Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 1.2 For \( n \geq 5 \), there exist non-standard lattices in \( {\mathbb{R}}^{n} \) . | Proof For \( n \geq 5 \), let\n\n\[ \n{\Lambda }_{n} = \{ \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right) \in {\mathbb{Z}}^{n} \mid {a}_{1} \equiv \cdots \equiv {a}_{n}{\;(\operatorname{mod}\;2)}\} ; \n\]\n\nthen \( {\Lambda }_{n} \) is a discrete additive subgroup of \( {\mathbb{R}}^{n} \) not contained in any \( \lef... | Yes |
Proposition 2.1 For \( n \geq 3 \), there exists non-standard lattices in \( {\mathbb{R}}^{n} \) with \( {L}^{1} \) -norm. | Proof We can show that the lattice \( {\Lambda }_{n} \) constructed in the proof of Theorem 1.2 are non-standard with \( {L}^{1} \) -norm for \( n \geq 3 \) . Similar as the proof in Theorem 1.2, the successive minima of \( {\Lambda }_{n} \) is \( {\lambda }_{1} = \cdots = {\lambda }_{n} = 2 \), but any vector in \( {\... | Yes |
Proposition 1.1 Let \( R \) be any ring. Then \( \mathop{\prod }\limits_{{i \in I}}{N}_{i} \in {\mathcal{C}}_{n} \) if and only if each \( {N}_{i} \in {\mathcal{C}}_{n} \) , \( i \in I \) . | Proof Let \( N = \mathop{\prod }\limits_{{i \in I}}{N}_{i} \) . If each \( {N}_{i} \in {\mathcal{C}}_{n} \), then for any \( F \in {\mathcal{F}}_{n} \), we have\n\n\[ \n{\operatorname{Ext}}_{R}^{1}\left( {F, N}\right) = \mathop{\prod }\limits_{{i \in I}}{\operatorname{Ext}}_{R}^{1}\left( {F,{N}_{i}}\right) = 0.\n\]\n\n... | Yes |
Lemma 1.1 Let \( R \) be right \( n \) -coherent. Then\n\n(1) \( {\mathcal{F}}_{n} \) is a resolving class (i.e., \( {\mathcal{F}}_{n} \) contains projective modules, is closed under extensions and if whenever \( 0 \rightarrow {F}^{\prime } \rightarrow F \rightarrow {F}^{\prime \prime } \rightarrow 0 \) is exact with \... | Proof (1) Clearly, the class of projective modules \( \mathcal{P} \) is contained in \( {\mathcal{F}}_{n} \) . Let\n\n\[ 0 \rightarrow {F}^{\prime } \rightarrow F \rightarrow {F}^{\prime \prime } \rightarrow 0 \]\n\nbe a short exact sequence and \( M \) any \( n \) -presented right \( R \) -module. If \( {F}^{\prime },... | Yes |
Proposition 2.1 Let \( R \) be right \( n \) -coherent. Then \( {\mathcal{F}}_{n} \) is covering. | Proof \( \; \) By Lemma 2.3, we only need to prove that \( {\mathcal{F}}_{n} \) is closed under pure quotient modules.\n\n\[ 0 \rightarrow S \rightarrow F \rightarrow F/S \rightarrow 0 \]\n\nbe a pure exact sequence with \( F \in {\mathcal{F}}_{n} \) . Then we get a split exact sequence\n\n\[ 0 \rightarrow {\left( F/S\... | Yes |
Proposition 2.2 Let \( R \) be right \( n \) -coherent. Then \( \left( {{\mathcal{F}}_{n},{\mathcal{C}}_{n}}\right) \) forms a cotorsion pair. | Proof \( \; \) Since \( \left( {{}^{ \bot }\left( {\mathcal{F}}_{n}^{ \bot }\right) ,{\mathcal{C}}_{n}}\right) \) forms a cotorsion pair, we must show \( {}^{ \bot }\left( {\mathcal{F}}_{n}^{ \bot }\right) = {\mathcal{F}}_{n}.\;{\mathcal{F}}_{n}{ \subseteq }^{ \bot }\left( {\mathcal{F}}_{n}^{ \bot }\right) \) is clear.... | Yes |
Theorem 2.1 Let \( R \) be right \( n \) -coherent. Then the cotorsion pair \( \left( {{\mathcal{F}}_{n},{\mathcal{C}}_{n}}\right) \) is cogenerated by a set. | Proof Let \( F \in {\mathcal{F}}_{n} \) and \( x \in F \) be any element of \( F \) . By Lemma 2.2 there exists a pure submodule \( {S}_{0} \) of \( F \) such that \( x \in {S}_{0} \) and \( \left| {S}_{0}\right| \leq \aleph \left( {\aleph \geq \left| R\right| }\right) \) . Then consider \( F/{S}_{0} \) and choose any ... | Yes |
Corollary 2.1 Let \( R \) be right \( n \) -coherent. Then \( {\mathcal{C}}_{n} \) is enveloping. | Proof Over right \( n \) -coherent rings, we get that the cotorsion pair \( \left( {{\mathcal{F}}_{n},{\mathcal{C}}_{n}}\right) \) have enough projectives and enough injectives by Theorem 2.1 and Lemma 2.1. By the proof of Proposition 2.1, we know that \( {\mathcal{F}}_{n} \) is closed under pure quotient modules. Sinc... | Yes |
Theorem 3.1 Let \( R \) be right \( n \) -coherent. Then \( {\mathcal{F}}_{n} \) is preenveloping. | Proof Since \( R \) is right \( n \) -coherent, \( {\mathcal{F}}_{n} \) is closed under direct products by \( \lbrack 3 \), Theorem 3.1]. For any \( R \) -module \( M \), by Lemma 2.2, there is a cardinal number \( {\aleph }_{\alpha } \) such that for any \( R \) -homomorphism \( f : M \rightarrow L \) with \( L \in {\... | No |
Proposition 3.1 Let \( R \) be right \( n \) -coherent. Then \( {\operatorname{Hom}}_{R}\left( {-, - }\right) \) is left balanced on \( {}_{R}\mathfrak{M}{ \times }_{R} \) \( \mathfrak{M} \) by \( {\mathcal{F}}_{n} \times {\mathcal{F}}_{n} \) . | Proof By Proposition 2.1 and Theorem 3.1, every \( R \) -module \( M \) has an \( {\mathcal{F}}_{n} \) -precover and \( {\mathcal{F}}_{n} \) -preenvelope over right \( n \) -coherent rings. So for any \( R \) -module \( M \), there is a complex\n\n\[ 0 \rightarrow M \rightarrow {F}^{0} \rightarrow {F}^{1} \rightarrow \... | Yes |
Proposition 3.2 Let \( R \) be right \( n \) -coherent and \( m \) a nonnegative integer. Then the following statements are equivalent for an \( R \) -module \( M \) .\n\n(1) left \( {\mathcal{F}}_{n} \) - \( \dim M \leq m \) .\n\n(2) \( {\operatorname{Tor}}_{m + n + k}^{R}\left( {N, M}\right) = 0 \) for any \( n \) -p... | Proof (1) \( \Rightarrow \) (2). Since left \( {\mathcal{F}}_{n} - \dim M \leq m \), there exists an exact sequence\n\n\[ 0 \rightarrow {F}_{m} \rightarrow \cdots \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow M \rightarrow 0 \]\n\nsuch that each \( {F}_{i}\left( {i = 0,1,\cdots, m}\right) \in {\mathcal{F}}_{n} \)... | Yes |
Proposition 3.3 Let \( R \) be right \( n \) -coherent and \( m \geq 2 \) . Then the following statements are equivalent.\n\n(1) left \( {\mathcal{F}}_{n} - \dim M \leq m \) .\n\n(2) \( {\operatorname{Ext}}_{m + k}^{R}\left( {N, M}\right) = 0 \) for all \( N{ \in }_{R}\mathfrak{M} \) and integers \( k \geq - 1 \) .\n\n... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) . Let\n\n\[ 0 \rightarrow {F}_{m} \rightarrow \cdots \rightarrow {F}_{1} \rightarrow {F}_{0} \rightarrow M \rightarrow 0 \]\n\nbe a left \( {\mathcal{F}}_{n} \) -resolution of \( M \) . Then\n\n\[ 0 \rightarrow {\operatorname{Hom}}_{R}\left( {N,{F}_{m}}\right) \ri... | Yes |
Theorem 3.2 Let \( R \) be right \( n \) -coherent and \( m \geq 2 \) . Then the following statements are equivalent.\n\n(1) gl right \( {\mathcal{F}}_{n} \) - \( {\dim }_{R}\mathfrak{M} \leq m - 2 \) .\n\n(2) gl left \( {\mathcal{F}}_{n} - {\dim }_{R}\mathfrak{M} \leq m \) .\n\n(3) \( {\operatorname{Ext}}_{m - 1}^{R}\... | Proof This follows from Propositions 3.3 and 3.4. | No |
Theorem 2.2 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra, \( \alpha \) a surjection and \( \mathrm{Z}\left( L\right) \) the center of \( L \) . Then \( \left\lbrack {\mathrm{C}\left( L\right) ,\mathrm{{QC}}\left( L\right) }\ri... | Proof Assume that \( {D}_{1} \in {\mathrm{C}}_{{\alpha }^{k}}\left( L\right) ,{D}_{2} \in {\mathrm{{QC}}}_{{\alpha }^{s}}\left( L\right) \) . For all \( {x}_{1} \in L \), since \( \alpha \) is surjection, \( \forall {x}_{2},\cdots ,{x}_{n} \in L,\exists {x}_{2}^{\prime },\cdots ,{x}_{n}^{\prime } \in L \), such that \(... | Yes |
