)\circ\sigma \notag \\
&=\bigl((\rho\boxtimes\pi)(b)\bigr)T_{\pi\rho}. \notag
\end{align}
Finally, if $R$ is triangular, by definition $\sigma\circ R=R_{21}=R^{-1}$.
Therefore,
\begin{align}
T_{\rho\pi}T_{\pi\rho}(\zeta_1\otimes\zeta_2)&=\bigl((\pi\otimes\rho)
(R_{21})\bigr)\bigl((\pi\otimes\rho)(R)\bigr)(\zeta_1\otimes\zeta_2) \notag \\
&=\bigl((\pi\otimes\rho)(R_{21}R)\bigr)(\zeta_1\otimes\zeta_2)
=(\zeta_1\otimes\zeta_2). \notag
\end{align}
Since this is true for arbitrary $\zeta_1\in{\mathcal H}_{\pi}$ and
$\zeta_2\in{\mathcal H}_{\rho}$, we have: $T_{\rho\pi}T_{\pi\rho}=I_
{{\mathcal H}_{\pi}\otimes{\mathcal H}_{\rho}}$.
\end{proof}
In \cite[\S6]{BJKp2}, we showed that the ``extended'' Hopf $C^*$-algebra
$(\tilde{A},\tilde{\Delta})$ has a quasitriangular quantum universal $R$-matrix
$R\in M(\tilde{A}\otimes\tilde{A})$. From this the following Corollary is immediate.
\begin{cor}
For the Hopf $C^*$-algebra $(\tilde{A},\tilde{\Delta})$, any two representations
$\tilde{\pi}$ and $\tilde{\rho}$ of $\tilde{A}$ will satisfy:
$$
\tilde{\pi}\boxtimes\tilde{\rho}\cong\tilde{\rho}\boxtimes\tilde{\pi}.
$$
By Proposition \ref{equivitp}, the operator $T_{\tilde{\pi}\tilde{\rho}}=
\sigma\circ(\tilde{\pi}\otimes\tilde{\rho})(R)$ is an intertwining operator
for this equivalence.
\end{cor}
Unlike $(\tilde{A},\tilde{\Delta})$, however, the Hopf $C^*$-algebra $(A,\Delta)$
does not have its own quantum $R$-matrix $R_A\in M(A\otimes A)$. Even at the
classical, Lie bialgebra level (studied in \cite[\S1]{BJKp2}), we can see that
the Poisson structures we consider cannot be obtained from any classical $r$-matrix.
Because of this, the result like the above Corollary is not automatic for
$(A,\Delta)$. Even so, we plan to show in the below that the representations
of $(A,\Delta)$ still satisfy the quasitriangular type property.
For this purpose and for possible future use, we are going to calculate here
the inner tensor product representations of our Hopf $C^*$-algebra $(A,\Delta)$.
Since it is sufficient to consider the inner tensor products of irreducible
representations, let us keep the notation of the previous section and let
$\{\pi_{p,q}\}_{(p,q)\in\mathbb{R}^{2n}}$ and $\{\pi_r\}_{r\in\mathbb{R}}$ be
the irreducible representations of $A$. Similarly, let $\{\tilde{\pi}_s\}_
{s\in\mathbb{R}}$, $\{\tilde{\pi}_{p,q}\}_{(p,q)\in\mathbb{R}^{2n}}$,
$\{\pi_{r,s}\}_{(r,s)\in\mathbb{R}^2}$ be the irreducible representations
of $\tilde{A}$. For convenience, we will calculate the inner tensor product
representations at the level of our dense subalgebra of functions, ${\mathcal A}$.
Let $f\in{\mathcal A}$ and consider $\Delta f$. To carry out our calculations,
it is convenient to regard $\Delta f$ also as a continuous function. By using
the definition of $\Delta$ (given in Theorem 3.2 of \cite{BJKp2}) and by using
Fourier transform purely formally with the Fourier inversion theorem, it is not
difficult to realize $\Delta f$ as a function in the $(x,y,r)$ variables:
\begin{align}
&\Delta f(x,y,r,x',y',r') \notag\\
&=\int f(x',y',r+r')\bar{e}\bigl[p\cdot(e^{\lambda r'}x'-x)+q\cdot
(e^{\lambda r'}y'-y)\bigr]\,dpdq. \notag
\end{align}
Note that in the $(p,q,r)\,(\in G)$ variables, it is just:
$$
\Delta f(p,q,r,p',q',r')=f(e^{\lambda r'}p+p',e^{\lambda r'}q+q',r+r'),
$$
which more or less reflects the multiplication law on $G$.
We can now explicitly calculate the inner tensor products of irreducible
representations of ${\mathcal A}$, by $(\pi\boxtimes\rho)(f)=(\pi\otimes\rho)
(\Delta f)$. We first begin with 1-dimensional representations.
\begin{prop}
For two 1-dimensional representations $\pi_{p,q}$ and $\pi_{p',q'}$
of $A$, we have: $\pi_{p,q}\boxtimes\pi_{p',q'}=\pi_{p+p',q+q'}$.
From this, it follows that:
$$
\pi_{p,q}\boxtimes\pi_{p',q'}=\pi_{p+p',q+q'}=\pi_{p',q'}\boxtimes
\pi_{p,q}.
$$
\end{prop}
\begin{proof}
We have for any $f\in{\mathcal A}$,
\begin{align}
(\pi_{p,q}\boxtimes\pi_{p',q'})(f)&=\int f(x',y',0)\bar{e}\bigl[\tilde{p}
\cdot(x'-x)+\tilde{q}\cdot(y'-y)\bigr] \notag \\
&\qquad \bar{e}[p\cdot x+q\cdot y]\bar{e}[p'\cdot x'+q'\cdot y']
\,d\tilde{p}d\tilde{q}dxdydx'dy' \notag \\
&=\int f(x,y,0)\bar{e}\bigl[(p+p')\cdot x+(q+q')\cdot y\bigr]\,dxdy
\notag \\
&=\pi_{p+p',q+q'}(f). \notag
\end{align}
\end{proof}
For other cases involving infinite dimensional (irreducible) representations,
the equivalence between the inner tensor products is not so apparent. However,
the inner tensor products of $\pi_{p,q}$ and $\pi_{r}$ has a property of being
equivalent to the infinite dimensional representation $\pi_r$ itself. So in
this case, equivalence between the inner tensor products follows rather easily.
\begin{prop}
Consider a 1-dimensional representation $\pi_{p,q}$ and an infinite
dimensional representation $\pi_r$ of $A$. Their inner tensor product
is equivalent to the irreducible representation $\pi_r$. We thus have:
$$
\pi_r\boxtimes\pi_{p,q}\cong\pi_r\cong\pi_{p,q}\boxtimes\pi_r.
$$
\end{prop}
\begin{proof}
For $\xi\in{\mathcal H}_r\otimes{\mathcal H}_{p,q}\cong L^2(\mathbb{R}^n)
\otimes\mathbb{C}\cong L^2(\mathbb{R}^n)$ and for $f\in{\mathcal A}$, we
have:
\begin{align}
\bigl((\pi_r\boxtimes\pi_{p,q})(f)\xi\bigr)(u)&=\int f(x',y',r)\bar{e}
\bigl[\tilde{p}\cdot(x'-x)+\tilde{q}\cdot(y'-y)\bigr] \notag \\
&\qquad\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,y)\bigr]\bar{e}[p\cdot x'
+q\cdot y']\xi(u+x)\,d\tilde{p}d\tilde{q}dxdydx'dy' \notag \\
&=\int f(x,y,r)\bar{e}[p\cdot x+q\cdot y]\bar{e}\bigr[\eta_{\lambda}(r)
\beta(u,y)\bigr]\xi(u+x)\,dxdy. \notag
\end{align}
Similarly for $\xi\in{\mathcal H}_{p,q}\otimes{\mathcal H}_r\cong
L^2(\mathbb{R}^n)$,
\begin{align}
\bigl((\pi_{p,q}\boxtimes\pi_r)(f)\xi\bigr)(u)&=\int f(x',y',r)\bar{e}
\bigl[\tilde{p}\cdot(e^{\lambda r}x'-x)+\tilde{q}\cdot(e^{\lambda r}y'-y)\bigr]
\notag \\
&\qquad\bar{e}[p\cdot x+q\cdot y]\bar{e}\bigr[\eta_{\lambda}(r)
\beta(u,y')\bigr]\xi(u+x')\,d\tilde{p}d\tilde{q}dxdydx'dy' \notag \\
&=\int f(x,y,r)\bar{e}\bigl[e^{\lambda r}p\cdot x+e^{\lambda r}q\cdot y
\bigr]\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,y)\bigr]\xi(u+x)\,dxdy.
\notag
\end{align}
Before proving $\pi_r\boxtimes\pi_{p,q}\cong\pi_{p,q}\boxtimes\pi_r$,
let us first show the equivalence $\pi_r\boxtimes\pi_{p,q}\cong\pi_r$.
This equivalence is suggested by the corresponding result at the level
of Heisenberg Lie group representation theory, which is obtained by
using the standard analysis via ``characters'' \cite{Ki}, \cite{CG}.
In our case, the equivalence is established by the intertwining operator
$S:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ defined by: $S\xi(u)=\bar{e}
(p\cdot u)\xi\left(u-\frac{q}{\eta_{\lambda}(r)}\right)$. Indeed for
$f\in{\mathcal A}$,
\begin{align}
S\bigl((\pi_r\boxtimes\pi_{p,q})(f)\bigr)\xi(u)&=\int \bar{e}(p\cdot u)
f(x,y,r)\bar{e}[p\cdot x+q\cdot y] \notag \\
&\qquad\bar{e}\left[\eta_{\lambda}(r)\beta\left(u-\frac{q}{\eta_{\lambda}(r)},
y\right)\right]\xi\left(u-\frac{q}{\eta_{\lambda}(r)}+x\right)\,dxdy, \notag
\end{align}
and
$$
\bigl(\pi_r(f)\bigr)S\xi(u)=\int f(x,y,r)\bar{e}\bigl[\eta_{\lambda}(r)
\beta(u,y)\bigr]\bar{e}\bigl[p\cdot(u+x)\bigr]\xi\left(u+x-\frac{q}
{\eta_{\lambda}(r)}\right)\,dxdy.
$$
We thus have: $S\bigl((\pi_r\boxtimes\pi_{p,q})(f)\bigr)=\bigl(\pi_r(f)
\bigr)S$, proving the equivalence: $\pi_r\boxtimes\pi_{p,q}\cong\pi_r$.
It is easy to check that $S^{-1}$ gives the intertwining operator for the
equivalence: $\pi_{r}\cong\pi_r\boxtimes\pi_{p,q}$.
Meanwhile, from the explicit calculations given at the beginning of the
proof, it is apparent that we have: $\pi_{p,q}\boxtimes\pi_r=\pi_r\boxtimes
\pi_{e^{\lambda r}p,e^{\lambda r}q}$. We thus obtain the equivalence:
$\pi_{p,q}\boxtimes\pi_r\cong\pi_r$, via the intertwining operator similar
to the above $S$, replacing $p$ and $q$ with $e^{\lambda r}p$ and
$e^{\lambda r}q$. Combining these results, we can find the intertwining
operator $T:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ between $\pi_r
\boxtimes\pi_{p,q}$ and $\pi_{p,q}\boxtimes\pi_r$, obtained by multiplying
the respective intertwining operators for the equivalences $\pi_r\boxtimes
\pi_{p,q}\cong\pi_r$ and $\pi_r\cong\pi_{p,q}\boxtimes\pi_r$. By
straightforward calculation, we have the following expression for $T$:
$$
T\xi(u)=\bar{e}\left[p\cdot u-e^{\lambda r}p\cdot\left(u-\frac{q}
{\eta_{\lambda}(r)}+\frac{e^{\lambda r}q}{\eta_{\lambda}(r)}\right)
\right]\xi\left(u-\frac{q}{\eta_{\lambda}(r)}+\frac{e^{\lambda r}q}
{\eta_{\lambda}(r)}\right).
$$
It is clear that $T^{-1}$ gives the intertwining operator between $\pi_{p,q}
\boxtimes\pi_r$ and $\pi_r\boxtimes\pi_{p,q}$.
\end{proof}
So far nothing very interesting has happened, in the sense that the results
are similar to those of the Heisenberg group representation theory. However,
a breakdown of this analogy occurs when we consider inner tensor products of
two infinite dimensional representations $\pi_r$ and $\pi_{r'}$. Let us first
prove the equivalence of $\pi_r\boxtimes\pi_{r'}$ and $\pi_{r'}\boxtimes\pi_{r}$.
\begin{prop}
Consider a pair $(\pi_r,\pi_r')$ of two infinite dimensional irreducible
representations of $A$. Then we have:
$$
\pi_r\boxtimes\pi_{r'}\cong\pi_{r'}\boxtimes\pi_r,
$$
where the equivalence between them is given by the intertwining operator
$T_{\pi_r\pi_{r'}}:L^2(\mathbb{R}^{2n})\to L^2(\mathbb{R}^{2n})$ defined by
$$
T_{\pi_r\pi_{r'}}\xi(v,u)=(e^{\frac{-\lambda r}2})^n(e^{-\frac
{\lambda r'}2})^n\,\xi\bigl(e^{-\lambda r'}u+(e^{\lambda r'}
-e^{-\lambda r'})e^{-\lambda r}v,e^{-\lambda r}v\bigr).
$$
\end{prop}
\begin{proof}
Let $f\in{\mathcal A}$. For $\xi\in{\mathcal H}_r\otimes{\mathcal H}_{r'}
\cong L^2(\mathbb{R}^{2n})$, we have:
\begin{align}
&\bigl((\pi_r\boxtimes\pi_{r'})(f)\xi\bigr)(u,v) \notag \\
&=\int f(x',y',r+r')\bar{e}\bigl[\tilde{p}\cdot(e^{\lambda r'}x'-x)
+\tilde{q}\cdot(e^{\lambda r'}y'-y)\bigr] \notag \\
&\qquad\bar{e}\bigr[\eta_{\lambda}(r)\beta(u,y)\bigr]\bar{e}\bigr[
\eta_{\lambda}(r')\beta(v,y')\bigr]\xi(u+x,v+x')
\,d\tilde{p}d\tilde{q}dxdydx'dy' \notag \\
&=\int f(x,y,r+r')\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,e^{\lambda r'}y)
\bigr]\bar{e}\bigl[\eta_{\lambda}(r')\beta(v,y)\bigr]
\xi(u+e^{\lambda r'}x,v+x)\,dxdy. \notag
\end{align}
Similarly for $\xi\in{\mathcal H}_{r'}\otimes{\mathcal H}_{r}\cong
L^2(\mathbb{R}^{2n})$, by interchanging the roles of $r$ and $r'$,
\begin{align}
\bigl((\pi_{r'}\boxtimes\pi_{r})(f)\xi\bigr)(v,u)&=\int f(x,y,r+r')\bar{e}
\bigl[\eta_{\lambda}(r')\beta(v,e^{\lambda r}y)\bigr]\bar{e}\bigl[\eta_{\lambda}(r)
\beta(u,y)\bigr] \notag \\
&\qquad\xi(v+e^{\lambda r}x,u+x)\,dxdy. \notag
\end{align}
To prove the equivalence between $\pi_{r}\boxtimes\pi_{r'}$ and $\pi_{r'}
\boxtimes\pi_{r}$, it is useful to recall the fact that any two representations
$\tilde{\pi}$ and $\tilde{\rho}$ of the ``extended'' Hopf $C^*$--algebra
$(\tilde{A},\tilde{\Delta})$ satisfy: $\tilde{\pi}\boxtimes\tilde{\rho}\cong
\tilde{\rho}\boxtimes\tilde{\pi}$. In particular, we would have:
$\tilde{\pi}_{r,0}\boxtimes\tilde{\pi}_{r',0}\cong\tilde{\pi}_{r',0}
\boxtimes\tilde{\pi}_{r,0}$. Its intertwining operator is:
$T_{\tilde{\pi}_{r,0}\tilde{\pi}_{r',0}}=\sigma\circ\bigl((\tilde{\pi}_{r,0}
\otimes\tilde{\pi}_{r',0})(R)\bigr)$, by Corollary of Proposition \ref{equivitp}.
By restriction to $A$ (in the sense of Definition \ref{restriction}), we obtain:
$$
(\tilde{\pi}_{r,0}|_A)\boxtimes(\tilde{\pi}_{r',0}|_A)\cong(\tilde{\pi}
_{r',0}|_A)\boxtimes(\tilde{\pi}_{r,0}|_A),
$$
which, by Proposition \ref{decomposition}, is just:
$\pi_{r}\boxtimes\pi_{r'}\cong\pi_{r'}\boxtimes\pi_{r}$.
To find the intertwining operator for this equivalence, let us find an
explicit expression for the operator $T_{\tilde{\pi}_{r,0}\tilde{\pi}_{r',0}}$.
Recall first that by equation (6.3) and Definition 6.3 of \cite{BJKp2}, the
quantum $R$-matrix for $(\tilde{A},\tilde{\Delta})$ is considered as a continuous
function defined by:
\begin{align}
R(p,q,r,s,p',q',r',s')&=\Phi(p,q,r,s,p',q',r',s')\Phi'(p,q,r,s,p',q',r',s')
\notag \\
&=\bar{e}\bigl[\lambda(rs'+r's)\bigr]\bar{e}\bigl[2\lambda(e^{-\lambda r'})
p\cdot q'\bigr]. \notag
\end{align}
We then calculate $(\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(R)$ as an
operator on $\tilde{\mathcal H}_{r,0}\otimes\tilde{\mathcal H}_{r',0}\cong
L^2(\mathbb{R}^{2n})$. By a straightforward calculation, we obtain for
$\xi\in L^2(\mathbb{R}^{2n})$,
\begin{align}
&\bigl((\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(\Phi)\xi\bigr)(u,v)
\notag \\
&=\int \bar{e}\bigl[\lambda(rs'+r's)\bigr]e[p\cdot x+q\cdot y+sw+p'\cdot
x'+q'\cdot y'+s'w'] \notag \\
&\qquad(e^{-\frac{w}2})^n(e^{-\frac{w'}2})^n\,\xi(e^{-w}u+e^{-w}x,
e^{-w'}v+e^{-w'}x') \notag \\
&\qquad\bar{e}\bigl[\eta_{\lambda}(r)\beta(u,y)\bigr]\bar{e}
\bigl[\eta_{\lambda}(r')\beta(v,y')\bigr]\,dpdqdsdp'dq'ds'dxdydwdx'dy'dw'
\notag \\
&=(e^{-\frac{\lambda r}2})^n(e^{-\frac{\lambda r'}2})^n\,\xi(e^{-\lambda r'}
u,e^{-\lambda r}v). \notag
\end{align}
Similarly,
$$
\bigl((\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(\Psi)\xi\bigr)(u,v)
=\xi(u+2\lambda e^{-\lambda r'}\eta_{\lambda}(r')v,v).
$$
Since $\eta_{\lambda}(r')=\frac{e^{2\lambda r'}-1}{2\lambda}$, we thus have:
\begin{align}
\bigl((\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(R)\xi\bigr)(u,v)
&=\bigl((\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(\Phi)\bigr)
\bigl((\tilde{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(\Psi)\bigr)\xi(u,v)
\notag \\
&=(e^{\frac{-\lambda r}2})^n(e^{-\frac{\lambda r'}2})^n\xi
\bigl(e^{-\lambda r'}u+(e^{\lambda r'}-e^{-\lambda r'})e^{-\lambda r}v,
e^{-\lambda r}v\bigr). \notag
\end{align}
By applying the flip $\sigma$, we therefore obtain:
\begin{align}
T_{\tilde{\pi}_{r,0}\tilde{\pi}_{r',0}}\xi(v,u)&=\sigma\circ\bigl((\tilde
{\pi}_{r,0}\otimes\tilde{\pi}_{r',0})(R)\bigr)\xi(v,u) \notag \\
&=(e^{\frac{-\lambda r}2})^n(e^{-\frac{\lambda r'}2})^n\xi
\bigl(e^{-\lambda r'}u+(e^{\lambda r'}-e^{-\lambda r'})e^{-\lambda r}v,
e^{-\lambda r}v\bigr). \notag
\end{align}
Define $T_{\pi_r\pi_{r'}}$ by $T_{\pi_r\pi_{r'}}=T_{\tilde{\pi}_{r,0}
\tilde{\pi}_{r',0}}$. Then it is a straightforward calculation to show
that $T_{\pi_r\pi_{r'}}$ is an intertwining operator between $\pi_{r}\boxtimes
\pi_{r'}$ and $\pi_{r'}\boxtimes\pi_{r}$. For $f\in{\mathcal A}$ and
$\xi\in L^2(\mathbb{R}^{2n})$, we have:
\small
\begin{align}
&T_{\pi_r\pi_{r'}}\bigl((\pi_{r}\boxtimes\pi_{r'})(f)\bigr)\xi(v,u)
\notag \\
&=\int f(x,y,r+r')\bar{e}\bigl[\eta_{\lambda}(r)\beta(e^{-\lambda r'}u
+(e^{\lambda r'}-e^{-\lambda r'})e^{-\lambda r}v,e^{\lambda r'}y)\bigr]
\bar{e}\bigl[\eta_{\lambda}(r')\beta(e^{-\lambda r}v,y)\bigr] \notag \\
&\quad\quad\quad(e^{\frac{-\lambda r}2})^n(e^{-\frac{\lambda r'}2})^n
\xi(e^{-\lambda r'}u+(e^{\lambda r'}-e^{-\lambda r'})e^{-\lambda r}v
+e^{\lambda r'}x,e^{-\lambda r}v+x)\,dxdy \notag \\
&=\bigl((\pi_{r'}\boxtimes\pi_{r})(f)\bigr)T_{\pi_r\pi_{r'}}\xi(v,u).
\notag
\end{align}
\normalsize
Since $\xi$ is arbitrary, it follows that:
$$
T_{\pi_r\pi_{r'}}\bigl((\pi_{r}\boxtimes\pi_{r'})(f)\bigr)=\bigl((\pi_{r'}
\boxtimes\pi_{r})(f)\bigr)T_{\pi_r\pi_{r'}}.
$$
\end{proof}
Observe that by interchanging $r$ and $r'$, we are able to find the expression
for the intertwining operator $T_{\pi_{r'}\pi_{r}}$:
$$
T_{\pi_{r'}\pi_{r}}\xi(u,v)=(e^{\frac{-\lambda r}2})^n(e^{-\frac
{\lambda r'}2})^n\,\xi\bigl(e^{-\lambda r}v+(e^{\lambda r}-e^{-\lambda r})
e^{-\lambda r'}u,e^{-\lambda r'}u\bigr).
$$
For $f\in{\mathcal A}$, we will have: $T_{\pi_r'\pi_{r}}\bigl((\pi_{r'}
\boxtimes\pi_{r})(f)\bigr)=\bigl((\pi_{r}\boxtimes\pi_{r'})(f)\bigr)
T_{\pi_r'\pi_{r}}$.
However, note that unlike in the ordinary group representation theory or in
the cases equipped with ``triangular'' quantum $R$-matrices, we no longer have:
$T_{\pi_{r'}\pi_{r}}T_{\pi_r\pi_{r'}}=I$. We instead have:
\small
\begin{align}
T_{\pi_{r'}\pi_{r}}T_{\pi_{r}\pi_{r'}}\xi(u,v)=(e^{-\lambda r})^n
(e^{\lambda r'})^n\,\xi\bigl(&u-(1-e^{-2\lambda r'})e^{-2\lambda r}u+
(e^{\lambda r'}-e^{-\lambda r'})e^{-2\lambda r}v, \notag \\
&\quad e^{-2\lambda r}v+
(1-e^{-2\lambda r})e^{-\lambda r'}u\bigr), \notag
\end{align}
\normalsize
which is clearly not the identity operator. Let us summarize our results
in the following theorem:
\begin{theorem}
Given any two representations $\pi$ and $\rho$ (acting on the Hilbert spaces
${\mathcal H}_{\pi}$ and ${\mathcal H}_{\rho}$) of our Hopf $C^*$-algebra
$(A,\Delta)$, their inner tensor products ``commutes'' (i.\,e. $\pi\boxtimes
\rho$ and $\rho\boxtimes\pi$ are equivalent). However, the intertwining
operators between them behave in an interesting way, in the sense that we have:
$T_{\rho\pi}T_{\pi\rho}\ne I_{{\mathcal H}_{\pi}\otimes{\mathcal H}_{\rho}}$,
in general.
\end{theorem}
This theorem means that the category of representations of $(A,\Delta)$
is essentially a ``quasitriangular monoidal category''. This is a typical
characteristic of the category of ``braids'' (in knot theory). In recent
years, representation theory of quantum groups led to the developments and
discoveries of some useful knot invariants, like Jones polynomials or HOMFLY
polynomials, especially in connection with the existence of quantum universal
$R$-matrices (For more discussion about these topics, see \cite{Bir} or
\cite[\S15]{CP}.).
In our case, it is interesting to point out that $(A,\Delta)$ possesses
the quasitriangular type property, without the existence of its own quantum
$R$-matrix $R_A$. Meanwhile, since there have been only a handful of examples
so far of non-compact, $C^*$-algebraic quantum groups possessing the property
of quasitriangularity, having these examples $(A,\Delta)$ and $(\tilde{A},
\tilde{\Delta})$ would benefit the study of non-compact quantum groups and
its development.
\bibliography{ref}
\bibliographystyle{amsplain}
\end{document}
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_1<<_2<<
Little Boy Drinks Milk Directly From Cow’s Udder [VIDEO]
A boy in the Cambodian village of Koak Roka has gained worldwide attention for suckling milk directly from a cow’s udders, Reuters reports.
Mishka the Talking Siberian Husky Profiled on ‘Today’ [VIDEO]
Heroic Crowd Lifts Burning Car Off Motorcyclist in Incredible Footage [VIDEO]
>>IMAGE.
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It’s difficult to maintain my election frenzy. The falling leaves are so beautiful. I keep getting distracted by their grace and whimsy. Much better than Zoloft.
We have become election news junkies. I probably devote four hours daily glued to the computer, the TV and the newspapers.
You, too? We are not alone.
Researchers have been checking regularly with 20,000 registered voters. Recent findings: 61 percent of the voters had browsed the Net for political info in the last week, 50 percent has succumbed to political e-mails and within the past 24 hours, 53 percent had shared some low talk about a candidate.
Beer has been an election night staple for as long as I have been voting. This year, however, both hands will be full. One with the remote, the other with the mouse. I told you we were hooked.
This year, we’re keeping track of election stats by more than merely “Breaking News” on the cable news shows. We’re following raw data via media outlets that were not around just four years ago. Think of it. YouTube did not exist the last election. Facebook was mainly Ivy League. No Huffington Post. I’m not sure about when Politico ginned up. News aggregators. E-mail alerts. Weather out west.
Which is the relevant news source anymore? The answer: there is no single source now that each of us has control of the information joy stick. No longer will we be content to be spoon-fed by Brian Williams, Charlie Gibson and Katie Couric. Full disclosure: I’ll miss Katie more than the others.
If you think election coverage is snappy this year, just wait until the election four years from now. Chances are, we will be watching one interactive screen that combines data streams, video streams, and perhaps newspapers, too. Plus probably some device or application we haven’t thought of yet. One day in the future, we may even be able to vote from home.
Here’s the good news: That should leave the other hand free for the beer.
