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Proposition 45.4. Suppose that \( \left( {G, H,\psi }\right) \) is a Gelfand triple, and let \( \left( {\pi, V}\right) \) be an irreducible representation of \( G \) . Then there exists at most one space \( \mathcal{M} \) of functions on \( G \) satisfying\n\n\[ M\left( {hg}\right) = \psi \left( h\right) M\left( g\righ... | Proof. This is just the Frobenius reciprocity. The space of functions satisfying (45.3) is \( {\operatorname{Ind}}_{H}^{G}\left( \psi \right) \), so \( \mathcal{M} \), if it exists, is the image of an element of \( {\operatorname{Hom}}_{G}\left( {V,{\operatorname{Ind}}_{H}^{G}\left( \psi \right) }\right) \) . This is o... | Yes |
Theorem 45.3. Let \( H \) be a closed subgroup of the compact group \( G \) . Let \( \mathcal{H} \) be the subring of \( C\left( G\right) \) consisting of functions that are both left- and right-invariant under \( H \) . If \( \mathcal{H} \) is commutative, then \( {V}^{H} \) is at most one-dimensional for every irredu... | Proof. Let \( \xi ,\eta \in {V}^{H} \) . For \( g \in G \), let\n\n\[{\phi }_{\xi ,\eta }\left( g\right) = \overline{\langle \pi \left( g\right) \xi ,\eta \rangle }\]\n\nwhere \( \langle \) , \( \rangle {isaninvariantinnerproductonV}\left( {Proposition2.1}\right) .{Itiseasyto} \) see that \( {\phi }_{\xi ,\eta } \in \m... | Yes |
Proposition 45.5. With \( G = \mathrm{{SO}}\left( {n + 1}\right), H = \mathrm{{SO}}\left( n\right) \), and \( K = \mathrm{{SO}}\left( 2\right) \) embedded as explained above, every double coset in \( H \smallsetminus G/H \) has a representative in \( K \) . | Proof. Let \( g \in G \) . Write the last column of \( g \) in the form\n\n\[ \left( \begin{matrix} b{v}_{1} \\ b{v}_{2} \\ \vdots \\ b{v}_{n} \\ a \end{matrix}\right) = \left( \begin{matrix} {bv} \\ a \end{matrix}\right) ,\;v = \left( \begin{matrix} {v}_{1} \\ \vdots \\ {v}_{n} \end{matrix}\right) \]\n\nwhere \( {b}^{... | Yes |
Theorem 45.4. The subgroup \( \mathrm{{SO}}\left( n\right) \) of \( \mathrm{{SO}}\left( {n + 1}\right) \) is a Gelfand subgroup. | Proof. With \( G = \mathrm{{SO}}\left( {n + 1}\right), H = \mathrm{{SO}}\left( n\right) \), and \( K = \mathrm{{SO}}\left( 2\right) \) embedded as explained above, we exhibit an involution of \( G \), namely\n\n\[ g \mapsto \left( \begin{array}{ll} {I}_{n} & \\ & - 1 \end{array}\right) {}^{t}g\left( \begin{array}{ll} {... | Yes |
Theorem 45.5. Let \( \left( {\pi, V}\right) \) be an irreducible representation of \( \mathrm{O}\left( {n + 1}\right) \) . Then there exists at most one subspace of \( {L}^{2}\left( {S}^{n}\right) \) that is invariant under the action of \( \mathrm{O}\left( {n + 1}\right) \) and affords a representation isomorphic to \... | Proof. Let \( \phi : V \rightarrow {L}^{2}\left( {S}^{n}\right) \) be an intertwining operator. It is sufficient to show that \( \phi \) is uniquely determined up to a constant multiple. The \( \mathrm{O}\left( {n + 1}\right) \) - equivariance of \( \phi \) amounts to the formula\n\n\[ \phi \left( {\pi \left( g\right) ... | Yes |
Proposition 45.6. If \( g \in \mathrm{U}\left( n\right) \), then there exist \( {k}_{1} \) and \( {k}_{2} \in \mathrm{O}\left( n\right) \) such that \( {k}_{1}g{k}_{2} \) is diagonal. | Proof. Let \( x = {g}^{t}g \) . This is a unitary symmetric matrix. By Proposition 28.2, there exists \( {k}_{1} \in \mathrm{O}\left( n\right) \) such that \( {k}_{1}x{k}_{1}^{-1} \) is diagonal. It is unitary, so its diagonal entries have absolute value 1. Taking their square roots, we find a unitary diagonal matrix \... | Yes |
Theorem 45.6. The group \( \mathrm{O}\left( n\right) \) is a Gelfand subgroup of \( \mathrm{U}\left( n\right) \) . | Proof. Let \( G = \mathrm{U}\left( n\right) \) and \( H = \mathrm{O}\left( n\right) \), and let \( \mathcal{H} \) be the ring of Theorem 45.3. The transpose involution of \( G \) preserves \( H \) and thus induces an involution of \( \mathcal{H} \) . By Proposition 45.6, every double coset in \( H \smallsetminus G/H \)... | Yes |
Lemma 46.1. Let \( J \) be any subset of \( \sum \) . Then there exist integers \( {k}_{1},\ldots ,{k}_{r} \) such that the subgroup of \( {S}_{k} \) generated by \( J \) is \( {S}_{{k}_{1}} \times \cdots \times {S}_{{k}_{r}} \) . | Proof. If \( J \) contains \( \left( {1,2}\right) ,\left( {2,3}\right) ,\ldots ,\left( {{k}_{1} - 1,{k}_{1}}\right) \), then the subgroup they generate is the symmetric group \( {S}_{{k}_{1}} \) acting on \( \left\{ {1,\ldots ,{k}_{1}}\right\} \) . Taking \( {k}_{1} \) as large as possible, assume that \( J \) omits \(... | Yes |
(i) Let \( J \subseteq \sum \) . Then\n\n\[ \n{M}_{J} = \mathop{\bigcup }\limits_{{w \in {W}_{J}}}{B}_{J}w{B}_{J}\;\text{ (disjoint). }\n\] | For (i), we have (46.3) for suitable \( {k}_{i} \) . Now \( {B}_{J} \) is the direct product of the Borel subgroups of these \( \mathrm{{GL}}\left( {{k}_{i}, F}\right) \), and \( {W}_{J} \) is the direct product (46.1). Part (i) follows directly from the Bruhat decomposition for \( \mathrm{{GL}}\left( {k, F}\right) \) ... | No |
Proposition 46.2. Let \( H \) be a group, and let \( {M}_{1} \) and \( {M}_{2} \) be subgroups of \( H \) . Then in the character ring of \( H \), the inner product of the characters induced from the trivial characters of \( {M}_{1} \) and \( {M}_{2} \), respectively, is equal to the number of double cosets in \( {M}_{... | Proof. By the geometric form of Mackey's theorem (Theorem 32.1), the space of intertwining maps from \( {\operatorname{Ind}}_{{M}_{1}}^{H}\left( 1\right) \) to \( {\operatorname{Ind}}_{{M}_{2}}^{H}\left( 1\right) \) is isomorphic to the space of functions \( \Delta : H \rightarrow \operatorname{Hom}\left( {\mathbb{C},\... | Yes |
Proposition 46.3. As a virtual representation, the alternating character \( {\mathbf{e}}_{k} \) of \( {S}_{k} \) admits the following expression: | Proof. We recall that \( {\mathbf{e}}_{k} = {\mathbf{s}}_{\lambda } \), where \( \lambda \) is the partition \( \left( {1,\ldots ,1}\right) \) of \( K \) . The right-hand side of (35.10) gives \[ {\mathbf{e}}_{k} = \left| \begin{matrix} {\mathbf{h}}_{1} & {\mathbf{h}}_{2} & {\mathbf{h}}_{3} & \cdots & {\mathbf{h}}_{k} ... | Yes |