Theorem 2.3 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra over a field \( \mathbb{F} \) . If \( \operatorname{char}\mathbb{F} \neq n - 1 \), then \( \operatorname{ZDer}\left( L\right) = \mathrm{C}\left( L\right) \cap \operatorn... | Proof Assume that \( D \in {\mathrm{C}}_{{\alpha }^{k}}\left( L\right) \cap {\operatorname{Der}}_{{\alpha }^{k}}\left( L\right) \) . For all \( {x}_{1},{x}_{2},\cdots ,{x}_{n} \in L \), we have\n\n\[ D\left( \left\lbrack {{x}_{1},{x}_{2},\cdots ,{x}_{n}}\right\rbrack \right) = \mathop{\sum }\limits_{{i = 1}}^{n}\left\l... | Yes |
Proposition 2.2 If \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) is a multiplicative \( n \) -Hom-Nambu-Lie algebra with the operation \( {D}_{1} \bullet {D}_{2} = {D}_{1}{D}_{2} + {D}_{2}{D}_{1} \) for all elements \( {D}_{1},{D}_{2} \in \mathcal{U} \), then the triple \( \left( {... | Proof Similar to [1]. | No |
Proposition 2.3 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra with the operation \( {D}_{1} \bullet {D}_{2} = {D}_{1}{D}_{2} + {D}_{2}{D}_{1} \) for all elements \( {D}_{1},{D}_{2} \in \mathrm{{QC}}\left( L\right) \) . Then the... | Proof We only need to show that \( {D}_{1} \bullet {D}_{2} \in \mathrm{{QC}}\left( L\right) \) . For any \( {D}_{1} \in {\mathrm{{QC}}}_{{\alpha }^{\mathrm{k}}}\left( L\right) ,{D}_{2} \in \) \( {\mathrm{{QC}}}_{{\alpha }^{\mathrm{s}}}\left( L\right) \) and \( {x}_{1},{x}_{2},\cdots ,{x}_{n} \in L \), we have\n\n\[ \le... | Yes |
If \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) is a multiplicative \( n \) -Hom-Nambu-Lie algebra over \( \mathbb{F} \), then we have\n\n(1) If char \( \mathbb{F} \neq 2 \), then \( \mathrm{{QC}}\left( L\right) \) is a Hom-Lie algebra with \( \left\lbrack {{D}_{1},{D}_{2}}\right\... | Proof \( \left( 1\right) \left( \Leftarrow \right) \) For all \( {D}_{1},{D}_{2} \in \mathrm{{QC}}\left( L\right) \), we have \( {D}_{1}{D}_{2} \in \mathrm{{QC}}\left( L\right) \) and \( {D}_{2}{D}_{1} \in \mathrm{{QC}}\left( L\right) \), so \( \left\lbrack {{D}_{1},{D}_{2}}\right\rbrack = {D}_{1}{D}_{2} - {D}_{2}{D}_{... | Yes |
Proposition 3.1 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra over \( \mathbb{F} \) and \( t \) an indeterminate. We define \( \breve{T} \mathrel{\text{:=}} \{ \sum \left( {x \otimes t + y \otimes {t}^{n}}\right) \subset L \oti... | Proof For all \( {x}_{1},{x}_{2},\cdots ,{x}_{n},\;{y}_{1},{y}_{2},\cdots ,{y}_{n} \in L \) and \( {i}_{1},{i}_{2},\cdots ,{i}_{n},\;{i}_{1}^{\prime },{i}_{2}^{\prime },\cdots ,{i}_{n}^{\prime } \in \{ 1, n\} , \) we have\n\n\[ \left\lbrack {{x}_{1} \otimes {t}^{{i}_{1}},\cdots ,{x}_{p} \otimes {t}^{{i}_{p}},\cdots ,{x... | Yes |
Proposition 4.1 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra with a surjection \( \alpha \) . If \( L \) has no nonzero Hom-ideals \( I, J \) with \( \left\lbrack {I, J, L,\cdots, L}\right\rbrack = 0 \), i.e., \( L \) is prime... | Proof First id \( \in \mathrm{C}\left( L\right) \) . If there exist \( 0 \neq \psi \in {\mathrm{C}}_{{\alpha }^{k}}\left( L\right) ,\;0 \neq \varphi \in {\mathrm{C}}_{{\alpha }^{s}}\left( L\right) \) such that \( {\psi \varphi } = 0, \) then there exist \( x, y,{x}^{\prime },{y}^{\prime } \in L \) such that \( \psi \le... | Yes |
Proposition 4.2 Let \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) be a multiplicative \( n \) -Hom-Nambu-Lie algebra over a field \( \mathbb{F} \) . Then the following results hold:\n\n(1) If \( \alpha \) is a surjection, then \( L \) is indecomposable (cannot be written as the dir... | \( \mathrm{{Proof}}\left( 1\right) \left( \Rightarrow \right) \mathrm{{If}}\;\mathrm{{there}}\;\mathrm{{exists}}\;\psi \in {\mathrm{C}}_{{\alpha }^{k}}\left( L\right) \;\mathrm{{being}}\;\mathrm{{an}}\;\mathrm{{idempotent}}\;\mathrm{{and}}\;\mathrm{{satisfies}}\;\psi \neq 0,\mathrm{{id}},\mathrm{{then}} \) \( {\psi }^{... | Yes |
Proposition 4.3 If \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) is a multiplicative \( n \) -Hom-Nambu-Lie algebra over a field \( \mathbb{F} \) with \( \alpha \) a surjection, \( I \) is an \( \alpha \) -invariant subspace of \( L \), then \( {Z}_{L}\left( I\right) \) is invarian... | Proof For any \( \psi \in {\mathrm{C}}_{{\alpha }^{k}}\left( L\right) \) and \( {x}_{1} \in {\mathrm{Z}}_{L}\left( I\right) ,\forall {x}_{2} \in I,{x}_{3},\cdots ,{x}_{n} \in L \), there are \( {x}_{2}^{\prime },\cdots ,{x}_{n}^{\prime } \) such that \( {x}_{2} = {\alpha }^{k}\left( {x}_{2}^{\prime }\right) ,\cdots ,{x... | Yes |
Theorem 4.2 Let \( \left( {{L}_{1},\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,{\alpha }_{1}}\right) \) and \( \left( {{L}_{2},\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,{\alpha }_{2}}\right) \) be two \( n \) -Hom-Nambu-Lie algebras over field \( \mathbb{F} \) with \( {\alpha }_{1} \) a surjection. Le... | Proof (1) It is easy to see that \( {\pi }_{\mathrm{{End}}} \) is a Hom-algebra homomorphism. In fact, for all \( f, g \in \) \( {\operatorname{End}}_{\mathbb{F}}\left( {{L}_{1},\operatorname{Ker}\pi }\right) \), we have \( \pi \left( {fg}\right) = \left( {\bar{f}\pi }\right) g = \bar{f}\bar{g}\pi \), so \( {\pi }_{\ma... | Yes |
Theorem 5.1 If \( \left( {L,\left\lbrack {\cdot ,\cdots , \cdot }\right\rbrack ,\alpha }\right) \) is a central simple \( n \) -Hom-Nambu-Lie algebra over an algebraically closed field \( \mathbb{F}, A \) is a unital associative algebra and \( \widetilde{L} = L \otimes A \), then \( \mathrm{C}\left( \widetilde{L}\right... | Proof From Theorem 4.2, it is easy to verify that a central simple \( n \) -Hom-Nambu-Lie algebra over an algebraically closed field \( \mathbb{F} \) is an \( n \) -Nambu-Lie algebra. So the discussion of the tensor product of a central simple \( n \) -Hom-Nambu-Lie algebra and an associative algebra with unit element ... | No |
Theorem 2.1 Let \( f\left( x\right) \in {L}_{\Phi }^{ * }\lbrack 0,\infty ),\;\varphi \left( x\right) \) be a step weight function and \( w\left( x\right) = {x}^{a}{\left( 1 + x\right) }^{b} \) be a Jacobi weight function. Then for some constants \( C \) and \( {t}_{0} \), we get\n\n\[ \n{C}^{-1}{\omega }_{r,\varphi }{... | Proof Since \( {\bar{K}}_{r,\varphi }{\left( f,{t}^{r}\right) }_{w,\Phi } \geq {K}_{r,\varphi }{\left( f,{t}^{r}\right) }_{w,\Phi } \), we only need to prove the upper estimate. We can split the third term of \( {\bar{K}}_{r,\varphi }{\left( f,{t}^{r}\right) }_{w,\Phi } \) as\n\n\[ \n\parallel w{g}^{\left( r\right) }{\... | No |
Lemma 3.1 \\[ {}^{\\left\\lbrack 5\\right\\rbrack } \\]\n\n\\[ 1 = {J}_{n,0}\\left( x\\right) > {J}_{n,1}\\left( x\\right) > \\cdots > {J}_{n, k}\\left( x\\right) > {J}_{n, k + 1}\\left( x\\right) > \\cdots > 0; \\] | \\[ \\begin{array}{l} {p}_{n,0}^{\\prime }\\left( x\\right) = - n{p}_{n,0}\\left( x\\right) ,\\;{p}_{n, k}^{\\prime }\\left( x\\right) = n\\left( {{p}_{n, k - 1}\\left( x\\right) - {p}_{n, k}\\left( x\\right) }\\right) ,\\;k = 1,2,\\cdots ; \\end{array} \\]\n\n\\[ {J}_{n,0}^{\\prime }\\left( x\\right) = 0,\\;{J}_{n, k}... | Yes |