Monday, November 3, 2008
National Politics
News on Aging
Geriatric Medicine News
Senior Health Insurance News
Social Security & Medicare News
Posts From Other Geezer Blogs
2 comments:
Sounds like my night as well. The Internet is just too informative to rely on talking heads to dish out the same old stuff.
I have no feel for the results, but a lot of fear...just like everyone on either side.
I wish we could have folks running that nobody in our country would be afraid of. Probably never in the cards.
Will it be "Twilight Zone" with Obama.or "It's a Wonderful Life" with Sarah Palin. The Shadow Knows!
---Goose
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TITLE: An abelian tower of a finite group admits a cyclic refinement — Proposition I.3.1, Lang's 'Algebra'
QUESTION [7 upvotes]: Recently I've been looking through Lang's Algebra, and I encountered a problem in the proof of Proposition 3.1 in Chapter I Groups.
Proposition 3.1. Let $G$ be a finite group. An abelian tower of $G$ admits a cyclic refinement. Let $G$ be a finite solvable group. Then $G$ admits a cyclic tower, whose last element is $\{e\}$.
Proof. The second assertion is an immediate consequence of the first, and it clearly suffices to prove that if $G$ is finite, abelian, then $G$ admits a cyclic tower. We use induction on the order of $G$. Let $x$ be an element of $G$. We may assume that $x \neq e$. Let $X$ be the cyclic group generated by $x$. Let $G' = G/X$. By induction, we can find a cyclic tower in $G'$, and its inverse image is a cyclic tower in $G$ whose last element is $X$. If we refine this tower by inserting $\{e\}$ at the end, we obtain the desired cyclic tower.
I don't understand why it suffices to prove that if $G$ is finite, abelian, then $G$ admits a cyclic tower. In the statement of the proposition $G$ is not assumed to be an abelian group.
Moroever, even assuming that we do prove that if $G$ is finite, abelian, then $G$ admits a cyclic tower, I don't see how can we use this in proving Proposition 3.1.
Maybe this question is very easy, but currently I can't understand it. Any help would be appreciated.
REPLY [1 votes]: This answer is also an attempt at Lamport's method of writing structured proofs.[1], [2]
Proposition 3.1.
Assume: $G$ is a finite group having the abelian tower
$$
G = G_0 \supset G_1 \supset G_2 \supset \dotsb \supset G_m.\tag{$*$}
$$
Prove: $(*)$ has a cyclic refinement.
Proof.
It suffices to assume that $m = 1$.
Proof: If each $G_i \supset G_{i+1}$ has a cyclic refinement, so does $(*)$ by concatenating these cyclic towers.
Let $f \colon G \to G/G_1$ be the canonical map.
It suffices to assume that $G$ is abelian and $G_1 = \{e\}$.
Proof: Since $G/G_1$ is abelian by hypothesis, $G/G_1 \supset \{ \bar{e} \}$ has a cyclic refinement by assumption. The pullback by $f$ of this cyclic tower is a cyclic refinement of $G \supset G_1$.
Let $P(n)$ be the statement that for a finite abelian group $G$ with $|G| = n$, $G \supset \{e\}$ has a cyclic refinement.
Case: $n = 1$
Proof: Then $G = \{e\}$, and $\{e\} \supset \{e\}$ is already cyclic, so there is nothing to show.
Assume that there exists $n \geq 1$ such that $P(m)$ is true for all $m \leq n$. Then, $P(n+1)$ is true.
4.1. Pick $x \in G$ such that $x \neq e$.
Proof: Since $|G| = n + 1$ and $n \geq 1$, there exists such an $x$.
Let $X = \langle x \rangle$, and $p \colon G \to G/X$ be the canonical map.
4.2. $G/X \supset \{ \bar{e} \}$ has a cyclic refinement.
Proof: Since $m = |G/X| < |G| = n + 1$, and $P(m)$ is true by assumption.
4.3. QED
Proof: The pullback by $p$ of the cyclic tower obtained in step 4.2. is a cyclic refinement of $G \supset X$. Since $X$ is cyclic, this is also a cyclic refinement of $G \supset \{e\}$.
QED
Proof: By the principle of strong mathematical induction: step 3. proves the base case and step 4. proves the induction step.
$\blacksquare$
| 134,308
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Server Monitoring Authors: Carmen Gonzalez, Yeshim Deniz, Liz McMillan, Pat Romanski, Ken Schwaber
News Feed Item
Over 50% of Windows respondents use Java Architecture for Content
Management
Alfresco
Software, Inc.
Open Source ECM and Enterprise 2.0
In an open source ECM environment:
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The results for infrastructure usage are consistent with the previous
Open Source Barometer surveys:
.
The full Alfresco Open Source Barometer survey (including a Slideshare
link) is available on:
Visit the Open Source Barometer page in Facebook:.
Alfresco is a trademark of Alfresco Software Inc in the US and other
countries. All other names and trademarks are the property of their
respective owners.
Published November 11, 2008.
| 96,329
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For the third time in history, the Rollins women's soccer team has been named Sunshine State Conference champions.
The No. 24 Tars clinched their first title since 2009 when they beat Nova SE 3-0 on Saturday.
"We are so proud to be the 2012 SSC champions," head coach Alicia Milyak Schuck said to Rollins College Athletics. "All of the conference games were extremely competitive and we had to battle for every point we obtained because of the impressive performances the other SSC teams put forth in the games."
| 273,174
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Dispatch Editor Ben Marrison was nice enough to post a reply to my letter on his own blog over at the Dispatch website.
The piece had some interesting fallout.
A few people who I expected would laugh off the whole thing were quite offended, and those I thought might be offended were actually pretty lighthearted about the whole thing.
I was pleased to learn, through the whole process, that the Dispatch is actually aware of some of their problems, and they have retained a crew of people to help them with their social media - and other - skills.
So, thanks to you, Ben Marrison, for taking my critique with a good dose of humor - you had some funny digs on your post that only close readers would have caught - and for acknowledging the road ahead. It really is my hope that our paper can figure out a way to become great. It would be amazing if we could lead the country in local papers who are doing something great.
I realize we are at a crossroads here, especially with print media, although I feel that we aren't really that far away from the model becoming online only. My parents are in their 60s, and they read most of their news online. I will admit they have always been closer to the cutting edge of technology than a lot of others their age, but it shows that there is a lot of room to develop your online presence.
I would encourage the Dispatch to see their criticism - not just here, but everywhere - as "an opportunity for improvement" (ha! That's one of those corporate phrases I hate but still love to use). Your city wants to support you. We want to root for you. There really isn't a better place from which to start.
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TITLE: Cohomologically minimal spaces
QUESTION [3 upvotes]: Let $X$ be a compact connected Hausdorff topological space.
We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property:
"For every proper subset $A\subset X$ with the inclusion map $i:A \to X$, $i^*$ is NOT a ring isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(A,\mathbb{Z})$"
In the other word, $A$ is the only subset of $A$ whose inclusion gives a ring isomorphism cohomology.
Examples; All closed manifolds (see A closed manifold with a subset with the same ring cohomology )
Non Examples: Closed disc, figure 8,...etc.
My question
Is the product of two CM spaces, a CM space?
REPLY [5 votes]: Let me say that $X$ has property $CM'$ if for every field $F$ and every point $a\in X$, the map $H_*(X\setminus a;F)\to H_*(X;F)$ is injective but not surjective. Then orientable closed manifolds have $CM'$, and $CM'$ implies $CM$. Now suppose that $X$ and $Y$ both have $CM'$. After noting that
$$ (X\times Y)\setminus (a,b) = ((X\setminus a)\times Y) \cup
(X\times (Y\setminus b))
$$
we can use the Kunneth isomorphism and the Mayer-Vietoris sequence to see that $X\times Y$ also has $CM'$. (I switched to using homology to ensure that Kunneth works in a straightforward way even if the spaces have infinitely generated homology.) I think that a similar approach can be made to work for non-orientable manifolds if we modify the definition of $CM'$ to distinguish between fields that do or do not have characteristic $2$. Thus, to make progress you should investigate whether there are any spaces that have $CM$ but not $CM'$.
Incidentally, contrary to your list of non-examples, I think that the figure 8 has $CM'$ (and therefore $CM$).
| 124,785
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It’s not being negative if it’s the truth, right?
When I moved to New York, over two years ago, I knew the city would be rough and ready but I didn’t know in what way. Although I visited New York a number of times before my move in 2016, holiday time here is a sharp contrast to actually making a life here.
I hope this helps anyone thinking or planning on moving to New York, or perhaps it’ll make for an interesting read for those of you who are happy living outside of this mecca too!
Rent is colossal and you’ll probably live in a shoebox
BusinessInsider says “the average rent for an apartment in Manhattan remains exorbitant at $3,667”. While that’s definitely on the higher end of the scale, for anything less, you’re looking at a very small space. Not only that but to secure the apartment, you’ll need to pay your broker (usually 1-month’s rent) the first and last months rent PLUS a security deposit. You’re looking at around thousands and thousands just to move in. OUCH.
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The streets are really, really dirty
Rats don’t even phase me anymore, I see them every day! People are dirty, the subways are dirty and you see stuff on a daily basis you just wish you could unsee. The glamour does not live on the streets of Manhattan. I carry hand sanitizer everywhere I go and wash my hands more than ever. I even find I wash clothes a lot more since I sat on the Subway that day.
Working hours are really long and vacation days are extremely limited
You find yourself working harder and longer because of this city. Because that’s what people do. You are entitled to ZERO holidays in the US, meaning they are doing you a favor by offering you a measly few days off a year. Because of the competition, the mentality is that if you’re not willing to do the work, someone else will.
$7 coffee will be ok
I met a co-worker in an elevator and she commented on my Starbucks. I told her “I can’t do this every day, these coffees cost me $5.” “That’s all??”, she said and went on to describe how her coffee costs her $8. So yes, the coffee machine at work is my friend and Starbucks trips are only for days of desperation.
The weather is intense AF
The summer heat is like something from a nightmare. It’s sweaty and sticky and I hate it. I realized living in New York that I’m a cold weather creature. I don’t tan well and I have so much hair it feels like a blanket is on my head. So I don’t think I’m built for this. In the winter, the snow will bury you inside your apartment and the temperatures will decrease to -15 °C. Ouch.
Manhattan doesn’t do groceries
One thing that really frustrates me is that it’s so hard to pick up fresh produce here. In Ireland (or any other country I feel), and you stop in a small Tesco/Centra/petrol station, and you can find basic fruit/veg/milk in there. Well, everything in the stores here come in packets. Although NYC is known for its convenience, getting groceries is terribly inconvenient, not to mention extremely expensive. Count yourself lucky if you live by a grocery store like Trader Joe’s or Whole Foods. It’s New York gold!
It’s really as magical as they say
Above all, the pro’s always outweigh the con’s here. The magic springs up on you when you least expect it and really makes it all worthwhile. While it’s a tougher, more expensive way to live, the return you get from Manhattan is just priceless.
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TITLE: How do we know what the integral of $\sin (x)$ is?
QUESTION [8 upvotes]: Since I started more or less formally learning the foundations of calculus, I naturally came across the Riemann definition of the integral. At first I considered this definition intuitive but not really useful since to obtain a closed form expression, one needed to add an infinite amount of values. Later on, an exercise prompted me to calculate a Riemann Integral, and from the definition and the expression for the sum of squares, I was able to calculate the limit with nothing more than I had learned at school.
This was a revelation for me, since so far I had considered the definition a mere formalism. Now I knew how it gave results. The next integral I tried to calculate this way was, for obvious reasons $\sqrt {1-x^2}$. Unfortunately, I found the sum intractable and gave up.
I started to question the usefulness of the definition again. If it only works for simple functions like polynomials, how did we ever find out that the integral of $\sin (x)$ is $-\cos (x)$? Did we use the Riemann definition or did we just say "the derivative of $-\cos (x)$ is $\sin (x)$ and therefore its integral must be $-\cos (x)$?
I would like to get some insight into the theory as well as the history that led to the tables of integrals we have today
REPLY [3 votes]: Expanding the comments of Doug M and benguin: this is a very simplified version of the history.
Gregory/Barrow/Newton proved (more or less) the Fundamental Theorem of Calculus with integral = area under the curve.
About the formulas $\sin' = \cos$, $\cos' = -\sin$, is really difficult to say who proved first. Maybe Roger Cotes? See The calculus of the trigonometric functions for details and also How were derivatives of trigonometric functions first discovered?
Also very interesting: Some History of the Calculus of the
Trigonometric Functions includes the proof by Archimedes of our formula
$$\int_0^\pi\sin = 2$$
than can be easily generalized (Archimedes dont't do this) to
$$\int_0^\alpha\sin x\,dx = 1 - \cos\alpha.$$
The section about Pascal is equally interesting.
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In order to further study the properties of the SPDE model \eqref{eq:lSPDE}, we require a more explicit characterization of the solution, in order to compute various quantities of interest and estimate model coefficients from observations. A useful approach is to look for a { finite dimensional realization} of the infinite-dimensional process $u$:
\begin{df}[Finite dimensional realizations]\label{df:fdr} A process $u= (u_t)_{t\geq 0}$ taking
values in an (infinite-dimensional) function space $E$ is said to
admit a {\it finite dimensional realization} of dimension $d\in
\N$ if there exists an $\R^{d}$-valued stochastic process $Z=(Z^1,...,Z^d)$
and a map
\[ \phi: \R^{d} \to E\quad{\rm such\ that}\quad \forall t\geq 0, \quad u_t = \phi(Z_t).\]
\end{df}
Availability of a finite dimensional realization for the SPDE~\eqref{eq:lSPDE} makes simulation, computation and estimation problems more tractable, especially if the
process $Z$ is a low-dimensional Markov process.
Existence of such finite-dimensional realizations for stochastic PDEs have been investigated for SPDEs arising in filtering \cite{levine1991} and interest rate modelling \cite{filipovic2003,gaspar2006}.
We will now show that finite dimensional realizations may indeed be
constructed for a class of SPDEs which includes \eqref{eq:lSPDE}, and
use this representation to perform an analytical study of these models.
\subsection{Homogeneous equations}
We now consider a more general class of linear homogeneous evolution equations
with multiplicative noise taking values in a real separable Hilbert space $(H, \langle \cdot,\cdot\rangle_H)$. Typically, $H$ will be a function space such as $L^2(I)$ for some interval $I\subset \R$. We consider the following class of evolution equations:
\begin{gather}
\label{eq:gSPDE}
\begin{split}
\d u_t &= A u_{t-} \d t + u_{t-} \d X_t,\qquad t>0,\\
u_0 &= h_0 \in H.
\end{split}
\end{gather}
where $X$ is a real c\`{a}dl\`{a}g semimartingale whose jumps satisfy $\Delta X_t>-1$ a.\,s. and
$A\colon \dom(A) \subset H \to H$ a linear operator on $H$ whose adjoint we denote by $A^*$. We assume
that $\dom(A)\subset H$ is dense, and $A$ is closed. Since $A$ is closed we have that
also $\dom(A^*)\subset H$ is dense and that $A^{**} = A$
\cite[Theorem VII.2.3]{yosida1995functional}.
\begin{df}\label{df:weak_solution}
An adapted $H$-valued stochastic process
$(u_t)$ is an \emph{(analytical) weak solution} of~\eqref{eq:gSPDE} with
initial condition $h_0$ if, for all $\varphi \in \dom(A^*)$, $[0,\infty) \ni t\mapsto \langle u_t, \varphi\rangle_H \in \R$ is c\`{a}dl\`{a}g a.\,s. and for each $t\geq 0$, a.\,s.
\[\langle u_t,\varphi \rangle_{H} - \langle h_0, \varphi \rangle_{H} = \int_0^t \langle u_{s-}, A^*\varphi \rangle_{H} \d s+ \int_0^t \langle u_{s-}, \varphi \rangle_{H} \d X_s.\]
\end{df}
The case $X\equiv 0$ corresponds to a notion of weak
solution for the PDE:
\begin{equation}
\label{eq:gPDE}
\forall t>0, \ddt g_t = A g_t\qquad
g_0 = h_0.
\end{equation}
That is, for all $\varphi \in \dom(A^*)$,
\begin{equation}
\label{eq:12}
\langle g_t, \varphi\rangle_{H} - \langle h_0,\varphi\rangle_{H}= \int_0^t \langle g_s, A^* \varphi \rangle_{H} \d s,
\end{equation}
where the integral on the right hand side is assumed to exist.\footnote{Note that this slightly differs from the classical
formulation of weak solutions for PDEs.} In
particular, this yields that $[0,\infty) \ni t\mapsto \langle g_t,
\varphi\rangle_H\in \R$ is continuous.
\begin{rmk}
By considering bid and ask side separately, we can
bring~\eqref{eq:lSPDE} into the form of \eqref{eq:gSPDE}, where $X$
is a Brownian motion and $A$ is given by $A := \eta\Delta \pm
\beta\nabla + \alpha \operatorname{\Id}$ on $H:= L^2(I)$, $I :=
(0,L)$ or $I:= (-L,0)$, with domain
\[\operatorname{dom}(A):= H^2(I)\cap H^1_0(I),\]
where $H^1_0(I)$ is the closure in $H^1(I)$ of test functions with compact support in $I$.
\end{rmk}
Denote by $Z_t=\mathcal{E}_t(X)$ the stochastic exponential of $X$. We recall the following useful lemma:
\begin{lem}\label{lem:ExpRec}
Let
\begin{equation}
\label{eq:10}
Y_t := -X_t + \left[ X, X\right]_t^c + \sum_{s\leq t} \frac{(\Delta X_s)^2}{1 + \Delta X_s},\qquad t\geq 0,
\end{equation}
Then, $\mathcal{E}_t(X) \mathcal E_t(Y) = 1$ almost surely, for all $t\geq 0$. Moreover,
\begin{equation}
\label{eq:11}
\left[ X, Y\right] = -\left[X,X\right]^c- \sum_{s\leq \cdot} \frac{(\Delta X_s)^2}{1+\Delta X_s}.
\end{equation}
\end{lem}
\begin{proof}
The first part is shown e.\,g. in \cite[Lemma 3.4]{karatzas2007numeraire}.
Recall that
$
\left[ X,X\right]_t = \left[X, X\right]^c_t + \sum_{s\leq t} (\Delta X_s)^2
$
so
\begin{align*}
\left[X,Y\right]_t &= - \left[X, X\right]^c_t - \sum_{s\leq t} \frac{(\Delta X_s)^2 + (\Delta X_s)^3}{1+\Delta X_s} + \sum_{s\leq t} \frac{(\Delta X_s)^3}{1+ \Delta X_s}\\
&=- \left[X, X\right]^c_t - \sum_{s\leq t} \frac{(\Delta X_s)^2}{1+\Delta X_s}.\qedhere
\end{align*}
\end{proof}
\begin{thm}\label{thm:homo}
Let $Z := \mathcal{E}(X)$, $h_0 \in H$. Then every
weak solution of~\eqref{eq:gSPDE} is of the form
$$u_t:= Z_tg_t,\qquad t\geq 0$$ where $g$ is a weak solution of~\eqref{eq:gPDE}.
\end{thm}
\begin{rmk}
In particular, the SPDE~\eqref{eq:gSPDE} admits a two dimensional realization in the sense of Definition~\ref{df:fdr} with factor process $(t,\mathcal{E}_t(X))$ and $\phi(t,y):= y g_t$.
\end{rmk}
\begin{proof}
Set $u_t := g_t Z_t$, $t\geq 0$, and for $\varphi \in
D(A^*)$ write $B^\varphi_t:= \langle g_t,\varphi \rangle_{H}$,
$C^\varphi_t := B^\varphi_t Z_t = \langle u_t,\varphi \rangle_{H}$. Since $t\mapsto \langle g_t,\varphi \rangle_H$ is continuous and $Z$ is scalar and c\`{a}dl\`{a}g, we get that $t\mapsto \langle u_t, \varphi \rangle_H$ is c\`{a}dl\`{a}g. Note that $B^\varphi$ is of
finite variation and $Z$ is a semimartingale, so that also $C^\varphi$ is a
semimartingale. Moreover, by It\=o product rule and since $B^\varphi$ is of
finite variation and continuous,
\begin{equation}
\label{eq:10}
\d C^\varphi_t = B^\varphi_{t} \d Z_t + Z_{t-} \d B^\varphi_t
= B^\varphi_{t-} Z_{t-} \d X_t + \langle u_{t-}, A^* \varphi
\rangle_{H} \d t,
\end{equation}
which is~\eqref{eq:gSPDE}.
Now, let $u$ be a solution of \eqref{eq:gSPDE} and set
\begin{equation*}
Y := -X + \left[ X, X\right]^c + J,\qquad J :=\sum_{s\leq \cdot} \frac{(\Delta
X_s)^2}{1+ \Delta X_s},
\end{equation*}
and $Z_t := \mathcal{E}_t(Y)$, $t\geq 0$. Recall that by Lemma~\ref{lem:ExpRec}
we have $Z_t\mathcal{E}_t(X) = 1$ for all $t\geq 0$.
Set $g_t := Z_t u_t$, and, as above, fix $\varphi \in \dom(A^*)$ and
write $B^\varphi_t:= \langle u_tZ_t, \varphi\rangle_{H} = \langle g_t,\varphi
\rangle_{H}$ and $C^\varphi_t := \langle u_t,\varphi \rangle_{H}$. By Ito's
product rule and Lemma~\ref{lem:ExpRec},
\begin{align}\nonumber
\d B^\varphi_t &= C^\varphi_{t-} \d Z_t + Z_21{t-} \d C^\varphi_t + \d \left[ C^\varphi, Z \right]_t\\
\nonumber &= C^\varphi_{t-}Z_{t-} \d Y_t + Z_{t-} \langle u_{t-},
A^*\varphi \rangle_{H} \d t + C^\varphi_{t-}Z_{t-} \d X_t + C^\varphi_{t-}
Z_{t-} \d \left[ X,Y\right]_t\\
\nonumber &= \langle g_{t-},
A^*\varphi \rangle_{H} \d t
+ B^\varphi_{t-} \left(\d \left[ X,X\right]^c_t + \d J_t\right)
- B^\varphi_{t-} \left( \d \left[X,X\right]^c_t + \d
J_t\right)\\
\nonumber &= \langle g_{t-},
A^*\varphi \rangle_{H} \d t.
\end{align}
Thus, $g$ is a weak solution of~\eqref{eq:gPDE}.
\end{proof}
\begin{ex}
Let $A$ be the generator of a strongly continuous semigroup
$(S_t)_{t\geq 0}$. Then, for $h_0\in H$ define
\[ g_t := S_t h_0,\quad t\geq 0, \]
which is a weak solution of~\eqref{eq:gPDE}. By Theorem~\ref{thm:homo}.
\[u_t := \mathcal{E}_t(X) S_t h_0, \qquad t\geq 0, \]
is a weak solution of~\eqref{eq:gSPDE}.
\end{ex}
\begin{rmk}
If $h_0$ is an eigenfunction of $A$ with eigenvalue $\nu$, then,
$g_t = e^{\nu t} h_0$ is the unique locally $H$-integrable solution
of~\eqref{eq:gPDE}, and the unique solutiuon of \eqref{eq:gSPDE} is given by
\[ u_t := h_0 e^{\nu t} \mathcal{E}_t (X).\]
\end{rmk}
\subsection{Inhomogeneous equations}
We keep the assumptions on $A$, $h_0$ and $X$ from the previous
section and let $f\in H$. We now consider the inhomogeneous
linear evolution equations
\begin{gather}
\label{eq:gSPDEinh}
\begin{split}
\d u_t &= \left[ A u_{t} + \alpha f \right] \d t + u_{t-} \d X_t,\qquad t\geq 0,\\
u_0 &= h_0.
\end{split}
\end{gather}
\begin{df}
A weak solution of~\eqref{eq:gSPDEinh} is an adapted
$H$-valued stochastic process $u$ such that for all $\varphi \in
\dom(A^*)$ the mapping $[0,\infty)\ni t\mapsto \iprod{u_t}{\varphi}{H}$
is c\`{a}dl\`{a}g and
\[\langle u_t,\varphi \rangle_{H} - \langle h_0, \varphi \rangle_{H} =
\int_0^t \langle u_{s-}, A^*\varphi \rangle_{H} \d s+ \int_0^t
\langle u_{s-}, \varphi \rangle_{H} \d X_s + t\alpha \langle
f,\varphi\rangle_H,\quad t\geq 0,\]
almost surely. In particular, all the integrals are assumed to exist.
\end{df}
We exclude the cases $\alpha = 0$ or $f\equiv 0$ which correspond to
the homogeneous case discussed above. Let us first consider
the case where $A$ admits at least one eigenfunction.
\begin{thm}\label{thm:inhomo}
Suppose that $f\in \dom(A)$ is an eigenfunction for $A$ with eigenvalue $\lambda \in \R$, let $z_0 >0$ and $Z$ be the solution of
\begin{equation}
\label{eq:Z}
\d Z_t = \left(\lambda Z_{t-}
+ \alpha\right) \d t + Z_{t-} \d X_t,\quad t\geq 0, \quad Z_0 = z_0.
\end{equation}
Then:
\begin{enumerate}[label=(\roman*)]
\item\label{inhomo:i:1factor}
The stochastic process defined by $u_t = Z_t f$, $t\geq 0$, is a
solution of \eqref{eq:gSPDEinh} with initial condition $h_0 := z_0
f$.
\item\label{inhomo:i:2factor} Let, in addition, $h_0\in H$ be such that there exists a
weak solution $g = (g_t)_{t\geq 0}$ of the deterministic equation
\begin{equation}
\label{eq:detPDEh0f}
\ddt g_t = A g_t,\; t\geq 0\,\qquad g_0 = h_0 -z_0 f.
\end{equation}
Then, $u_t:= g_t\mathcal{E}_t(X) + f Z_t$ is a solution
of~\eqref{eq:gSPDEinh} with initial condition $h_0$.
\item\label{inhomo:i:2factor-nec} Let $h_0\in H$ be such that there exists a weak solution
$u = (u_t)_{t\geq 0}$ of~\eqref{eq:gSPDEinh} with initial condition
$h_0$. Then, $g := (u - f Z)\mathcal{E}(X)^{-1}$, is a weak solution
of~\eqref{eq:detPDEh0f}.
\end{enumerate}
\end{thm}
\begin{rmk}
Let $(Z_t^1)_{t\geq 0}$ and $(Z_t^2)_{t\geq 0}$ be given by
\eqref{eq:Z} with respective initial data $z_1$, $z_2>0$, $z_1\neq
z_2$. Then, in fact $Z_t^2 - Z_t^1 = (z_2-z_1) \mathcal{E}_t(X)$,
which is consistent with choosing different values for $z_0$ in \ref{inhomo:i:2factor}.
\end{rmk}
\begin{proof}
Part~\ref{inhomo:i:1factor} follows by direct a computation: Let $\varphi \in H$, then for $t\geq 0$,
\begin{multline}
\label{eq:15}
d\iprod{u_t}{\varphi}{H} = \iprod{f}{\varphi}{H} dZ_t = \\
=\iprod{f}{\varphi}{H}\left(\lambda Z_{t-} + \alpha\right)dt +
\iprod{f}{\varphi}{H} Z_{t-} d X_t\\
=\left[\iprod{u_{t-}}{A^*\varphi}{H} + \alpha \iprod{f}{\phi}{H} \right] dt + \iprod{u_{t-}}{\phi}{H} dX_t.