Theorem 46.2. As a virtual representation, the Steinberg representation \( {\mathbf{e}}_{k}\left( q\right) \) of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) admits the following expression:\n\n\[ \n{\mathbf{e}}_{k}\left( q\right) = \mathop{\sum }\limits_{{J \subseteq \sum }}{\left( -1\right) }^{\left| J\right... | Proof. This follows immediately from Proposition 46.3 on applying the mapping of Theorem 46.1. | No |
Proposition 46.4. Suppose that \( S \) is any subset of \( \Phi \) such that if \( \alpha \in S \), then \( - \alpha \notin S \), and if \( \alpha ,\beta \in S \) and \( \alpha + \beta \in \Phi \), then \( \alpha + \beta \in S \) . Let \( {U}_{S} \) be the set of \( g = \left( {g}_{ij}\right) \) in \( \mathrm{{GL}}\lef... | Proof. Let \( \widetilde{S} \) be the set of \( \left( {i, j}\right) \) such that the root \( {\alpha }_{ij} \in S \) . Translating the hypothesis on \( S \) into a statement about \( \widetilde{S} \), if \( \left( {i, j}\right) \in \widetilde{S} \) we have \( i < j \), and\n\n\[ \text{if both}\left( {i, j}\right) \tex... | Yes |
Proposition 46.5. Let \( F = {\mathbb{F}}_{q} \) be finite, and let \( w \in W \) . We have\n\n\[ \left| {U}_{w}^{ - }\right| = {q}^{l\left( w\right) } \] | Proof. By Propositions 20.2 and 20.5, the cardinality of \( S = {\Phi }^{ + } \cap {w}^{-1}{\Phi }^{ - } \) is \( l\left( w\right) \), so this follows from the definition of \( {U}_{S} \) . | Yes |
Proposition 46.6. Let \( w \in W \) . The multiplication map \( {U}_{w}^{ + } \times {U}_{w}^{ - } \rightarrow U \) is bijective. | Proof. We will prove this if \( F \) is finite, the only case we need. In this case \( {U}_{w}^{ + } \cap {U}_{w}^{ - } = \{ 1\} \) by definition since the sets \( {\Phi }^{ + } \cap w{\Phi }^{ - } \) and \( {\Phi }^{ + } \cap w{\Phi }^{ + } \) are disjoint. Thus, if \( {u}_{1}^{ + }{u}_{1}^{ - } = {u}_{2}^{ + }{u}_{2}... | Yes |
Proposition 46.7. Let \( F = {\mathbb{F}}_{q} \) be finite, and let \( w \in W \) . The order of \( {BwB}/B \) is \( {q}^{l\left( w\right) } \) . | Proof. We will show that \( {u}^{ - } \mapsto {u}^{ - }{wB} \) is a bijection \( {U}_{w}^{ - } \rightarrow {BwB}/B \) . The result then follows from Proposition 46.5.\n\nNote that every right coset in \( {BwB}/B \) is of the form \( {bwB} \) for some \( b \in B \) . Using Proposition 46.6, we may write \( b \in B \) un... | Yes |
Proposition 46.8. Let \( w,{w}^{\prime } \in W \) such that \( l\left( {w{w}^{\prime }}\right) = l\left( w\right) + l\left( {w}^{\prime }\right) \) . Then\n\n\[{\phi }_{w{w}^{\prime }} = {\phi }_{w}{\phi }_{{w}^{\prime }} \] | Proof. By Proposition 27.1, we have \( \mathcal{C}\left( {w{w}^{\prime }}\right) = \mathcal{C}\left( w\right) \mathcal{C}\left( {w}^{\prime }\right) \) . Therefore \( {\phi }_{w} * {\phi }_{{w}^{\prime }} \) is supported in \( \mathcal{C}\left( {w{w}^{\prime }}\right) \) and is hence a constant multiple of \( {\phi }_{... | Yes |
Proposition 46.9. Let \( s \in W \) be a simple reflection. Then\n\n\[{\phi }_{s} * {\phi }_{s} = q{\phi }_{1} + \left( {q - 1}\right) {\phi }_{s}\] | Proof. By (27.2), we have \( \mathcal{C}\left( s\right) \mathcal{C}\left( s\right) \subseteq \mathcal{C}\left( 1\right) \cup \mathcal{C}\left( s\right) \) . Therefore, there exist constants \( \lambda \) and \( \mu \) such that \( {\phi }_{s} * {\phi }_{s} = \lambda {\phi }_{1} + \mu {\phi }_{s} \) . Evaluating both si... | Yes |
Proposition 46.10. If \( q = 1 \), the Hecke ring \( {\mathcal{H}}_{k}\left( 1\right) \) is isomorphic to the group ring of \( {S}_{k} \) . | Proof. This is clear from Theorem 25.1 since if \( q = 1 \) the defining relations of the ring \( {\mathcal{H}}_{k}\left( 1\right) \) coincide with the Coxeter relations presenting \( {S}_{k} \) . | Yes |
Proposition 46.11. Suppose that \( w \in W \) with \( l\left( w\right) = r \), and suppose that \( w = {s}_{1}\cdots {s}_{r} = {s}_{1}^{\prime }\cdots {s}_{r}^{\prime } \) are distinct decompositions of minimal length into simple reflections. Then\n\n\[ \n{f}_{{s}_{1}} * \cdots * {f}_{{s}_{r}} = {f}_{{s}_{1}^{\prime }}... | Proof. Let \( B \) be the braid group generated by \( {u}_{{\alpha }_{i}} \) parametrized by the simple roots \( {\alpha }_{i} \), with \( n\left( {{u}_{{\alpha }_{i}},{u}_{{\alpha }_{j}}}\right) \) equal to the order (2 or 3) of \( {s}_{{\alpha }_{i}}{s}_{{\alpha }_{j}} \) . Let \( {s}_{i} = {s}_{{\beta }_{i}} \) and ... | Yes |
Theorem 46.4. Let \( q \) be a prime power. Then the Hecke algebra \( {\mathcal{H}}_{k}\left( q\right) \) is isomorphic to the convolution ring of \( B \) -bi-invariant functions on \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) , where \( B \) is the Borel subgroup of upper triangular matrices in \( \mathrm{{GL... | Proof. It follows from Propositions 46.8 and 46.9 that (46.11)-(46.13) are all satisfied by the elements \( {\phi }_{w} \) in the ring \( \mathcal{H} \) of \( B \) -bi-invariant functions on \( \mathrm{{GL}}\left( {n,{\mathbb{F}}_{q}}\right) \), so there exists a homomorphism \( {\mathcal{H}}_{k}\left( q\right) \righta... | Yes |
Theorem 47.1 (Selberg, Langlands). Let \( r + t = k \) . Let \( P \) and \( Q \) be the parabolic subgroups of \( \mathrm{{GL}}\left( k\right) \) with Levi factors \( \mathrm{{GL}}\left( r\right) \times \mathrm{{GL}}\left( t\right) \) and \( \mathrm{{GL}}\left( t\right) \times \) \( \mathrm{{GL}}\left( r\right) \), res... | \[ M\left( s\right) : \operatorname{Ind}\left( {{\pi }_{1} \otimes {\pi }_{2} \otimes {\delta }_{P}^{s}}\right) \rightarrow \operatorname{Ind}\left( {{\pi }_{2} \otimes {\pi }_{1} \otimes {\delta }_{Q}^{-s}}\right) \] such that the functional equation \[ E\left( {g,{f}_{s}, s}\right) = E\left( {g, M\left( s\right) {f}_... | Yes |
Proposition 47.1. Let \( \left( {\pi, V}\right) \) be a cuspidal representation of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) . If \( U \) is the unipotent radical of a standard maximal parabolic subgroup of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) and if \( \eta : V \rightarrow \mathbb{C} \) is... | Proof. Choose an invariant inner product \( \langle \) , \( \rangle {onV} \) . There exists a vector \( y \in V \) such that \( \eta \left( v\right) = \langle v, y\rangle \) . Then\n\n\[ \langle v,\pi \left( u\right) y\rangle = \left\langle {\pi {\left( u\right) }^{-1}v, y}\right\rangle = \eta \left( {\pi {\left( u\rig... | Yes |