Theorem 3.1 (Direct theorem) Let \( f \in {L}_{\Phi }^{ * }\lbrack 0,\infty ),\;w\left( x\right) = {x}^{a}{\left( 1 + x\right) }^{b},\;\alpha \geq 1,\;\varphi \left( x\right) = \sqrt{x}, \) \( a \geq 0, a + b \geq 0,\Psi \in {\Delta }_{2} \) . Then\n\n\[ \n{\begin{Vmatrix}w\left( {S}_{n,\alpha }\left( f\right) - f\righ... | Proof By Lemma 3.3, Lemma 3.5 and Theorem 2.1, we have\n\n\[ \n{\begin{Vmatrix}w\left( {S}_{n,\alpha }\left( f\right) - f\right) \end{Vmatrix}}_{\Phi } \leq {\begin{Vmatrix}w{S}_{n,\alpha }\left( f - g\right) \end{Vmatrix}}_{\Phi } + \parallel w\left( {f - g}\right) {\parallel }_{\Phi } + {\begin{Vmatrix}w\left( {S}_{n... | Yes |
Theorem 2.1 Let \( 0 < p \leq q < \infty ,1 < {s}_{i} < p, i = 1,2 \) and let \( u \) and \( v \) be weight functions on \( {\mathbb{R}}_{ + }^{2} \) . Then the inequality\n\n\[ \n{\begin{Vmatrix}{G}_{2}f\end{Vmatrix}}_{{L}^{q}\left( {{\mathbb{R}}_{ + }^{2}, u}\right) } \leq {C}_{p, q}^{ * }\parallel f{\parallel }_{{L}... | Proof Assume that (2.2) holds. If we use the definition \( w\left( {{x}_{1},{x}_{2}}\right) \) as in (0.7) and let\n\n\[ \ng\left( {{x}_{1},{x}_{2}}\right) = {f}^{p}\left( {{x}_{1},{x}_{2}}\right) v\left( {{x}_{1},{x}_{2}}\right) \n\]\n\nin (2.1), then the inequality (2.1) is equivalent to the following inequality\n\n\... | Yes |
Lemma 2.8 Suppose that \( 1 < {p}_{0} < \infty \) and \( \omega \in {A}_{{p}_{0}} \), then we have\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}{\mu }_{\Omega }\left( f\right) {\left( x\right) }^{{p}_{0}}\omega \left( x\right) \mathrm{d}x \leq {C}_{0}{\int }_{{\mathbb{R}}^{n}}f{\left( x\right) }^{{p}_{0}}\omega \left( x\right) \m... | Proof Denote \( {\mu }_{m, s} \) as the following Marcinkiewicz integral operator\n\n\[ \n{\mu }_{m, s}\left( f\right) \left( x\right) = {\left\{ {\left. {\int }_{0}^{\infty }{\left| {\int }_{\left| {x - y}\right| \leq t}\frac{{Y}_{m, s}\left( {\left( x - y\right) }^{\prime }\right) }{{\left| x - y\right| }^{n - 1}}f\l... | Yes |
Theorem 3.2 Let \( k \in {\mathbb{N}}_{0},{q}_{1}\left( \cdot \right) ,{q}_{2}\left( \cdot \right) \in {\mathcal{P}}^{0}\left( {\mathbb{R}}^{n}\right), p\left( \cdot \right) ,\alpha \left( \cdot \right) \in {\mathcal{P}}_{0}^{\log }\left( {\mathbb{R}}^{n}\right) \cap {\mathcal{P}}_{\infty }^{\log }\left( {\mathbb{R}}^{... | Proof \( {Foranyk} \in {\mathbb{N}}_{0}, \) we consider the norm \( \parallel {\left( \frac{{2}^{{k\alpha }\left( \cdot \right) }\left| {{\mu }_{\Omega }\left( f\right) }\right| {\chi }_{k}}{\lambda }\right) }^{q\left( \cdot \right) }{\parallel }_{{L}^{\frac{p\left( \cdot \right) }{q\left( \cdot \right) }}}, \) since\n... | Yes |
Theorem 0.2 Let \( \left( {{M}^{m}, g}\right) \) be a compact biharmonic Lagrangian submanifold in the complex projective space \( {\mathbb{{CP}}}^{m}\left( {4k}\right) \) . Then \( {\left| \mathbf{H}\right| }_{\min }^{2} \leq \frac{k\left( {m + 3}\right) }{m}. \) | In this paper, \( {\left| \mathbf{H}\right| }_{\min }^{2} \) denotes the minimum of \( {\left| \mathbf{H}\right| }^{2} \) . The proof depends on the maximum principle, and cannot be extended to the non-compact case. However, when \( M \) is complete, we can prove similar results under suitable conditions on \( M \) . T... | No |
Proposition 2.1 \( {}^{\left\lbrack 4,9\right\rbrack } \) Let \( \left( {{M}^{m}, g}\right) \) be a Lagrangian submanifold in \( {\mathbb{{CP}}}^{m}\left( {4k}\right) \) . Then it is biharmonic if and only if\n\n\[ \bar{\Delta }\mathbf{H} - k\left( {m + 3}\right) \mathbf{H} = 0. \] | Proof By the curvature formula (5),\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{R}^{N}(\mathbf{H},{e}_{i}){e}_{i} = k\mathop{\sum }\limits_{{i = 1}}^{m}\{ \langle {e}_{i},{e}_{i}\rangle \mathbf{H} - \langle \mathbf{H},{e}_{i}\rangle {e}_{i} + \langle J{e}_{i},{e}_{i}\rangle J\mathbf{H} - \langle J\mathbf{H},{e}_{i}\rangl... | Yes |
Theorem 2.1 Let \( \left( {{M}^{m}, g}\right) \) be a compact biharmonic Lagrangian submanifold in a complex projective space \( \;{\mathbb{{CP}}}^{m}\left( {4k}\right) .\; \) Then \( \;{\left| \mathbf{H}\right| }_{\min }^{2} \leq \frac{k\left( {m + 3}\right) }{m}. \) | Proof If \( {\left| \mathbf{H}\right| }^{2} > \frac{k\left( {m + 3}\right) }{m} \), then by Lemma 2.1, we have\n\n\[ \n\frac{1}{2}\Delta {\left| \mathbf{H}\right| }^{2} \geq \left\{ {m{\left| \mathbf{H}\right| }^{2} - k\left( {m + 3}\right) }\right\} {\left| \mathbf{H}\right| }^{2} + {\left| {\nabla }^{ \bot }\mathbf{H... | Yes |
Theorem 2.2 Let \( \\left( {{M}^{m}, g}\\right) \) be a complete biharmonic Lagrangian submanifold in a complex projective space \( {\\mathbb{{CP}}}^{m}\\left( {4k}\\right) \). If the Ricci curvature of \( M \) is bounded from below, then \( {\\left| \\mathbf{H}\\right| }_{\\min }^{2} \\leq \\frac{k\\left( {m + 3}\\rig... | Proof We first show that for any small \( \\varepsilon > 0 \), we have\n\n\[ \n{\\left| \\mathbf{H}\\right| }_{\\min }^{2} < \\frac{k\\left( {m + 3}\\right) }{m - \\varepsilon } \n\]\n\n(9)\n\nIndeed, if \( {\\left| \\mathbf{H}\\right| }^{2} \\geq \\frac{k\\left( {m + 3}\\right) }{m - \\varepsilon } \), then by Lemma 2... | Yes |
Theorem 1.3 \( {}^{\left\lbrack {20}\right\rbrack } \) For large enough \( k \), if \( m \geq C{k}^{2} \) where \( C \) is a constant, then we can construct an explicit RIP matrix of order \( k \) . | In the proof, DeVore gave the constructions of matrices \( \Phi \) using finite fields theory to satisfy RIP. However, DeVore’s approach showed that the rows of explicit matrix is of order \( \mathcal{O}\left( {k}^{2}\right) \) , which implied that we need more rows than that of random methods. To overcome the natural ... | Yes |
Theorem 2. \( {5}^{\left\lbrack 5\right\rbrack }\; \) Let \( 1 \leq k \leq \frac{n}{2} \) . There exists a measurement matrix \( \Phi \in {\mathbb{R}}^{m \times n} \) such that \( {\delta }_{k} + {\theta }_{k, k} = 1 \) and for some \( k \) -sparse signals \( {\beta }_{1},{\beta }_{2} \in {\mathbb{R}}^{n} \) with \( {\... | In the paper [6], the authors proved the following results. To prove the theorem, they developed a key technical lemma which states that any point in the given polytope can be decomposed as a convex combination of sparse vectors. They called it sparse representation of a polytope. | No |
Theorem 2. \( {\mathbf{6}}^{\left\lbrack 6\right\rbrack } \) Consider the signal recovery model (0.3) with \( \parallel z{\parallel }_{2} \leq \varepsilon \) . Suppose that \( {\widetilde{x}}^{{\ell }_{2}} \) is the minimizer of (0.4) with \( {\mathcal{B}}^{{\ell }_{2}}\left( \eta \right) \) for some \( \eta \geq \vare... | \[ \begin{aligned} \parallel {\widetilde{x}}^{{\ell }_{2}} - x{\parallel }_{2} & \leq \frac{\sqrt{2\left( {1 + {\delta }_{tk}}\right) }}{1 - \sqrt{\frac{t}{t - 1}}{\delta }_{tk}}\left( {\varepsilon + \eta }\right) + \left( {\frac{\sqrt{2}{\delta }_{tk} + \sqrt{t\left( {\sqrt{\frac{t - 1}{t}} - {\delta }_{tk}}\right) {\... | Yes |