\end{multline}
Similarly, we obtain that any solution $u$ of~\eqref{eq:gSPDEinh}
with initial data $h_0\in H$ can be written as
\[ u = u^{\circ, (h_0-z_0f)} + u^{(z_0f)}\]
where $u^{\circ,(h_0-z_0 f)}$ is the solution of the
homogeneous problem~\eqref{eq:gSPDE} with initial data $h_0-z_0 f$
and $u^{(z_0f)}$ is a solution of~\eqref{eq:gSPDEinh} with initial data
$z_0f$. Then, part (i) and Theorem~\ref{thm:homo} finish the proof of
(ii) and (iii).
\end{proof}
The following result shows that $Z$ is known explicitly and extends \cite[Proposition 21.2]{kallenberg} which covers linear SDEs driven by continuous semimartingales.
\begin{prop}\label{prop:Zexp}
Let
\[ Y_t:= -X_t + [X,X]^c_t + \sum_{s\leq t} \frac{\Delta X_s^2}{1+
\Delta X_s},\qquad t\geq 0.\]
Then, the unique solution of~\eqref{eq:Z} is given by
\begin{equation}
\label{eq:Zexplicit}
Z_t:= \mathcal{E}_t(X)e^{\lambda t}\left( Z_0 + \alpha \int_0^t e^{-\lambda s} \mathcal{E}_{s-}(Y) \d s\right),\qquad t\geq 0.
\end{equation}
\end{prop}
\begin{proof}
First recall from Lemma~\ref{lem:ExpRec} that $\mathcal{E}_t(Y)\mathcal{E}_t(X) = 1$ and define
\begin{align*}
A_t := Z_0 + \alpha \int_0^t e^{-\lambda s} \mathcal{E}_{s-}(Y) \d s,\\
B_t := \mathcal{E}_t(X)e^{\lambda t},\qquad t\geq 0.
\end{align*}
Let $Z$ be given by~\eqref{eq:Zexplicit}, then $Z_t = A_t B_t$ and by Ito product rule
\begin{align*}
\d Z_t &= A_{t-}\d B_t + B_{t-} \d A_t + \d\, [A, B]_t \\
&= \lambda A_{t-}B_{t-} \d t + A_{t-}B_{t-} \d X_t + \alpha e^{-\lambda t}\mathcal{E}_{t-}(Y)B_{t-} \d t\\
&= \left(\lambda Z_t+\alpha\right) \d t + Z_{t-} \d X_t. \qedhere
\end{align*}
\end{proof}
To analyze the assumption that $f$ is an eigenfunction
of $A$ in more detail we now focus on the case $X = \sigma W$ for a real Brownian
motion $W$ and a constant $\sigma>0$. Then, we will consider {\it regular} two-dimensional realizations of the form $u_t = \Phi(t,Y_t)$, where
\begin{enumerate}[label=(\alph*)]
\item $Y$ is a diffusion process with state space $J \subseteq \R$, satisfying
\[\d Y_t = b(Y_t) \d t + a(Y_t) \d W_t,\]
for Borel measurable functions $b$, $a\colon J\to \R$, where $J$ has non-empty interior, $a(y) > 0$ for all $y \in J$ and $1/a$ is locally integrable on $J$.
\item $\Phi \colon [0,\infty) \times J \to \dom(A)$ such that for
all $\varphi \in \dom(A^*)$, the maps defined by
$\Phi^\varphi(t,y) := \langle \Phi(t,y), \varphi\rangle$, $t\geq 0$, $y\in J$, are in $C^{1,2}(\R_{\geq 0}\times J; \R)$.
\end{enumerate}
Examples of such regular two-dimensional realizations are given by
Theorem~\ref{thm:inhomo}.(i).
\begin{thm}\label{thm:inhomo_fdr}
Let $X_t = \sigma W_t$, $t\geq 0$, for $\sigma>0$ and a real Brownian
motion $W$, and assume that \eqref{eq:gSPDEinh} admits a regular finite-dimensional realization $u_t= \Phi(t,Y_t)$, $t\geq 0$. Then $f$ is an eigenfunction of $A$
for some eigenvalue $\lambda \in\mathbb{R}$, and there exists
an invertible transformation $h\colon J\to \R_+$ such that for $t\geq
0$, almost surely
$$ Z_t = h(Y_t), \qquad u_t = \Phi(t,h^{-1}(Z_t)) = f Z_t, $$
where $Z$ is given by \eqref{eq:Z}.
\end{thm}
\begin{proof}
Let $\varphi \in \dom(A^*)$, and
\begin{equation}
\label{eq:18}
\Phi^\varphi(t,Y_t) := \iprod{\Phi(t,Y_t)}{\varphi}{}.
\end{equation}
Then, Ito formula yields
\begin{multline}\label{eq:dux}
\d \iprod{u_t}{\varphi}{} = d\Phi^\varphi(t,Y_t)=\\
= \left(\partial_t \Phi^\varphi(t,Y_t) + \partial_y
b(Y_t)\Phi^\varphi(t,Y_t) + \frac{1}{2}a^2(Y_t) \partial_{yy}
\Phi^\varphi(t,Y_t) \right) dt + \\
+ a(Y_t)\partial_y \Phi^\varphi(t,Y_t) dW_t.
\end{multline}
Comparing the martingale term with \eqref{eq:gSPDEinh}, we see that $\Phi^\varphi$ satisfies the ODE
\[\partial_y \Phi^\varphi(t,Y_t) = \frac{\sigma \Phi^\varphi(t,Y_t)}{a(Y_t)},\]
$dt \otimes d\PP$-a.\,e. and hence, $\Phi^\varphi$ must be of the form
\[\Phi^\varphi(t,y) = g^\varphi(t) h(y) = g^\varphi(t) \exp\left(\int_{y_0}^y \frac{\sigma d\eta}{a(\eta)}\right),\qquad t\geq 0,\, y\in J,\]
for some $g^\varphi \in C^{1}(\R_{\geq 0})$ and $y_0$ in the
interior of $J$. The regularity property of the representation
guarantees that $h$ is well-defined and strictly monotone
increasing. We stress that $h$ is in fact independent of $\varphi
\in \dom(A^*)$.
Setting $Z_t = h(Y_t)$, we see that $Z$ satisfies
\[dZ_t = m(Z_t) dt + \sigma Z_t dW_t\]
for the drift function $m = (b h') \circ h^{-1} + \frac{1}{2}(a^2
h'') \circ h^{-1}$.
Note that for each $t\geq 0$, the mapping
$\varphi \mapsto g^\varphi(t)$ is linear continuous from
$\dom(A^*)\subset H$ into $\R$. Since $\dom(A^*)\subset H$ is dense,
by Riesz representation theorem for each $t\geq 0$ there exists $g(t)\in H$ such that
\begin{equation}
\label{eq:17}
\iprod{g(t)}{\varphi}{} = g^\varphi(t).
\end{equation}
Since $\Phi^\varphi(t,y) = g^\varphi(t) h(y)$, $g^\varphi$ is differentiable and \eqref{eq:dux} becomes, for $\varphi \in \dom(A^{*})$,
\[d\iprod{u_t}{\varphi}{} = \left( Z_t \partial_t g^\varphi(t) + g^\varphi(t) m(Z_t) \right) dt + g^\varphi(t) Z_t dW_t.\]
Comparing the drift terms with \eqref{eq:gSPDEinh} yields for $t \geq 0$, $\varphi \in \dom(A^*)$ and $z \in h(J)$,
\begin{equation}\label{eq:redux0}
z\left( \iprod{g(t)}{A^*\varphi}{} - \partial_t g^\varphi(t)\right) + \alpha\iprod{f}{\varphi}{} = m(z) g^\varphi(t).
\end{equation}
Evaluating at two different points $z_0, z_1 \in h(J)$ and subtracting we obtain that
\[( \iprod{g(t)}{A^*\varphi}{} - \partial_t \iprod{g(t)}{\varphi}{}) \cdot (z_1 - z_0) = \iprod{g(t)}{\varphi}{} \cdot (m(z_1) - m(z_0)),\]
for all $t \in \R_{\geq 0}$, $\varphi \in \dom(A^*)$ and $z_0,z_1 \in h(J)$.
We conclude that there exists a constant $\lambda \in \R$ such that
\begin{align}
\iprod{g(t)}{A^* \varphi}{} - \partial_t \iprod{g(t)}{\varphi}{} &= \lambda\iprod{g(t)}{\varphi}{}\label{eq:Alambda}\\
\intertext{and}
m(z_1) - m(z_0) &= \lambda (z_1 - z_0).\label{eq:az}
\end{align}
Thus $m$ must be of the form $m(z) = \lambda z + c$ for $c:= m(0)$. Inserting into \eqref{eq:redux0} we obtain that
\[\alpha \iprod{f}{\varphi}{} = c \iprod{g(t)}{\varphi}{},\qquad
\forall \varphi \in \dom(A^*).\]
Since $\dom(A^*)\subset H$ is dense the equation holds for all $\varphi \in H$. Due to the
assumption that $\alpha \neq 0$ and $f$ is non-zero, also $c \neq
0$ and we get $g(t) = \frac{\alpha}{c} f$. In particular, $g(t)$ is
independent of $t$ and \eqref{eq:Alambda} yields
\begin{equation*}
\langle f, A^*\varphi \rangle = \left\langle \tfrac{\lambda c}{\alpha}
f,\varphi\right\rangle \qquad \forall \varphi \in \dom(A^*).
\end{equation*}
This means that $f\in \dom(A^{**})$. Since $A=A^{**}$, see e.\,g. \cite[Theorem
VII.2.3]{yosida1995functional}, we have $f\in \dom(A)$ and
\begin{equation*}
\langle f, A^* \varphi \rangle = \langle A f,\varphi \rangle = \lambda \langle
f,\varphi\rangle, \qquad \forall \varphi \in \dom(A^*).
\end{equation*}
By density of $\dom(A^*)$ in $H$ this yields that $Af = \lambda f$, i.e. $f$ must be an eigenfunction of $A$ with eigenvalue $\lambda$.
Putting everything together, we have shown that $u_t = \frac{\alpha}{c} f Z_t$ where
\[dZ_t = \left(\lambda Z_t + c \right) dt + \sigma Z_t dW_t.\]
Rescaling $Z$ by $\frac{\alpha}{c}$ concludes the proof.
\end{proof}
\subsection{Linear SDEs \& Pearson diffusions}\label{sec:pearson}
Let again $X_t = \sigma W_t$ for some $\sigma>0$ and a real Brownian
motion $W$. The factor processes $Z$ appearing above are then special cases of the linear SDE
\begin{equation}\label{eq:linearSDE}
dZ_t = (aZ_t + c) dt + (bZ_t + d) dW_t, \quad t\geq 0, \quad Z_0 = z_0,
\end{equation}
studied e.g. in \cite[Ch.~4]{Kloeden1992} or
\cite[Prop. 21.2]{kallenberg}. Well-known special cases are the
geometric Brownian motion ($c=d=0$) and the Ornstein-Uhlenbeck-process
($b=0$). Relevant in our context is the less common case $d=0$, on
which we focus now. Also in this case, recall from Proposition~\ref{prop:Zexp}, the SDE is explicitly solvable, with solution given by
\begin{equation}
Z_t = X_t \left(Z_0 + c\int_0^t X_s^{-1} ds\right),\quad t\geq 0,
\end{equation}
where
\begin{equation}
X_t = \exp\left((a -
\frac{b^2}{2})t + bW_t\right), \quad t\geq 0. \label{eq:gBM_toinv}
\end{equation}
Solutions of \eqref{eq:linearSDE} have also been studied in the
context of reciprocal gamma diffusions (see e.g. the `Case~4' in
\cite{forman2008pearson}) or also Pearson diffusions. These are generalizations of
\eqref{eq:linearSDE} that allow for a square-root term in the
diffusion coefficient.
\begin{prop}\label{prop:pearson_ergodicity}
Assume that $z_0>0$, $a<0$ and $c>0$. Then, $Z$ has unique invariant distribution
$\varpi$, which is an Inverse Gamma distribution with shape parameter $1-\frac{2a}{b^2}$ and scale
parameter $\frac{b^2}{2c}$ and, for any bounded measurable function
$\phi \colon (0,\infty)\to \R$,
\[\lim_{t \to \infty} \E{\phi(Z_t)} = \lim_{t\to\infty} \frac1{t} \int_0^t \phi(Z_s)\d s= \int_0^\infty \phi(x)\varpi(\d
x).\]
\end{prop}
\begin{proof}
First, note that
\[ s'(x) := x^{2\frac{c}{b^2}} e^{2 \frac{c}{b^2 x}},\qquad m(\d x):=
x^{-2(1+ \frac{c}{b^2})} e^{-2\frac{c}{b^2 x}} \d x,\quad x\in
(0,\infty)\]
define a scale density and speed measure for $Z$. Then, one can easily
verify that $Z$ is strictly positive and recurrent on $(0,\infty)$, see
e.\,g. \cite[Prop. 5.5.22]{karatzas2012brownian}. Moreover, $m((0,\infty))
<\infty$ and so the unique invariant distribution of $Z$ is
\begin{equation}
\label{eq:7}
\varpi(A) := \frac{m(A)}{m((0,\infty))}.
\end{equation}
The remaining results then follow from e.\,g. \cite[II.35]{borodin2012handbook} or
\cite[X.3.12]{ry}.
\end{proof}
$\mu(t) := \EE Z_t$, $t\geq 0$, satsifies the ODE
\begin{equation*}
\ddt \mu(t) = a \mu(t) + c,\; t>0,\;\qquad \mu(0) = Z_0.
\end{equation*}
Thus,
\begin{equation}
\label{eq:meanPearson}
\mu(t) = \left(Z_0 + \frac{c}{a} \right)e^{a t} -\frac{c}{a}.
\end{equation}
\begin{rmk}\label{rmk:autocorr}
Let $a<0$, $c>0$ and $(Z_t)$ be the stationary solution of
\[ \d Z_t = \left( aZ_t + c\right) \d t + b Z_t\d W_t,\]
that is, $Z_0$ is chosen distributed according to inverse gamma distribution with shape parameter $1-\frac{2a}{b^2}$ and scale parameter $\frac{b^2}{2c}$. Then, as shown in \cite{bibby2005diffusion}, the autocorrelation function of $(Z_t)$ is given by
\begin{equation}
\label{eq:autocorr}
r(t) := \operatorname{Corr}(Z_{s+t}, Z_s) = e^{at},\qquad s,\,t\geq 0.
\end{equation}
\end{rmk}
To study price dynamics it is also useful to examine the reciprocal process $Y=1/Z$:
\begin{prop}\label{prop:logistic}
Let $Z$ be the stochastic process given by~\eqref{eq:linearSDE} with $d=0$.
Then, $Y_t := Z_t^{-1}$ is the unique solution of the stochastic differential equation
\begin{equation}
\label{eq:logistic}
\d Y_t =- Y_t (a -b^2 + c Y_t) \d t - b Y_t \d W_t,\qquad Y_0 = z_0^{-1}.
\end{equation}
In particular, with $X$ given in~\eqref{eq:gBM_toinv},
\begin{equation}
\label{eq:6}
Y_t = \mathcal{E}_t(-b W_. -a (.)) \left(Z_0 + c \int_0^tX_s^{-1}\d s\right)^{-1},\quad t\geq 0.
\end{equation}
\end{prop}
\begin{proof}
In fact, It\={o}'s formula yields
\begin{multline}
\d Z_t^{-1} = -Z^{-2}_t \left( c+ a Z_t\right) \d t - Z_t^{-2} b Z_t^{1} \d W_t + Z_t^{-3} b^2 Z_t^2 \d t \\
= -Z_t^{-1} \left( a- b^2 + c Z_t^{-1} \right) \d t - b Z_t^{-1} \d W_t.\qedhere
\end{multline}
\end{proof}
When $a<b^2$, ~\eqref{eq:logistic} is called the stochastic logistic differential equation.
\subsection{Positivity, stationarity and martingale property}
Let us first come back to the linear homogeneous situation. On
average, market makers do not accumulate inventory, which suggests to consider the baseline case in which the order arrival and cancellation process $X$ is a (local)
martingale.
\begin{cor}\label{cor:locmartingale}
Let $M$ be a local martingale with $\Delta M>-1$ a.\,s., and let $u_t$ be a weak solution of the homogeneous equation \eqref{eq:gSPDE} with $X = M$.
\begin{enumerate}[label=(\roman*)]
\item The solution $u_t$ of \eqref{eq:gSPDE} is a local martingale,
if and only if the initial condition $h_0$ is $A-$harmonic: $h_0\in \dom(A)$ and $Ah_0 = 0$.
\item If $\mathcal{E}(M)$ is a martingale and $Ah_0 = 0$, then $(u_t)_{t\geq 0}$ is a martingale.
\end{enumerate}
\end{cor}
\begin{proof}
If $h_0 \in \dom(A)$ and $Ah_0 = 0$, then clearly $u_t = h_0 \mathcal{E}_t(M)$ is a local martingale. Conversely, assume that $u_t$ is a local martingale. Thus, for all $\varphi\in \dom(A^*)$, $\left( \langle z_t,A^*\varphi\rangle\right)_{t\geq 0}$ is a local martingale. Write $u_t = g_t \mathcal{E}_t(M)$, where $g$ solves \eqref{eq:gPDE}, then
\[ \d \langle u_t, \varphi \rangle = \langle g_t ,A^*\varphi\rangle \mathcal{E}_t(M) \d t + \langle u_{t-} , \varphi \rangle \d M_t.\]
Since $\mathcal{E}_t(M) >0$ almost surely, this means that $\langle g_t, A^*\varphi \rangle = 0$ for all $t\geq 0$. Hence, the mapping
\[ \dom(A^*) \ni \varphi \mapsto \langle g_t, A^*\varphi \rangle = 0,\]
is continuous, and thus $g_t \in \dom(A^{**}) = \dom(A)$. Since $\dom(A^*)\subset H$ is dense, we get that $Ag_t = 0$ and $g_t = h_0$ for all $t\geq 0$. The second part is immediate.
\end{proof}
For the long time behavior we again switch to the Brownian motion case. From the discussion in the previous subsection we directly obtain:
\begin{cor}\label{eq:stationarity_inh}
Let $X= \sigma W$ for a Brownian motion $W$ and $\sigma >0$, and let $u$ be the solution of the inhomogeneous equation \eqref{eq:gSPDEinh}, where $f$ is an eigenfunction of $-A$ with eigenvalue $\nu$ and $h_0 = z_0f$, for some $z_0>0$. If $\nu > 0$ and $\alpha > 0$, then
$$u_t \mathop{\Rightarrow}^{t\to \infty}\quad f Z_\infty$$ where $Z_\infty$ has an Inverse Gamma distribution with shape parameter $1 + 2 \frac{\nu}{\sigma^2}$ and scale parameter $\frac{\sigma^2}{2\alpha}$.
\end{cor}
\begin{rmk}\label{rmk:invgamma}
The Inverse Gamma distribution has a regularly varying right tail with tail index $1 + 2 \nu$ in this case: the $k$-th moment of $\mathbb{E}(Z_\infty^k)<\infty$ if and only if $k < 1 + 2\nu$.
\end{rmk}
So far, we have set aside the requirement that the positivity constraint for $u$. By Theorem~\ref{thm:homo} this reduces to analysis of the deterministic equation. We now return to the case of second-order elliptic operators:
\begin{ass}\label{ass:elliptic}
Let $I\subset \R$ be and interval and suppose that $A$ is a uniformly elliptic operator of the form
\[Au(x) = \eta(x) \Delta u(x) + \beta(x) \nabla u(x) + \alpha(x) u(x), \qquad x\in I,\]
with Dirichlet boundary conditions, and where $\eta, \beta$ and $\alpha$ are smooth and bounded coefficients, and in particular $\eta(x) \geq \underline{\eta} >0$ for all $x\in I$.
\end{ass}
For this class of operators, the strong parabolic maximum principle (cf. \cite[Sec.~7.1]{Evans2010}) implies that the solution $g_t(x)$ of \eqref{eq:gPDE} remains positive, whenever the initial condition $h_0$ is positive. In addition, it is known that there exists a real simple eigenvalue $\lambda_1$, the principal eigenvalue of $A$, such that all other eigenvalues satisfy $\Re(\lambda) \le \lambda_1$ and such that the eigenfunction $f$ associated to $\lambda_1$ is positive on $I$ (cf. \cite[Sec.~6.5]{Evans2010}). Note that the factor process $Z_t$ has state space $(0,\infty)$ both in Theorem~\ref{thm:homo} and \ref{thm:homo}. Thus, we obtains the following corollary.
\begin{cor}[Positivity]
Under Assumption~\ref{ass:elliptic} the following holds:
\begin{enumerate}[label=(\roman*)]
\item If $h_0$ is positive on $I$, then also the solution $g_t$ of \eqref{eq:gPDE} and the solution $u_t$ of \eqref{eq:gSPDE} are a.s. positive on $I$.
\item If $f$ is the principal eigenfunction of $A$, then the finite-dimensional realization $u_t = f Z_t$ of \eqref{eq:gSPDEinh} is a.s. positive on $I$.
\end{enumerate}
\end{cor}
This simple result thus guarantees the existence of a solution with the correct sign, thereby avoiding recourse to `reflected' solutions as in \cite{hambly2018} and considerably simplifying the analysis of our model.
| 12,423
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Google Fiber app comes to iPhone, adds ability to schedule and watch DVR recordings.
| 354,494
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Do you have a septic line in your crawl space? Install a trap in it from above, fill the trap with water, then you can install your drain line.
Can anyone with a Mitsubishi confirm the actual max wattage, in heating and cooling mode? Looking to order one soon and I'm starting to figure out logistics - like if I'll put the one in my most-used room on my UPS panel so I can run it from batteries or Prius (or my larger genset when it is running).
I would like to know also! FYI the Sanyo 9,000 Btu can easily use 1750 watts in max heat mode. I really do not know anyone who can run that kind of power off batteries, very long! I have 2 freinds who have 96KWH banks and even they run heat pumps only when the sun is out offgrid.
"we go where the power lines don't"
Do you have a model (or BTUh) level in mind? My Sanyo 24,000 BTUh units can use 280w to 2.5kw
depending on the weather.
Check the Power Input spec. (abt 8 lines down).
Because these are inverter units and vary the amount of power being used, every few minutes,
There won't be any fixed number you can use.
This winter, when it was pretty cold, my Sanyos were each using 400 to 600w each.. Abt 24 kWh per day.
That certainly was a long read
I have been interested in a 120 Volt Sanyo Split System for years now. I won't have much use for the heat side of it, but it sure does look COOL to me.
This thread has been very enlightening !!
If and when I do it, it will only be for a master bedroom, so it can be used via a GenSet during an extended outage.
Nice thread everyone !!
Where Ever You Go, There You Are
Well, I took the plunge! My new Mitsubishi 12000 BTU, 23 SEER, mini-split is on the way. The location is all selected and I've ordered a 25' line-set/electrical wire/drain line that should be just the right size for the locations we've planned. My husband called an HVAC guy to do the final inspection, testing, and hook-up for the freon tubing.
FYI for those asking about wattage: The web site where I bought my unit specifies that the heating wattage is 950 and cooling wattage is 930 for the unit I purchased. If you go up to a 17 SEER unit with 24000 BTUs the wattage goes up to 2330 for heating and 2270 for cooling, very similar to XRinger's Sanyo unit.
I'm excited and a little nervous about this installation due to the interesting location we have chosen. Should be an interesting project for sure.
Kelly
Keep in mind that the power specs are average with cycling between different amounts during an hour based on what you program! The average is very useful for grid tie but the peak power is where offgrid folks get mislead. Even running the heat pump in float with full sun there still will be conversion loss from the inverter to add to the load.
"we go where the power lines don't"
I'm looking at the 9 and 12 k BTU Mitsu units. I checked the specs online for both of them before posting my question - I just want to get a real-world peak measurement.
Don't be nervous Kelly. They are a dream come true.
Bookmarks
| 279,854
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TITLE: Infiniteness of set of primes such $f$ have root $\mod p$
QUESTION [5 upvotes]: Let $f \in \mathbb{Z}[x]$ be non constant. How to prove that exists infinitely many primes such $f$ have root in $\mathbb{Z/_{(p)}}$. I spent much time, but with no benefits.
REPLY [7 votes]: By contradiction using the same argument as Euclid' proof of infinitude of primes
First you can assume that $f$ has no root in $\Bbb Z$ otherwise it's obvious that $f$ has a root in every $\Bbb Z_p$
Take $g(x)=\frac{f(xf(0))}{f(0)}$ (this is a well defined polynomial over $\Bbb Z$ ), and we have also $g(0)=1$
Now assume that $g$ has a root modulo only finitely primes : $p_1,\cdots,p_k$, this means in other words that the only primes which are allowed to divide $g(n)$ for any integer $n$ are only $p_1,\cdots,p_k$
Consider $m=\prod_{i=1}^kp_i$, Let $t\neq 0 $ be an integer we have $g(tm)\equiv g(0)\equiv 1\mod m$ so no prime from $p_1,\cdots,p_k$ divides $g(tm)$ but the only primes which can divide $g(x)$ are $p_1,\cdots,p_k$ hence $g(tm)=1$ for all values of $t\neq 0$ and finally $g$ is a constant because it's a polynomial and here we have a contradiction as the polynomial $g$ is assumed to be non-constant.
| 131,798
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Utilizing Chair Massage to Address One Woman’s Health in Ghana, West Africa: A Case Report
AWARD YEAR: 2015
Gold Award Winner
Cathy Meryanos
Conway, South Carolina
Thank you to Massage Warehouse for for their generous gift to our gold case report winner!
Background and Objectives: There is limited access to healthcare in rural Ghana and virtually no rehabilitative services available. This case study presents a unique opportunity to utilize chair massage in addressing women’s health in rural Africa, particularly when it comes to muscle pain and fatigue from heavy labor. The objective of this study is to determine the results of chair massage as a strategy to reduce neck, shoulder, and back pain and increase range of motion.
Case Presentation: The patient is a 63-year-old Ghanaian female, who was struck by a public transport van while carrying a 30-50 pound load on her head. The accident resulted in a broken right humerus and soft tissue pain. A traditional medicine practitioner set the bone. There was no post-accident rehabilitation available. At the time of referral, she presented complaints of shoulder, elbow, and wrist pain. In addition, she was unable to raise her right hand to her mouth for food intake.
Results: The results of this case study include an increase in range of motion, as well as diminished pain in the right shoulder, elbow, and hand. There was also a decrease in muscle hypertonicity in the thoracic and cervical areas, and a profound increase in quality of life for the patient.
Discussion: This case study illustrates how therapeutic chair massage was utilized to address a common health concern for one woman in rural Africa. It also demonstrates pre-exiting myoskeletal issues and pain may be eliminated with massage intervention. Massage therapy may be important to ameliorating certain types of health problems in remote rural villages in low income countries.
| 55,354
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TITLE: This is my bucket experiment. I use this experiment to explain lift. Am I right?
QUESTION [1 upvotes]: Hang the bucket on a long rope. Rotate the bucket to tighten the rope. Then fill the bucket with water and close the lid (shown in the blue line). At this time, the bucket and water are in a static state, and the isobaric surface in the water is plane.
Release the bucket and the torsion of the rope makes the bucket rotate.
After rotating for a period of time, water also rotates because of its viscosity. At this time, the isobaric surface in water is no longer a plane, but a parabolic surface: the pressure at the wall of the bucket is high, and the pressure at the center of the bucket is low.