Proposition 47.2. Every irreducible representation \( \left( {\pi, V}\right) \) of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) is a constituent in some representation \( {\pi }_{1} \circ \cdots \circ {\pi }_{m} \) with the \( {\pi }_{i} \) cuspidal. | Proof. If \( \pi \) is cuspidal, then we may take \( m = 1 \) and \( {\pi }_{1} = \pi \) . There is nothing to prove in this case.\n\nIf \( \pi \) is not cuspidal, then there exists a decomposition \( k = r + t \) such that the space \( {V}^{U} \) of \( U \) -fixed vectors is nonzero, where \( U \) is the group (47.3).... | Yes |
Theorem 47.2. The multiplication in \( \mathcal{R}\left( q\right) \) is commutative. | Proof. We will frame our proof in terms of characters rather than representations, so in this proof elements of \( {\mathcal{R}}_{k}\left( q\right) \) are generalized characters of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \). We make use of the involution \( \iota : \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\ri... | Yes |
Theorem 47.4. Suppose that \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{r}}\right) \) is an ordered partition of \( k \) , and let \( {\pi }_{\lambda } = \otimes {\pi }_{i} \) be a cuspidal representation of \( {M}_{\lambda } \) . Suppose that no \( {\pi }_{i} \cong {\pi }_{j} \) . Then \( {\pi }_{1} \circ \... | Proof. By Theorem 47.3, the dimension of the space of intertwining operators of \( \operatorname{Ind}\left( {\pi }_{\lambda }\right) \) to itself is one, and it follows that this space is irreducible. The last statement is also clear from Theorem 47.3. | Yes |
Theorem 47.5 (Howlett and Lehrer). Let \( {\pi }_{0} \) be a cuspidal representation of \( \mathrm{{GL}}\left( {l,{\mathbb{F}}_{q}}\right) \) . Then the endomorphism ring \( \operatorname{End}\left( {\pi }_{0}^{\circ t}\right) \) is naturally isomorphic to \( {\mathcal{H}}_{t}\left( {q}^{l}\right) \) . | Proof. Proofs may be found in Howlett and Lehrer [80] and Howe [74]. | No |
Corollary 47.1. There exists a natural bijection between the set of partitions \( \lambda \) of \( t \) and the irreducible constituents \( {\sigma }_{\lambda \left( \pi \right) } \) of \( {\pi }_{0}^{\circ t} \) . The multiplicity of \( {\sigma }_{\lambda \left( \pi \right) } \) in \( {\pi }_{0}^{\text{ot }} \) equals... | Proof. The multiplicity of \( {\sigma }_{\lambda \left( \pi \right) } \) in \( {\pi }_{0}^{\circ t} \) equals the multiplicity of the corresponding module of \( {\mathcal{H}}_{t}\left( {q}^{l}\right) \) . By Exercise 46.5, this is the degree of \( {\mathbf{s}}_{\lambda } \) . | No |
Theorem 47.6. Let \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{r}}\right) \) be an ordered partition of \( k \), and let \( {\lambda }_{i} = {l}_{i}{t}_{i} \) . Let \( {\pi }_{i} \) be a cuspidal representation of \( \mathrm{{GL}}\left( {{l}_{i},{\mathbb{F}}_{q}}\right) \), with no two \( {\pi }_{i} \) isomorphic. Let... | Proof. We note the following general principle: \( \chi \) is a character of any group, and if \( \chi = \sum {d}_{i}{\chi }_{i} \) is a decomposition into subrepresentations such that\n\n\[ \langle \chi ,\chi \rangle = \sum {d}_{i}^{2} \]\n\nthen the \( {\chi }_{i} \) are irreducible and mutually nonisomorphic. Indeed... | Yes |
Theorem 47.7 (Green). Let \( G \) be a finite group, and let \( \rho : G \rightarrow \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) be a representation. Let \( f \in \mathbb{Z}\left\lbrack {{X}_{1},\ldots ,{X}_{k}}\right\rbrack \) be a symmetric polynomial with integer coefficients. Let \( \theta : {\overline{\math... | Proof. First, we reduce to the following case: \( \theta : {\overline{\mathbb{F}}}_{q}^{ \times } \rightarrow {\mathbb{C}}^{ \times } \) is injective and \( f\left( {{X}_{1},\ldots ,{X}_{k}}\right) = \sum {X}_{i} \) . If this case is known, then by replacing \( \rho \) by its exterior powers we get the same result for ... | Yes |
Lemma 47.1. Let \( \left\{ {{N}_{1},{N}_{2},\ldots }\right\} \) be a sequence of numbers, and for each \( {N}_{k} \) let \( {X}_{k} \) be a set of cardinality \( {N}_{k}\left( {X}_{k}\right. \) disjoint). Let \( {\sum }_{k} \) be the following set. An element of \( {\sum }_{k} \) consists of a 4-tuple \( \left( {\left\... | Proof. By induction on \( k \), we may assume that the cardinalities \( {N}_{1},\ldots ,{N}_{k - 1} \) are determined by the \( {M}_{k} \) . Let \( {M}_{k}^{\prime } \) be the cardinality of the set of equivalence classes of \( \left( {\left\{ {r}_{i}\right\} ,\left\{ {d}_{i}\right\} ,\left\{ {\lambda }^{i}\right\} ,\l... | Yes |
Theorem 47.8. The number of cuspidal representations of \( \mathrm{{GL}}\left( {k,{\mathbb{F}}_{q}}\right) \) equals the number of irreducible monic polynomials of degree \( k \) over \( {\mathbb{F}}_{q} \) . | Proof. We can apply the lemma with \( {X}_{k} \) either the set of cuspidal representations of \( {S}_{k} \) or with the set of monic irreducible polynomials of degree \( k \) over \( {\mathbb{F}}_{q} \) . We will show that in the first case, \( {M}_{k} \) is the number of irreducible representations of \( \operatornam... | Yes |
Proposition 47.4. Assume that \( \theta \) can be extended to a character \( \theta : {\overline{\mathbb{F}}}_{q}^{ \times } \rightarrow \) \( {\mathbb{C}}^{ \times } \) that is injective. Then the inner product \( \left\langle {{\chi }_{k},{\chi }_{k}}\right\rangle = k \) . | Proof. We will first prove that this is true for \( q \) sufficiently large, then show that it is true for all \( q \) . We will use \ | No |
Theorem 47.9. Assume that \( \theta \) is an injective character \( \theta : {\overline{\mathbb{F}}}_{q}^{ \times } \rightarrow {\mathbb{C}}^{ \times } \) . For each \( k \) there exists a cuspidal \( {\sigma }_{k} = {\sigma }_{k,\theta } \) of \( \operatorname{GL}\left( {k,{\mathbb{F}}_{q}}\right) \) such that if \( g... | Proof. By induction, we assume the existence of \( {\sigma }_{k} \) and the decomposition of \( {\chi }_{k} \) as stated for \( k < n \), and we deduce them for \( k = n \) .\n\nWe will show first that\n\n\[ \n\left\langle {{\chi }_{n},{\sigma }_{k} \circ {1}_{n - k}}\right\rangle = {\left( -1\right) }^{k - 1}.\n\]\n\n... | Yes |