Theorem 2. \( {8}^{\left\lbrack {47}\right\rbrack } \) Consider the signal recovery model with \( \parallel z{\parallel }_{2} \leq \varepsilon \) . Suppose that \( {\widetilde{x}}^{{\ell }_{2}} \) is the minimizer of \( \left( {0.4}\right) \) with \( {\mathcal{B}}^{{\ell }_{2}}\left( \eta \right) \) for some \( \eta \g... | \[ \parallel {\widetilde{x}}^{{\ell }_{2}} - x{\parallel }_{2} \leq \frac{\max \left( {\sqrt{t},1}\right) \sqrt{{2t}\left( {1 + {\delta }_{tk}}\right) }}{t - \left( {4 - t}\right) {\delta }_{tk}}\left( {\varepsilon + \eta }\right) \] \[ + 2\sqrt{2}\left( {\frac{2{\delta }_{tk} + \sqrt{\left\lbrack {t - \left( {4 - t}\r... | Yes |
Example 1.1 Let \( A = \left( \begin{matrix} B & M \\ 0 & C \end{matrix}\right) \) be a triangular matrix ring, where \( B \) and \( C \) are two rings, and \( M \) is a \( B - C \) bimodule. We identify the ring \( B \) as the pre-additive category \( \mathcal{B} \) with objects \( B \) and a zero object 0, and \( \ma... | \[ \mathcal{C}\xrightarrow[{i}^{!}]{{i}_{!}}\mathcal{A},\;\mathcal{A}\xrightarrow[{j}_{ * }]{{j}^{ * }}\mathcal{B} \] where \( {i}_{!} \) and \( {j}_{ * } \) are the canonical inclusions. It is worth mentioning that the pre-additive category \( \mathcal{B} \) is moreover equivalent to the localization \( \mathcal{A}\le... | Yes |
Theorem 2.1 Let \( \mathcal{C},\mathcal{D},\mathcal{A},\mathcal{B} \) be four pre-additive categories, where \( \mathcal{C} \) and \( \mathcal{D} \) are skeletally small. Given two adjoint pairs\n\n\[ \mathcal{C}\xrightarrow[S]{T}\mathcal{D},\;\mathcal{A}\xrightarrow[G]{F}\mathcal{B} \]\n\nthen the induced functors \( ... | Proof Let \( \eta : {1}_{\mathcal{A}} \rightarrow {GF} \) and \( \varepsilon : {FG} \rightarrow {1}_{\mathcal{B}} \) be the unit and the counit of \( \left( {F, U}\right) \) such that\n\n\[ \left( {\varepsilon \circ F}\right) \left( {F \circ \eta }\right) = {1}_{F},\;\left( {G \circ \varepsilon }\right) \left( {\eta \c... | Yes |
Corollary 2.2 Let \( A = \left( \begin{matrix} B & M \\ 0 & C \end{matrix}\right) \) be a triangular matrix ring, where \( B \) and \( C \) are two rings, and \( M \) is a \( B - C \) a bimodule. Then we have an abelian recollement situation of their module categories  is dense and full, and the pre-additive category \( \mathcal{B} \) is the quoti... | Yes |
Lemma 1.6 Let \( G \) be a simple \( {K}_{3} \) -group. Then \( \left| G\right| ,{\mathrm{{ON}}}_{1}\left( G\right) ,{\mathrm{{ONC}}}_{1}\left( G\right) \) and \( {o}_{2}\left( G\right) \) are as in Table 1, where \( {o}_{2}\left( G\right) \) denotes the second largest element order of \( G \) . | Table 1 The \( \left| G\right| ,{\mathrm{{ON}}}_{1}\left( G\right) ,{\mathrm{{ONC}}}_{1}\left( G\right) \), and \( {o}_{2}\left( G\right) \) of simple \( {K}_{3} \) -groups\n\n<table><thead><tr><th>\( G \)</th><th>\( \left| G\right| \)</th><th>\( {\mathrm{{ON}}}_{1}\left( G\right) \)</th><th>\( {\mathrm{{ONC}}}_{1}\lef... | Yes |
Lemma 1.7 Let a finite group \( G \) have \( m \) cyclic subgroups of order \( {o}_{1}\left( G\right), g \) be an element of order \( {o}_{1}\left( G\right) \) of \( G \), and \( l = \left| {G : {N}_{G}\left( {\langle g\rangle }\right) }\right| \) be the length of the conjugacy class of cyclic subgroup \( \langle g\ran... | Proof For the element \( g \), the following equation always holds,\n\n\[ \left| G\right| = \left| {G : {N}_{G}\left( {\langle g\rangle }\right) }\right| \cdot \left| {{N}_{G}\left( {\langle g\rangle }\right) : {C}_{G}\left( {\langle g\rangle }\right) }\right| \cdot \left| {{C}_{G}\left( {\langle g\rangle }\right) }\ri... | Yes |
Theorem 2.3 Let \( G \) be a group and \( M \) be one of the following simple \( {K}_{3} \) -groups: \( {A}_{5} \) , \( {A}_{6},{L}_{2}\left( 8\right) ,{L}_{2}\left( {17}\right) ,{L}_{3}\left( 3\right) \) . Then \( G \cong M \) if and only if \( {\mathrm{{ONC}}}_{1}\left( G\right) = {\mathrm{{ONC}}}_{1}\left( M\right) ... | Proof We only need to prove the sufficiency. And the proof will be made through a case by case analysis.\n\nCase 1: If \( M = {A}_{5} \), then \( G \cong M \) .\n\nWhile \( M = {A}_{5} \), from Table 1, we have \( {\operatorname{ONC}}_{1}\left( G\right) = {\operatorname{ONC}}_{1}\left( {A}_{5}\right) = \{ 5;{24};5\} \)... | Yes |
Theorem 2.4 Let \( G \) be a nonsolvable group. Then \( G \cong {U}_{3}\left( 3\right) \) if and only if \( {\mathrm{{ONC}}}_{1}\left( G\right) = \) \( {\mathrm{{ONC}}}_{1}\left( {{U}_{3}\left( 3\right) }\right) \) . | Proof We only need to prove the sufficiency. From Table 1, we have \( {\operatorname{ONC}}_{1}\left( G\right) = \) \( {\mathrm{{ONC}}}_{1}\left( {{U}_{3}\left( 3\right) }\right) \; = \;\{ {12};{1008};{12}\} .\; \) Clearly, \( \;\left| G\right| \;|\;{2}^{6} \cdot {3}^{3} \cdot 7.\; \) As \( \;G\; \) is not a solvable gr... | Yes |
For \( 0 < p \leq \infty \) and \( \lambda \geq 0 \), the quasi-norm \( \parallel f{\parallel }_{{\mathrm{{LM}}}_{p,\lambda }^{\left\{ {x}_{0}\right\} }} \) is equivalent to the quasi-norm \( \parallel f{\parallel }_{{\dot{B}}_{p,\lambda }^{\{ {x}_{0}\} }}. \) | Proof Let \( 0 < p \leq \infty ,\lambda \geq 0 \) and \( f \in {\operatorname{LM}}_{p,\lambda }^{\left\{ {x}_{0}\right\} }\left( {\mathbb{R}}^{n}\right) \) . Then it follows that\n\n\[ \parallel f{\parallel }_{{\dot{B}}_{p,\lambda }^{\{ {x}_{0}\} }} \leq \mathop{\sup }\limits_{{k \in \mathbf{Z}}}{\left( {2}^{k}\right) ... | Yes |
Theorem 2.1 \\( \\; \\) Let \\( {x}_{0} \\in {\\mathbb{R}}^{n},0 < \\alpha < {mn} \\) and \\( 1 \\leq {p}_{i} < \\frac{mn}{\\alpha } \\) with \\( \\frac{1}{p} = \\mathop{\\sum }\\limits_{{i = 1}}^{m}\\frac{1}{{p}_{i}},\\frac{1}{{q}_{i}} = \\frac{1}{{p}_{i}} - \\frac{\\alpha }{mn} \\) and \\( \\frac{1}{q} = \\mathop{\\s... | Proof In order to simplify the proof, we consider only the situation when \\( m = 2 \\) . Actually, a similar procedure works for all \\( m \\in \\mathbb{N} \\) . Thus, without loss of generality, it is sufficient to show that the conclusion holds for \\( {T}_{\\alpha }^{\\left( 2\\right) }\\left( \\overrightarrow{f}\\... | Yes |
Corollary 2.2 \( \; \) Let \( {x}_{0} \in {\mathbb{R}}^{n},0 < \alpha < {mn}\; \) and \( 1 \leq {p}_{i} < \frac{mn}{\alpha }\; \) with \( \frac{1}{p} = \mathop{\sum }\limits_{{i = 1}}^{m}\frac{1}{{p}_{i}},\frac{1}{{q}_{i}} = \frac{1}{{p}_{i}} - \frac{\alpha }{mn} \) and \( \frac{1}{q} = \mathop{\sum }\limits_{{i = 1}}^... | Remark 2.2 Note that, in the case of \( m = 1 \), Corollary 2.2 has been proved in [16-17]. | No |
Theorem 3.1 \\( \\; \\) Let \\( {x}_{0} \\in {\\mathbb{R}}^{n},0 < \\alpha < {mn} \\), and \\( 1 \\leq {p}_{i} < \\frac{mn}{\\alpha } \\) with \\( \\frac{1}{q} = \\mathop{\\sum }\\limits_{{i = 1}}^{m}\\frac{1}{{p}_{i}} + \\mathop{\\sum }\\limits_{{i = 1}}^{m}\\frac{1}{{q}_{i}} - \\frac{\\alpha }{n} \\) and \\( {b}_{i} ... | Proof As in the proof of Theorem 2.1, we consider only the situation when \\( m = 2 \\) . Actually, a similar procedure works for all \\( m \\in \\mathbb{N}. \\) Thus, without loss of generality, it is sufficient to show that the conclusion holds for \\( {T}_{\\alpha ,\\left( {{b}_{1},{b}_{2}}\\right) }^{\\left( 2\\rig... | Yes |