Why?
Because of the rotation, the water in the center of the bucket expands, so the pressure decreases, while the water in the wall of the bucket is compressed, so the pressure increases.
The lift of the wing is the same. The airflow at the top of the wing tends to be away from the wing along the normal direction, so the airflow is expanded and the pressure is reduced. The airflow at the bottom of the wing tends to approach the wing along the normal direction, so the airflow is compressed and the pressure increases. One high, one low, there is pressure difference, so it produces lift.
I use air instead of water, then fill the air into a bucket deep enough and large enough, and then rotate the bucket, the air in the bucket will form a parabolic surface. What does this mean?
REPLY [0 votes]: The atmosphere is open, and it has no walls of buckets. The centripetal force at the top of the wing is generated by the tendency of the airflow away from the aircraft. The centripetal force in the bucket is generated by the tendency of the flow of water away from the center. The tendency to stay away causes depression, so there is centripetal force. Why is there a tendency to stay away? In buckets, because the center is not in line with the direction of the flow. On the wing, because of the angle of attack, the center and the direction of the air flow are not collinear.
When air rotates with the bucket, isobaric surfaces are formed in the bucket. These isobaric surfaces are parabolic. The water in the bucket also forms an isobaric surface.The faster the air rotates, the lower the pressure in the center of the water.So low pressure can be produced without viscosity, and low pressure can be produced with rotation.So your theory that you have to rely on viscosity to produce low pressure is wrong.
| 8,003
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The Aspire Breeze 2 is an all-in-one device perfect for those looking for a small, but super effective vape to help transition away from smoking.
The Breeze 2 comes in a great range of colours and features a compact and ergonomic design, so that no matter how you hold or grip it, it feels good in the hand. The breeze has a built-in refillable tank that holds 3ml of e-juice. The U-tech designed coil allows for a good flavourful vape.
The Aspire Breeze 2 is a perfect mini starter kit standing at just 96mm tall but with a built-in 1000mAh battery.
The Breeze 2 tank is built into the side of the device, and access to it is easy. Simply click the release button to remove the top section, and the new and improved filling method will have you vaping again quickly. The mouthpiece comes with a removable cap meaning you can protect the tank from dust and other elements while it's in your pocket.
The Breeze 2 charges via the included Micro USB charging cable from any USB power output.
| 311,990
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CHAPEL HILL — North Carolina linebacker Travis Hughes has been suspended from all team activities after campus police charged him with assault on Thursday, UNC football team spokesman Kevin Best said.
Police charged Hughes Thursday morning after an altercation with a parking attendant on campus, Best said. Details of the altercation werent immediately available, but Best said UNC coach Larry Fedora suspended Hughes immediately when he learned that Hughes had been charged.
Randy Young, a spokesman for the UNC Department of Public Safety, didnt immediately return a call seeking more details about the alleged assault.
Hughes started 11 games for UNC last season, and played in 12. He was fifth on the team in tackles, with 76, and he would be the Tar Heels second-leading returning tackler.
Hughes also had 5.5 tackles for loss and one sack last season. He missed the Tar Heels victory against Cincinnati in the Belk Bowl because of an injury.
This isnt the first time that Hughes has been in trouble. He was charged in Durham County last January for possession of drug paraphernalia.
Fedora didnt find out about that until August, not long after preseason practice had started, and he said at the time that Hughes faced discipline. He played in the season-opening loss at South Carolina.
Carter: 919-829-8944; Twitter: @_andrewcarter
| 185,321
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Get Satisfaction provides businesses a third-party web-based customer support service. The service got an HTML5 mobile makeover this week. Businesses with paid plans now have a mobile Get Satisfaction presence for customers who use an iPhone or iPod touch.
Introducing Get Satisfaction Mobile
Visiting.
| 191,825
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Tonight I took Ava to her 1st Parent & Tot swimming lesson at The Grove in Inver Grove Heights. I thought for sure she would love the water, but I don't think either of us realized how chilly the water was (84 degrees? I don't think so!) Brrrrrr.....!
Ava immediately started doing a cry/whine and tried to crawl up my chest. For the remaining 40 minutes, I barely got her to release her hold on me to play in the water. There were babies a few months younger than her that were having a ball being paddled around. I'm not going to fret too much about it, since this was her 1st time in the water, and she didn't know what to expect. We even went into the kiddie fun pool afterwards where it was much warmer, and she still wasn't sure about it.
I'm not a water fan myself, and I was hoping that my children wouldn't be like me in that aspect - I really want them to be comfortable in the water. Keep your fingers crossed for next Monday!
One thing to add - the hardest part of the whole night was trying to change the wet Mommy and the wet baby into dry clothes with no Daddy around to assist! My hats off to single parents everywhere - wow!
3 comments:
That does seem pretty chilly! I wouldn't like it either. Hopefully she'll get a little more comfortable next week. That sounds fun!
Violet started swim class last week - she, on the other hand, loves it - but our water is freezing too! I was sure she'd hate it, but she doesn't! Our biggest challenge is getting mommy and baby changed too!
I'm sure she'll start to like it more as she becomes more familar and comfortable with it. Ryan and Owen just finished a month of swimming lessons and Ryan has turned into a little fishie now!! I'll be posting more about it on our blog in the next day or so.
| 212,064
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Active8 help young people and adults with physical disabilities or inpairments. They are looking for students to help on the second weekend of each month with young people aged 14-16. This will include helping them move around, cooking and generally socialising with them and getting them involved with outdoor activities.
Active8 are also after volunteers to assist 16-30 year olds with disabilities. The purpose is to generally boost independence and will require helping once a month by assisting people to go to pub quizzes, gigs, shopping trips or sport.
Other voluntary positions are also available:
For more information, please see the website or email communityaction@fxu.org.uk.
| 169,450
|
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Improve quality control and ensure safe operation in the coal mine
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:: $T_0$ Topological Spaces
:: by Mariusz \.Zynel and Adam Guzowski
environ
vocabularies XBOOLE_0, FUNCT_1, SUBSET_1, RELAT_1, TARSKI, PRE_TOPC, TOPS_2,
RCOMP_1, EQREL_1, STRUCT_0, RELAT_2, BORSUK_1, ORDINAL2, CARD_3,
CLASSES1, T_0TOPSP, FUNCT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, CLASSES1, RELAT_2, RELAT_1, FUNCT_1,
RELSET_1, PARTFUN1, FUNCT_2, STRUCT_0, PRE_TOPC, TOPS_2, BORSUK_1,
EQREL_1;
constructors SETFAM_1, RFINSEQ, TOPS_2, BORSUK_1, CLASSES1;
registrations XBOOLE_0, FUNCT_1, PARTFUN1, FUNCT_2, STRUCT_0, PRE_TOPC,
BORSUK_1, EQREL_1, RELSET_1;
requirements SUBSET, BOOLE;
definitions TOPS_2, RELAT_2, RELSET_1;
equalities STRUCT_0;
expansions TOPS_2, PRE_TOPC;
theorems FUNCT_1, FUNCT_2, EQREL_1, RELAT_1, TOPS_2, BORSUK_1, TARSKI,
RELSET_1, XBOOLE_0, RELAT_2, PARTFUN1, ORDERS_1, CLASSES1;
schemes FUNCT_2, RELSET_1;
begin
::
:: Preliminaries
::
theorem Th1:
for X,Y being non empty set, f being Function of X,Y holds for A
being Subset of X st for x1,x2 being Element of X holds x1 in A & f.x1=f.x2
implies x2 in A holds f"(f.:A) = A
proof
let X,Y be non empty set;
let f be Function of X,Y;
let A be Subset of X;
assume
A1: for x1,x2 being Element of X holds x1 in A & f.x1=f.x2 implies x2 in A;
for x being object st x in f"(f.:A) holds x in A
proof
let x be object;
assume
A2: x in f"(f.:A);
then f.x in f.:A by FUNCT_1:def 7;
then ex x0 being object st x0 in X & x0 in A & f.x = f.x0 by FUNCT_2:64;
hence thesis by A1,A2;
end;
then A c= f"(f.:A) & f"(f.:A) c= A by FUNCT_2:42,TARSKI:def 3;
hence thesis by XBOOLE_0:def 10;
end;
:: Homeomorphic TopSpaces
definition
let T,S be TopStruct;
pred T,S are_homeomorphic means
ex f being Function of T,S st f is being_homeomorphism;
end;
:: Open Function
definition
let T,S be TopStruct;
let f be Function of T,S;
attr f is open means
for A being Subset of T st A is open holds f.:A is open;
correctness;
end;
::
:: Indiscernibility Relation
::
definition
let T be non empty TopStruct;
func Indiscernibility(T) -> Equivalence_Relation of the carrier of T means
:
Def3: for
p,q being Point of T holds [p,q] in it iff for A being Subset of T st
A is open holds p in A iff q in A;
existence
proof
defpred X[set,set] means for A being Subset of T st A is open holds $1 in
A iff $2 in A;
consider R being Relation of the carrier of T,the carrier of T such that
A1: for p,q being Element of T holds [p,q] in R iff X[p,q] from
RELSET_1:sch 2;
A2: R is_transitive_in the carrier of T
proof
let x,y,z be object;
assume that
A3: x in the carrier of T & y in the carrier of T & z in the carrier of T and
A4: [x,y] in R and
A5: [y,z] in R;
reconsider x9 = x, y9 = y, z9 = z as Element of T by A3;
for A being Subset of T st A is open holds x9 in A iff z9 in A
proof
let A be Subset of T;
assume
A6: A is open;
then x9 in A iff y9 in A by A1,A4;
hence thesis by A1,A5,A6;
end;
hence thesis by A1;
end;
R is_reflexive_in the carrier of T
proof
let x be object;
A7: for A being Subset of T st A is open holds x in A iff x in A;
assume x in the carrier of T;
hence thesis by A1,A7;
end;
then
A8: dom R = the carrier of T & field R = the carrier of T by ORDERS_1:13;
R is_symmetric_in the carrier of T
proof
let x,y be object;
assume that
A9: x in the carrier of T & y in the carrier of T and
A10: [x,y] in R;
for A being Subset of T st A is open holds y in A iff x in A by A1,A9,A10
;
hence thesis by A1,A9;
end;
then reconsider R as Equivalence_Relation of the carrier of T by A8,A2,
PARTFUN1:def 2,RELAT_2:def 11,def 16;
take R;
let p,q be Point of T;
thus [p,q] in R implies for A be Subset of T st A is open holds p in A iff
q in A by A1;
assume for A being Subset of T st A is open holds p in A iff q in A;
hence thesis by A1;
end;
uniqueness
proof
let R1,R2 be Equivalence_Relation of the carrier of T;
assume that
A11: for p,q being Point of T holds [p,q] in R1 iff for A being Subset
of T st A is open holds p in A iff q in A and
A12: for p,q being Point of T holds [p,q] in R2 iff for A being Subset
of T st A is open holds p in A iff q in A;
let x,y be Point of T;
[x,y] in R1 iff for A being Subset of T st A is open holds x in A iff
y in A by A11;
hence thesis by A12;
end;
end;
::
:: Indiscernibility Partition
::
definition
let T be non empty TopStruct;
func Indiscernible(T) -> non empty a_partition of the carrier of T equals
Class Indiscernibility(T);
coherence;
end;
::
:: T_0 Reflex of TopSpace
::
definition
let T be non empty TopSpace;
func T_0-reflex(T) -> TopSpace equals
space Indiscernible(T);
correctness;
end;
registration
let T be non empty TopSpace;
cluster T_0-reflex(T) -> non empty;
coherence;
end;
::
:: Function from TopSpace to its T_0 Reflex
::
definition
let T be non empty TopSpace;
func T_0-canonical_map T -> continuous Function of T,T_0-reflex T equals
Proj Indiscernible T;
coherence;
end;
theorem Th2:
for T being non empty TopSpace, V being Subset of T_0-reflex(T)
holds V is open iff union V in the topology of T
proof
let T be non empty TopSpace;
let V be Subset of T_0-reflex(T);
A1: V is Subset of Indiscernible(T) by BORSUK_1:def 7;
thus V is open implies union V in the topology of T
by A1,BORSUK_1:27;
assume union V in the topology of T;
then V in the topology of space Indiscernible(T) by A1,BORSUK_1:27;
hence thesis;
end;
theorem Th3:
for T being non empty TopSpace, C being set holds C is Point of
T_0-reflex(T) iff ex p being Point of T st C = Class(Indiscernibility(T),p)
proof
let T be non empty TopSpace;
set TR = T_0-reflex(T);
set R = Indiscernibility(T);
let C be set;
hereby
assume C is Point of TR;
then C in the carrier of TR;
then C in Indiscernible(T) by BORSUK_1:def 7;
hence ex p being Point of T st C = Class(R,p) by EQREL_1:36;
end;
assume ex p being Point of T st C = Class(R,p);
then C in Class R by EQREL_1:def 3;
hence thesis by BORSUK_1:def 7;
end;
theorem Th4:
for T being non empty TopSpace, p being Point of T holds (
T_0-canonical_map(T)).p = Class(Indiscernibility(T),p)
proof
let T be non empty TopSpace;
let p be Point of T;
set F = T_0-canonical_map(T);
set R = Indiscernibility(T);
F.p in the carrier of T_0-reflex(T);
then F.p in Indiscernible(T) by BORSUK_1:def 7;
then consider y being Element of T such that
A1: F.p = Class(R,y) by EQREL_1:36;
p in Class(R,y) by A1,BORSUK_1:28;
hence thesis by A1,EQREL_1:23;
end;
theorem Th5:
for T being non empty TopSpace, p,q being Point of T holds (
T_0-canonical_map(T)).q = (T_0-canonical_map(T)).p iff [q,p] in
Indiscernibility(T)
proof
let T be non empty TopSpace;
let p,q be Point of T;
set F = T_0-canonical_map(T);
set R = Indiscernibility(T);
hereby
assume F.q = F.p;
then q in F.p by BORSUK_1:28;
then q in Class(R,p) by Th4;
hence [q,p] in R by EQREL_1:19;
end;
assume [q,p] in R;
then Class(R,q) = Class(R,p) by EQREL_1:35;
then F.q = Class(R,p) by Th4;
hence thesis by Th4;
end;
theorem Th6:
for T being non empty TopSpace, A being Subset of T st A is open
holds for p,q being Point of T holds p in A & (T_0-canonical_map(T)).p = (
T_0-canonical_map(T)).q implies q in A
proof
let T be non empty TopSpace;
let A be Subset of T such that
A1: A is open;
set F=T_0-canonical_map(T);
let p,q be Point of T;
assume that
A2: p in A and
A3: F.p = F.q;
A4: F.p = Class(Indiscernibility(T),p) by Th4;
q in F.p by A3,BORSUK_1:28;
then [q,p] in Indiscernibility(T) by A4,EQREL_1:19;
hence thesis by A1,A2,Def3;
end;
theorem Th7:
for T being non empty TopSpace, A being Subset of T st A is open
for C being Subset of T st C in Indiscernible(T) & C meets A holds C c= A
proof
let T be non empty TopSpace;
let A be Subset of T such that
A1: A is open;
set R = Indiscernibility(T);
let C be Subset of T;
assume that
A2: C in Indiscernible(T) and
A3: C meets A;
consider y being object such that
A4: y in C and
A5: y in A by A3,XBOOLE_0:3;
consider x being object such that
x in the carrier of T and
A6: C = Class(R,x) by A2,EQREL_1:def 3;
for p being object st p in C holds p in A
proof
let p be object;
[y,x] in R by A6,A4,EQREL_1:19;
then
A7: [x,y] in R by EQREL_1:6;
assume
A8: p in C;
then [p,x] in R by A6,EQREL_1:19;
then [p,y] in R by A7,EQREL_1:7;
hence thesis by A1,A5,A8,Def3;
end;
hence thesis by TARSKI:def 3;
end;
theorem Th8:
for T being non empty TopSpace holds T_0-canonical_map(T) is open
proof
let T be non empty TopSpace;
set F = T_0-canonical_map(T);
for A being Subset of T st A is open holds F.:A is open
proof
set D = Indiscernible(T);
A1: F = proj D by BORSUK_1:def 8;
let A be Subset of T such that
A2: A is open;
A3: for C being Subset of T st C in D & C meets A holds C c= A by A2,Th7;
set A9 = F.:A;
A9 is Subset of D by BORSUK_1:def 7;
then F"A9 = union A9 by A1,EQREL_1:67;
then A = union A9 by A1,A3,EQREL_1:69;
then union A9 in the topology of T by A2;
hence thesis by Th2;
end;
hence thesis;
end;
Lm1: for T being non empty TopSpace, x,y being Point of T_0-reflex(T) st x <>
y ex V being Subset of T_0-reflex(T) st V is open & ( x in V & not y in V or y
in V & not x in V )
proof
let T be non empty TopSpace;
set S = T_0-reflex(T);
set F = T_0-canonical_map(T);
assume not (for x,y being Point of S st not x = y ex V being Subset of S st
V is open & ( x in V & not y in V or y in V & not x in V ));
then consider x,y being Point of S such that
A1: x <> y and
A2: for V being Subset of S st V is open holds x in V iff y in V;
reconsider x,y as Point of space Indiscernible(T);
consider p being Point of T such that
A3: F.p = x by BORSUK_1:29;
consider q being Point of T such that
A4: F.q = y by BORSUK_1:29;
for A being Subset of T st A is open holds p in A iff q in A
proof
let A be Subset of T such that
A5: A is open;
F is open by Th8;
then
A6: F.:A is open by A5;
reconsider F as Function of the carrier of T, the carrier of S;
hereby
assume p in A;
then x in F.:A by A3,FUNCT_2:35;
then F.q in F.:A by A2,A4,A6;
then
ex x being object st x in the carrier of T & x in A & F.q = F.x by FUNCT_2:64;
hence q in A by A5,Th6;
end;
assume q in A;
then y in F.:A by A4,FUNCT_2:35;
then F.p in F.:A by A2,A3,A6;
then ex x being object st x in the carrier of T & x in A & F.p = F.x by
FUNCT_2:64;
hence thesis by A5,Th6;
end;
then [p,q] in Indiscernibility(T) by Def3;
hence contradiction by A1,A3,A4,Th5;
end;
::
:: Discernible TopStruct
::
definition
let T be TopStruct;
redefine attr T is T_0 means
:Def7:
T is empty or for x,y being Point of T
st x <> y holds ex V being Subset of T st V is open & ( x in V & not y in V or
y in V & not x in V );
compatibility;
end;
registration
cluster T_0 non empty for TopSpace;
existence
proof
set T = the non empty TopSpace;
take T_0-reflex(T);
for x,y being Point of T_0-reflex(T) st x <> y holds ex V being Subset
of T_0-reflex(T) st V is open & ( x in V & not y in V or y in V & not x in V )
by Lm1;
hence thesis;
end;
end;
::
:: T_0 TopSpace
::
definition
mode T_0-TopSpace is T_0 non empty TopSpace;
end;
theorem
for T being non empty TopSpace holds T_0-reflex(T) is T_0-TopSpace
proof
let T be non empty TopSpace;
for x,y being Point of T_0-reflex(T) st not x = y ex A being Subset of
T_0-reflex(T) st A is open & ( x in A & not y in A or y in A & not x in A ) by
Lm1;
hence thesis by Def7;
end;
::
:: Homeomorphism of T_0 Reflexes
::
theorem
for T,S being non empty TopSpace st ex h being Function of T_0-reflex(
S),T_0-reflex(T) st h is being_homeomorphism & T_0-canonical_map(T),h*
T_0-canonical_map(S) are_fiberwise_equipotent holds T,S are_homeomorphic
proof
let T,S be non empty TopSpace;
set F = T_0-canonical_map(T), G = T_0-canonical_map(S);
set TR = T_0-reflex(T), SR = T_0-reflex(S);
given h being Function of SR,TR such that
A1: h is being_homeomorphism and
A2: F,h*G are_fiberwise_equipotent;
consider f being Function such that
A3: dom f = dom F and
A4: rng f = dom (h*G) and
A5: f is one-to-one and
A6: F = h*G*f by A2,CLASSES1:77;
A7: dom f = the carrier of T by A3,FUNCT_2:def 1;
A8: h is continuous by A1;
A9: h is one-to-one by A1;
reconsider f as Function of T,S by A4,A7,FUNCT_2:def 1,RELSET_1:4;
take f;
thus
A10: dom f = [#] T & rng f = [#] S by A4,FUNCT_2:def 1;
A11: rng h = [#] TR by A1;
A12: [#]SR <> {};
A13: for A being Subset of S st A is open holds f"A is open
proof
set g=(h*G);
let A be Subset of S;
set B=g.:A;
A14: h" is continuous by A1;
assume
A15: A is open;
A16: for x1,x2 being Element of S holds x1 in A & g.x1=g.x2 implies x2 in A
proof
let x1,x2 be Element of S;
assume that
A17: x1 in A and
A18: g.x1=g.x2;
h.(G.x1) = g.x2 by A18,FUNCT_2:15;
then h.(G.x1) = h.(G.x2) by FUNCT_2:15;
then G.x1 = G.x2 by A9,FUNCT_2:19;
hence thesis by A15,A17,Th6;
end;
G is open by Th8;
then G.:A is open by A15;
then
A19: (h")"(G.:A) is open by A12,A14,TOPS_2:43;
A20: h is onto by A11,FUNCT_2:def 3;
h.:(G.:A) = (h qua Function")"(G.:A) by A9,FUNCT_1:84;
then h.:(G.:A) is open by A9,A19,A20,TOPS_2:def 4;
then
A21: (h*G).:A is open by RELAT_1:126;
[#]T_0-reflex T <> {};
then
A22: F"B is open by A21,TOPS_2:43;
F"B = f"(g"(g.:A)) by A6,RELAT_1:146;
hence thesis by A22,A16,Th1;
end;
A23: dom h = [#] SR by A1;
A24: for A being Subset of T st A is open holds (f" qua Function of S,T)"A
is open
proof
set g = h"*F;
let A be Subset of T;
set B = g.:A;
assume
A25: A is open;
A26: for x1,x2 being Element of T holds x1 in A & g.x1=g.x2 implies x2 in A
proof
let x1,x2 be Element of T;
assume that
A27: x1 in A and
A28: g.x1=g.x2;
h".(F.x1) = g.x2 by A28,FUNCT_2:15;
then
A29: h".(F.x1) = h".(F.x2) by FUNCT_2:15;
h" is one-to-one by A11,A9,TOPS_2:50;
then F.x1 = F.x2 by A29,FUNCT_2:19;
hence thesis by A25,A27,Th6;
end;
F = h*(G*f) by A6,RELAT_1:36;
then g = (h"*h)*(G*f) by RELAT_1:36;
then g = (id the carrier of SR)*(G*f) by A23,A11,A9,TOPS_2:52;
then g*f" = G*f*f" by FUNCT_2:17;
then g*f" = G*(f*f") by RELAT_1:36;
then g*f" = G*(id the carrier of S) by A5,A10,TOPS_2:52;
then G = g*f" by FUNCT_2:17;
then
A30: G"B = (f")"(g"B) by RELAT_1:146;
F is open by Th8;
then F.:A is open by A25;
then
A31: h"(F.:A) is open by A11,A8,TOPS_2:43;
B = (h").:(F.:A) by RELAT_1:126;
then G"B = G"(h"(F.:A)) by A11,A9,TOPS_2:55;
then G"B is open by A12,A31,TOPS_2:43;
hence thesis by A26,A30,Th1;
end;
thus f is one-to-one by A5;
[#]S <> {};
hence f is continuous by A13,TOPS_2:43;
[#]T <> {};
hence thesis by A24,TOPS_2:43;
end;
::
:: Properties of Continuous Mapping from TopSpace to its T_0 Reflex
::
theorem Th11:
for T being non empty TopSpace, T0 being T_0-TopSpace, f being
continuous Function of T,T0 holds for p,q being Point of T holds [p,q] in
Indiscernibility(T) implies f.p = f.q
proof
let T be non empty TopSpace;
let T0 be T_0-TopSpace;
let f be continuous Function of T,T0;
let p,q be Point of T;
set p9 = f.p, q9 = f.q;
assume that
A1: [p,q] in Indiscernibility(T) and
A2: not f.p = f.q;
consider V being Subset of T0 such that
A3: V is open and
A4: p9 in V & not q9 in V or q9 in V & not p9 in V by A2,Def7;
set A = f"V;
[#]T0 <> {};
then
A5: A is open by A3,TOPS_2:43;
reconsider f as Function of the carrier of T, the carrier of T0;
q in the carrier of T;
then
A6: q in dom f by FUNCT_2:def 1;
p in the carrier of T;
then p in dom f by FUNCT_2:def 1;
then not (p in A iff q in A) by A4,A6,FUNCT_1:def 7;
hence contradiction by A1,A5,Def3;
end;
theorem Th12:
for T being non empty TopSpace, T0 being T_0-TopSpace, f being
continuous Function of T,T0 holds for p being Point of T holds f.:Class(
Indiscernibility(T),p) = {f.p}
proof
let T be non empty TopSpace;
let T0 be T_0-TopSpace;
let f be continuous Function of T,T0;
let p be Point of T;
set R = Indiscernibility(T);
for y being object holds y in f.:Class(R,p) iff y in {f.p}
proof
let y be object;
hereby
assume y in f.:Class(R,p);
then consider x being object such that
A1: x in the carrier of T and
A2: x in Class(R,p) and
A3: y = f.x by FUNCT_2:64;
[x,p] in R by A2,EQREL_1:19;
then f.x = f.p by A1,Th11;
hence y in {f.p} by A3,TARSKI:def 1;
end;
assume y in {f.p};
then
A4: y = f.p by TARSKI:def 1;
p in Class(R,p) by EQREL_1:20;
hence thesis by A4,FUNCT_2:35;
end;
hence thesis by TARSKI:2;
end;
::
:: Factorization
::
theorem
for T being non empty TopSpace, T0 being T_0-TopSpace, f being
continuous Function of T,T0 ex h being continuous Function of T_0-reflex(T),T0
st f = h*T_0-canonical_map(T)
proof
let T be non empty TopSpace;
let T0 be T_0-TopSpace;
let f be continuous Function of T,T0;
set F = T_0-canonical_map(T);
set R = Indiscernibility(T);
set TR = T_0-reflex(T);
defpred X[object,object] means ex D1 being set st D1 = $1 & $2 in f.:D1;
A1: for C being object st C in the carrier of TR
ex y being object st y in the carrier of T0 & X[C,y]
proof
let C be object;
assume C in the carrier of TR;
then consider p being Point of T such that
A2: C = Class(R,p) by Th3;
A3: f.p in {f.p} by TARSKI:def 1;
reconsider C as set by TARSKI:1;
f.:C = {f.p} by A2,Th12;
hence thesis by A3;
end;
ex h being Function of the carrier of TR,the carrier of T0 st for C
being object st C in the carrier of TR holds X[C,h.C]
from FUNCT_2:sch 1(A1);
then consider h being Function of the carrier of TR,the carrier of T0
such that
A4: for C being object st C in the carrier of TR holds X[C,h.C];
A5: for p being Point of T holds h.Class(R,p) = f.p
proof
let p be Point of T;
Class(R,p) is Point of TR by Th3;
then X[Class(R,p),h.Class(R,p)] by A4;
then h.Class(R,p) in f.:Class(R,p);
then h.Class(R,p) in {f.p} by Th12;
hence thesis by TARSKI:def 1;
end;
reconsider h as Function of TR,T0;
A6: [#]T0 <> {};
for W being Subset of T0 st W is open holds h"W is open
proof
let W be Subset of T0;
assume W is open;
then
A7: f"W is open by A6,TOPS_2:43;
set V = h"W;
for x being object holds x in union V iff x in f"W
proof
let x be object;
hereby
assume x in union V;
then consider C being set such that
A8: x in C and
A9: C in V by TARSKI:def 4;
consider p being Point of T such that
A10: C = Class(R,p) by A9,Th3;
x in the carrier of T by A8,A10;
then
A11: x in dom f by FUNCT_2:def 1;
[x,p] in R by A8,A10,EQREL_1:19;
then
A12: C = Class(R,x) by A8,A10,EQREL_1:35;
h.C in W by A9,FUNCT_1:def 7;
then f.x in W by A5,A8,A12;
hence x in f"W by A11,FUNCT_1:def 7;
end;
assume
A13: x in f"W;
then f.x in W by FUNCT_1:def 7;
then
A14: h.Class(R,x) in W by A5,A13;
Class(R,x) is Point of TR by A13,Th3;
then
A15: Class(R,x) in V by A14,FUNCT_2:38;
x in Class(R,x) by A13,EQREL_1:20;
hence thesis by A15,TARSKI:def 4;
end;
then union V = f"W by TARSKI:2;
then union V in the topology of T by A7;
hence thesis by Th2;
end;
then reconsider h as continuous Function of TR,T0 by A6,TOPS_2:43;
set H = h*F;
for x being object st x in the carrier of T holds f.x = H.x
proof
let x be object;
assume
A16: x in the carrier of T;
then Class(R,x) in Class R by EQREL_1:def 3;
then
A17: Class(R,x) in the carrier of TR by BORSUK_1:def 7;
x in dom F & F.x = Class(R,x) by A16,Th4,FUNCT_2:def 1;
then
A18: (h*F).x = h.Class(R,x) by FUNCT_1:13;
X[Class(R,x),h.Class(R,x)] by A4,A17;
then H.x in f.:Class(R,x) by A18;
then H.x in {f.x} by A16,Th12;
hence thesis by TARSKI:def 1;
end;
hence thesis by FUNCT_2:12;
end;
| 29,485
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Author: Chase Shê
8 Comments
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Annie Yi’s Son Who Once Drew Negative Attention for Wearing Drag Explains His Clothing Choices
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| 116,989
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There is one thing that people should never have a conversation about, money. Today at work, it got very heated when the subject of money and what income and bills we all had to pay come up in conversation. So much so, that one of the girls got quite upset. If the same subject comes up next week, I think I will just tell them all to shut up.