Lemma 48.1. In the characterization of \( {\mathfrak{C}}_{\left( d\right) } \) it is only necessary to impose the condition (48.6) at integers \( 0 < i < r + s \) such that \( {d}_{i + 1} = {d}_{i} > {d}_{i - 1} \) . | Proof. If \( {d}_{i + 1} > {d}_{i} \) and \( \dim \left( {V \cap {F}_{i + 1}}\right) \geq {d}_{i + 1} \), then since \( V \cap {F}_{i} \) has codimension at most 1 in \( V \cap {F}_{i + 1} \) we do not need to assume \( \dim \left( {V \cap {F}_{i}}\right) \geq {d}_{i} \) . If \( {d}_{i} = {d}_{i - 1} \) and \( \dim \le... | Yes |
Proposition 48.1. The image in \( {\mathcal{G}}_{r, s} \) of the Schubert cell \( \mathcal{C}\left( w\right) \) corresponding to the partition \( \lambda \) (the diagram of which, we have noted, must fit in an \( r \times s \) box) is \( {\mathfrak{C}}_{\left( d\right) } \), where the integer sequence \( \left( {{d}_{0... | Proof. We will prove this for the open cell. The image of \( \mathcal{C}{\left( w\right) }^{ \circ } \) in \( {\mathcal{G}}_{r, s} \) consists of all spaces \( {bw}{F}_{r} \) with \( b \in B \), so we must show that, with \( {d}_{i} \) as in (48.7), we have\n\n\[ \n\dim \left( {{bw}{F}_{r} \cap {F}_{i}}\right) = {d}_{i... | Yes |
Lemma 1.1. An automorphism \( \varphi \) of \( {\mathbb{B}}_{n} \) is a unitary transformation of \( {\mathbb{C}}^{n} \) if and only if \( \varphi \left( 0\right) = 0 \) . | Proof. Assume that \( \varphi \) is an automorphism of \( {\mathbb{B}}_{n} \) with \( \varphi \left( 0\right) = 0 \) . Fix any complex number \( \lambda \) with \( \left| \lambda \right| = 1 \) and consider the holomorphic mapping \( F : {\mathbb{B}}_{n} \rightarrow {\mathbb{B}}_{n} \) defined by\n\n\[ F\left( z\right)... | Yes |
For each \( a \in {\mathbb{B}}_{n} \) the mapping \( {\varphi }_{a} \) satisfies\n\n\[1 - {\left| {\varphi }_{a}\left( z\right) \right| }^{2} = \frac{\left( {1 - {\left| a\right| }^{2}}\right) \left( {1 - {\left| z\right| }^{2}}\right) }{{\left| 1-\langle z, a\rangle \right| }^{2}},\;z \in {\mathbb{B}}_{n},\]\n\nand\n\... | Proof. The case \( a = 0 \) is obvious. So we assume that \( a \neq 0 \) .\n\nSince \( a - {P}_{a}\left( z\right) \) and \( {Q}_{a}\left( z\right) \) are perpendicular in \( {\mathbb{C}}^{n} \), we have\n\n\[{\left| a - {P}_{a}\left( z\right) - {s}_{a}{Q}_{a}\left( z\right) \right| }^{2} = {\left| a - {P}_{a}\left( z\r... | Yes |
Theorem 1.4. Every automorphism \( \varphi \) of \( {\mathbb{B}}_{n} \) is of the form\n\n\[ \varphi = U{\varphi }_{a} = {\varphi }_{b}V \]\n\nwhere \( U \) and \( V \) are unitary transformations of \( {\mathbb{C}}^{n} \), and \( {\varphi }_{a} \) and \( {\varphi }_{b} \) are involutions. | Proof. Suppose \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) and \( a = {\varphi }^{-1}\left( 0\right) \) . Then the automorphism \( \psi = \varphi \circ {\varphi }_{a} \) satisfies \( \psi \left( 0\right) = 0 \) . By Lemma 1.1, there exists a unitary transformation \( U \) of \( {\mathbb{C}}^{n} \... | Yes |
Corollary 1.5. Every \( \varphi \) in \( \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) extends to a homeomorphism of \( {\mathbb{S}}_{n} \) . | Proof. It is obvious that every unitary transformation in \( \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) induces a homeomorphism of \( {\mathbb{S}}_{n} \) . By Lemma 1.2, every involution \( {\varphi }_{a} \) also extends to a homeomorphism on \( {\mathbb{S}}_{n} \) . | No |
Lemma 1.6. If we identify linear transformations of \( {\mathbb{C}}^{n} \) with \( n \times n \) matrices via the standard basis of \( {\mathbb{C}}^{n} \), then for every \( a \in {\mathbb{B}}_{n} - \{ 0\} \) we have\n\n\[ \n{\varphi }_{a}^{\prime }\left( 0\right) = - \left( {1 - {\left| a\right| }^{2}}\right) {P}_{a} ... | Proof. For any \( a \in {\mathbb{B}}_{n}, a \neq 0 \), we can write \n\n\[ \n{\varphi }_{a}\left( z\right) = \left( {a - {P}_{a}\left( z\right) - {s}_{a}{Q}_{a}\left( z\right) }\right) \mathop{\sum }\limits_{{k = 0}}^{\infty }\langle z, a{\rangle }^{k} \n\] \n\n\[ \n= a + a\langle z, a\rangle - \left( {{P}_{a} + {s}_{a... | Yes |
Lemma 1.7. For each \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) we have\n\n\[ \n{J}_{R}\varphi \left( z\right) = {\left( \frac{1 - {\left| a\right| }^{2}}{{\left| 1-\langle z, a\rangle \right| }^{2}}\right) }^{n + 1}, \n\]\n\n(1.11)\n\nwhere \( a = {\varphi }^{-1}\left( 0\right) \) . | Proof. For any fixed \( a \) and \( z \) in \( {\mathbb{B}}_{n} \) with \( a \neq 0 \), we let \( w = {\varphi }_{a}\left( z\right) \) and consider the automorphism\n\n\[ \nU = {\varphi }_{w} \circ {\varphi }_{a} \circ {\varphi }_{z} \]\n\nSince \( U\left( 0\right) = 0 \), Lemma 1.1 shows that \( U \) is a unitary. Rew... | Yes |
Lemma 1.8. The measures \( v \) and \( \sigma \) are related by\n\n\[ \n{\int }_{{\mathbb{B}}_{n}}f\left( z\right) {dv}\left( z\right) = {2n}{\int }_{0}^{1}{r}^{{2n} - 1}{dr}{\int }_{{\mathbb{S}}_{n}}f\left( {r\zeta }\right) {d\sigma }\left( \zeta \right) .\n\] | Proof. Let \( {dV} = d{x}_{1}d{y}_{1}\cdots d{x}_{n}d{y}_{n} \) be the actual Lebesgue measure in \( {\mathbb{C}}^{n} \) (before normalization), where we identify each \( {z}_{k} \) with \( {x}_{k} + i{y}_{k} \) . Similarly, let \( {dS} \) be the surface measure on \( {\mathbb{S}}_{n} \) before normalization. Then the ... | Yes |
Lemma 1.9. Suppose \( f \) is a function on \( {\mathbb{S}}_{n} \) that depends only on \( {z}_{1},\cdots ,{z}_{k} \), where \( 1 \leq k < n \) . Then \( f \) can be regarded as defined on \( {\mathbb{B}}_{k} \) and\n\n\[ \n{\int }_{{\mathbb{S}}_{n}}{fd\sigma } = \left( \begin{matrix} n - 1 \\ k \end{matrix}\right) {\i... | Proof. For the purpose of this proof let \( {P}_{k} \) denote the orthogonal projection from \( {\mathbb{C}}^{n} \) onto \( {\mathbb{C}}^{k} \) . Then\n\n\[ \n{\int }_{{\mathbb{S}}_{n}}{fd\sigma } = {\int }_{{\mathbb{S}}_{n}}f \circ {P}_{k}{d\sigma }\n\]\n\nBy an approximation argument, it suffices for us to prove the ... | Yes |
Lemma 1.10. For \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) we have\n\n\[{\int }_{{\mathbb{S}}_{n}}{fd\sigma } = {\int }_{{\mathbb{S}}_{n}}{d\sigma }\left( \zeta \right) \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {{e}^{i\theta }\zeta }\right) {d\theta }\]\n\n(1.14)\n\nand if \( 1 < k < n \), then\n\n\... | Proof. It is obvious that\n\n\[{\int }_{{\mathbb{S}}_{n}}{fd\sigma } = {\int }_{{\mathbb{S}}_{n}}f\left( {{e}^{i\theta }\zeta }\right) {d\sigma }\left( \zeta \right)\]\n\nfor all \( 0 \leq \theta \leq {2\pi } \) . Integrate with respect to \( \theta \in \\left\\lbrack {0,{2\pi }}\\right\\rbrack \) and apply Fubini’s th... | Yes |