Theorem 3.2 (Our main result) Let \( {x}_{0} \in {\mathbb{R}}^{n},\;0 < \alpha < {mn}, \) and \( 1 \leq {p}_{i} < \frac{mn}{\alpha } \) for \( i = 1,2,\cdots, m \) such that \( \frac{1}{q} = \mathop{\sum }\limits_{{i = 1}}^{m}\frac{1}{{p}_{i}} + \mathop{\sum }\limits_{{i = 1}}^{m}\frac{1}{{q}_{i}} - \frac{\alpha }{n} \... | Proof The statement of Theorem 3.2 follows by Theorem 3.1 and (1.3) in the same manner as in the proof of Theorem 2.2. | No |
Lemma 1.3 \( {}^{\left\lbrack {42}\right\rbrack }\; \) Let \( m \geq 2,\;{p}_{1},\cdots ,{p}_{m} \in \left( {0,\infty }\right) \) and \( p \in \left( {0,\infty }\right) \) with \( \frac{1}{p} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}}.\; \) If \( {\omega }_{1},\cdots ,{\omega }_{m} \in {A}_{\infty } \), the... | \[ \mathop{\prod }\limits_{{j = 1}}^{m}{\left( {\omega }_{j}\left( B\right) \right) }^{\frac{p}{{p}_{j}}} \leq C{\nu }_{\overrightarrow{\omega }}\left( B\right) \] | Yes |
Theorem 1.1 Let \( 1 \leq s \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty ,\;j = 1,\cdots, m \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}},\frac{1}{p} = \) \( \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}},\frac{1}{\alpha } = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{\al... | Proof Let \( 1 \leq s \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \) and \( {f}_{j} \in {\left( {L}^{{q}_{j}}\left( {\omega }_{j}\right) ,{L}^{{p}_{j}}\right) }^{{\alpha }_{j}}, j = 1,\cdots, m \) . Fix \( B \mathrel{\text{:=}} B\left( {y, r}\right) \), then we have that for almost every \( x \in B\left( {y, r}\right) \)... | Yes |
Theorem 1.2 Let \( 1 \leq s < {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty, j = 1,\cdots, m \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}},\frac{1}{p} = \) \( \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}},\frac{1}{\alpha } = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{\alpha }... | Proof For every \( j \in \{ 1,\cdots, m\} \), we write \( \left| {{\mathcal{T}}_{{b}_{j}}\left( \overrightarrow{f}\right) \left( x\right) }\right| \) as\n\n\[ \left| {{\mathcal{T}}_{{b}_{j}}\left( \overrightarrow{f}\right) \left( x\right) }\right| \leq \left| {{\mathcal{T}}_{{b}_{j}}\left( {\overrightarrow{f}}^{0}\righ... | Yes |
Theorem 1.3 Let \( 1 \leq s < {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty, j = 1,\cdots, m \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}} \) , \( \frac{1}{p} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}},\frac{1}{\alpha } = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{\alpha... | Proof Without loss of generalization, we only consider the case \( m = 2 \) . Write\n\n\[ \left| {{\mathcal{T}}_{\Pi \overrightarrow{b}}\left( \overrightarrow{f}\right) \left( x\right) }\right| \leq \left| {{\mathcal{T}}_{\Pi \overrightarrow{b}}\left( {{f}_{1}^{0},{f}_{2}^{0}}\right) \left( x\right) }\right| + \mathop{... | Yes |
Theorem 2.1 Let \( 1 \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty ,\;j = 1,\cdots, m \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}},\;\frac{1}{p} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}}, \) \( \frac{1}{\alpha } = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{\alpha ... | Proof By Theorems 1.1,1.2 and 1.3, it suffices to verify that \( g\left( \overrightarrow{f}\right) \left( x\right) \) satisfies the condition (1.1). For any ball \( B = B\left( {y, r}\right) \subset {\mathbb{R}}^{n} \), we split\n\n\[ {f}_{j} = {f}_{j}^{0} + {f}_{j}^{\infty } \mathrel{\text{:=}} {f}_{j}{\chi }_{2B} + {... | Yes |
Theorem 2.3 Let \( \lambda > {2m} + 1,\;0 < \gamma < \min \{ \frac{n\left( {\lambda - {2m}}\right) }{2},\delta \} ,\;1 \leq s \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty ,\;j = 1,\cdots, m \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}},\frac{1}{p} = \mathop{\sum }\limits_{{j = 1... | Proof In fact, we can decompose \( {g}_{\lambda }^{ * }\left( \overrightarrow{f}\right) \left( x\right) \) into the sum of multilinear area integrals of Lusin \( {S}_{{2}^{j}}\left( \overrightarrow{f}\right) \left( x\right) \) as follows,\n\n\[ \n{g}_{\lambda }^{ * }\left( \overrightarrow{f}\right) \left( x\right) \leq... | No |
Theorem 2.5 Let \( {T}_{\lambda } \) be the operator defined in Definition 2.3 with the kernel satisfying the integral condition of C-Z type II with \( \lambda > {2m} \) and assume that \( {T}_{\lambda } \) can be extended to a bounded operator for some \( 1 \leq {r}_{1},\cdots ,{r}_{m} < \infty \) with \( \frac{1}{r} ... | Proof By Minkowski’s inequality and the condition of C-Z type II, we obtain\n\n\[ \left| {{T}_{\lambda }\left( {\overrightarrow{f}}^{\mathcal{D}}\right) \left( x\right) }\right| \leq {\int }_{{\left( {\mathbf{R}}^{n}\right) }^{m}}{\left( {\iint }_{{\mathbf{R}}_{ + }^{n + 1}}{\left( \frac{t}{t + \left| z\right| }\right)... | Yes |
Theorem 2.6 Assume that \( \mu \) can be extended to a bounded operator for some \( 1 \leq \) \( {\theta }_{1},\cdots ,{\theta }_{m} \) with \( \frac{1}{\theta } = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{\theta }_{j}} < \infty . \) Let \( 1 \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty , \) \( j = 1,\cdots... | Proof Let \( x \in B = B\left( {y, t}\right) . \) Since there exists at least one \( {y}_{j} \in B\left( {x, t}\right) \cap \left( {{2}^{k + 1}B \backslash {2}^{k}B}\right) \neq \varnothing \) for \( k = 1,2,\cdots \), we have\n\n\[ t \geq \mathop{\min }\limits_{{j = 1,\cdots, m}}\left| {{y}_{j} - x}\right| \geq {2}^{k... | Yes |
Theorem 2.7 Suppose that for some \( 1 \leq {s}_{1},\cdots ,{s}_{m} < \infty \) with \( \frac{1}{s} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{s}_{j}},{\mu }_{k} \) is bounded from \( {L}^{{s}_{1}}\left( {\mathbb{R}}^{n}\right) \times \cdots \times {L}^{{s}_{m}}\left( {\mathbb{R}}^{n}\right) \) . Let \( 1 \leq {q}_... | Proof By [46, Theorem 1.3], we know that \( {K}_{t} \) satisfies the integral smooth condition of C-Z type I with \( 0 < \gamma \leq \min \left\{ {\delta ,{\gamma }_{0}}\right\} \), therefore there exists a positive constant \( {C}_{3} > 1 \) such that\n\n\[ \n{\left( {\int }_{0}^{\infty }{\left| {K}_{t}\left( x,\overr... | Yes |
Lemma 3. \( {1}^{\left\lbrack {36}\right\rbrack } \) Let \( m \geq 2,{q}_{1},\cdots ,{q}_{m} \in \lbrack 1,\infty ), q \in \left( {0,\infty }\right) \) with \( \frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}} \) , \( {\omega }_{1}^{{q}_{1}},\cdots ,{\omega }_{m}^{{q}_{m}} \in {A}_{\infty } \) and \( {... | \[ \mathop{\prod }\limits_{{j = 1}}^{m}{\left( {\int }_{B}{\omega }_{j}^{{q}_{j}}\left( x\right) \mathrm{d}x\right) }^{\frac{1}{{q}_{j}}} \leq C{\left( {\int }_{B}{\nu }_{\overrightarrow{\omega }}^{q}\left( x\right) \mathrm{d}x\right) }^{\frac{1}{q}}. \] | Yes |
Lemma 3.2 \( {}^{\left\lbrack 5\right\rbrack }\; \) Let \( \;0 < \gamma < {mn},\;{p}_{1},\cdots ,{p}_{m} \in \lbrack 1,\infty ),\;\frac{1}{p} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{p}_{j}}\; \) and \( \;\frac{1}{q} = \frac{1}{p} - \frac{\gamma }{n}.\; \) Then \( \overrightarrow{\omega } = \left( {{\omega }_{1},... | \[ \left\{ \begin{array}{l} {\nu }_{\overrightarrow{\omega }}^{q} \in {A}_{mq}, \\ {\omega }_{j}^{-{p}_{j}^{\prime }} \in {A}_{m{p}_{j}^{\prime }},\;j = 1,\cdots, m, \end{array}\right. \] where \( {\nu }_{\varpi } = \mathop{\prod }\limits_{{j = 1}}^{m}{\omega }_{j}. \) | Yes |