Anyway, I have made a valentine's card tonight which is something I only ever make when I have an order for one.
The materials used are:
White card stock
Red card stock
Pink card stock
Digital patterned paper
Lattice embossing folder
Tonic First Crush Affections Insert die set
Red stitched ribbon
Pearl heart button
Pearls
I am entering this card into the following challenges:
Really Reasonable Ribbon Blog - Red And Pink
CAS(E) This Sketch - Sketch
CAS On Sunday - Hearts
Sweet Stampin' Challenge Blog - Spots And Dots
Die Cuttin Divas - Anything Goes
Tuesday Throwdown - Love Notes
Inkspirational - Polka Dots
Crafting With Dragonflies Challenge Blog - Valentine's Day
Crafty Cardmakers - Love, Love, Love
Creatalicious Challenges - Anything Goes
{PIN}spriational Challenges - Photo Inspiration
That's all for now.
Teresa
x
| 399,605
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Service Animals.
Activities of Daily Living (ADL’s)—Activities of daily living include, but are not limited to, the things we do on a regular basis to meet our physical, psychological, financial, and social needs. Examples of activities are:
Standards of Behavior for Service Animals-These insure the animal is under control and calm while working:
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\begin{document}
\bibliographystyle{abbrv}
\maketitle
{\noindent\bf Abstract.} We study open quantum random walks (OQRW) for which the underlying graph is a lattice, and the generators of the walk are translation-invariant. Using the results obtained recently in \cite{CP1}, we study the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process. We study in detail the case of homogeneous OQRWs on a lattice, with internal space $\h=\cc^2$.
\section{Introduction}
Open quantum random walks (OQRW in short) were defined by Attal \textit{et al.} in~\cite{APSS}.
They seem to be a good quantum analogue of Markov chains, and, as such, are a very promising tool to model many physical problems (see \cite{CP1} and the references therein).
In the paper \cite{CP1}, we described the notions of irreducibility and aperiodicity for OQRWs, and derived, in particular, convergence properties for irreducible, or irreducible and aperiodic, OQRWs. In the same way as for classical Markov chains, those convergence results assumed the existence of an invariant state.
In the present paper we focus on translation-invariant OQRWs on a lattice, which attracted special attention in many recent papers (see, for instance, \cite{AGS,BBT,KY,LardSouza,SP2,XY}).
As we have shown in \cite{CP1}, these OQRWs do not have an invariant state so that most of the convergence results from \cite{CP1} are useless. We will show, however, that there exists an auxiliary map which allows to characterize many properties of the homogeneous open quantum random walk. With the help of the Perron-Frobenius theorem described in \cite{CP1}, we can obtain results about the \textit{quantum trajectory} associated with the process. More precisely, if the quantum trajectory is given by the position and state couple $(X_p,\rho_p)$, we obtain a central limit theorem and a large deviation principle for the position process $(X_p)_p$.
The immediate physical application of our results is quantum measurements, or more precisely, repeated indirect measurements. In that framework, a given system $\mathcal S$ interacts sequentially with external systems $\mathcal R_p$, $p=1,2,\ldots$ representing measuring devices, and after each interaction, a measurement is done on $\mathcal R_p$. Then the above sequence $(X_p)_p$ represents the sequence of measurement outcomes, and the sequence $(\rho_p)_p$ represents the state of the physical system $\mathcal S$ after the first $p$ measurements (we refer the reader to section 6 of \cite{AGS})). Our results immediately give a law of large numbers, a central limit theorem and a large deviation principle for the statistics of the measurements $(X_p)_p$.
We will pay specific attention to the application of our results to the case where the internal state space of the particle (what is sometimes called the \emph{coin space}) is two-dimensional. This will allow us to illustrate the full structure of homogeneous OQRWs, and in particular the notions of irreducibility, period, as well as the Baumgartner-Narnhofer (\cite{BN}) decompositions discussed previously in~\cite{CP1}.
Of the above cited articles, some give a central limit theorem for the position~$(X_p)_p$ associated with an OQRW on~$\zz^d$. The most general result so far is given in \cite{AGS}, and its proof is based on a central limit theorem for martingales and the K\"ummerer-Maassen ergodic theorem (see \cite{KuMa}). Our proof is based on a completely different strategy, using a computation of the Laplace transform, and uses an irreducibility assumption which does not appear in existing central limit results. We will show, however, that the irreducibility assumption can be dropped in some situations, and that our central limit theorem contains the result of \cite{AGS}, but yields more general formulas.
In addition, we can prove a large deviation principle for the position process~$(X_p)_p$ associated with an homogeneous OQRW on a lattice. The technique we used, based on the application of the Perron-Frobenius theorem to a suitable deformed positive map, goes back (to the best of our knowledge) to \cite{HMO}. None of the articles cited above proves a large deviation principle. As we were completing this paper, however, we learnt of the recent article \cite{vHG}, which proves a similar result. We comment on this in section \ref{section_cltldp}.
The structure of the present paper is the following: in section \ref{section_OQRWs} we recall the main definitions of open quantum random walks
specialized to the case where the underlying graph is a lattice in $\rr^d$, and define the auxiliary map of an open quantum random walk.
In section~\ref{section_CPmaps} we recall standard results about irreducibility and period of completely positive maps. In section \ref{section_irreducibility} we characterize irreducibility and period of the open quantum random walk and its auxiliary map. In section \ref{section_cltldp} we state our main results: the central limit theorem and the large deviation principle. In section \ref{section_ZdC2} we specialize to the situation where the underlying graph is $\zz^d$ and the internal state space is $\cc^2$, and characterize each situation in terms of the transition operators.
In section \ref{section_examples} we study explicit examples, and compare our theoretical results to simulations.
\section{Homogeneous open quantum random walks}
\label{section_OQRWs}
In this section we recall basic results and notations about open quantum random walks. We essentially follow the notation of \cite{CP1}, but specialize to the homogeneous case. For a more detailed exposition we refer the reader to \cite{APSS}.
We consider a Hilbert space $\mathfrak h$ and a locally finite lattice $V\subset \rr^d$, which we assume contains $0$, and is positively generated by a set $S\neq\{0\}$, in the sense that any $v$ in $V$ can be written as $s_1+\ldots+s_n$ with $s_1,\ldots, s_n \in S$. In particular, $V$ is an infinite subgroup of $\rr^d$. The canonical example is $V=\zz^d$, with $S=\{\pm v_1,\ldots, \pm v_d\}$ where $(v_1,\ldots,v_d)$ is the canonical basis of $\rr^d$.
We denote by $\mathcal H$ the Hilbert space $\mathcal H= \mathfrak h \otimes\cc^V$. We view $\mathcal H$ as describing the degrees of freedom of a particle constrained to move on $V$: the ``$V$-component" describes the spatial degrees of freedom (the position of the particle) while $\mathfrak h$ describes the internal degrees of freedom of the particle. According to quantum mechanical canon, we describe the state of the system as a positive, trace-class operator $\rho$ on $\H$ with trace one. Precisely, such an operator will be called a state.
\smallskip
We consider a map on the space $\mathcal I _1(\mathcal H)$ of trace-class operators, given by
\begin{equation}\label{eq_OQRW}
\mathfrak M \, : \, \rho \mapsto \sum_{j\in V}\, \sum_{s\in S}
\left(L_{s} \otimes \ketbra{j+s}{j}\right) \,\rho \,\left(L_{s}^* \otimes \ketbra{j}{j+s}\right)
\end{equation}
where the $L_{s}$, $s\in S$, are operators acting on $\h$ satisfying
\begin{equation}\label{eq_stochastic}
\sum_{s\in S} L_{s}^* \, L_{s}=\id.
\end{equation}
The $L_{s}$ are thought of as encoding both the probability of a transition by the vector $s$, and the effect of that transition on the internal degrees of freedom. Equation \eqref{eq_stochastic} therefore encodes the ``stochasticity" of the transitions.
\begin{remark}
The map $\M$ defined above is a special case of a quantum Markov chain, as introduced by Gudder in \cite{Gudder}. See Section 8 of \cite{CP1} for more comments.
\end{remark}
We associate with the OQRW $\M$ the auxiliary map $\L$ on the space $\mathcal I_1(\mathfrak h)$ of trace-class operators on $\h$ defined by
\begin{equation}\label{eq_OQRWaux}
\mathfrak L \, : \, \rho \mapsto\sum_{s\in S} L_{s}\,\rho \, L_{s}^*.
\end{equation}
Both \eqref{eq_OQRW} and \eqref{eq_OQRWaux} define trace-preserving (TP) maps, which are completely positive (CP), \textit{i.e.} for any $n$ in $\nn^*$, the extensions $\M\otimes \id$ and $\L\otimes\id$ to $\mathcal I_1(\mathcal H)\otimes \mathcal B(\cc^n)$ and $\mathcal I_1(\h)\otimes \mathcal B(\cc^n)$, respectively, are positive. In particular, such a map transforms states (understood here as positive elements of $\mathcal I_1(\mathcal H)$ with trace one) into states. A completely-positive, trace-preserving map will be called a CP-TP map. We will call a map $\M$, as defined by \eqref{eq_OQRW}, an open quantum random walk, or OQRW; and we will call $\L$ the auxiliary map of~$\M$.
To be more precise, we should call such an $\M$ an \emph{homogeneous} OQRW, but will drop the adjective homogeneous in the rest of this paper.
Let us recall that the topological dual $\mathcal I_1(\mathcal H)^*$ can be identified with $\mathcal B(\mathcal H)$ through the duality
\[(\rho,X)\mapsto \tr(\rho\, X).\]
\begin{remark}\label{remark_TPnormone}
When $\Phi=\M$ or $\Phi=\L$,
the adjoint $\Phi^*$ is a positive, unital (\textit{i.e.}~$\Phi^*(\id)=\id$) map on $\mathcal B (\mathcal H)$ (respectively $\mathcal B (\h)$), and by the Russo-Dye theorem (\cite{RD}) one has $\|\Phi^*\|=\|\Phi^*(\id)\|$ where the latter is the operator norm on $\mathcal B (\mathcal H)$ (respectively $\mathcal B (\mathcal H)$). This implies that trace-preserving positive maps have norm one, and in particular $\|\M\|=~1$ and $\|\L\|=1$.
\end{remark}
\begin{remark}\label{remark_classical} As noted in \cite{APSS}, classical Markov chains can be written as open quantum random walks. In the present case, if we have a subgroup $V$ of $\rr^d$ generated by a set $S$, and a Markov chain on $V$ with translation-invariant transition matrix $P=(p_{i,j})_{i,j\in V}$ induced by the law $(p_s)_{s\in S}$ on $S$, in the sense that $p_{i,j}=0$ if $j-i\not\in S$ and $p_{i,j}=p_{j-i}$ otherwise, then taking $\h=\cc$ and $L_{s}=\sqrt{p_{s}}$
induces the Markov chain with transition matrix $P$. This OQRW is called the minimal dilation of the Markov chain (see \cite{CP1} for a discussion of minimal and non-minimal dilations). Note that in this case the reduced map $\L$ is trivial:~$\L=1$.
\end{remark}
A crucial remark is that, for any initial state $\rho$ on $\mathcal H$, which is therefore of the form
\[ \rho=\sum_{i,j\in V} \rho(i,j)\otimes \ketbra ij ,\]
the evolved state $\mathfrak M(\rho)$ is of the form
\begin{equation}\label{eq_Mn}
\mathfrak M (\rho) = \sum_{i\in V} \mathfrak M (\rho,i) \otimes |i\rangle \langle i | ,
\ \mbox{ where }\ \M(\rho,i)=\sum_{s\in S} L_{s}\, \rho(i-s,i-s)\, L_{s}^*.
\end{equation}
Each $\mathfrak M(\rho,i)$ is a positive, trace-class operator on $\mathfrak h$ and
$ \sum_{i\in V} \tr\,\mathfrak M(\rho,i) =1$.
We notice that off-diagonal terms $\rho(i,j)$, for $i\neq j$, do not appear in $\M(\rho)$, and $\M(\rho)$ itself is diagonal. For this reason, from now on, we will only consider states of the form $\rho=\sum_{i\in V}\rho(i)\otimes \ketbra ii$. Equation \eqref{eq_Mn} remains valid, replacing $\rho(i,i)$ by~$\rho(i)$.
\smallskip
We now describe the (classical) processes of interest associated with $\M$. We begin with an informal discussion of these processes and their laws, and will only define the underlying probability space at the end of this section. We start from a state of the form $\rho=\sum_{i\in V}\rho(i) \otimes \ketbra ii$. We evolve $\rho$ for a time $p$, obtaining the state $\M^p(\rho)$ which, according to the previous discussion, is of the form
\[\M^p(\rho)=\sum_{i\in V} \M^p(\rho,i)\otimes\ketbra ii.\]
We then make a measurement of the position observable. According to standard rules of quantum measurement, we obtain the result $i\in V$ with probability $\tr\,\M^p(\rho,i)$. Therefore, the result of this measurement is a random variable $Q_p$, with law $\pp(Q_p=i)= \tr\,\mathfrak M^p(\rho,i)$ for $i\in V$. In addition, if the position $Q_p=i\in V$ is observed, then the state is transformed to $\frac{\mathfrak M^p(\rho,i)}{\tr \mathfrak M^p(\rho,i)}$. This process $(Q_p,\frac{\M^p(\rho,Q_p)}{\tr\,\M^p(\rho,Q_p)})$ we call the process ``without measurement", to emphasize the fact that virtually only one measurement is done, at time $p$. Notice that, in practice, two values of this process at times $p<p'$ cannot be considered simultaneously as the measure at time $p$ perturbs the system, and therefore subsequent measurements.
Now assume that we make a measurement at every time $p\in\nn$, applying the evolution by $\M$ between two measurements. Again assume that we start from a state $\rho$ of the form $\sum_{i\in V}\rho(i)\otimes |i\rangle \langle i|$. Suppose that at time $p$, the position was measured at $X_p=j$ and the state (after the measurement) is $\rho_p \otimes \ketbra {j}{j}$. Then, after the evolution, the state becomes
\[\M(\rho_p \otimes \ketbra {j}{j})= \sum_{s\in S} L_{s}\, \rho_p\, L_{s}^* \otimes \ketbra {j+s}{j+s},\]
so that a measurement at time $p+1$ gives a position $X_{p+1}=j+s$ with probability $\tr\, L_{s}\, \rho_p\, L_{s}^*$, and then the state becomes $\rho_{p+1}\otimes\ketbra {j+s}{j+s}$ with $\rho_{p+1}=\frac{L_{s}\, \rho_p\, L_{s}^*}{\tr\, L_{s}\, \rho_p\, L_{s}^*}$. The sequence of random variables $(X_p,\rho_p)$ is therefore a Markov process with transitions defined by
\begin{equation}\label{eq_Markov}
\pp\Big((X_{p+1},\rho_{p+1})=(j+s,\frac{L_{s}\,\sigma\, L_{s}^*}{\tr(L_{s}\sigma L_{s}^*)})
\Big| (X_p,\,\rho_p)=(j,\sigma)\Big)=\tr (L_{s}\,\sigma\, L_{s}^*),\end{equation}
for any $j\in V$, $s\in S$ and $\sigma\in{\cal I}_1(\h)$ and initial law $\pp\big((X_0,\rho_0)=(i,\frac{\rho(i)}{\tr\rho(i)})\big)=\tr\rho(i).$
Note that the sequence $X_0=i_0$, \ldots, $X_p=i_p$ is observed with probability
\begin{equation} \label{eq_probatraj}
\pp(X_0=i_0, \ldots, X_p=i_p)=\tr\,\big(L_{s_p}\ldots L_{s_1}L_{i_0}\,\rho(i_0)\, L_{i_0}^*L_{s_1}^*\ldots L_{s_p}^*\big)
\end{equation}
if $i_1-i_0=s_1$,\ldots $i_p-i_{p-1}=s_p$ belong to $S$, and zero otherwise. In addition, this sequence completely determines the state $\rho_p$:
\begin{equation}\label{eq_rhon} \rho_p = \frac{L_{s_p}\ldots L_{s_1}\,\rho(i_0)\, L_{s_1}^*\ldots L_{s_p}^*}{\tr\,L_{s_p}\ldots L_{s_0}\,\rho(i_0)\, L_{s_0}^*\ldots L_{s_p}^*}.
\end{equation}
As emphasized in \cite{APSS}, this implies that, for every $p$, the laws of $X_p$ and $Q_p$ are the same, \textit{i.e.} \[ \pp(X_p=i)=\pp(Q_p=i)\quad \forall i\in V.\]
We now construct a probability space to carry the processes just described. Fixing an open quantum random walk $\M$ on $V$ defined by operators $(L_s)_{s\in S}$ we define the set $\Omega=V^\nn$, equipped with the $\sigma$-field generated by cylinder sets. An element of $\Omega$ is denoted by $\omega=(\omega_k)_{k\in \nn}$ and we denote by $(X_p)_{p\in\nn}$ the coordinate maps. For any state $\rho$ on $\H$ of the form $\rho=\sum_{i\in V}\rho(i)\otimes \ketbra ii$, we define a probability $\pp^{(p)}_\rho$ on $V^{p+1}$ by formula \eqref{eq_probatraj}. One easily shows, using the stochasticity property \eqref{eq_stochastic}, that the family $(\pp_\rho^{(p)})_p$ is consistent, and can therefore be extended uniquely to a probability $\pp_\rho$ on $\Omega$. We denote by $\rho_p$ the random variable
\[ \rho_p = \frac{L_{X_p- X_{p-1}}\ldots L_{X_1-X_{0}} \,\rho(X_0)\,L^*_{X_1- X_{0}} \ldots L^*_{X_p- X_{p-1}}}{\tr(L_{X_p- X_{p-1}}\ldots L_{X_1- X_{0}} \,\rho(X_0)\,L^*_{X_1-X_{0}} \ldots L^*_{X_p- X_{p-1}})}.\]
We will also denote $Q_p=X_p$, but will only use the notation $Q_p$ when we consider ``non-measurement" experiments, and in particular will never consider an event implying simultaneously outcomes $Q_p$ and $Q_{p'}$ for $p\neq p'$. These processes reproduce the behaviour of the measurement outcomes and of the associated resulting states. In particular, equation \eqref{eq_Markov} above holds in a mathematical sense with $\pp_\rho$ replacing $\pp$. From now on, we will usually drop the $\rho$ in $\pp_\rho$.
\section{Irreducibility and period: general results}
\label{section_CPmaps}
In this section we focus on the general notions of irreducibility and period for a completely positive (CP) map $\Phi$ on $\mathcal I_1(\mathcal K)$, where $\mathcal K$ is a separable Hilbert space which, in practice, will be either $\h$ or $\H$. We assume $\Phi$ is given in the form
\begin{equation}\label{eq_defKraus}
\Phi(\rho)=\sum_{\kappa\in K} A_\kappa \rho A_\kappa^*
\end{equation}
where $K$ is a countable set, and the series $\sum_{\kappa\in K} A_\kappa^* A_\kappa$ is strongly convergent. This is the case for operators such as $\M$ or $\L$ and we actually know from the Kraus theorem that this is the case for any completely positive $\Phi$, see \cite{Kraus} or \cite{NieChu}, where this is called the operator-sum representation.
We recall that such a map is automatically bounded as a linear map on $\mathcal I_1(\mathcal K)$ (see \textit{e.g.} Lemma~2.2 in \cite{Sch}), so that it is also weak-continuous. In most practical cases, we will additionally assume that $\|\Phi\|=1$; this will be the case, in particular, if~$\Phi$ is trace-preserving.
\smallskip
We give various equivalent definitions of the notion of irreducibility for $\Phi$ which was originally defined by Davies in \cite{Dav}. Note that this original definition holds for $\Phi$ positive, but for simplicity, we discuss it only for maps $\Phi$ which are completely positive (CP), and therefore have a Kraus decomposition \eqref{eq_defKraus}. The equivalence between the different definitions, as well as the relevant references, are discussed in \cite{CP1}. We recall some standard notations: an operator $X$ on $\mathcal K$ is called positive, denoted $X\geq0$, if, for $\phi\in \mathcal K$, one has $\langle \phi, X\, \phi\rangle \geq 0$. It is called strictly positive, denoted $X>0$, if, for $\phi\in \mathcal K\setminus\{0\}$, one has $\langle\phi, X\, \phi\rangle>0$.
\begin{defi}\label{defi_irreducibility}
The CP map $\Phi$ is called irreducible if one of the following equivalent conditions hold:
\begin{itemize}
\item the only orthogonal projections $P$ reducing $\Phi$, \textrm{i.e.} such that~$\Phi\big(P\mathcal I_1(\mathcal K)P\big) \subset P\mathcal I_1(\mathcal K)P$, are $P=0$ and $\id$,
\item for any $\rho\geq 0$, $\rho\neq 0$ in $\mathcal I_1(\mathcal K)$, there exists $t$ such that~$\e^{t \Phi} (\rho)>~0$,
\item for any non-zero $\phi\in\mathcal K$, the set
$ \cc[A] \, \phi $ is dense in $\mathcal K$, where $\cc[A]$ is the set of polynomials in $A_\kappa, \, \kappa\in K$,
\item the only subspaces of $\mathcal K$ that are invariant by all operators $A_\kappa$ are $\{0\}$ and~$\mathcal K$.
\end{itemize}
\end{defi}
\begin{remark}
Note also that the notion of irreducibility is strongly related to the notion of subharmonic projection, again see \cite{CP1}.
\end{remark}
We will also need on occasion the notion of regularity, which is evidently stronger than irreducibility:
\begin{defi}\label{defi_regularity}
The CP map $\Phi$ is called $N$-regular if one of the equivalent conditions hold:
\begin{itemize}
\item for any $\rho\geq 0$, $\rho\neq 0$ in $\mathcal I_1(\mathcal K)$, one has $\Phi^N (\rho)>~0$,
\item for any non-zero $\phi\in\mathcal K$, the set
$\{ A_{\kappa_1}\ldots A_{\kappa_N}\, \phi\,|\, \kappa_1,\ldots,\kappa_N\in K \}$ is total in $\mathcal K$.
\end{itemize}
The map $\Phi$ is called regular if it is $N$-regular for some $N$ in $\nn^*$.
\end{defi}
\begin{remark}\label{remark_regularity}
The following properties are quite immediate:
\begin{itemize}
\item If $\Phi$ is irreducible, then $\vee_{\kappa\in K} {\rm Ran} A_\kappa={\mathcal K}$ (while the converse is not true).
\item If $\vee_{\kappa\in K} {\rm Ran} A_\kappa={\mathcal K}$ and $\sigma$ is a faithful state, then $\Phi(\sigma)$ is faithful.
Indeed, we can write $\sigma=\sum_j\sigma_j |u_j\rangle\langle u_j|$, with $\sigma_j>0$ and $(u_j)_j$ an orthonormal basis for $\mathcal K$.
Then $\Phi(\sigma)=\sum_{j,\kappa}\sigma_j |A_\kappa u_j\rangle\langle A_\kappa u_j|$, and the conclusion easily follows.
\item If $\vee_{\kappa\in K} {\rm Ran} A_\kappa={\mathcal K}$ and $\Phi$ in $N$-regular, $N\ge 1$, then $\Phi$ is $(N+n)$\,--\,regular for any $n\ge 0$.
This is an immediate consequence of the previous point.
\end{itemize}
\end{remark}
The following proposition, which is a Perron-Frobenius theorem for positive maps on $\mathcal I_1(\mathcal K)$, essentially comes from \cite{EHK} (for the finite dimensional case) and~\cite{Sch} (for the infinite dimensional case). To state it in sufficient generality, we need to recall the definition of the spectral radius of a map $\Phi$:
\[r(\Phi)=\sup \{|\lambda|, \, \lambda\in \mathrm{Sp}\,\Phi\}\]
where $\mathrm{Sp}\,\Phi$ is the spectrum of $\Phi$.
\begin{prop}\label{prop_Schrader}
Assume a CP map $\Phi$ on $\mathcal I_1(\mathcal K)$ has an eigenvalue $\lambda$ of modulus $r(\Phi)$, with eigenvector $\rho$, and
either $\mathrm{dim}\, \mathcal K <\infty$ or $r(\Phi)=\|\Phi\|$.
Then:
\begin{itemize}
\item $|\lambda|$ is also an eigenvalue, with eigenvector $|\rho|$,
\item if $\Phi$ is irreducible, then $\mathrm{dim}\,\mathrm{Ker}\,(\Phi-\lambda\,\id)=1$.
\end{itemize}
In particular, if $\Phi$ is irreducible and has an eigenvalue of modulus $r(\Phi)$,
then~$r(\Phi)$ is an eigenvalue with geometric multiplicity one, with an eigenvector that is a strictly positive operator.