Lemma 1.11. Suppose \( m = \left( {{m}_{1},\cdots ,{m}_{n}}\right) \) is a multi-index of nonnegative integers and \( \alpha > - 1 \) . Then\n\n\[{\int }_{{\mathbb{S}}_{n}}{\left| {\zeta }^{m}\right| }^{2}{d\sigma }\left( \zeta \right) = \frac{\left( {n - 1}\right) !m!}{\left( {n - 1 + \left| m\right| }\right) !}\]\n\n... | Proof. We identify \( {\mathbb{C}}^{n} \) with \( {\mathbb{R}}^{2n} \) using the real and imaginary parts of a complex number, and denote the usual Lebesgue measure on \( {\mathbb{C}}^{n} \) by \( {dV} \) . If the Euclidean volume of \( {\mathbb{B}}_{n} \) is \( {c}_{n} \), then \( {c}_{n}{dv} = {dV} \) .\n\nWe evaluat... | Yes |
Theorem 1.12. Suppose \( c \) is real and \( t > - 1 \) . Then the integrals\n\n\[ \n{I}_{c}\left( z\right) = {\int }_{{\mathbb{S}}_{n}}\frac{{d\sigma }\left( \zeta \right) }{{\left| 1-\langle z,\zeta \rangle \right| }^{n + c}},\;z \in {\mathbb{B}}_{n}, \n\]\n\nand\n\n\[ \n{J}_{c, t}\left( z\right) = {\int }_{{\mathbb{... | Proof. Let \( \lambda = \left( {n + c}\right) /2 \) . Then\n\n\[ \n\frac{1}{{\left| 1-\langle z,\zeta \rangle \right| }^{n + c}} = {\left| \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{\Gamma \left( {k + \lambda }\right) }{k!\Gamma \left( \lambda \right) }\langle z,\zeta {\rangle }^{k}\right| }^{2}. \n\]\n\nFor any fi... | Yes |
Proposition 1.13. Suppose \( \\alpha \) is real and \( f \) is in \( {L}^{1}\\left( {{\\mathbb{B}}_{n}, d{v}_{\\alpha }}\\right) \) . Then\n\n\\[ \n{\\int }_{{\\mathbb{B}}_{n}}f \\circ \\varphi \\left( z\\right) d{v}_{\\alpha }\\left( z\\right) = {\\int }_{{\\mathbb{B}}_{n}}f\\left( z\\right) \\frac{{\\left( 1 - {\\lef... | Proof. By Theorem 1.4 there exists a unitary transformation \( U \) such that \( \\varphi = {\\varphi }_{a}U \) , where \( a = \\varphi \\left( 0\\right) \) . Since the measure \( d{v}_{\\alpha } \) is invariant under the action of unitary transformations, we may as well assume that \( \\varphi = {\\varphi }_{a} \) . I... | Yes |
Proposition 1.14. Suppose neither \( n + \alpha \) nor \( n + \alpha + t \) is a negative integer. Then the operator \( {R}^{\alpha, t} \) is the unique continuous linear operator on \( H\left( {\mathbb{B}}_{n}\right) \) satisfying\n\n\[ \n{R}^{\alpha, t}\left( \frac{1}{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + ... | Proof. The series\n\n\[ \n\frac{1}{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \alpha }} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{\Gamma \left( {n + 1 + k + \alpha }\right) }{k!\Gamma \left( {n + 1 + \alpha }\right) }\langle z, w{\rangle }^{k}\n\]\n\nand\n\n\[ \n\frac{1}{{\left( 1-\langle z, w\rangle \righ... | Yes |
Proposition 1.15. Suppose \( N \) is a positive integer and \( \alpha \) is a real number such that \( n + \alpha \) is not a negative integer. Then \( {R}^{\alpha, N} \), as an operator acting on \( H\left( {\mathbb{B}}_{n}\right) \), is a linear partial differential operator of order \( N \) with polynomial coefficie... | Proof. Fix \( w \in {\mathbb{B}}_{n} \), replace the numerator of \n\n\[ \n\frac{1}{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \alpha + N}} \n\] \n\nby \n\n\[ \n{\left( 1-\langle z, w\rangle +\langle z, w\rangle \right) }^{N} \n\] \n\nand apply the binomial formula. Then \n\n\[ \n\frac{1}{{\left( 1-\langle z, w\r... | Yes |
Proposition 1.16. Suppose \( f \) is twice differentiable in \( {\mathbb{B}}_{n} \) . Then\n\n\[ \widetilde{\Delta }\left( {f \circ \varphi }\right) = \left( {\widetilde{\Delta }f}\right) \circ \varphi \]\n\nfor all \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) . | Proof. Fix \( z \in {\mathbb{B}}_{n} \) and \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) . Let \( a = \varphi \left( z\right) \) . Then the automorphism\n\n\[ U = {\varphi }_{a} \circ \varphi \circ {\varphi }_{z} \]\n\nfixes the origin and hence is a unitary by Lemma 1.1. It is easy to see that\n\... | Yes |
Proposition 1.17. If \( f \) is twice differentiable in \( {\mathbb{B}}_{n} \), then\n\n\[\n\left( {\widetilde{\Delta }f}\right) \left( z\right) = 4\left( {1 - {\left| z\right| }^{2}}\right) \mathop{\sum }\limits_{{i, j = 1}}^{n}\left( {{\delta }_{ij} - {z}_{i}{\bar{z}}_{j}}\right) \frac{{\partial }^{2}f}{\partial {z}_... | Proof. Fix \( z \in {\mathbb{B}}_{n} \) and write\n\n\[{\varphi }_{z}\left( w\right) = \left( {{\varphi }_{1}\left( w\right) ,\cdots ,{\varphi }_{n}\left( w\right) }\right) ,\;w \in {\mathbb{B}}_{n}.\n\]\n\nBy the chain rule,\n\n\[\n\left( {\widetilde{\Delta }f}\right) \left( z\right) = \Delta \left( {f \circ {\varphi ... | Yes |
Proposition 1.18. For \( z \in {\mathbb{B}}_{n} \) the matrices \( A\left( z\right) \) and \( B\left( z\right) \) have the following properties:\n\n(a) \( B\left( z\right) = \left\lbrack {\left( {1 - {\left| z\right| }^{2}}\right) I + A\left( z\right) }\right\rbrack /{\left( 1 - {\left| z\right| }^{2}\right) }^{2} \), ... | Proof. Since\n\n\[ \log K\left( {z, z}\right) = - \left( {n + 1}\right) \log \left( {1 - {\left| z\right| }^{2}}\right) \]\n\nwe have\n\n\[ \frac{\partial }{\partial {z}_{j}}\log K\left( {z, z}\right) = \left( {n + 1}\right) \frac{{\bar{z}}_{j}}{1 - {\left| z\right| }^{2}} \]\n\nfor \( j = 1,\cdots, n \) . It follows t... | Yes |
Proposition 1.19. The Bergman matrix is invariant under automorphisms, that is,\n\n\[ B\left( z\right) = {\left( {\varphi }^{\prime }\left( z\right) \right) }^{ * }B\left( {\varphi \left( z\right) }\right) {\varphi }^{\prime }\left( z\right) \]\n\nfor all \( z \in {\mathbb{B}}_{n} \) and \( \varphi \in \operatorname{Au... | Proof. Without loss of generality we may assume that \( \varphi = {\varphi }_{a} \) for some \( a \in {\mathbb{B}}_{n} \) . In this case, it follows from (1.5) and (1.11) that the Bergman kernel satisfies\n\n\[ K\left( {z, z}\right) = {\left| {J}_{C}\varphi \left( z\right) \right| }^{2}K\left( {\varphi \left( z\right) ... | Yes |
Proposition 1.20. The Bergman metric is invariant under automorphisms, that is,\n\n\[ \n\beta \left( {\varphi \left( z\right) ,\varphi \left( w\right) }\right) = \beta \left( {z, w}\right) \n\]\n\nfor all \( z, w \in {\mathbb{B}}_{n} \) and \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) . | Proof. This follows easily from Proposition 1.19 and the definition of the Bergman metric. | No |