Theorem 3.1 Let \( 1 \leq \theta < {p}_{j} \leq {\alpha }_{j} < {s}_{j} \leq \infty ,\;\frac{1}{{q}_{j}} = \frac{1}{{p}_{j}} - \frac{\gamma }{mn},\;\frac{1}{q} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{q}_{j}},\;\frac{1}{s} = \mathop{\sum }\limits_{{j = 1}}^{m}\frac{1}{{s}_{j}} \) and \( \begin{aligned} \frac{1}{{... | Proof Letting \( x \in B\left( {y, r}\right) \), in view of (3.1), we have\n\n\[ \left| {{\mathcal{T}}_{\gamma }\left( \overrightarrow{f}\right) \left( x\right) }\right| \leq \left| {{\mathcal{T}}_{\gamma }\left( {\overrightarrow{f}}^{0}\right) \left( x\right) }\right| + \mathop{\sum }\limits_{{\left( {{d}_{1},\cdots ,... | No |
Theorem 3.3 Let \( m \in {\mathbb{N}}_{ + },\;0 < \gamma < {mn},\;{\beta }_{j} > {mn},\;1 \leq {\alpha }_{j} < {\beta }_{j},\;\frac{1}{\left( {\alpha }_{j},{\beta }_{j}\right) } = \frac{1}{{\alpha }_{j}} - \frac{1}{{\beta }_{j}}, \) \( 1 \leq {q}_{j} \leq {\alpha }_{j} < {p}_{j} \leq \infty ,\;\frac{1}{\beta } = \frac{... | Proof Fix \( y \in {\mathbb{R}}^{n} \) and \( r > 0 \) . Set \( B = B\left( {y, r}\right) \) and let \( x \in B\left( {y, r}\right) \) . We decompose \( {f}_{j} = {f}_{j}^{0} + {f}_{j}^{\infty } \mathrel{\text{:=}} {f}_{j}{\chi }_{{2B}\left( {y, r}\right) } + {f}_{j}{\chi }_{{2B}{\left( y, r\right) }^{c}} \) for \( j =... | Yes |
Theorem 2.1 Let \( X \) be a Banach space. If every point \( x \in S\left( X\right) \) is a weak denting point of \( U\left( X\right) \), then \( X \) is strictly convex. | Proof Suppose that \( X \) is not strictly convex. Then, there exist three points \( {x}_{0},{x}_{1},{x}_{2} \in \) \( S\left( X\right) \) which are different from each other, such that \( {x}_{0} = \frac{{x}_{1} + {x}_{2}}{2} \), and there exists \( f \in {X}^{ * } \) such that\n\n\[ f\left( {x}_{0}\right) = a,\;f\lef... | Yes |
Theorem 2.3 Let \( X \) be a Banach space. If \( X \) is weakly locally uniformly convex then for every \( x \in S\left( X\right) \), the weak neighborhood \( {U}^{w} \) of the origin and \( f \in {S}_{x} \), there exists \( \delta = \delta \left( {x,{U}^{w}}\right) > \) 0 such that \( F\left( {f,\delta }\right) \subse... | Proof Suppose that the theorem is not true. Then there exists \( {x}_{0} \in S\left( X\right) \), the weak neighborhood \( {U}_{0}^{w} \) of the origin and \( {f}_{0} \in {S}_{{x}_{0}} \), such that \( F\left( {{f}_{0},\frac{1}{n}}\right) \nsubseteq {x}_{0} + {U}_{0} \) for any \( n \in \mathbb{N} \) . Thus, there exis... | Yes |
Theorem 2.4 Let \( X \) be a reflexive Banach space. If \( {X}^{ * } \) is weakly locally uniformly smooth, then every \( x \in S\left( X\right) \) is a weakly locally uniformly denting point of \( U\left( X\right) \) . | Proof Since \( {X}^{ * } \) is weakly locally uniformly smooth, by Lemma 1.3, \( X \) is weakly locally uniformly convex, so by Theorem 2.3, for every \( x \in S\left( X\right) \), the weak neighborhood \( {U}^{w} \) of the origin and \( f \in {S}_{x} \), there exists \( \delta = \delta \left( {x,{U}^{w}}\right) > 0 \)... | Yes |
Theorem 2.5 Let \( X \) be a reflexive Banach space. If \( U\left( X\right) \) is a \( K - w \) dentable set, then every point \( x \in S\left( X\right) \) is a \( K - w \) strongly exposed point of \( U\left( X\right) \) . | Proof Suppose that the theorem is not true. Then there exists \( {x}_{0} \in S\left( X\right) \) which is not the \( K \) - \( w \) strongly exposed point of \( S\left( X\right) \), which implies that for every \( f \in {S}_{{x}_{0}} \) with \( \dim {A}_{f} \leq K \) , there exists \( \delta > 0 \) such that although \... | Yes |
Lemma 1.1 Suppose that \( \left( {A, G,\alpha }\right) \) is an inversely pro-Banach algebra dynamical system where \( A \) has a bounded approximate left identity. Then \( {L}^{1}\left( {G, A,\alpha }\right) \) has a bounded approximate left identity. | Proof Let \( \left\{ {a}_{i}\right\} \) be a bounded approximate left identity of \( A \), and \( \left\{ {h}_{i}\right\} \) be an approximate left identity of \( {L}^{1}\left( G\right) \) . For all \( p \in m\left( A\right) \), since\n\n\[ \n{N}_{p}\left( {{h}_{i} \otimes {a}_{i}}\right) = {\int }_{G}p\left( {{h}_{i}\... | Yes |
Theorem 1.1 Let \( \left( {A, G,\alpha }\right) \) be an inversely pro-Banach algebra dynamical system, where \( A \) has a bounded approximate left identity.\n\n(i) Suppose that \( \left( {\pi, U}\right) \) is a non-degenerate continuous covariant representation of \( \left( {A, G,\alpha }\right) \) on a Banach space ... | Proof (i) For \( f \in {C}_{c}\left( {G, A,\alpha }\right) \), we define\n\n\[ \pi \rtimes U\left( f\right) = {\int }_{G}\pi \left( {f\left( s\right) }\right) {U}_{s}\mathrm{\;d}\mu \left( s\right) \]\n\nFrom the vector-valued Fubini Theorem, it is not difficult to verify\n\n\[ \pi \rtimes U\left( {f * g}\right) = \pi ... | Yes |
Corollary 0.1 Under the assumption above, the range of \( ▱ \) denoted by \( \mathcal{R}\left( ▱\right) \) is closed. \( {L}^{2}\left( {V,{A}^{p, q}}\right) \) admits the strong orthogonal decompostion: | \[ {L}^{2}\left( {V,{A}^{p, q}}\right) = \mathcal{R}\left( ▱\right) \oplus {\mathbb{H}}_{\left( p, q\right) }\left( V\right) = {\overline{\partial \partial }}^{ * }\left( {\operatorname{Dom}▱}\right) \oplus {\overline{\partial }}^{ * }\overline{\partial }\left( {\operatorname{Dom}▱}\right) \oplus {\mathbb{H}}_{\left( p... | Yes |
Proposition 1.1 For \( N \) a compact complex manifold, \( V \subset N \) an analytic subvariety of dimension \( d \), and \( \phi \) a differential form of degree \( {2d} - 1 \) ,\n\n\[{\int }_{V}\mathrm{\;d}\phi = 0\] | For the proof, we refer to [4]. | No |
Proposition 1.3 For any \( s \in {C}^{\infty }\left( {V, E}\right) \cap {L}^{2}\left( {V, E}\right) \), we have\n\n\[ \operatorname{Re}\left( {{D}^{ * }{Ds}, s}\right) \geq {\left| s\right| }_{\varepsilon }\left( {\Delta {\left| s\right| }_{\varepsilon }}\right) \]\n\nwhere \( {\left| s\right| }_{\varepsilon }^{2} \mat... | Proof Since\n\n\[ ▱\left( {\frac{1}{2}{\left| s\right| }_{\varepsilon }^{2}}\right) = - \mathop{\sum }\limits_{{i, j}}^{n}\frac{1}{2}{g}^{\bar{j}i}\frac{{\partial }^{2}}{\partial {z}^{i}\partial {\bar{z}}^{j}}\left( {s, s}\right) = - \mathop{\sum }\limits_{{i, j}}^{n}\frac{1}{2}{g}^{\bar{j}i}\frac{\partial }{\partial {... | Yes |
Theorem 2.1 Let \( \bar{P} \) be the \( {L}^{2} \) closure of \( P \) . Denote the heat kernel of \( P \) by \( {\mathrm{e}}^{-t\bar{P}} \) . Then the domination of heat kernel holds:\n\n\[ \left| {{\mathrm{e}}^{-t\bar{P}}s}\right| \leq {\mathrm{e}}^{-{tl}}{\mathrm{e}}^{-{t\Delta }}\left| s\right| \] \n\nfor any \( s \... | The idea is as follows. One can use Proposition 1.3 to prove that, for any \( s \in {C}^{\infty \left( {V, E}\right) } \cap \) \( {L}^{2}\left( {V, E}\right) \), the inequality of resolvent holds:\n\n\[ \left| {{\left( \Delta + l + \nu \right) }^{-1}s}\right| \geq \left| {{\left( P + \nu \right) }^{-1}s}\right| \] \n\n... | Yes |