\end{prop}
\begin{remark}\label{rem_finitedim}
When $\Phi$ is a completely positive, trace-preserving map, one has $\|\Phi\|=1$, so that the conclusion applies if $\lambda$ is of modulus $1$. In \cite{CP1}, this was enough, since we applied this result to the operator $\M$. In section \ref{section_cltldp} we will also need to apply it to a deformation of the operator $\L$, which will no longer be trace-preserving.
\end{remark}
\begin{remark}
The previous proposition gives in particular uniqueness and faithfulness of the invariant state, when it exists, for an irreducible map $\Phi$.
As one can expect, the converse result holds: if $\Phi$ admits a unique invariant state and that state is faithful, then $\Phi$ is irreducible (see \cite{CP1}, Section 7).
\end{remark}
\medskip
We now turn to the notion of period for positive maps. We will denote by~$\submod$,~$\addmod$ the substraction and addition \emph{modulo} $d$.
\begin{defi}\label{def_period}
Let $\Phi$ be a CP, trace-preserving, irreducible map and consider a resolution of the identity
$(P_0,\ldots,P_{d-1})$, \emph{i.e.} a family of orthogonal
projections such that $\sum_{j=0}^{d-1} P_j=\id$. One says that $(P_0,\ldots,P_{d-1})$
is $\Phi$-cyclic if $P_j A_\kappa= A_\kappa P_{j\submod 1}$ for $j=0,\ldots,d-1$ and any $k$.
The supremum of all $d$ for which there exists a $\Phi$-cyclic resolution of identity $(P_0,\ldots,P_{d-1})$ is called the \textrm{period} of $\Phi$.
If $\Phi$ has period~$1$ then we call it aperiodic.
\end{defi}
\begin{remark}
If $\mathrm{dim}\,\mathcal K$ is finite then the period is always finite.
\end{remark}
The following proposition is the analog of a standard result for classical Markov chains:
\begin{prop}\label{prop-periodic-blocks}
Assume $\Phi$ is completely positive, irreducible, with finite period $d$, and denote by $P_0,\ldots,P_{d-1}$ a cyclic decomposition of $\Phi$. Then :
\begin{enumerate}
\item we have the relation $
\Phi(P_i\, \rho \, P_j)=P_{i\addmod 1}\, \Phi(\rho) \, P_{j\addmod 1},
$
\item
for any $j=0,\ldots, d-1$, the restriction $\Phi^d_j$ of $\Phi^d$ to $P_j{\cal I}_1({\mathcal K})P_j$ is irreducible aperiodic,
\item if $\Phi$ has an invariant state $\rhoinv$, then $\Phi^d_j$ has a unique invariant state
$\rhoinv_j\overset{\mathrm{def}}=d\times P_j\rhoinv P_j$.
\end{enumerate}\end{prop}
\pre
\begin{enumerate}
\item The first relation is obvious, and shows that $P_j{\cal I}_1({\mathcal K})P_j$ is stable by $\Phi^d$.
\item Consider a state $P_j\rho P_j$ in $P_j{\cal I}_1({\mathcal K})P_j$. By irreducibility of $\Phi$, $\e^{t\Phi}(P_j\rho P_j)$ is faithful, so~$P_j\e^{t\Phi}(P_j \rho P_j)P_j$ is faithful in $\mathrm{Ran}\,P_j$. But by the relation in point $1$,
\[P_j\e^{t\Phi}(P_j\rho P_j)P_j =\sum_{n=0}^\infty \frac{t^{dn}}{(dn)!}\, \Phi^{dn}(P_j \rho P_j) = \sum_{n=0}^\infty \frac{t^{dn}}{(dn)!}\, (\Phi_j^{d})^n(P_j \rho P_j).\]
This shows that $\Phi_j^d$ is irreducible. Now, if $\Phi_j^d$ has a cyclic decomposition of identity $(P_{j,0},\ldots, P_{j,\delta-1})$ then by the commutation relations this induces a cyclic decomposition of identity for $\Phi$ with $d\times \delta$ elements. Therefore,~$\delta=1$.
\item The invariance of $\rhoinv_j$ is trivial by point 1, and the irreducibility of $\Phi_j^d$ implies the unicity of the invariant state. By remark 4.8 in \cite{CP1}, $\tr(P_j \rhoinv P_j)$ does not depend on $j$, so it is $1/d$.
\fin
\end{enumerate}
The following results were originally proved by Fagnola and Pellicer in \cite{FP} (with partial results going back to \cite{EHK} and \cite{Gro}). We recall that the point spectrum of an operator is its set of eigenvalues, and that we denote by $\mathrm{Sp}_{pp}\Phi^*$ the point spectrum of $\Phi^*$.
\begin{prop}\label{prop_aperiodicCPTP}
If $\Phi$ is an irreducible, completely positive, trace-preserving map on~$\mathcal I_1(\mathcal K)$ and has finite period $d$ then:
\begin{itemize}
\item the set $\mathrm{Sp}_{pp}\Phi^* $, is a subgroup of the circle group $\mathbb T$,
\item the primitive root of unity $\e^{\i 2\pi/d}$ belongs to $\mathrm{Sp}_{pp}\Phi^*$ if and only if $\Phi$ is~$d$-periodic.
\end{itemize}
\end{prop}
An immediate consequence is the following:
\begin{prop}\label{prop_cvgCPTPaperiodic}
If a completely positive, trace-preserving map $\Phi$ on $\mathcal I_1(\mathcal K)$ is irreducible and aperiodic with invariant state $\rhoinv$, and $\mathcal K$ is finite-dimensional then
\begin{itemize}
\item $\mathrm{Sp_{pp}}\,\Phi\cap\mathbb T = \{1\}$,
\item for any $\rho\in \mathcal I_1(\mathcal K)$ one has $\Phi^p(\rho)\rightarrow \rhoinv$ as $p\to\infty$.
\end{itemize}
\end{prop}
\section{Irreducibility and period of $\M$ and $\L$}\label{section_irreducibility}
Now we turn to the case where the operator $\Phi$ is an open quantum random walk~$\M$ generated by $L_s$, $s\in S$, or the auxiliary map $\L$ as defined by \eqref{eq_OQRWaux}.
We will study irreducibility and periodicity properties of both operators $\M$ and $\L$ and mutual relations. We will first explain why we focus on a study of $\L$, when~$\M$ should intuitively be the object of interest.
For any $v$ in $V$ we denote
\[\mathcal P_\ell(v)= \{\pi=(s_1,\ldots, s_\ell)\in S^{\ell}\, |\, \sum_{p=1}^\ell s_p = v\}\]
and, in addition, we consider
\[\mathcal P(v)=\cup_{\ell\geq 1}\mathcal P_\ell(v),
\qquad
\mathcal P_\ell = \cup_{v\in V} \mathcal P_\ell(v) \qquad \mathcal P=\cup_{\ell\in \nn}\mathcal P_\ell = \cup_{v\in V} \mathcal P(v).\]
In analogy with \cite{CP1}, we use the notation
\[
L_\pi = L_{s_\ell}\cdots L_{s_1},
\qquad
\mbox{ for } \pi=(s_1,\ldots, s_\ell)\in \mathcal P_\ell.
\]
We remark that the notations for the paths and the set of paths are slightly different from our previous paper \cite{CP1} since we can use homogeneity, which allows us to drop the dependence on the particular starting point.
The irreducibility of $\L$ and $\M$ are easily characterized in terms of paths. This is true in general for OQRWs (see \cite{CP1}, Proposition 3.9 in particular), but the following characterization for $\M$ is specific to homogeneous OQRWs.
\begin{lemma}\label{lemma_irreducLM}
Let $\M$ be an open quantum random walks defined by transition operators $L_s$, $s\in S$, and $\L$ its auxiliary map.
\begin{enumerate}
\item The operator $\L$ is irreducible if and only if, for any $x\neq 0$ in $\h$, the set~$\{L_\pi x,\, |\, \pi \in \mathcal P\}$ is total in~$\h$.
\item
The operator $\M$ is irreducible if and only if, for any $x\neq 0$ in $\h$ and $v$ in~$V$, the set $\{L_\pi x,\, |\, \pi \in \mathcal P(v)\}$ is total in $\h$.
\end{enumerate}
\end{lemma}
\pre
This lemma is proven by a direct application of Definition \ref{defi_irreducibility} (third condition). For $\L$, it is immediate. For $\M$, one sees easily that irreducibility amounts to the fact that, for any $x\otimes \ket {w}$, the set $\{L_\pi x \otimes \ket{v+w}, \pi\in\mathcal P(v)\}$ is dense in~$\h$ for any $v\in V$ (see the details in Proposition 3.9 of \cite{CP1}) and this is equivalent to the statement above.
\fin
\smallskip
This lemma obviously implies the following result:
\begin{coro}\label{coro_ML}
If $\M$ is irreducible, then $\L$ is irreducible.
\end{coro}
One can, however, prove a more explicit criterion for irreducibility of $\M$ than Lemma \ref{lemma_irreducLM}. For consistency we rephrase here one of the equivalent definitions in Definition \ref{defi_irreducibility}:
\begin{prop}\label{prop_caracM}
The operator $\L$ is irreducible if and only if the operators
\[\{L_{s},\ s\in S\}\]
have no invariant closed subspace in common, apart from $\{0\}$ and $\h$.
The operator $\M$ is irreducible if and only if the operators
\[\{L_{\pi_0},\ \pi_0\in \mathcal P(0)\}\]
have no invariant closed subspace in common, apart from $\{0\}$ and $\h$.
\end{prop}
\pre
The characterization for $\L$ immediately follows from the last condition in Definition \ref{defi_irreducibility}. If $\M$ is not irreducible then for some $v\in V$, the closed space
\[ \h_v=\overline{\mathrm{Vect}\{L_\pi x,\, \pi \in \mathcal P(v)\}} \]
is different from $\h$. Since the concatenation of any $\pi_0\in \mathcal P(0)$ with $\pi\in\mathcal P(v)$ gives an element of $\mathcal P(v)$, the space $\h_v$ must be $L_{\pi_0}$-invariant. Conversely, if all operators $L_{\pi_0}$ have an invariant subspace $\mathfrak h'$ in common, then for any $x\in \mathfrak h'$, the set $\{L_\pi x,\, |\, \pi \in \mathcal P(0)\}$ is contained in $\mathfrak h'$ and $\M$ is not irreducible. \fin
Proposition \ref{prop_caracM} allows us to construct examples of OQRWs such that $\L$ is irreducible, but not $\M$.
\begin{example}\label{ex_LirrMnotirr}
Let $d=1$ and
\[L_+=\begin{pmatrix}0 & a_+ \\ b_+ & 0 \end{pmatrix}\qquad L_-=\begin{pmatrix}0 & a_- \\ b_- & 0 \end{pmatrix}\]
with $a_+, a_-, b_+, b_-$ positive, with $a_+^2+b_+^2=a_-^2+b_-^2=1$ and $a_+ b_- \neq a_- b_+$. Then, by Proposition \ref{prop_caracM},
$\L$ is irreducible, but $\M$ is not, since the vectors of the canonical basis are eigenvectors for any $L_{\pi}$, $\pi\in\mathcal P(0)$ (see also Proposition \ref{reducibility-C2}).
\end{example}
The following proposition is proved in \cite{CP1}. We reprove it here.
\begin{prop}\label{prop_inexistenceetatinv}
Assume $\M$ is irreducible. Then it does not have an invariant state.
\end{prop}
\pre
By Corollary \ref{coro_ML}, $\L$ is irreducible, so it has a unique invariant state $\rhoinv$ on~$\h$, which is faithful. Assume $\M$ has an invariant state; by irreducibility it is unique. Since $\M$ is translation-invariant, any translation of that state would be also invariant, so by unicity the invariant state is translation-invariant. It must then be of the form $\sum_{i\in V}\rhoinv\otimes \ketbra ii$, but this has infinite trace, a contradiction.
\fin
All ergodic convergence results for $\M$ given in \cite{CP1} assume the existence of an invariant state.
This is similar to the situation for classical Markov chains; however, some interesting asymptotic properties of $\M$ can be studied in the absence of an invariant state, and this includes large deviations or central limit theorems. As we will see, such properties can be derived from the study of $\L$. This is why, in the study of homogeneous OQRWs, the focus shifts from $\M$ to~$\L$.
\smallskip
To avoid discussing trivial cases, in the rest of this paper we will usually make the following assumption, which by Remark \ref{remark_regularity} automatically holds as soon as $\L$ (or $\M$) is irreducible:
\begin{center}
\textbf{Assumption H1:} one has the equality $ \overline{\bigvee_{s\in S} \mathrm{Ran}\,L_{s}}=\h$.
\end{center}
\smallskip
This assumption is a natural one, since after just one step, even in the reducible case, the system is effectively restricted to the space $\bigvee_{s\in S} \mathrm{Ran}\,L_{s} $. More precisely, for any positive operator $\rho$ on $\h$, one has, for any $s$,
\[\mathrm{supp}\, L_s\, \rho\, L_s^* \subset \mathrm{supp}\, \L(\rho)\subset \overline{\bigvee_{s\in S} \mathrm{Ran}\,L_{s}}.\]
Note that we have not given results equivalent to Lemma \ref{lemma_irreducLM} for the notion of regularity. We do this here:
\begin{lemma}\label{lemma_regularLM}
The operator $\L$ is $N$-regular if and only if for any $x\neq 0$ in $\h$, the set $\{L_\pi x,\, |\, \pi \in \mathcal P_N\}$ is total in~$\h$.
The operator $\M$ can never be regular.
\end{lemma}
\pre
This is obtained by direct application of Definition \ref{defi_regularity}, that shows the criterion for $\L$. It also shows that $\M$ is $N$-regular if and only if for any $x\neq 0$ in $\h$, any $v$ in $V$, the set $\{L_\pi x,\, |\, \pi \in \mathcal P_N(v)\}$ is total in~$\h$. However, if the distance from the origin to $v$ is larger than $N$, then $\mathcal P_N(v)$ is empty. \fin
\smallskip
One could be tempted to consider a weaker version of regularity for $\L$ where the index $N$ can depend on $\rho$. The following result shows that, if $\h$ is finite-dimensional, this is not weaker than regularity:
\begin{lemma}\label{lemma_regularity}
Assume $\h$ is finite-dimensional. If for every $\rho\geq 0$ in $\mathcal I_1(\h)\setminus\{0\}$, there exists $N>0$ such that $\L^N(\rho)$ is faithful, then there exists $N_0$ such that $\L$ is $N_0$-regular.
\end{lemma}
\pre
First observe that $\L$ is necessarily irreducible and so assumption H1 must hold.
Besides, the current assumption implies that, for any $x$ in $\h$, there exists $N_x>0$ such that $\L^{N_x}(\ketbra xx)$ is faithful. Since faithfulness of $\L^{N_x}(\ketbra xx)$ is equivalent to the existence of a family $\pi_1,\ldots, \pi_{\mathrm{dim}\,\h}$ of paths of length $N_x$, such that the determinant of $(L_{\pi_1} x, \ldots, L_{\pi_{\mathrm{dim}\,\h}} x)$ is nonzero, there exist open subsets $B_x$ of the unit ball, such that $x\in B_{x_0}$ implies that $\L^{N_{x_0}}(\ketbra xx)$ is faithful. By compactness of the unit ball, there exists a finite covering by $B_{x_1}\cup\ldots\cup B_{x_p}$. Remark \ref{remark_regularity} then implies that if we let~$N_0=\sup_{i=1,\ldots,p}N_{x_i}$ one has $\L^{N_0}(\ketbra xx)$ faithful for any nonzero $x$. This implies that $\L$ is $N_0$-regular.
\fin
\medskip
We now turn to the notion of period for $\L$ and $\M$. By Definition \ref{def_period}, a resolution of identity $(p_0, \ldots, p_{d-1})$ of $\h$ will be $\L$-cyclic if and only if
\begin{equation*}
p_j L_s = L_s p_{j\submod 1}\quad \mbox{for }j=0,\ldots,d-1 \mbox{ and any }s\in S.
\end{equation*}
Consequently, by Proposition \ref{prop-periodic-blocks}, we have
\begin{equation} \label{eq_commutationLu}
\L(p_j \, \rho \, p_j) = p_{j\addmod 1}\, \L(\rho) \, p_{j\addmod 1}.
\end{equation}
\begin{remark}\label{remark_dsmallerdimh}
Since the $p_j$ sum up to $\id_\h$, the period of $\L$ cannot be greater than $\mathrm{dim}\,\h$, a feature which will be extremely useful when $\mathrm{dim}\,\h$ is small.
\end{remark}
On the other hand, as we observed in \cite{CP1}, a resolution of identity $(P_0, \ldots, P_{d-1})$ of $\H$ will be $\M$-cyclic if and only if it is of the form
\begin{equation}\label{eq_cyclicity}
P_k=\sum_{i\in V}P_{k,i}\otimes \ketbra ii \quad\mbox{with}\quad
P_{k,i} L_s = L_{s} P_{k\submod 1,i+s}.
\end{equation}
\begin{remark}
The cyclic resolutions for $\M$ are translation invariant, in the sense that, if $P_k=\sum_{i\in V}P_{k,i}\otimes \ketbra ii $, $k=0,...d-1$, is a cyclic resolution for $\M$, then also $P'_k=\sum_{i\in V}P_{k,i+v}\otimes \ketbra ii $, $k=0,...d-1$, is a cyclic resolution for any $v$.
\end{remark}
We will, however, make little use for a cyclic resolution of identity for $\M$ in this paper. On the other hand, the periodicity of $\L$ can be an easy source of information on $\M$:
\begin{prop}\label{prop-M-period}
We have the following properties:
\begin{enumerate}
\item The period of $\M$, when finite, is even.
\item If $\L$ is irreducible and has even period $d$, then $\M$ is reducible.
\end{enumerate}
\end{prop}
\pre
\begin{enumerate}
\item Assume that $(P_0,\ldots,P_{d-1})$ is a $\M$-cyclic resolution of identity associated with $\M$. As we observed above, the $P_k$ are of the form
\[P_k=\sum_{i\in V}P_{k,i}\otimes \ketbra ii \quad\mbox{with}\quad
P_{k,i} L_s = L_{s} P_{k\submod 1,i+s}.\]
Then if we call $i$ in $V$ odd or even depending on the parity of its distance to the origin, define
\[P_{k,\textrm{odd}}=\sum_{i\, \textrm{odd}}P_{k,i}\otimes \ketbra ii \quad \mbox{and} \quad P_{k,\textrm{even}}=\sum_{i\, \textrm{even}}P_{k,i}.\]
Then $(P_{0,\textrm{odd}}, P_{1,\textrm{even}}, P_{2,\textrm{odd}},\ldots)$ is a $\M$-cyclic resolution of identity.
\item Denote by $(p_0,\ldots,p_{d-1})$ a cyclic resolution of identity associated with $\L$. Define
\[p_{\mathrm{odd}}=\sum_{k\,\mathrm{odd}} p_k\qquad p_{\mathrm{even}}=\sum_{k\,\mathrm{even}} p_k.\]
It is obvious from relations \eqref{eq_cyclicity} that $\mathrm{Ran}\, p_{\mathrm{odd}}$ and $\mathrm{Ran}\, p_{\mathrm{even}}$ are nontrivial invariant spaces for any $L_{\pi_0}$, $\pi_0\in\mathcal P(0)$. We conclude by Proposition \ref{prop_caracM}. \fin
\end{enumerate}
\smallskip
Last, we give an analogue of a classical property of Markov chains with finite state space:
\begin{lemma}\label{lemma_irrapereg}
If $\h$ is finite-dimensional, then the map $\L$ is irreducible and aperiodic if and only if it is regular.
\end{lemma}
\pre
If $\L$ is irreducible and aperiodic, then by Proposition \ref{prop_cvgCPTPaperiodic} for any state $\rho$ on $\h$, one has $\L^p(\rho)\underset{n\to\infty} {\longrightarrow}\rhoinv$ so that $\L^p(\rho)$ is faithful for large enough $p$. By Lemma \ref{lemma_regularity}, this implies the regularity of $\L$. Conversely, if $\L$ is regular, then it is irreducible, and for any projection $p$, the operator $\L^N(p)$ is faithful, so that $p$ cannot be a member of a cyclic resolution of identity unless $p=\id$.
\fin
\section{Central Limit Theorem and Large Deviations }\label{section_cltldp}
The Perron-Frobenius theorem for CP maps allows us to obtain a large deviations principle and a central limit theorem for the position process $(X_p)_{p\in\nn}$ (or, equivalently, for the process $(Q_p)_{p\in\nn}$) associated with an open quantum random walk $\M$ and an initial state $\rho$ (see section \ref{section_OQRWs}). In most of our statements, we assume for simplicity that $\L$ is irreducible. We discuss extensions of our results at the end of this section.
Before going into the details of the proof, we should mention that, as we were completing the present article, we learnt about the recent paper \cite{vHG}, which proves a large deviation result for empirical measures of outputs of quantum Markov chains, which can be viewed as the ``steps" $(X_p-X_{p-1})_p$ taken by an open quantum random walk. This result is similar to the statement in our Remark \ref{remark_Sanov}, and implies a level-1 large deviation result for the position $(X_p)_p$ when the OQRW is irreducible and aperiodic. In addition, the statement in \cite{vHG} extends to a large deviations principle for empirical measures of $m$-tuples of $(X_p-X_{p-1})_p$. Our (independent) result, however, treats the case where the OQRW is irreducible but not aperiodic, and can be extended beyond the irreducible case.
For the proofs of this section, it will be convenient to introduce some new notations. For $u$ in $\rr^d$ we define $L_{s}^{(u)}= \e^{\braket us /2}L_s$, and denote $\L_u$ the map induced by the $L_{s}^{(u)}$, $s\in S$: for $\rho$ in $\mathcal I_1(\h)$,
\[\L_u(\rho)=\sum_s L_{s}^{(u)}\rho L_{s}^{(u)*}.
\]
The operators $\L_u$ will be useful in order to treat the moment generating functions of the random variables $(X_p)$:
\begin{lemma}\label{def_Lu}
For any $u$ in $\rr^d$ one has
\begin{equation}\label{mgf}
\ee(\exp\,\braket u {X_p-X_0}) = \sum_{i_0\in V} \tr\big(\L_{u}^p(\rho(i_0))\big).
\end{equation}
\end{lemma}
\pre
For any $k$ in $\nn^*$ let $S_k=X_{k+1}-X_k$ and consider $u\in\rr^d$. Then we have
\begin{eqnarray*}
&& \hspace{-2em}\ee(\exp\,\braket u{X_p-X_0})\\
&=& \sum_{i_0\in V}\sum_{s_1,\ldots,s_p\in S^p} \hspace{-1em}\pp(X_0=i_0,S_1=s_1,\ldots, S_p=s_p)\, \exp\,\braket{u}{s_1+\ldots+s_p} \\
&=& \sum_{i_0\in V}\sum_{s_1,\ldots,s_p\in S^p} \hspace{-1em}\tr(L_{s_p}\ldots L_{s_1}\, \rho(i_0)\, L_{s_1}^*\ldots L_{s_p}^*)\, \exp\,\braket{u}{s_1+\ldots+s_p} \end{eqnarray*}
and this gives formula \eqref{mgf}.
\fin
\begin{remark}\label{remark_XpQp}
One also has
\[ \ee(\exp\,\braket u{X_p})= \ee(\exp\,\braket u{Q_p})=\sum_{i_0\in V} \exp\braket{u}{i_0}\,\tr\big(\L_{u}^p(\rho(i_0))\big).\]
This will allow us to give results analogous to Theorem \ref{theo_ldp} and \ref{theo_clt} for the process~$(Q_p)_p$. Note that considering $X_p$ or $X_p-X_0$ is essentially equivalent, but as we remarked in section \ref{section_OQRWs}, $Q_p$ and $Q_0$ cannot be considered simultaneously.
\end{remark}
\smallskip
The following lemma describes the properties of the largest eigenvalue of $\L_u$:
\begin{lemma}\label{lemma_analyticitylambdau}
Assume that $\h$ is finite-dimensional and $\L$ is irreducible. For any $u$ in $\rr$, the spectral radius $\lambda_u\overset{\mathrm{def}}=r(\L_u)$ of $\L_u$ is an algebraically simple eigenvalue of $\L_u$, and has an eigenvector $\rho_u$ which is a strictly positive operator, and we can normalize it to be a state. In addition, the map $u\mapsto \lambda_u$ can be extended to be analytic in a neighbourhood of~$\rr^d$.
\end{lemma}
\smallskip
\pre
By Lemma \ref{lemma_irreducLM}, if $\L$ is irreducible, then so is any $\L_u$ for $u\in \rr^d$. Proposition~\ref{prop_Schrader}, applied here specifically to an Hilbert space of finite dimension, gives the first sentence except for the algebraic simplicity of the eigenvector $\lambda_u$, as it implies only the geometric simplicity.
If we can prove that, for all $u$ in $\rr^d$, the eigenvalue $\lambda_u$ is actually algebraically simple then the theory of perturbation of matrix eigenvalues (see Chapter II in \cite{Kato}) will give us the second sentence.
Now, in order to prove the missing point, consider the adjoint $\L_u^*$ of $\L_u$ on $\mathcal B(\h)$, which in this finite-dimensional setting, can be identified, with $\mathcal I_1(\h)$. It is easy to see from Definition \ref{defi_irreducibility} that $\L_u^*$ is irreducible. Its largest eigenvalue is $\lambda_u$, with eigenvector $M_u$, which, by Proposition \ref{prop_Schrader}, is invertible. We can consider the map
$$
\widetilde \L _u : \rho\mapsto \frac1{\lambda_u}\,M_u^{1/2} \L_u( M_u^{-1/2}\, \rho \, M_u^{-1/2}) M_u^{1/2}.
$$
This $\widetilde \L _u$ is clearly completely positive, and is trace-preserving since $\widetilde \L _u^*(\id)=\id$.
Proposition \ref{prop_Schrader} shows that $\widetilde \L _u$ has $1$ as a geometrically simple eigenvalue, with a strictly positive eigenvector $\widetilde \rho _u$. Then $1$ must also be algebraically simple, otherwise there exists $\eta_u$ such that $\widetilde \L _u(\eta_u)= \eta_u + \widetilde\rho_u$, but taking the trace of this equality yields $\tr(\widetilde\rho _u)=0$, a contradiction. This implies that $\L_u$ has $\lambda_u$ as a algebraically simple eigenvalue.
\fin
\smallskip
We can now state our large deviation result:
\begin{theo}\label{theo_ldp}
Assume that $\h$ is finite-dimensional and that $\L$ is irreducible. Then the process $(\frac1p(X_p-X_0))_{p\in\nn^*}$ associated with $\M$ satisfies a large deviation principle with a good rate function~$I$. Explicitly, there exists a lower semicontinuous mapping $I:\mathbb R^d\to\mathbb [0, + \infty]$ with compact level sets $\{x\, |\, I(x)\leq \alpha\}$, such that, for any open $G$ and closed~$F$ with $G\subset F \subset \mathbb R^d$, one has
\begin{eqnarray*}
&&-\inf_{x\in G} I(x) \leq \liminf_{p\to\infty}\frac1p \log P(\frac {X_p-X_0}{p}\in G)
\\&& \qquad\qquad\qquad
\leq\limsup_{p\to\infty}\frac1p \log P(\frac {X_p-X_0}{p}\in F)\leq -\inf_{x\in F} I(x).