Proposition 1.21. If \( z \) and \( w \) are points in \( {\mathbb{B}}_{n} \), then\n\n\[ \beta \left( {z, w}\right) = \frac{1}{2}\log \frac{1 + \left| {{\varphi }_{z}\left( w\right) }\right| }{1 - \left| {{\varphi }_{z}\left( w\right) }\right| } \]\n\nwhere \( {\varphi }_{z} \) is the involutive automorphism of \( {\m... | Proof. By invariance, we only need to prove the result for \( w = 0 \) .\n\nFix a point \( z \in {\mathbb{B}}_{n} \) and let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{B}}_{n} \) be a piecewise smooth curve from 0 to \( z \) . Dividing the interval \( \left\lbrack {0,1}\right\rbrack \) into a fini... | Yes |
For \( z \) and \( w \) in \( {\mathbb{B}}_{n} \) let\n\n\[ \rho \left( {z, w}\right) = \left| {{\varphi }_{z}\left( w\right) }\right| \]\n\nThen \( \rho \) is a metric on \( {\mathbb{B}}_{n} \) . Moreover, \( \rho \) is invariant under automorphisms, that is,\n\n\[ \rho \left( {\varphi \left( z\right) ,\varphi \left( ... | Proof. A calculation shows that\n\n\[ \rho \left( {z, w}\right) = \tanh \beta \left( {z, w}\right) \]\n\n(1.40)\n\nfor all \( z, w \in {\mathbb{B}}_{n} \) . The invariance of \( \rho \), which can be checked directly, is thus a\n\nconsequence of the invariance of \( \beta \) .\n\nIt remains to prove that \( \rho \) is ... | No |
Lemma 1.23. For any \( z \in {\mathbb{B}}_{n} \) and \( r > 0 \) we have\n\n\[ v\left( {D\left( {z, r}\right) }\right) = \frac{{R}^{2n}{\left( 1 - {\left| z\right| }^{2}\right) }^{n + 1}}{{\left( 1 - {R}^{2}{\left| z\right| }^{2}\right) }^{n + 1}}, \]\n\nwhere \( R = \tanh \left( r\right) \) . In particular, for any \(... | Proof. By Proposition 1.21, each \( D\left( {0, r}\right) \) is actually a Euclidean ball of radius \( R = \tanh \left( r\right) \) centered at the origin. Since the Bergman metric is invariant under automorphisms of \( {\mathbb{B}}_{n} \), we have\n\n\[ D\left( {z, r}\right) = {\varphi }_{z}\left( {D\left( {0, r}\righ... | Yes |
Lemma 1.24. For any real \( \alpha \) and positive \( r \) there exist constants \( C > 0 \) and \( c > 0 \) such that\n\n\[ c{\left( 1 - {\left| z\right| }^{2}\right) }^{n + 1 + \alpha } \leq {v}_{\alpha }\left( {D\left( {z, r}\right) }\right) \leq C{\left( 1 - {\left| z\right| }^{2}\right) }^{n + 1 + \alpha } \]\n\nf... | Proof. Let \( R = \tanh \left( r\right) \) again and make a change of variables according to Proposition 1.13. We obtain\n\n\[ {v}_{\alpha }\left( {D\left( {z, r}\right) }\right) = {c}_{\alpha }{\int }_{D\left( {z, r}\right) }{\left( 1 - {\left| z\right| }^{2}\right) }^{\alpha }{dv}\left( z\right) \]\n\n\[ = {c}_{\alph... | Yes |
Proposition 1.26. The invariant Green’s function \( G \) has the following asymptotic behavior.\n\n(a) As \( \left| z\right| \rightarrow {0}^{ + } \), we have\n\n\[ G\left( z\right) \sim \log \frac{1}{\left| z\right| } \]\n\n\[ \text{for}n = 1\text{, and} \]\n\n\[ G\left( z\right) \sim \frac{1}{{\left| z\right| }^{{2n}... | Proof. This is elementary and we leave it as an exercise. | No |
Corollary 1.29. If \( \alpha > - 1 \) and \( f \) is subharmonic in \( {\mathbb{B}}_{n} \), then\n\n\[ f\left( a\right) \leq {\int }_{{\mathbb{B}}_{n}}f\left( {a + {rz}}\right) d{v}_{\alpha }\left( z\right) \]\n\n(1.50)\n\nfor all \( a \in {\mathbb{B}}_{n} \) and \( 0 \leq r < 1 - \left| a\right| \) . | Proof. This follows from (1.49) and integration in polar coordinates. | No |
Corollary 1.30. If \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) and \( 0 < p < \infty \), then \( \log \left| f\right| \) and \( {\left| f\right| }^{p} \) are both subharmonic in \( {\mathbb{B}}_{n} \) . | Proof. Fix a point \( a \in {\mathbb{B}}_{n} \) . If \( f\left( a\right) = 0 \), then obviously,\n\n\[ \log \left| {f\left( a\right) }\right| \leq {\int }_{{\mathbb{S}}_{n}}\log \left| {f\left( {a + {r\zeta }}\right) }\right| {d\sigma }\left( \zeta \right) \]\n\nand\n\n\[ {\left| f\left( a\right) \right| }^{p} \leq {\i... | Yes |
Theorem 2.1. Suppose \( 0 < p < \infty \) and \( \alpha > - 1 \) . Then\n\n\[ \left| {f\left( z\right) }\right| \leq \frac{\parallel f{\parallel }_{p,\alpha }}{{\left( 1 - {\left| z\right| }^{2}\right) }^{\left( {n + 1 + \alpha }\right) /p}} \]\n\nfor all \( f \in {A}_{\alpha }^{p} \) and \( z \in {\mathbb{B}}_{n} \) . | Proof. If \( f \) is any holomorphic function in \( {\mathbb{B}}_{n} \), then \( {\left| f\right| }^{p} \) is subharmonic, so by Corollary 1.29,\n\n\[ {\left| f\left( 0\right) \right| }^{p} \leq {\int }_{{\mathbb{B}}_{n}}{\left| f\left( w\right) \right| }^{p}d{v}_{\alpha }\left( w\right) \]\n\nThis proves the desired r... | Yes |
Theorem 2.2. If \( \alpha > - 1 \) and \( f \in {A}_{\alpha }^{1} \), then\n\n\[ f\left( z\right) = {\int }_{{\mathbb{B}}_{n}}\frac{f\left( w\right) d{v}_{\alpha }\left( w\right) }{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \alpha }} \] \n\nfor all \( z \in {\mathbb{B}}_{n} \) . | Proof. Let \( f \in {A}_{\alpha }^{1} \) . By the mean value property for holomorphic functions,\n\n\[ f\left( 0\right) = {\int }_{{\mathbb{S}}_{n}}f\left( {r\zeta }\right) {d\sigma }\left( \zeta \right) ,\;0 \leq r < 1. \]\n\nThis together with integration in polar coordinates shows that\n\n\[ f\left( 0\right) = {\int... | Yes |
Corollary 2.3. Suppose \( \alpha > - 1, t > 0 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . If neither \( n + \alpha \) nor \( n + \alpha + t \) is not a negative integer, then\n\n\[ \n{R}^{\alpha, t}f\left( z\right) = \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{\int }_{{\mathbb{B}}_{n}}\frac{f\left(... | Proof. For any fixed \( r \in \left( {0,1}\right) \) the dilation \( {f}_{r} \), defined by \( {f}_{r}\left( z\right) = f\left( {rz}\right) \), belongs to both \( {A}_{\alpha }^{1} \) and \( {A}_{\alpha + t}^{1} \) . So, according to Theorem 2.2,\n\n\[ \n{f}_{r}\left( z\right) = {\int }_{{\mathbb{B}}_{n}}\frac{{f}_{r}\... | Yes |
Lemma 2.4. Suppose \( p > 0,\alpha > - 1,0 < r < 1 \), and \( m = \left( {{m}_{1},\cdots ,{m}_{n}}\right) \) is a multi-index of nonnegative integers. Then there exists a positive constant \( C \) such that\n\n\[ \left| {\frac{{\partial }^{m}f}{\partial {z}^{m}}\left( z\right) }\right| \leq C\parallel f{\parallel }_{p,... | Proof. Fix some \( \delta \in \left( {r,1}\right) \) and apply Theorem 2.2 in the special case \( \alpha = 0 \) . We\n\nobtain\n\[ f\left( {\delta z}\right) = {\int }_{{\mathbb{B}}_{n}}\frac{f\left( {\delta w}\right) {dv}\left( w\right) }{{\left( 1-\langle z, w\rangle \right) }^{n + 1}},\;z \in {\mathbb{B}}_{n}. \]\n\n... | Yes |