For any Kähler subvariety \( V \), if the Weitzenböck curvature operator \( \mathcal{N} \) is bounded from below on degree \( p + q \), which means that there exists a constant \( l \), such that\n\n\[ \left( {\mathcal{N}\varphi ,\phi }\right) \geq l\left( {\varphi ,\phi }\right) \]\n\nfor any \( \left( {p + q}\right) ... | Proof By the Bochner-Kodaria formula (1.4) and (1.6), we have\n\n\[ ▱\varphi = - \operatorname{Tr}\nabla \bar{\nabla }\varphi - {R\varphi } + \operatorname{Ric}\varphi \]\n\nand\n\n\[ \overline{▱}\varphi = - \operatorname{Tr}\bar{\nabla }\nabla \varphi - {R\varphi } + \overline{\operatorname{Ric}}\varphi \]\n\nThen acc... | Yes |
Corollary 2.1 Under the assumption of Theorem 2.2, the range of \( ▱ \) is closed. | Proof Let \( {\mathbb{H}}_{\left( p, q\right) }\left( V\right) \) denote the space of harmonic forms on \( V \), i.e., \( {\mathbb{H}}_{\left( p, q\right) }\left( V\right) = \operatorname{Ker}\left( ▱\right) \) . We see that \( \;\square \; \) is bounded away from zero on the orthogonal complement \( {\left( {\mathbb{H... | Yes |
Corollary 2.2 Under the assumption of Theorem 2.2, \( {L}^{2}\left( {V,{A}^{p, q}}\right) \) admits the strong orthogonal decomposition\n\n\[ \n{L}^{2}\left( {V,{A}^{p, q}}\right) = \mathcal{R}\left( ▱\right) \oplus {\mathbb{H}}_{\left( p, q\right) }\left( V\right) = \partial {\overline{\partial }}^{ * }\left( {\operat... | \( {\mathrm{}\mathbf{{Proof}}}\mathrm{{Because}}\mathcal{R}\left( ▱\right) = {\left( {\mathbb{H}}_{\left( p, q\right) }\left( V\right) \right) }^{ \bot } \) and \( \mathcal{R}\left( {\overline{\partial }{\overline{\partial }}^{ * }}\right) \bot \left( {{\bar{\partial }}^{ * }\overline{\partial }}\right) , \) the decomp... | Yes |
Corollary 2.3 For any Kähler subvariety \( V \) with lower bounded Ricci curvature, we have that for \( \left( {0, q}\right) \) forms, \( ▱ \) has closed image. Moreover, \( {L}_{\left( 0, q\right) }^{2}V \) admits the strong Hodge decomposition: | Proof Using the formula (1.8), the proof is similar to Theorem 2.2, Corollary 2.1 and Corollary 2.2. | No |
Lemma 2.1 The following facts are well known for \( \parallel \cdot {\parallel }_{{L}^{p\left( \cdot \right) }} \) | (1) \( \parallel f{\parallel }_{{L}^{p\left( \cdot \right) }} \geq 0;\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }} = 0 \Leftrightarrow f = 0 \) ;\n(2) \( \parallel {cf}{\parallel }_{{L}^{p\left( \cdot \right) }} = \left| c\right| \parallel f{\parallel }_{{L}^{p\left( \cdot \right) }} \) for any complex number ... | Yes |
Lemma 2.2 \( {2}^{\left\lbrack 5\right\rbrack } \) (Hölder’s inequality) Let \( p, q, s \in \mathcal{P}\left( \Omega \right) \) be such that \( \frac{1}{s\left( \cdot \right) } = \frac{1}{p\left( \cdot \right) } + \frac{1}{q\left( \cdot \right) } \). Then | \[ \parallel {fg}{\parallel }_{{L}^{s\left( \cdot \right) }} \leq 2\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }}\parallel g{\parallel }_{{L}^{q\left( \cdot \right) }}. \] | No |
Lemma 2.3 \( {}^{\left\lbrack 5\right\rbrack } \) (Norm conjugate formula) Let \( p \in \mathcal{P}\left( \Omega \right) \) . Then\n\n\[ \frac{1}{2}\parallel f{\parallel }_{{L}^{p\left( \cdot \right) }} \leq \mathop{\sup }\limits_{{g \in {L}^{{p}^{\prime }\left( \cdot \right) },\parallel g{\parallel }_{{L}^{{p}^{\prime... | Thus we have\n\[ \parallel f{\parallel }_{{L}_{\omega }^{p\left( \cdot \right) }} \approx \sup \left\{ {{\int }_{\Omega }\left| f\right| \left| g\right| \mathrm{d}P : g \in {L}_{{\omega }^{-1}}^{{p}^{\prime }\left( \cdot \right) },\parallel g{\parallel }_{{L}_{{\omega }^{-1}}^{{p}^{\prime }\left( \cdot \right) }} \leq ... | No |
Theorem 3.1 Let \( p \in \mathcal{P}\left( \Omega \right) \) and \( \omega \in {\mathcal{W}}_{p\left( \cdot \right) } \) . For any \( f = \left( {f}_{n}\right) \in {H}_{\omega, p\left( \cdot \right) }^{\sigma } \), there exist a sequence \( {\left( {a}^{k}\right) }_{k \in \mathbb{Z}} \) of \( \left( {1,\infty }\right) ... | Proof Assume \( f \in {H}_{\omega, p\left( \cdot \right) }^{\sigma } \) . Let us consider the stopping times for all \( k \in \mathbb{Z} \),\n\n\[ {\tau }_{k} = \inf \left\{ {n \in \mathbb{N} : {\sigma }_{n + 1}\left( f\right) > {2}^{k}}\right\} ,\;\inf \varnothing = \infty . \]\n\nFor each stopping time \( {\tau }_{k}... | Yes |
Theorem 3.2 Let \( p \in \mathcal{P}\left( \Omega \right) \) and \( \omega \in {\mathcal{W}}_{p\left( \cdot \right) } \) . For any \( f = \left( {f}_{n}\right) \in {H}_{\omega, p\left( \cdot \right) }^{\sigma } \), there exist a sequence \( {\left( {a}^{k}\right) }_{k \in \mathbb{Z}} \) of \( \left( {2,2}\right) \) -at... | Proof For all \( k \in \mathbb{Z} \), define \( {F}_{k} = \left\{ {\sigma \left( f\right) > {2}^{k}}\right\} ,{\tau }_{k} = \inf \left\{ {n \geq 0 : 2{E}_{n}{\chi }_{{F}_{k}} > 1}\right\} \).\n\nLet\n\n\[ {u}_{k} = 8 \cdot {2}^{k + 1} \cdot {\begin{Vmatrix}{\chi }_{\left\{ {\tau }_{k} < \infty \right\} }\end{Vmatrix}}_... | Yes |
Theorem 3.4 Let \( p \in \mathcal{P}\left( \Omega \right) \) and \( \omega \in {\mathcal{W}}_{p\left( \cdot \right) } \) . For any \( f = \left( {f}_{n}\right) \in {\mathrm{{PH}}}_{\omega, p\left( \cdot \right) } \), there exist a sequence \( {\left( {a}^{k}\right) }_{k \in \mathbf{Z}} \) of \( \left( {2,\infty }\right... | We sketch the proofs of Theorems 3.3, 3.4 and omit the details since they are similar to that of Theorem 3.1. Let \( {\tau }_{k} = \inf \left\{ {n \in \mathbb{N} : {\lambda }_{n} > {2}^{k}}\right\} \), where \( {\left( {\lambda }_{n}\right) }_{n \geq 0} \) is the sequence in the definitions of \( {\mathrm{{SPH}}}_{\ome... | No |
Theorem 4.1 Given a sublinear operator \( T \), suppose for some \( {p}_{0} \in \left( {0,\infty }\right) \) that\n\n\[{\int }_{\Omega }{\left| Tf\left( x\right) \right| }^{{p}_{0}}\omega \left( x\right) \mathrm{d}P \leq {c}_{1}{\int }_{\Omega }{\left| f\left( x\right) \right| }^{{p}_{0}}\omega \left( x\right) \mathrm{... | Proof Define \( s \in \mathcal{P}\left( \Omega \right) \) by \( s \mathrel{\text{:=}} \frac{p}{{p}_{0}} \) and let \( h \in {L}^{{s}^{\prime }\left( \cdot \right) }\left( \Omega \right) \) . By assumption, we have \( \parallel {Mh}{\parallel }_{{L}^{{s}^{\prime }\left( \cdot \right) }} \leq A\parallel h{\parallel }_{{L... | Yes |
Corollary 4.1 Given a sublinear operator \( T \), and suppose for some fixed \( 0 < {p}_{0} < {q}_{0} < \infty \) that \[ {\left( {\int }_{\Omega }{\left| Tf\left( x\right) \right| }^{{q}_{0}}\omega \left( x\right) \mathrm{d}P\right) }^{\frac{1}{{q}_{0}}} \leq {c}_{1}{\left( {\int }_{\Omega }{\left| f\left( x\right) \r... | Proof The proof is similar to [2, Theorem 1.8], and we omit it. | No |