\end{eqnarray*}
\end{theo}
\begin{remark}\label{remark_LDPXp}
If we add the assumption that $X_0$ has an everywhere defined moment generating function, {\it e.g.} that the initial state $\rho$ satisfies
$\ee(\exp\,\braket u{X_0})= \sum_{i_0\in V} \e^{\braket u{i_0}}\tr\rho(i_0)< \infty$ for all $u$ in $\rr^d$, then this theorem also holds for $(X_p)_p$ or equivalently $(Q_p)_p$ in place of $(X_p-X_0)_p$.
\end{remark}
\begin{remark}\label{remark_GvH}
Using the techniques detailed in \cite{vHG}, it is possible, for any $m$ in $\nn$, to extend the above theorem and obtain a full large deviation principle for the sequence of $(m+1)$-tuples $\frac1p(X_p-X_0, X_{p+1}-X_1,\ldots, X_{p+m}-X_m)_p$, or (under the same condition as in Remark \ref{remark_LDPXp}) for $(X_p,\ldots,X_{p+m})_p$.
\end{remark}
\pre
We start with equation \eqref{mgf}. Since $\mathfrak h$ is finite-dimensional, if $\rho(i_0)$ is faithful, then, with $r_{u,i_0}=\inf\mathrm{Sp}(\rho(i_0))>0$ and $s_{u,i_0}=\frac{\tr\rho(i_0)}{\inf\mathrm{Sp}(\rho_u)}>0$ (where $\mathrm{Sp}(\sigma)$ denotes the spectrum of an operator $\sigma$),
\begin{equation}\label{eq_preuveldp1}
r_{u,i_0}\, \rho_u \leq \rho(i_0) \leq s_{u,i_0} \,\rho_u.
\end{equation}
Note that $r_{u,i_0}\leq \tr \rho(i_0)$ so that both $r_{u,i_0}$ and $s_{u,i_0}$ are summable along $i_0$. Consequently, we shall have
\begin{equation}\label{eq_preuveldp2}
r_{u,i_0} \, \lambda_u^p\,\rho_u \leq \L_u^p\big(\rho(i_0) \big)\leq s_{u,i_0} \,\lambda_u^p\, \rho_u .
\end{equation}
Using these bounds in relation \eqref{mgf}, we immediately obtain, for all $u\in\rr^d$,
\[
\lambda_u^p\,\sum_{i_0\in V} r_{u,i_0}\, \rho_u
\le \ee(\exp\,\braket u {X_p-X_0}) \le \lambda_u^p\,\sum_{i_0\in V} s_{u,i_0}\, \rho_u
\]
where the sums are finite and strictly positive; so that
\begin{equation}\label{eq_GE}\lim_{p\to\infty}\frac1p \, \log\ee(\exp\,\braket u {X_p}) = \log\lambda_u .\end{equation}
Now, if $\rho(i_0)$ is not faithful, but $\L$ is aperiodic, due to Proposition \ref{prop_cvgCPTPaperiodic}, then $\L^N(\rho(i_0))$ is faithful for large enough $N$, and \eqref{eq_preuveldp1} holds with $\L_u^N(\rho(i_0))$ in place of $\rho(i_0)$ and \eqref{eq_preuveldp2} holds with $(p-N)$ instead of $p$ in the exponents of $\lambda_u$. We still recover \eqref{eq_GE}.
Finally, if $\rho(i_0)$ is not faithful and $\L$ has period $d>1$, then, considering a cyclic decomposition of identity $(p_0,\ldots,p_{d-1})$, we can consider the single blocks of the form $p_j\rho(i_0)p_j$.
By Proposition \ref{prop-periodic-blocks}, $\L^d$ is irreducible aperiodic when restricted to each $p_j{\cal I}_1({\h})p_j$
and $\L_u^d(p_j \rho_u p_j) = \lambda_u^d p_{j} \rho_u p_{j}$.
Then, by the regularity of the restrictions of $\L^d$, using Remark \ref{remark_regularity} and the obvious extension of \eqref{eq_commutationLu} to $\L_u$, there exist $N\in {\mathbb N}$ and $r_{u,i_0},s_{u,i_0}>0$ such that, for any block $p_j\rho(i_0)p_j \neq 0$,
\begin{equation*}
r_{u,i_0}\, p_j\,\rho_u \,p_j \leq p_j\, \L_u^{dN}\rho(i_0)\,p_j \leq s_{u,i_0} \,p_j\, \rho_u\, p_j
\end{equation*}
and if $p=dN+r$, $r\in\{0,\ldots,d-1\} $,
\begin{equation*}
r_{u,i_0}\, \lambda_u^{p-dN}\,p_{j+r}\,\rho_u \,p_{j+r} \leq \L_u^p\big(p_j \,\rho(i_0)\,p_j \big)
\leq s_{u,i_0} \,\lambda_u^{p-dN}\, p_{j+r}\,\rho_u \,p_{j+r}.
\end{equation*}
Summing over $j$, we recover equation \eqref{eq_GE} again.
In any case, we obtain \eqref{eq_GE} for all $u\in\rr^d$. Lemma \ref{lemma_analyticitylambdau} shows that $u\mapsto \log\lambda_u$ is analytic on $\rr$. Applying the G\"artner-Ellis theorem (see \cite{DZ}) we obtain the bounds mentioned in Theorem \ref{theo_ldp}, with rate function
\[I(x) = \sup_{u\in\rr^d} \, \big(\langle u,x \rangle - \log\lambda_u\big).\ {\Box}\]
\begin{remark}\label{remark_Sanov}
If $\varphi$ is any function $S\to \rr$ and $S_p= \sum_{k=1}^p \varphi(X_k-X_{k-1})$ then the process~$(\frac{S_p}p)_{p\in\nn}$ also satisfies a large deviation principle, with rate function
\[I_\varphi(x) = \sup_{t\in\rr} \, \big(t\, x - \log\lambda_{t \varphi}\big)\]
where $\lambda_{t\varphi}$ is the largest eigenvalue of
\[\L_{t \varphi} : \rho \mapsto \sum_{s\in S}\e^{ t \varphi(s)} L_s \, \rho L_s^*. \]
This is shown by an immediate extension of the proofs of Lemma \ref{lemma_analyticitylambdau} and Theorem \ref{theo_ldp}, and yields a level-2 large deviation result for the process $(X_p-X_{p-1})$ (\emph{e.g.} using Kifer's theorem \cite{Kifer}).
\end{remark}
\begin{remark}
As noted in Remark \ref{remark_classical}, when $\M$ is the minimal dilation of a classical Markov chain with transition probabilities $(p_s)_{s\in S}$, the map $\L$ is trivial: it is just multiplication by $1$ on $\rr$. The maps $\L_u$, however, are not trivial: they are multiplication by
\[\lambda_u = \sum_{s\in S}\exp\braket us \, p_s.\]
We therefore recover the same rate function as in the classical case, see \textit{e.g.} section 3.1.1 of \cite{DZ}.
\end{remark}
\begin{remark}
The technique of applying the Perron-Frobenius theorem to a $u$-dependent deformation of the completely positive map defining the dynamics, goes back (to the best of our knowledge) to \cite{HMO}, and is a non-commutative adaptation of a standard proof for Markov chains.
\end{remark}
We denote by $c$ the map $c:\rr^d\ni u\mapsto \log\lambda_u$. As is well-known (see \textit{e.g.} section II.6 in \cite{Ellis}), the differentiability of $c$ at zero is related to a law of large numbers for the process $(X_p)_{p\in \nn}$. Similarly, the second order differential will be relevant for the central limit theorem.
\begin{coro}\label{coro_derivatives}
Assume that $\h$ has finite dimension and that $\L$ is irreducible. The function $c$ on $\rr^d$ is infinitely differentiable at zero. Denote by
\[\L_u' : \rho \mapsto \sum_{s\in S} \braket us \, L_s \rho L_s^* \quad \mbox{ and }\quad\L_u'' : \rho \mapsto \sum_{s\in S} {\braket us}^2 L_s \rho L_s^*.\]
Then, denoting $\lambda_u'\overset{\mathrm{def}}{=}{\frac\d{\d t}}_{|t=0} \lambda_{tu}$ and $\lambda_u''\overset{\mathrm{def}}{=}\frac{\d^2}{\d t^2}_{|t=0} \lambda_{tu}$, we have
\begin{equation}\label{eq_lambdaprime}
\lambda_u' =\tr\big(\L_u'(\rhoinv)\big)
\end{equation}
\begin{equation}\label{eq_lambdasec}
\lambda_u'' = \tr\big(\L''_u(\rhoinv)\big)+2\tr\big(\L'_u(\eta_u)\big)
\qquad
\end{equation}
where $\eta_u$ is the unique solution with trace zero of the equation
\begin{equation}\label{eq_etau}
\big(\id-\L\big)(\eta_u) = \L'_u(\rhoinv)-\tr\big(\L'_u(\rhoinv)\big)\,\rhoinv.
\end{equation}
This implies immediately that
\begin{equation}\label{eq_diffc}
\d c (0)\,(u)=\lambda_u' \qquad \d^2 c(0)\, (u,u)=\lambda_u''-\lambda_u'{}^2.
\end{equation}
\end{coro}
\pre
Lemma \ref{lemma_analyticitylambdau} shows that $c_u$ is infinitely differentiable at any $u\in \rr^d$.
In addition (again see Chapter II in \cite{Kato}), the largest eigenvalue $\lambda_u$ of $\L_u$ is an analytic perturbation of $\lambda_0=1$, and has an eigenvector $\rho_{u}$ which we can choose to be a state, and this $\rho_u$ is an analytic perturbation of $\rho_0$. Then one has
\[\lambda_{tu} = 1 + t \lambda'_u + \frac {t^2}2 \lambda_u'' + o(t^2)\]
\[ \rho_{tu} = \rhoinv + t \,\eta_u + \frac {t^2}2 \,\sigma_u+o(t^2)\]
\[ \L_{tu} = \L + t \,\L'_u + \frac{t^2}2\, \L''_u + o(t^2)\]
and since every $\rho_{tu}$ is a state then $\tr \,\eta_u=\tr\, \sigma_u=0$. Then the relation $\L_{tu}(\rho_{tu})=\lambda_{tu}\, \rho_{tu}$
yields
\[\L'_u(\rhoinv) + \L(\eta_u) = \eta_u + \lambda'_u\, \rhoinv\]
\[\frac12\L(\sigma_u)+ \L_u'(\eta_u)+ \frac12 \L_u''(\rhoinv)= \frac12 \sigma_u + \lambda_u'\, \eta_u + \frac12 \lambda_u '' \, \rhoinv. \]
Taking the trace of the first relation immediately yields relation \eqref{eq_lambdaprime}. In addition, it yields relation \eqref{eq_etau}. Since $\id - \L$ has kernel of dimension one, and range in the set of operators with zero trace, it induces a bijection on that state, so that \eqref{eq_etau} has a unique solution with trace zero. Then taking the trace of the second relation above, and using the fact that $\L$ is trace-preserving gives relation \eqref{eq_lambdasec}.
\fin
\begin{coro}\label{coro_lln}
Assume that $\h$ has finite dimension and $\L$ is irreducible, and let~$m=\sum_{s} \tr(L_s\rhoinv L_s^*) \, s$. Then the process $(\frac1p(X_p-X_0))_{p\in\nn}$ associated with $\M$ converges exponentially to $m$, \emph{i.e.} for any $\eps>0$ there exists $N>0$ such that, for large enough $p$,
\[\pp(\big\|\frac{X_p-X_0}p-m\big\|>\eps)\leq \exp -pN.\]
This implies the almost-sure convergence of $(\frac{X_p}p)_{p\in\nn}$ to $m$.
\end{coro}
\begin{remark}
The almost-sure convergence holds replacing~$X_p$ by $Q_p$.
\end{remark}
\pre
This is a standard result, see \emph{e.g.} Theorem II.6.3 and Theorem II.6.4 in \cite{Ellis}.
\fin
\begin{theo}\label{theo_clt}
Assume that $\h$ is finite-dimensional and $\L$ is irreducible. Denote by $m$ the quantity defined in Corollary \ref{coro_lln}, and by $C$ the covariance matrix associated with the quadratic form $u\mapsto \lambda_u'' - \lambda_u'{}^2$.
Then the position process~$(X_p)_{p\in\nn}$ associated with $\M$ satisfies
\[ \frac{X_p-p\,m}{\sqrt p} \underset{p\to\infty}\longrightarrow \mathcal N(0,C)\]
where convergence is in law.
\end{theo}
\begin{remark}
Again this result holds replacing $X_p$ by $Q_p$.
\end{remark}
\begin{remark}
The formulas for the mean and variance are the same as in \cite{AGS} when $V=\zz^d$ and $S=\{\pm v_i,\, i=1,\ldots,d\}$ ($v_1,\ldots,v_d$ is the canonical basis of $\rr^d$). This can be observed from the fact that, if $Y_u$ is the unique (up to a constant multiple of the $\id$) solution of equation
\[(\id-\L^*)(Y_u)= \sum_{s\in S}\braket us L_s^*L_s - \braket um \,\id,\]
(note that our $Y_u$ is the $L_l$ of \cite{AGS}) then
\[\tr\big(\L_u'(\eta_u)\big) = \tr\big(\L_u'(\rhoinv) Y_u\big) - \tr\big(\L_u'(\rhoinv)\big)\,\tr\big(\rhoinv Y_u\big)\]
and denoting $Y_i=Y_{v_i}$ we have
\begin{eqnarray*}
\braket u {Cu} = \sum_{i,j=1}^d u_i u_j &&\hspace{-2em}\Big(\ind_{i=j}\big(\tr (L_{+i}\rhoinv L_{+i}^*) +\tr (L_{-i}\rhoinv L_{-i}^*) \big)\\
&& +2\tr (L_{+i}\rhoinv L_{+i}^*\,Y_j) - 2\tr (L_{-i}\rhoinv L_{-i}^*\, Y_j)\\
&& - 2m_i \tr(\rhoinv \, Y_j)- m_im_j\Big)
\end{eqnarray*}
which leads to the formula for $C$ given in \cite{AGS}:
\begin{eqnarray*}
C_{i,j}= &&\hspace{-2em}\ind_{i=j}\big(\tr (L_{+i}\rhoinv L_{+i}^*) +\tr (L_{-i}\rhoinv L_{-i}^*) \big)\\
&&\hspace{-2em} +\big(\tr (L_{+i}\rhoinv L_{+i}^* \,Y_j) +\tr (L_{+j}\rhoinv L_{+j}^*\,Y_i)\big)\\
&&\hspace{-2em} \big( \tr (L_{-i}\rhoinv L_{-i}^*\, Y_j)+\tr (L_{-j}\rhoinv L_{-j}^*\, Y_i)\big)\\
&&\hspace{-2em} - \big(m_i \tr(\rhoinv \, Y_j)+m_j \tr(\rhoinv \, Y_i) \big)
- m_im_j.
\end{eqnarray*}
\end{remark}
\noindent \textbf{Proof of Theorem \ref{theo_clt}:}
Let us first consider the case where $\L$ is irreducible and aperiodic. Equation \eqref{mgf} implies
\[ \ee(\exp\braket u {X_p-X_0})=\sum_{i_0\in V} \tr\big(\L_u^p(\rho(i_0))\big). \]
Now, considering the Jordan form of $\L$ shows that, if \[\delta\overset{\mathrm{def}}=\sup\{|\lambda|,\lambda\in\mathrm{Sp}\,\L\setminus\{1\}\},\]
then $\delta<1$ and for $u$ in a real neighbourhood of $0$ and $p$ in $\nn$,
\begin{equation}\label{eq_Jordan} \L_u^p = \lambda_u^p \big(\varphi_u(\cdot)\, \rho_u + O((\delta+\varepsilon)^p)\big) \end{equation}
for some $\varepsilon$ such that $\delta+\varepsilon <1$, where $\varphi_u$ is a linear form on $\mathcal I_1(\h)$, analytic in $u$ and such that $\varphi_0=\tr$ and the $O((\delta+\varepsilon)^p)$ is in terms of the operator norm on $\mathcal I_1(\h)$. This implies
\begin{equation}\label{eq_mgf2}
\frac1p\log \sum_{i_0\in V}\tr (\L_u^p(\rho(i_0))) = \log \lambda_u + \frac1p \log \sum_{i_0\in V}\varphi_u(\rho(i_0)) + O((\delta+\eps)^p)
\end{equation}
for $u$ in the above real neighbourhood of the origin. This and Lemma \ref{lemma_analyticitylambdau} implies that the identity
\begin{equation}\label{eq_mgf3}
\lim_{p\to\infty}\frac 1p\log\ee(\exp\braket u {X_p-X_0}) = \log \lambda_u
\end{equation}
holds for $u$ in a neighbourhood of the origin. In addition, by equation \eqref{eq_mgf2} and Corollary \ref{coro_derivatives},
\[ \lim_{p\to\infty}\frac 1{ p} \big(\nabla \log\ee(\exp\braket u {X_p-X_0})-pm)=0 \qquad \lim_{p\to\infty}\frac 1p \nabla^2 \log\ee(\exp\braket u {X_p-X_0})=C.\]
By an application of the multivariate version of Bryc's theorem (see Appendix A.4 in \cite{QSM}), we deduce that
\[ \frac{X_p-X_0-p\,m}{\sqrt p} \underset{p\to\infty}\longrightarrow \mathcal N(0,C)\]
and this proves our statement in the case where $\L$ is irreducible aperiodic.
We now consider the case where $\L$ is irreducible with period $d$. Let $p_0,\ldots, p_{d-1}$ be a cyclic partition of identity; then, writing $p=qd+r$ we have for any $i_0 \in V$
\begin{eqnarray*}
\tr\big(\L_u^p(\rho(i_0))\big)&=& \sum_{j=0}^{d-1}\tr\big(p_j\,\L_u^{qd+r}(\rho(i_0)) \, p_j\big)\\
&=& \sum_{j=0}^{d-1}\tr\big(\L_u^{qd}(p_j\,\L_u^r(\rho(i_0)) \, p_j)\big)
\end{eqnarray*}
by a straightforward extension of \eqref{eq_cyclicity} to $\L_u$. By Proposition \ref{prop-periodic-blocks}, for any $j$, $r$ and the previous discussion, one has
\[\lim_{q\to\infty}\frac1{qd} \log \tr\big(\L_u^{qd}(p_j\,\L_u^r(\rho(i_0)) \, p_j)\big) = \log \lambda_u\]
and one can extend all terms in this identity so that it holds in a complex neighbourhood of the origin. This finishes the proof of our statement. \fin
\begin{remark}
The reader might wonder why we need to go through the trouble of considering relations \eqref{eq_Jordan} and \eqref{eq_mgf2} to derive the extension of \eqref{eq_mgf3} to complex $u$. This is because there is no determination of the complex logarithm that allows to consider $\log\ee(\exp\braket u {X_p-X_0})$ for complex $u$ and arbitrarily large $p$. This forces us to start by transforming $\frac1p\log\ee(\exp\braket u {X_p-X_0})$.
\end{remark}
\paragraph{Generalizations of Theorems \ref{theo_ldp} and \ref{theo_clt}}
We finish with a discussion of possible generalizations of Theorems \ref{theo_ldp} and \ref{theo_clt} beyond the case of irreducible $\L$. To this aim, we introduce the following subspaces of $\h$:
\begin{equation}\label{eq_defDR}
\mathcal D = \{\phi\in \h\, |\,\langle\phi,\L^p(\rho)\,\phi\rangle\underset{p\to\infty}{\longrightarrow}0 \mbox{ for any state }\rho\} \quad \mbox{and} \quad \R=\D^\perp.
\end{equation}
Alternatively, $\R$ can be defined as the supremum of the supports of $\L$-invariant states, and $\D$ as $\R^\perp$. Note in particular that $\mathrm{dim}\,\R\geq 1$ and $\R$ is invariant by all operators $L_s$, $s\in S$. These subspaces are the Baumgartner-Narnhofer decomposition of $\h$ associated with $\L$ (see \cite{BN} or \cite{CP1}).
Note that, in \cite{CP1}, we only considered the spaces $\D_\M$ and $\R_\M$ associated with $\M$ instead of $\L$. Here the decomposition for $\M$ plays no role and $\R_\M$ is equal to $\{0\}$.
The following result will replace the Perron-Frobenius theorem when $\L$ is not irreducible. The proof can be easily adapted from Proposition \ref{prop_Schrader} and Lemma~\ref{lemma_analyticitylambdau}.
\begin{prop}\label{prop_PFnonirr}
The following properties are equivalent:
\begin{enumerate}
\item the auxiliary map $\L$ has a unique invariant state $\rhoinv$,
\item the restriction $\L_{|\mathcal I_1(\R)}$ of $\L$ to $\R$ is irreducible,
\item the value $1$ is an eigenvalue of $\L$ with algebraic multiplicity one.
\end{enumerate}
If, in addition, $\L_{|\mathcal I_1(\R)}$ is aperiodic, then $1$ is the only eigenvalue of modulus one, and for any state $\rho$, one has $\L^p(\rho)\underset{p\to\infty}\longrightarrow \rhoinv$.
\end{prop}
This leads to an extension of Theorem \ref{theo_clt} to the cases
\begin{itemize}
\item where $\L_{|\mathcal I_1(\mathcal R)}$ is irreducible (even if $\R \neq\h$); by Proposition \ref{prop_PFnonirr}, this is equivalent to $\L$ having a unique invariant state;
\item when $\mathcal R=\h$.
\end{itemize}
With these two extensions, our central limit theorem has the same generality as the one given in \cite{AGS}: the first case is Theorem 5.2 of that reference, the second case is treated in Section $7$ in \cite{AGS}.
These extensions are proven observing that:
\begin{itemize}
\item by Proposition \ref{prop_PFnonirr}, the proof of Theorem \ref{theo_clt} can immediately be extended to the situation where $\L_{|\mathcal I_1(\R)}$ is irreducible aperiodic, and from there to the situation where $\L_{|\mathcal I_1(\R)}$ is irreducible periodic;
\item when $\mathcal R=\h$, it admits a decomposition $\R=\oplus_k \R_k$ (see \cite{CP1}), each $\mathcal I_1(\R_k)$ is stable by $\L$, the restrictions $\L_{|\mathcal I_1(\R_k)}$ are irreducible, and the non-diagonal blocks do not appear in a probability like \eqref{eq_probatraj}.
\end{itemize}
We have seen in \cite{CP1} that one can always decompose $\h$ into $\h=\D\oplus \bigoplus_{k\in K} \R_k$
with each $\R_k$ as discussed above. However, in the general case, we do not have a clear statement of Theorems \ref{theo_ldp} and \ref{theo_clt} because
if $\D$ is non-trivial and $\mathrm{card}\,K\geq 2$, it is difficult to control how the mass of $\rho_0$ will flow from $\D$ into the different components $\R_k$.
Last, remark that the proof of Theorem \ref{theo_ldp} relies on the fact that $\L_u$ is irreducible. This holds if $\L$ is irreducible; the converse, however, is not true, and $\R_\L$ may be different from $\R_{\L_u}$. The proof of Theorem \ref{theo_ldp} can be extended to derive a lower large deviation bound in the case when $\R=\h$ using the idea described above, but when $\L$ is not irreducible, the quantity $\lambda_u$ may not be analytic, in which case we \emph{a priori} obtain only the upper large deviation bound, see Example \ref{ex_ldpbreakdown}.
\section{Open quantum random walks with lattice $\zz^d$ and internal space $\cc^2$}
\label{section_ZdC2}
The goal of this section is to illustrate our various concepts, and give explicit formulas in the case where $V=\zz^d$ and $\h=\cc^2$. We start with a study of the operators $\L$ and $\M$, and a characterization of their (ir)reducibility and of the associated decompositions of the state space in this specific situation.
We begin in Proposition \ref{prop_caracL} with a classification of the possible situations depending on the dimension of $\R$ (as defined in \eqref{eq_defDR}) and its possible decompositions. Then, in Lemma \ref{lemma_diag} we characterize those situations in terms of the form of the operators $L_s$. Later on, we also consider the period.
To avoid discussing trivial cases, we will make a second assumption:
\begin{center}\textbf{Assumption H2:} the operators $L_s$ are not all proportional to the identity.\end{center}
This is equivalent to saying that we assume $\L\neq \id$.
\medskip
We start by discussing the possible forms of $\R$ and $\D$:
\begin{prop}\label{prop_caracL}
Consider the operators $L_s$, $s\in S$, defining the open quantum random walk $\M$, and suppose that assumptions \emph{H1} and \emph{H2} hold. Then we are in one of the following three situations.
\begin{enumerate}
\item If the $L_{s}$ have no eigenvector in common, then $\L$ is irreducible, there exists a unique $\L$-invariant state which is faithful, and one has
\[\R = \h \qquad \D=\{0\}.\]
\item If the $L_{s}$ have only one (up to multiplication) eigenvector $e_1$ in common, then $\L$ is not irreducible, the state $\ketbra {e_1}{e_1}$ (if $\|e_1\|=1$) is the unique~$\L$-invariant state, and for any nonzero vector $e_2\perp e_1$, one has
\[ \R = \cc\, e_1\qquad \D = \cc\, e_2.\]
\item If the $L_{s}$ have two linearly independent eigenvectors $e_1$ and $e_2$ in common, any invariant state is of the form $\rhoinv=t\,\ketbra {e_1}{e_1} + (1-t)\ketbra {e_2}{e_2}$ for $t\in[0,1]$, and one has
\[\R = \h = \cc\, e_1 \oplus \cc\, e_2 \qquad \D=\{0\}.\]
\end{enumerate}
\end{prop}
\pre
We recall that, by the fourth equivalent statement in Definition \ref{defi_irreducibility}, the map~$\L$ is irreducible if and only if the $L_{s}$ do not have a common, nontrivial, invariant subspace. If~$\h=\cc^2$ then this is equivalent to saying that the $L_{s}$ do not have a common eigenvector.
Now assume that $\L$ is not irreducible, so that the $L_{s}$ have a common norm one eigenvector $e_1$, with $L_{s}\,e_1=\alpha_{s}\, e_1$ for all $s$. Then $\ketbra {e_1}{e_1}$ is an invariant state. Complete $(e_1)$ into an orthonormal basis $(e_1,e_2)$. Then, if $\rho$ is an invariant state, $\rho=\sum_{i,j=1,2}\rho_{i,j}\,\ketbra{e_i}{e_j}$, and
\[ \L(\rho)= \sum_{i,j=1,2}\, \sum_{s\in S}\rho_{i,j}\,\ketbra{L_s e_i}{L_s e_j}.\]
Then
\[\rho_{2,2}=\braket{e_2}{\rho\, e_2}=\sum_{s\in S}\rho_{2,2}\,|\braket{e_2}{L_s e_2}|^2\]
so that either $\rho_{2,2}=0$ or $\sum_{s\in S}|\braket{e_2}{L_s e_2}|^2=1$; but, since $\sum_{s\in S}\|L_s e_2\|^2=1$, this is possible only if $e_2$ is an eigenvector of all $L_s$, $s\in S$.
Therefore, in situation 2, $\ketbra{e_1}{e_1}$ is the only invariant state. In situation~3, observe that if there existed an invariant state with $\rho_{1,2}=\overline{\rho_{2,1}}\neq 0$, then any state would be invariant and $\L$ would be the identity operator, a case we excluded.