For each \( p > 0 \) and \( \alpha > - 1 \) the weighted Bergman space \( {A}_{\alpha }^{p} \) is closed in \( {L}^{p}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) . | Proof. Suppose \( \left\{ {f}_{n}\right\} \) is a sequence in \( {A}_{\alpha }^{p} \) and\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{p,\alpha } = 0 \]\n\nfor some \( f \in {L}^{p}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) . Then some subsequence of \( \left\... | Yes |
Proposition 2.6. Suppose \( p > 0 \) and \( \alpha > - 1 \) . Then the set of polynomials is dense in \( {A}_{\alpha }^{p} \) . | Proof. Writing the ball as\n\n\[ \n{\mathbb{B}}_{n} = \{ z : \left| z\right| \leq 1 - \epsilon \} \bigcup \{ z : 1 - \epsilon < \left| z\right| < 1\} \]\n\nfor a sufficiently small positive \( \epsilon \), we easily prove that\n\n\[ \n\mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{\begin{Vmatrix}{f}_{r} - f\end{Vmat... | Yes |
For each \( \alpha > - 1 \) the reproducing kernel of \( {A}_{\alpha }^{2} \) is given by\n\n\[ \n{K}^{\alpha }\left( {z, w}\right) = \frac{1}{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \alpha }},\;z, w \in {\mathbb{B}}_{n}.\n\] | Proof. This follows from Theorem 2.2 and the uniqueness of the Riesz representation for a bounded linear functional on a Hilbert space. | No |
Lemma 2.8. Suppose \( \alpha > - 1 \) . Then the restriction of \( {P}_{\alpha } \) to \( {L}^{2}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) is the orthogonal projection from \( {L}^{2}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) onto \( {A}_{\alpha }^{2} \) . | Proof. Let \( P \) be the orthogonal projection from \( {L}^{2}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) onto \( {A}_{\alpha }^{2} \) . For \( f \in \) \( {L}^{2}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) and \( z \in {\mathbb{B}}_{n} \) the reproducing property of \( {K}^{\alpha } \) and the self-ad... | Yes |
Theorem 2.9. Suppose \( \left( {X,\mu }\right) \) is a measure space, \( 1 < p < \infty \), and \( 1/p + 1/q = 1 \) . For a nonnegative kernel \( H\left( {x, y}\right) \) consider the integral operator\n\n\[ \n{Tf}\left( x\right) = {\int }_{X}H\left( {x, y}\right) f\left( y\right) {d\mu }\left( y\right) .\n\]\n\nIf the... | Proof. Given a function \( f \in {L}^{p}\left( {X,\mu }\right) \), Hölder’s inequality gives\n\n\[ \n\left| {{Tf}\left( x\right) }\right| \leq {\left\lbrack {\int }_{X}H\left( x, y\right) h{\left( y\right) }^{q}d\mu \left( y\right) \right\rbrack }^{1/q}{\left\lbrack {\int }_{X}H\left( x, y\right) h{\left( y\right) }^{-... | Yes |
Theorem 2.11. Suppose \( - 1 < \alpha < \infty \) , \( - 1 < t < \infty \), and \( 1 \leq p < \infty \) . Then the operator \( {P}_{\alpha } \) is a bounded projection from \( {L}^{p}\left( {{\mathbb{B}}_{n}, d{v}_{t}}\right) \) onto \( {A}_{t}^{p} \) if and only if\n\n\[ p\left( {\alpha + 1}\right) > t + 1 \] | In particular, \( {P}_{\alpha } \) is a bounded projection from \( {L}^{p}\left( {{\mathbb{B}}_{n}, d{v}_{\alpha }}\right) \) onto \( {A}_{\alpha }^{p} \) if and only if \( p > 1 \), and \( {P}_{\alpha } \) is a bounded projection from \( {L}^{1}\left( {{\mathbb{B}}_{n}, d{v}_{t}}\right) \) onto \( {A}_{t}^{1} \) if an... | Yes |
Theorem 2.12. Suppose \( \alpha > - 1,\beta > - 1 \), and \( 1 < p < \infty \) . Then\n\n\[{\left( {A}_{\alpha }^{p}\right) }^{ * } = {A}_{\beta }^{q}\]\n\n(with equivalent norms) under the integral pairing\n\n\[\\langle f, g{\\rangle }_{\\gamma } = {\\int }_{{\\mathbb{B}}_{n}}f\\left( z\\right) \\overline{g\\left( z\\... | Proof. If \( g \\in {A}_{\\beta }^{q} \) and\n\n\[F\\left( f\\right) = \\langle f, g{\\rangle }_{\\gamma } = {c}_{\\gamma }{\\int }_{{\\mathbb{B}}_{n}}{\\left( 1 - {\\left| z\\right| }^{2}\\right) }^{\\alpha /p}f\\left( z\\right) \\overline{{\\left( 1 - {\\left| z\\right| }^{2}\\right) }^{\\beta /q}g\\left( z\\right) }... | Yes |
Lemma 2.13. If \( f \) is holomorphic in \( {\mathbb{B}}_{n} \), then\n\n\[{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2} = \left( {1 - {\left| z\right| }^{2}}\right) \left( {{\left| \nabla f\left( z\right) \right| }^{2} - {\left| Rf\left( z\right) \right| }^{2}}\right)\]\n\nfor all \( z \in {\mathbb{B}}_{n}... | Proof. For any holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) we have\n\n\[ \widetilde{\Delta }\left( {\left| f\right| }^{2}\right) \left( 0\right) = \Delta \left( {\left| f\right| }^{2}\right) \left( 0\right) = 4{\left| \nabla f\left( 0\right) \right| }^{2} = 4{\left| \widetilde{\nabla }f\left( 0\right) \right... | Yes |
Lemma 2.14. If \( f \) is holomorphic in \( {\mathbb{B}}_{n} \), then\n\n\[ \left( {1 - {\left| z\right| }^{2}}\right) \left| {{Rf}\left( z\right) }\right| \leq \left( {1 - {\left| z\right| }^{2}}\right) \left| {\nabla f\left( z\right) }\right| \leq \left| {\widetilde{\nabla }f\left( z\right) }\right| \]\n\nfor all \( ... | Proof. By the Cauchy-Schwarz inequality for \( {\mathbb{C}}^{n} \) ,\n\n\[ \left| {{Rf}\left( z\right) }\right| \leq \left| z\right| \left| {\nabla f\left( z\right) }\right| \leq \left| {\nabla f\left( z\right) }\right| . \]\n\nThis proves the first inequality. The second inequality follows from Lemma 2.13 and the fact... | Yes |
Lemma 2.15. Let \( 0 < p \leq 1 \) and \( \alpha > - 1 \) . Then\n\n\[{\int }_{{\mathbb{B}}_{n}}\left| {f\left( z\right) }\right| {\left( 1 - {\left| z\right| }^{2}\right) }^{\left( {n + 1 + \alpha }\right) /p - \left( {n + 1}\right) }{dv}\left( z\right) \leq \frac{\parallel f{\parallel }_{p,\alpha }}{{c}_{\alpha }}\]\... | Proof. Write\n\n\[ \left| {f\left( z\right) }\right| = {\left| f\left( z\right) \right| }^{p}{\left| f\left( z\right) \right| }^{1 - p}, \]\n\nand estimate the second factor using Theorem 2.1. The desired result follows. | No |
Theorem 2.17. Suppose \( \alpha > - 1, p > 0, N \) is a positive integer, and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then \( f \in {A}_{\alpha }^{p} \) if and only if the functions\n\n\[ \n{\left( 1 - {\left| z\right| }^{2}\right) }^{N}\frac{{\partial }^{m}f}{\partial {z}^{m}}\left( z\right) ,\;\left| m\rig... | Proof. The case \( N = 1 \) follows from the equivalence of (a) and (c) in Theorem 2.16. We prove the case \( N = 2 \) here; the general case can then be proved using the same idea and induction.\n\nSo we assume \( f \in {A}_{\alpha }^{p} \) . By the equivalence of (a) and (c) in Theorem 2.16, each function \( \partial... | Yes |