Theorem 4.2 Let \( p \in \mathcal{P}\left( \Omega \right) \) and \( \omega \in {\mathcal{W}}_{p\left( \cdot \right) } \). Then for any \( r > {s}_{\omega } \), there exists a constant \( c > 0 \) such that\n\n\[ \parallel {Mf}{\parallel }_{{L}_{{w}^{1/r}}^{{rp}\left( \cdot \right) }} \leq c\parallel f{\parallel }_{{L}_... | Proof According to the definition of \( {s}_{\omega } \), we have \( s > {s}_{\omega } \) satisfying \( s < r \) such that \( M \) is bounded on \( {L}_{{\omega }^{-1/\sigma }}^{{\left( sp\right) }^{\prime }\left( \cdot \right) } \). Similarly, we also define the Rubio de Francia operator \( R \) by ![4503b315-c0ff-4d9... | Yes |
Given prime \( p \geq 3 \) and integers \( m \geq 0,\;1 \leq k \leq p - 1 \), then for any \( m \geq t \geq k, \)\n\nwe have\n\[ \n{v}_{p}\left( {\left( \begin{matrix} m \\ t \end{matrix}\right) {p}^{t}}\right) \geq {v}_{p}\left( m\right) + k \n\]\n\nIn particular, given integers \( m \geq 0,1 \leq k \leq 4 \), then fo... | Proof Write \( m = {p}^{{v}_{p}\left( m\right) }{m}_{1} \) with \( p \nmid {m}_{1} \) . Write \( t = {p}^{{v}_{p}\left( t\right) }{t}_{1} \) with \( p \nmid {t}_{1} \) . We have\n\n\[ \n{v}_{p}\left( {\left( \begin{matrix} m \\ t \end{matrix}\right) {p}^{t}}\right) = {v}_{p}\left( {\frac{m}{t}\left( \begin{matrix} m - ... | Yes |
Lemma 2 Let \( m \) be a nonnegative integer. We have\n\n\[ \n{v}_{5}\left( {{2}^{m} + 1}\right) = \left\{ \begin{array}{ll} {v}_{5}\left( m\right) + 1, & \text{ if }m \equiv 2\left( {\;\operatorname{mod}\;4}\right) , \\ 0, & \text{ otherwise. } \end{array}\right.\n\] | Proof If \( m \equiv 0,1,3\;\left( {\text{mod }4}\right) \), then \( {2}^{m} + 1 \equiv 2,3,4\;\left( {\text{mod }5}\right) \), thus \( {v}_{5}\left( {{2}^{m} + 1}\right) = 0 \) . If \( m \equiv 2 \) \( \left( {\;\operatorname{mod}\;4}\right) \), then write \( m = {5}^{i}d \) with \( 5 \nmid d \) and \( i \geq 0 \) . T... | Yes |
Lemma 3 Let \( m \) be a nonnegative integer. We have \( {v}_{5}\left( {{6}^{m} + {3}^{m} + {2}^{m}}\right) = 0 \) . | Proof If \( m \) is odd, then\n\n\[ \n{6}^{m} + {3}^{m} + {2}^{m} \equiv 1 + {\left( -2\right) }^{m} + {2}^{m} \equiv 1\;\left( {\;\operatorname{mod}\;5}\right) .\n\]\n\nIf \( m \) is even, then \( m \equiv 0,2\left( {\;\operatorname{mod}\;4}\right) \). We have\n\n\[ \n{6}^{m} + {3}^{m} + {2}^{m} \equiv 1 + {2}^{m + 1}... | Yes |
Lemma 5 If \( m \equiv 1,13,17\left( {\;\operatorname{mod}\;{20}}\right) \), then\n\n\[ {v}_{5}\left( {{12}^{m} + {6}^{m} + {4}^{m} + {3}^{m}}\right) = 2. \] | Proof Write \( m = {20u} + {2v} - 1 \) with \( u \geq 0, v = 1,7,9 \) . Then \( m + 1 = 2\left( {{10u} + v}\right) \) . Noting that\n\n\[ {12}{T}_{m} = {12}^{m + 1} + 2 \cdot {6}^{m + 1} + 3 \cdot {4}^{m + 1} + 4 \cdot {3}^{m + 1} \]\n\n\[ \equiv {19}^{{10u} + v} + 2 \cdot {36}^{{10u} + v} + 3 \cdot {16}^{{10u} + v} + ... | Yes |
Proposition 2.1 Let \( N \) be a dyadic number \( \left( {N \geq 1\text{when}\epsilon = - 1}\right) \) and \( \phi \) be a Schwartz function whose Fourier transform according to \( x \) supports in the set \( \{ \left| \xi \right| \sim N\} \) . Then for \( \delta \left( r\right) = 2\left( {\frac{1}{2} - \frac{1}{r}}\ri... | Proof of Proposition 2.1 Write\n\n\[ \n{G}_{N}\left( {x, y, t}\right) = {\int }_{{\mathbb{R}}^{2}}{\chi }_{\left| \xi \right| \sim N}{\mathrm{e}}^{\mathrm{i}{t\omega }\left( {\xi ,\eta }\right) }{\mathrm{e}}^{\mathrm{i}\left( {{x\xi } + {y\eta }}\right) }\mathrm{d}\xi \mathrm{d}\eta .\n\]\n\nHere \( \omega \left( {\xi ... | Yes |
Theorem 2.1 There exist \( \left( {v,{G}_{1},1}\right) \) -OPD and \( \left( {v,{G}_{1},1}\right) \) -OCD for \( v \geq 9 \) . | Proof We can get the conclusion by Theorem 1.1, Lemmas 1.1, 1.2 and 2.2. The leave graphs \( {L}_{1} \) for these OPDs are as follows:\n\nTable 2 The leave graphs \( {L}_{1} \) for \( \left( {v,{G}_{1},1}\right) \) -OPDs for \( v \geq 9 \)\n\n \) -OPD and \( \\left( {v,{G}_{2},1}\\right) \) -OCD for \( v \\geq 9 \) . | Proof It is easy to prove by Theorem 1.1, Lemmas 1.1,1.2 and 2.3. The leave graphs \( {L}_{1} \) for these OPDs are as follows:\n\nTable 4 The leave graphs \( {L}_{1} \) for \( \\left( {v,{G}_{2},1}\\right) \) -OPDs for \( v \\geq 9 \)\n\n \) -OPD for \( w = {11},{12},\cdots ,{17} \) . | Proof Let \( \left( {w,{G}_{3},1}\right) - \mathrm{{OPD}} = \left( {X,\mathcal{B}}\right) \) .
\( w = {11} : \;X = \left( {{Z}_{3} \times {Z}_{3}}\right) \cup \{ a, b\} ,
\( \left( {{0}_{1}, a,{0}_{0},{1}_{1},{2}_{0},{0}_{2},{2}_{2},{2}_{1},{1}_{2}}\right) ,\left( {{2}_{2}, b,{1}_{0},{0}_{2},{2}_{1},{2}_{0},{0}_{1},... | Yes |
Theorem 2.3 There exist \( \left( {v,{G}_{3},1}\right) \) -OPD and \( \left( {v,{G}_{3},1}\right) \) -OCD for \( v \geq 9 \) . | Proof It is easy to prove by Theorem 1.1, Lemmas 1.1,1.2 and 2.4. The leave graphs \( {L}_{1} \) for these OPDs are as follows:\n\nTable 6 The leave graphs \( {L}_{1} \) for \( \left( {v,{G}_{3},1}\right) \) -OPDs for \( v \geq 9 \)\n\n is a planar graph with \( g\left( G\right) \geq 5 \) and \( \Delta \left( G\right) \geq {44} \), then \( {\chi }_{2}\left( G\right) \leq \) \( \Delta \left( G\right) + 4 \) . | ## 2 Proof of Theorem 0.2\n\nWe will prove Theorem 0.2 by contradiction. Let \( {G}^{ * } \) be a planar graph such that \( \Delta \left( {G}^{ * }\right) \geq \) \( {44}, g\left( {G}^{ * }\right) \geq 5 \) but \( {\chi }_{2}\left( {G}^{ * }\right) > \Delta \left( {G}^{ * }\right) + 4 \), i.e., \( {G}^{ * } \) is a cou... | Yes |
Lemma 2.1 \( \delta \left( G\right) \geq 2 \) . | Proof Suppose to the contrary that \( \delta \left( G\right) = 1 \) . Without loss of generality, let \( d\left( v\right) = 1 \) and \( {uv} \in E\left( G\right) \) . By the minimality of \( G,{G}^{\prime } = G - v \) has a 2-distance coloring \( \varphi \) . Since \( \left| {F\left( v\right) }\right| \leq D\left( v\ri... | No |
Lemma 2.2 There is no two adjacent 2-vertices. | Proof Assume to the contrary that \( {uv} \in E\left( G\right) \) and \( d\left( u\right) = d\left( v\right) = 2 \) . By the minimality of \( G,{G}^{\prime } = G - {uv} \) has a \( \left( {\Delta + 4}\right) \) -2-distance coloring \( \varphi \) . We erase the colors of \( u, v \) . Since \( \left| {F\left( v\right) }\... | Yes |
Lemma 2.3 A 3-vertex has at most one 2-neighbor. If \( v \) is a 3-vertex with exactly one 2-neighbor, then the other two neighbors of \( v \), namely \( {v}_{2},{v}_{3} \), satisfy \( d\left( {v}_{2}\right) + d\left( {v}_{3}\right) \geq \Delta + 3 \) . | Proof Assume to the contrary that \( v \) has two 2-neighbors \( {v}_{1},{v}_{2} \) . By the minimality of \( G \) , \( {G}^{\prime } = G - {v}_{1} \) is \( \left( {\Delta + 4}\right) - 2 \) -distance colorable. We erase the colors of \( v,{v}_{2} \) and recolor them. Since \( \left| {F\left( v\right) }\right| \leq \De... | Yes |
Lemma 2.4 3(1)-vertex is not adjacent to 3(1)-vertex. If a 3-vertex is adjacent to exactly one 2-vertex and one 3-vertex, then this 3-vertex must be a heavy 3(0)-vertex. | Proof Suppose to the contrary that \( 3\left( 1\right) \) -vertex \( v \) is adjacent to a \( 3\left( 1\right) \) -vertex \( {v}_{2} \) . Let \( {v}_{1},{v}_{21} \) be the 2-neighbors of \( v,{v}_{1} \) respectively. By the minimality of \( G,{G}^{\prime } = G - {v}_{1} \) is \( \left( {\Delta + 4}\right) \) - 2-distan... | Yes |
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