\fin
\begin{remark}
In situations $2$ and $3$ we recover the fact, proven in \cite{CP1} (and originally in \cite{BN}) that, if $\ketbra{e_1}{e_1}$ is an invariant state and $e_2\neq 0$ is in $e_1^\perp\cap\R$ then $\ketbra{e_2}{e_2}$ is an invariant state. The above proposition gives an explicit Baumgartner-Narnhofer decomposition of $\h$ (see \cite{BN} or sections 6 and 7 of \cite{CP1}).
In the case where $\h=\cc^2$, it turns out that $\R$ can always be written in a unique way as $\R=\bigoplus \R_k$ with $\L_{|\mathcal I_1(\R_k)}$ irreducible (except for the trivial case when $\L$ is the identity map). This is not true in general and is a peculiarity related to the low dimension of $\h$.
\end{remark}
\medskip
Next we study the explicit form of the operators $L_s$ in each of the situations described by Proposition \ref{prop_caracL}. We will use the standard notation that, for two families of scalars $(\alpha_s)_{s\in S}$ and $(\beta_s)_{s\in S}$, $\|\alpha\|^2$ is $\sum_{s\in S}|\alpha_s|^2$ and $\braket \alpha \beta$ is~$\sum_{s\in S}\overline{\alpha_s}\beta_s$.
\begin{lemma}\label{lemma_diag}
With the assumptions and notations of Proposition \ref{prop_caracL}:
\begin{itemize}
\item We are in situation $2$ if and only if there exists an orthonormal basis of~$\h=\cc^2$ in which
\[L_{s}=\begin{pmatrix}\alpha_{s} & \gamma_{s} \\ 0 & \beta_{s}\end{pmatrix}\]
for every $s$ with
\[\|\alpha\|^2 =\|\beta\|^2+\|\gamma\|^2=1, \qquad \braket\alpha\gamma=0,\]
\[\sup_{s\in S} |\beta_{s}| >0,\qquad \sup_{s\in S} |\gamma_{s}| >0,\]
\[\mbox{ there exist } s\neq s'\mbox{ in } S \mbox{ such that }(\alpha_s-\beta_s)\,\gamma_{s'}\neq (\alpha_{s'}-\beta_{s'})\,\gamma_{s}.\]
\item We are in situation $3$ if and only if there exists an orthonormal basis of~$\h=\cc^2$ in which
\[L_{s}=\begin{pmatrix}\alpha_{s} &0 \\ 0 & \beta_{s}\end{pmatrix}\]
for every $s$, with
\[\|\alpha\|^2 =\|\beta\|^2=1,\]
\[\mbox{there exists }s\mbox{ in } S \mbox{ such that } \alpha_s \neq \beta_s.\]\end{itemize}
\end{lemma}
\pre
This is immediate by examination.
\fin
\begin{remark}
In situation 2, let $\rho$ be any state. One has
\[ \braket {e_2} {\L^p(\rho)\, e_2} = \tr\big(\rho\, \L^{*\,p}(\ketbra {e_2}{e_2})\big) = \|\beta\|^{2p}\, \braket {e_2} {\rho\, e_2} \underset{p\to\infty}{\longrightarrow} 0 \]
by the observation that $ \|\beta\|^2 <1$. We recover the fact that $\D=\cc\, e_2$.
\end{remark}
\medskip
We now turn to the study of periodicity for the operator $\L$. We start with a simple remark:
\begin{remark}\label{rem_irrC2}
Whenever the operators $L_s$ have a common eigenvector $e$, then the restriction of $\L$ to $\mathcal I_1(\cc e)$ is aperiodic. In particular, if $\L$ is not irreducible but has a unique invariant state, then by necessity $\R$ is one-dimensional so that $\L_{|\R}$ must be aperiodic.
\end{remark}
In more generality, because $\mathrm{dim}\,\h=2$, by Remark \ref{remark_dsmallerdimh}, any irreducible~$\L$ has period either one or two. The following lemma characterizes those $L_s$ defining an operator $\L$ with period 2:
\begin{lemma}\label{lemma_Laperiodic}
The map $\L$ is irreducible periodic if and only if there exists a basis of $\h$ for which every operator $L_{s}$ is of the form $\begin{pmatrix} 0 & \gamma_{s}\\ \nu_{s}& 0\end{pmatrix}$. In that case, for any~$s\neq s'$, one has $\gamma_s \,\nu_{s'}\neq \gamma_{s'}\,\nu_s$ and $\|\gamma\|^2=\|\nu\|^2=1$, and the unique invariant state of $\L$ is $\frac12\,\id$.
\end{lemma}
\pre
If the period of $\L$ is two, then the cyclic partition of identity must be of the form $\ketbra {e_1}{e_1}, \ketbra {e_2}{e_2}$ and the cyclicity imposes the relations
\[L_{s} e_1 \in \cc\, e_2, \quad L_{s} e_2 \in \cc\, e_1 \quad \mbox{for any }s\in S.\]
This gives the form of the $L_s$. The condition $\sum_s|\gamma_s|^2=\sum_s|\nu_s|^2=1$ simply follows by the trace preservation property. Now observe that the eigenvalues of $L_s$ are solutions of $\lambda_s^2=\gamma_s \nu_s$. Fix one solution $\lambda_s$, the other being $-\lambda_s$. Then $\begin{pmatrix}x\\y\end{pmatrix}$ is an eigenvector if and only if $\gamma_s y=\pm \lambda_s \, x$.
Therefore, two operators $L_s$ and $L_{s'}$ have an eigenvector in common if and only if $\nu_s\,\lambda_{s'}=\pm\nu_{s'}\,\lambda_{s}$. This is easily seen to be equivalent to $\gamma_s \,\nu_{s'}= \gamma_{s'}\,\nu_s$. Last, one easily sees that the equation
\[\sum_{s\in S}L_s \begin{pmatrix}a&b\\c&d\end{pmatrix} L_s^* = \begin{pmatrix}a&b\\c&d\end{pmatrix}\]
is equivalent to $a=d$, $b=\braket \nu \gamma\, c$ and $c=\braket \gamma\nu \, b$. Moreover, $|\braket \gamma\nu|=1$ would imply that the vectors $(\gamma_s)_{s\in S}$ and $(\nu_s)_{s\in S}$ are proportional, which is forbidden by irreducibility. Therefore $a=d$ and $b=c=0$.
\fin
\medskip
The following theorem is a central limit theorem for all open quantum random walks satisfying H1 and H2. It gives more explicit expressions for the parameters of the limiting Gaussian, except when $\L$ is irreducible aperiodic, in which case the parameters of the Gaussian are given in Theorem~\ref{theo_clt}.
\begin{theo}\label{theo_hisc2}
Assume an open quantum random walk with $V=\zz^d$ and $\h=\cc^2$ satisfies assumptions \emph{H1}, \emph{H2}. Then there exist $m\in\cc^d$ and $C$ a $d\times d$ positive semi-definite matrix such that we have the convergence in law
\[ \frac{X_p-p\,m}{\sqrt p} \underset{p\to\infty}\longrightarrow \mathcal N(0,C).\]
Following the notation of Lemmas \ref{lemma_diag} and \ref{lemma_Laperiodic} we have:
\begin{itemize}
\item In situation 1, if $\L$ is periodic, consider two random variables $A$ and~$B$ with $\pp(A=s)=|\nu_s|^2$ and $\pp(B=s)=|\gamma_s|^2$. Then we have
\[m=\frac12(\ee(A)+\ee(B))\qquad C=\frac12(\mathrm{var}(A)+\mathrm{var}(B)).\]
\item In situation 2, consider a classical random variable $A$ with $\pp(A=s)=~|\alpha_s|^2.$ Then we have
\[m=\ee(A) \qquad C=\mathrm{var}(A).\]
\item In situation 3, consider two classical random variables $A$ and~$B$ with $\pp(A=s)=|\alpha_s|^2$ and $\pp(B=s)=|\beta_s|^2$, and denote
$p=\sum_{i\in V}\braket{e_1}{\rho(i)\,e_1}$, where $\rho$ is the initial state. Then we have
\[m=p\,\ee(A)+(1-p)\, \ee(B) \qquad C=p\,\mathrm{var}(A)+(1-p)\, \mathrm{var}(B).\]
\end{itemize}
\end{theo}
\pre
If $\L$ is irreducible periodic, for any $\sigma=\begin{pmatrix}\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{pmatrix}$, we have
$$
L_s \sigma L_s^* =
\begin{pmatrix}\sigma_{22} |\gamma_s|^2 & \sigma_{21}\gamma_s \bar\nu_s\\
\sigma_{12} \bar \gamma_s \nu_s & \sigma_{11} |\nu_s|^2 \end{pmatrix}.
$$
By direct examination of the equation $\L_u(\sigma)=\lambda_u \,\sigma$ we obtain
\begin{equation}\label{eq_lambdauirrap}
\lambda_u=\sqrt{\ee(\exp\braket u A)\,}\sqrt{ \ee(\exp\braket uB)\,}.
\end{equation}
We immediately deduce
\[\lambda_u' = \braket u {\frac12(\ee(A)+\ee(B))}\qquad \lambda_u''-\lambda_u'{}^2 = \braket u {\frac12(\mathrm{var}\,A+\mathrm{var}\,B)\,u}.\]
In situation 2, we can use the extension discussed at the end of section~\ref{section_cltldp} with $P_{\R}=\ketbra{e_1}{e_1}$, and apply the formulas of Theorem \ref{theo_clt} with $\L$ replaced by~$\L_{\mathcal I_1(\cc e_1)}$. We see easily that the largest eigenvalue of $\L_u$ is
\begin{equation}\label{eq_lambdausit2}
\lambda_u=\max(\sum_{s\in S}\e^{\braket us} |\alpha_s|^2, \sum_{s\in S}\e^{\braket us} |\beta_s|^2)
\end{equation}
and in a neighbourhood of zero, the first term is the largest, so that
\[\lambda_u'=\sum_{s\in S}\braket us\, |\alpha_s|^2 \quad \mbox{and}\quad \lambda_u'' = \sum_{s\in S} {\braket u s}^2\, |\alpha_s|^2.\]
In situation 3, we again use the extension discussed at the end of section~\ref{section_cltldp} with ${\R_1}=\cc{e_1}$ and $\R_2=\cc e_2$.
The limit parameters for each corresponding restriction are computed in the previous point and correspond to those for the random variables $A$ and $B$. Since for any initial state $\rho$, a probability $\pp(X_0~=~i_0, \ldots, X_n=i_n)$ equals
\[\braket{e_1}{\rho(i_0)\,e_1}\, \prod_{k=1}^{n}|\alpha_{i_k-i_{k-1}}|^2+\braket{e_2}{\rho(i_0)\,e_2}\, \prod_{k=1}^{n}|\beta_{i_k-i_{k-1}}|^2\]
and we recover the parameters given in the statement above.
\fin
\begin{remark}
The irreducible periodic case described above can be understood in terms of a classical random walk, in a similar way to situation 3. Indeed, call a site $i$ in $V$ odd or even depending on the parity of its distance to the origin. Then exchanging the order of the basis vectors $e_1$ and $e_2$ at odd sites only is equivalent to considering a non-homogeneous OQRW with
\[L_{i,i+s}=\begin{pmatrix}\nu_s&0\\ 0&\gamma_s\end{pmatrix}\quad \mbox{ if } i \mbox{ is even},\qquad
L_{i,i+s}=\begin{pmatrix}\gamma_s&0\\ 0&\nu_s\end{pmatrix}\quad \mbox{ if } i \mbox{ is odd}\]
(strictly speaking, such OQRWs do not enter into the framework of this article, but in the general case studied in \cite{CP1}).
Then, we define $(A_p)_{p\in \nn}$ and $(B_p)_{p\in \nn}$ to be two i.i.d. sequences with same law as $A$, $B$ respectively, and, if for example $X_0=0$ is even, we define a random variable $\pi$ to take the values $1$ and $2$ with probabilities $p=\braket{e_1}{\rho(i_0)\,e_1}$, $1-p$ respectively. Then, conditioned on $\pi=1$, the variable $X_p-X_0$ has the same law as $A_1+B_2+A_3+\ldots$ (where the sum stops at step~$p$). This explains the formulas given in Theorem~\ref{theo_hisc2} for situation~1, with $\L$ periodic, as well as the next proposition.
\end{remark}
\smallskip
For the case of irreducible, periodic $\L$ we also have a simpler explicit formula for the rate function of large deviations:
\begin{lemma}\label{lemma_ldpperiodicirr}
Assume an open quantum random walk with $V=\zz^d$ and $\h=\cc^2$ satisfies assumptions \emph{H1}, \emph{H2} and is irreducible periodic. Then, with the same notation as in Theorem \ref{theo_hisc2}, the position process $(X_p-X_0)_p$ satisfies a full large deviation principle, with rate function
\[c(u)=\frac12(\log\ee(\exp\braket uA)) + \log\ee(\exp\braket uB))).\]
\end{lemma}
\pre
This follows immediately from Theorem \ref{theo_ldp} and equation \eqref{eq_lambdauirrap} giving $\lambda_u$.
\fin
\begin{remark}\label{remark_LDPbreakdown}
In situation 2 of Lemma \ref{lemma_diag}, one sees that the largest eigenvalue $\lambda_u$ is given by \eqref{eq_lambdausit2}. For $u$ in a neighbourhood of zero, one has $\|\alpha_u\|>\|\beta_u\|$, but, if there exists $u$ such that $\|\alpha_u\|=\|\beta_u\|$, then $\lambda_u$ may not be differentiable and the large deviations principle may break down: see Example \ref{ex_ldpbreakdown}. A similar phenomenon can also appear in situation 3.
\end{remark}
\section{Examples}\label{section_examples}
\begin{example}
We consider the case $d=1$, $\h={\mathbb C}^2$, $S=\{-1,+1\}$. In this case we can characterize irreducibility and period of the open quantum random walk $\M$ through the transition matrices $L_s, \, s=\pm 1$. In this example, we denote $L_-=L_{-1}$, $L_+=L_{+1}$. We state the next two propositions without proofs, as these are lengthy. The extension of these statements to finite homogeneous open quantum random walks, as well as the proofs, will be given in a future note.
\begin{prop}\label{reducibility-C2}
{\rm Irreducibility.}
Define
\[W\overset{\mathrm{def}}{=} \{ \mbox{common eigenvectors of $L_+L_-$ and $L_-L_+$}\}.
\]
The homogeneous OQRW on ${\mathbb Z}$ is reducible if and only if one of the following facts holds
\begin{itemize}
\item $W$ contains an eigenvector of $L_-$ or $L_+$
\item $W=\cc e_0\, \cup\, \cc e_1\setminus\{0\},$ for some linearly independent vectors $e_0$ and $e_1$ satisfying $L_-e_0,L_+e_0\in \cc e_1$ and $L_-e_1,L_+e_1\in \cc e_0$.
\end{itemize}
\end{prop}
\begin{prop} {\rm Period.} Suppose that the open quantum random walk $\M$ is irreducible. Its period can only be $2$ or $4$. It is $4$ if and only if there exists an orthonormal basis of ${\mathbb C}^2$ such that the representation of the transition matrices in that basis is
$$
L_\varepsilon=\begin{pmatrix} a& 0\\ 0& b \end{pmatrix}, \qquad
L_{-\varepsilon}=\begin{pmatrix} 0 & c \\ d & 0 \end{pmatrix}
$$
for some $\varepsilon\in\{+,-\}$, where $a,b,c,d\in {\mathbb C}\setminus \{0\}$ are such that $|a|^2+|d|^2=|b|^2+|c|^2=1$.
\end{prop}
\end{example}
\begin{example}\label{ex_stdexample}
We consider the standard example from \cite{APSS}, which is treated in section 5.3 of \cite{AGS}. This open quantum random walk is defined by $V=\zz$, $\h=\cc^2$, and transition operators given in the canonical basis $e_1$, $e_2$ of $\cc^2$ by
\[L_+=\frac1{\sqrt 3}\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}\qquad L_-=\frac1{\sqrt 3}\begin{pmatrix}1 & 0 \\ -1 & 1\end{pmatrix}.\]
The only eigenvector of $L_+$ is $e_1$, the only eigenvector of $L_-$ is $e_2$, so that we are in situation 1 of Proposition \ref{prop_caracL} and $\L$ is irreducible. Again $L_+^2$ and $L_-^2$ have no eigenvector in common, so by Lemma \ref{lemma_Laperiodic}, we conclude that $\L$ is aperiodic (and therefore regular, by Lemma \ref{lemma_irrapereg}). We observe that $\rhoinv=\frac12\id$ is the invariant state of $\L$. We compute the quantities $m$ and $C\in\rr_+$ from Theorem \ref{theo_clt}:
\[m= \tr(L_+ L_+^*)-\tr(L_- L_-^*)=0.\]
To compute $C$ we need to find the solution $\eta$ of
\[(\id -\L)(\eta)= \frac16 \begin{pmatrix} \hphantom{,}1 & \hphantom{+}2 \\ \hphantom{,}2 & -1\end{pmatrix}\]
satisfying $\tr\,\eta=0$.
We find $\eta=\frac1{12}\begin{pmatrix} 5 & \hphantom{-}2 \\ 2 & -5\end{pmatrix}$, and we have
\begin{equation*}
C= \tr\big(L_+ \rhoinv L_+^*+ L_-\rhoinv L_-^*) + 2\,\tr\big(L_+ \eta L_+^*-L_-\eta L_-^*\big)=\frac89.
\end{equation*}
By Theorem \ref{theo_clt}, we have the convergence in law
\[\frac{X_p-X_0}{\sqrt p}\underset{p\to\infty}{\longrightarrow} \mathcal N(0, \frac89).\]
To verify the validity of this statement, in Figure \ref{figure_CLT1} we display the empirical cumulative distribution function of a 1000-sample of $\frac{X_p}{\sqrt{8p/9}}$ conditioned on~$X_0=~0$ for $p=10,100,1000$, and compare it to the cumulative distribution function of a standard normal variable.
\begin{remark}
In this and the following simulations, the initial state is assumed to be of the form $\rho(0)\otimes\ketbra 00$ and $\rho(0)$ is chosen randomly as $\frac{XX'}{\tr(XX')}$ where $X$ has independent entries with uniform law on $[0,1]$.
\end{remark}
\begin{figure*}[h]
\begin{center}
\includegraphics[width=1\textwidth]{FigureExample42.pdf}
\caption{C.D.F. of a 1000-sample of $(\frac{X_p}{\sqrt{8p/9}})_p$ with $X_0=0$, in Example \ref{ex_stdexample}.}
\label{figure_CLT1}
\end{center}
\end{figure*}
By Theorem \ref{theo_ldp}, the process $(\frac{X_p-X_0}p)_p$ satisfies a large deviation property with good rate function equal to the Legendre transform $I$ of $u\mapsto \log\lambda_u$, where $\lambda_u$ is the largest eigenvalue of $\L_u$. This map $\L_u$, written in the canonical basis of the set of two by two matrices, has basis
\[\frac13 \begin{pmatrix}
\e^{u}+\e^{-u} & \e^{u} & \e^{u} & \e^{u} \\
-\e^{-u} & \e^{u}+\e^{-u} & 0 & \e^{u}\\
-\e^{-u} & 0 & \e^{u}+\e^{-u} & \e^{u}\\
\e^{-u} & -\e^{-u} & -\e^{-u} & \e^{u}+\e^{-u}
\end{pmatrix} \]
and by a tedious computation, one shows that $\lambda_u$ equals
\[\frac13 \big(\e^{u}+\e^{-u}+(\e^{u}+\e^{-u}+\sqrt{\e^{2u}+\e^{-2u}+3})^{1/3} - (\e^{u}+\e^{-u}+\sqrt{\e^{2u}+\e^{-2u}+3})^{-1/3}\big).\]
As expected from Lemma \ref{lemma_analyticitylambdau}, this is a smooth and strictly convex function. Numerical computations prove that the rate function $I$ has the form displayed in Figure \ref{figure_ratefunction}.
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=0.6\textwidth]{ratefunction.pdf}
\caption{Rate function for $(\frac{X_p-X_0}p)_p$ in Example \ref{ex_stdexample} }
\label{figure_ratefunction}
\end{center}
\end{figure*}
\end{example}
\begin{example}\label{ex_Lnonirr}
We consider the open quantum random walk defined by $V=\zz$, $\h=\cc^2$, and transition operators given in the canonical basis $e_1$, $e_2$ of $\cc^2$ by
\[L_+=\begin{pmatrix}0 & \sqrt 3 /2 \\1/\sqrt2 & 0\end{pmatrix}\qquad L_-=\begin{pmatrix}0 & 1/2 \\ 1/\sqrt 2 & 0\end{pmatrix}.\]
From Lemma \ref{lemma_Laperiodic}, the map $\L$ is irreducible and $2$-periodic. Then according to Theorem \ref{theo_hisc2}, defining $A$ and $B$ to be random variables with values in $S$ satisfying
\[\pp(A=+1)=1/2,\qquad \pp(A=-1)=1/2,\]
\[\pp(B=+1)=3/4,\qquad \pp(B=-1)=1/4,\]
with mean, variance, and cumulant generating function
\[m_A=0,\quad C_A=1,\quad c_A(u)=\log(\e^{u}+\e^{-u})-\log 2\]
\[m_B=1/2,\quad C_B=3/4,\quad c_B(u)=\log(3\e^{u}+\e^{-u})-2\log 2\]
then with the notation of Theorem \ref{theo_hisc2}
\[m=(m_A+m_B)/2 = 1/4\qquad C=(C_A+C_B)/2=7/8\]
and one has the convergence in law
\[\frac{X_p- p/4}{\sqrt p}\underset{p\to\infty}{\longrightarrow} \mathcal N(0,\frac78).\]
Figure \ref{figure_CLT3} below displays the cumulative distribution function of a 1000-sample of $\frac{X_p-p/4}{\sqrt{7p/8}}$, conditioned on $X_0=0$, and the cumulative distribution function of a standard normal variable for $p=10,100,1000$.
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=1.0\textwidth]{FigureExample43.pdf}
\caption{C.D.F. of a 1000-sample of $\frac{X_p-p/4}{\sqrt{7p/8}}$ with $X_0=0$, in Example \ref{ex_Lnonirr}.}
\label{figure_CLT3}
\end{center}
\end{figure*}
In addition, the process $(\frac{X_p-X_0}p)_{p\in\nn}$ satisfies a large deviation property with a good rate function $I$ obtained as the Legendre transform of
\[c(u)=\frac12(c_A(u)+c_B(u))= \frac12\big(\log(\e^{u}+\e^{-u})+\log(3\e^{u}+\e^{-u})\big)-\frac32\log 2.\]
Explicitly, one finds that $I(t)=+\infty$ for $t\not\in]-1,1[$ and for $t\in]-1,+1[$:
\[I(t)= t\, u_t + \frac32\log 2 - \frac12\big(\log(\e^{u_t}+\e^{-u_t})+\log(3\e^{u_t}+\e^{-u_t})\big)\]
where $u_t=\frac12\, \log\,\frac{2t+\sqrt{t^2+3}}{3(1-t)}$. This rate function has the profile displayed in Figure \ref{figure_Lnonirr}.
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=0.6\textwidth]{ratefunctionExample44.pdf}
\caption{Rate function for $(\frac{X_p-X_0}p)_p$ in Example \ref{ex_Lnonirr}}
\label{figure_Lnonirr}
\end{center}
\end{figure*}
\end{example}
\begin{example}\label{ex_ldpbreakdown}
Consider the open quantum random walk defined by $V=\zz$, $\h=\cc^2$, and transition operators given in the canonical basis $e_1$, $e_2$ of $\cc^2$ by
\[L_+=\begin{pmatrix}\frac1{\sqrt 2} & \frac1{2\sqrt 2} \\ 0 & \frac{\sqrt 3}2\end{pmatrix}\qquad L_-=\begin{pmatrix}\frac1{\sqrt 2} & -\frac1{2\sqrt 2} \\ 0 & 0 \end{pmatrix}.\]
First observe that the map $\L$ is not irreducible in this case, as we are in situation~2 of Proposition \ref{prop_caracL}. A straightforward computation shows that the largest eigenvalue of $\L_u$ is
\[\lambda_u=\sup(\frac{\e^u+\e^{-u}}2, \frac {3\,\e^u}{4}).\]
For $u$ close to zero $\lambda_u$ is $\frac{\e^u+\e^{-u}}2$ so that $\lambda_u'=0$ and $\lambda_u''=1$ for $u=0$. We must therefore have
\[\frac{X_p-X_0}{\sqrt p} \underset{p\to\infty}{\longrightarrow} \mathcal N(0,1).\]
Figure \ref{figure_CLT4} below displays the cumulative distribution function of a 1000-sample of $\frac{X_p}{\sqrt{p}}$, conditioned on $X_0=0$, and the cumulative distribution function of a standard normal variable for $p=10,100,1000$.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=1.0\textwidth]{FigureExample57.pdf}
\caption{C.D.F. of a 1000-sample of $(\frac{X_p-X_0}{\sqrt{p}})_p$ with $X_0=0$, in Example \ref{ex_ldpbreakdown}.}
\label{figure_CLT4}
\end{center}
\end{figure*}
Due to the generalizations discussed at the end of Section \ref{section_cltldp}, we have $\R=\cc e_1$, $\D=\cc e_2$ and the central limit theorem holds: the behavior of the process~$(X_p)_p$, associated with $\L$, is the same as the one of the process $(\widetilde X_p)_p$ associated with the restriction $\L_{|\R}$.
As we commented previously, giving a large deviations result in this case is harder and we cannot use the general results we proved. A G\"artner-Ellis theorem could be applied by direct computation of the moment generating functions. In general, however, the rate function for the process $(X_p)_p$ will not coincide with the one for $(\widetilde X_p)_p$, since it will essentially depend on how much time the evolution spends in $\D$.
More precisely, for the transition matrices introduced above and taking the initial state $\rho=\ketbra {e_2}{e_2}\otimes \ketbra 0 0$, we have, by relation \eqref{eq_probatraj},
$$
P(X_n = n) = \tr(\ketbra {L_+^ne_2}{L_+^ne_2})
= \left(\frac{3}{4}\right)^n + \left(\frac{1}{8}\right)2^{1-n} \frac{\sqrt {3^n}-\sqrt {2^n}}{\sqrt 3- \sqrt 2}
$$
and consequently
$$\lim_n \frac 1n \log \ee[e^{uX_n}] \ge \log\big(\frac 34 e^u\big)\qquad \mbox{ for all } u,$$
while $\lim_n \frac 1n \log \ee[e^{u\widetilde X_n}] = \log\big(\frac {e^u+e^{-u}}{2}\big)$, which for $u>\log 2$ is smaller than the bound
$\log\big(\frac 34 e^u\big)$.
This clarifies the fact that the large deviations will not depend only on $\L_{|\R}$. Moreover, a second problem arises in this example, which is the lack of regularity of $\lambda_u$. Indeed, $\lambda_u$ is the supremum of two quantities which coincide for $u_0=\frac12\log2$, and $\log\lambda_u$ is not differentiable at $u_0$: the left derivative is equal to $\frac{\sqrt2}4$ and the right derivative to $\frac{3\sqrt2}4$. The Legendre transform of $\log\lambda_u$ is displayed in Figure \ref{figure_ldpbreakdown}, and we observe that it is not strictly convex.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=0.6\textwidth]{ratefunctionExample45.pdf}
\caption{Rate function for $(\frac{X_p-X_0}p)_p$ in Example \ref{ex_ldpbreakdown}}
\label{figure_ldpbreakdown}
\end{center}
\end{figure*}
\end{example}
\bibliography{biblio}
\end{document}
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