Lemma 2.18. Suppose neither \( n + s \) nor \( n + s + t \) is a negative integer. If \( \beta = s + N \) for some positive integer \( N \), then there exists a one-variable polynomial \( h \) of degree\n\n\( N \) such that\n\[ \n{R}^{s, t}\frac{1}{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \beta }} = \frac{h\lef... | Proof. Recall that\n\n\[ \n\frac{1}{{\left( 1-\langle z, w\rangle \right) }^{\lambda }} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{\Gamma \left( {k + \lambda }\right) }{k!\Gamma \left( \lambda \right) }\langle z, w{\rangle }^{k} \n\]\n\n(2.14)\n\nfor any \( \lambda \neq 0, - 1, - 2,\cdots \) . It follows from the... | Yes |
Lemma 2.20. For each \( r > 0 \) there exists a positive constant \( {C}_{r} \) such that\n\n\[ \n{C}_{r}^{-1} \leq \frac{1 - {\left| a\right| }^{2}}{1 - {\left| z\right| }^{2}} \leq {C}_{r} \]\n\n(2.17)\n\nand\n\n\[ \n{C}_{r}^{-1} \leq \frac{1 - {\left| a\right| }^{2}}{\left| 1-\langle a, z\rangle \right| } \leq {C}_{... | Proof. Given any two points \( a \) and \( z \) in \( {\mathbb{B}}_{n} \) with \( \beta \left( {a, z}\right) < r \), we can write \( z = \) \( {\varphi }_{a}\left( w\right) \) for some \( w \in {\mathbb{B}}_{n} \) with \( \beta \left( {0, w}\right) < r \) . It follows from Lemma 1.2 that\n\n\[ \n1 - {\left| z\right| }^... | Yes |
Corollary 2.21. Suppose \( - \infty < \alpha < \infty ,{r}_{1} > 0,{r}_{2} > 0 \), and \( {r}_{3} > 0 \) . Then there exists a constant \( C > 0 \) such that\n\n\[ \n{C}^{-1} \leq \frac{{v}_{\alpha }\left( {D\left( {z,{r}_{1}}\right) }\right) }{{v}_{\alpha }\left( {D\left( {w,{r}_{2}}\right) }\right) } \leq C \n\]\n\nf... | Proof. This follows from Lemmas 1.24 and 2.20. | No |
Lemma 2.22. Given any positive number \( R \) and natural number \( M \), there exists a natural number \( N \) such that every Bergman metric ball of radius \( r \), where \( r \leq R \) , can be covered by \( N \) Bergman metric balls of radius \( r/M \) . | Proof. Fix a Bergman metric ball \( D\left( {a, r}\right) \) with \( 0 < r \leq R \) . Let \( \delta = r/M \) and let \( \left\{ {D\left( {{a}_{k},\delta /2}\right) }\right\} \) be any finite covering of \( D\left( {a, r}\right) \), where each \( {a}_{k} \in D\left( {a, r}\right) \) . Set \( {a}_{1}^{\prime } = {a}_{1}... | Yes |
Theorem 2.23. There exists a positive integer \( N \) such that for any \( 0 < r \leq 1 \) we can find a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) with the following properties:\n\n(1) \( {\mathbb{B}}_{n} = { \cup }_{k}D\left( {{a}_{k}, r}\right) \).\n\n(2) The sets \( D\left( {{a}_{k}, r/4}\righ... | Proof. Fix any \( r \in (0,1\rbrack \) . It is easy to find a sequence \( \left\{ {a}_{k}\right\} \) such that\n\n\[{\mathbb{B}}_{n} = \mathop{\bigcup }\limits_{k}D\left( {{a}_{k}, r}\right)\]\n\nand that \( \beta \left( {{a}_{i},{a}_{j}}\right) \geq r/2 \) for all \( i \neq j \) ; see the first part of the proof of Le... | Yes |
Lemma 2.24. Suppose \( r > 0, p > 0 \), and \( \alpha > - 1 \) . Then there exists a constant \( C > 0 \) such that\n\n\[{\left| f\left( z\right) \right| }^{p} \leq \frac{C}{{\left( 1 - {\left| z\right| }^{2}\right) }^{n + 1 + \alpha }}{\int }_{D\left( {z, r}\right) }{\left| f\left( w\right) \right| }^{p}d{v}_{\alpha }... | Proof. Recall from Proposition 1.21 that \( D\left( {0, r}\right) \) is a Euclidean ball centered at the origin with Euclidean radius \( R = \tanh \left( r\right) \) . So the subharmonicity of \( {\left| f\right| }^{p} \) and Corollary 1.29 show that\n\n\[{\left| f\left( 0\right) \right| }^{p} \leq \frac{1}{{v}_{\alpha... | Yes |
Theorem 2.25. Suppose \( p > 0, r > 0,\alpha > - 1 \), and \( \mu \) is a positive Borel measure on \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) There exists a constant \( C > 0 \) such that\n\n\[{\int }_{{\mathbb{B}}_{n}}{\left| f\left( z\right) \right| }^{p}{d\mu }\left( z\right) \leq... | Proof. It is easy to see that (a) implies (b). In fact, setting\n\n\[f\left( z\right) = {\left( \frac{{\left( 1 - {\left| a\right| }^{2}\right) }^{n + \alpha + 1}}{{\left( 1-\langle z, a\rangle \right) }^{2\left( {n + 1 + \alpha }\right) }}\right) }^{1/p}\]\n\nin (a) immediately yields (b).\n\nIf (b) is true, then\n\n\... | Yes |
Theorem 2.26. Suppose \( p > 0, r > 0,\alpha > - 1 \), and \( \mu \) is a positive Borel measure on \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) Whenever \( \left\{ {f}_{k}\right\} \) converges ultra-weakly to 0 in \( {A}_{\alpha }^{p} \), we have\n\n\[ \mathop{\lim }\limits_{{k \righta... | Proof. The proof is similar to that of Theorem 2.25. We leave the details to the interested reader. | No |
Lemma 2.27. For any \( R > 0 \) and any real \( b \) there exists a constant \( C > 0 \) such that\n\n\[ \left| {\frac{{\left( 1-\langle z, u\rangle \right) }^{b}}{{\left( 1-\langle z, v\rangle \right) }^{b}} - 1}\right| \leq {C\beta }\left( {u, v}\right) \]\n\nfor all \( z, u \), and \( v \) in \( {\mathbb{B}}_{n} \) ... | Proof. If \( u \) and \( v \) satisfy \( \beta \left( {u, v}\right) \leq R \), we can write \( v = {\varphi }_{u}\left( w\right) \) with \( \left| w\right| \leq r \), where \( r = \tanh R \in \left( {0,1}\right) \) . Let \( {z}^{\prime } = {\varphi }_{u}\left( z\right) \) . Then by Lemma 1.3,\n\n\[ 1 - \langle z, u\ran... | Yes |
Lemma 2.28. For each \( k \geq 1 \) there exists a Borel set \( {D}_{k} \) satisfying the following conditions:\n\n(i) \( D\left( {{a}_{k}, r/4}\right) \subset {D}_{k} \subset D\left( {{a}_{k}, r}\right) \) for every \( k \) .\n\n(ii) \( {D}_{k} \cap {D}_{j} = \varnothing \) for \( k \neq j \) .\n\n(iii) \( {\mathbb{B}... | Proof. For any \( k \geq 1 \) let\n\n\[ {E}_{k} = D\left( {{a}_{k}, r}\right) - \mathop{\bigcup }\limits_{{j \neq k}}D\left( {{a}_{j}, r/4}\right) \]\n\nThen each \( {E}_{k} \) contains \( D\left( {{a}_{k}, r/4}\right) \) and is contained in \( D\left( {{a}_{k}, r}\right) \) . Also, \( \left\{ {E}_{k}\right\} \) covers... | Yes |
Corollary 2.33. Suppose \( \alpha > - 1 \) and \( p > 0 \) . If \( r \) and \( q \) are positive numbers such that \[ \frac{1}{p} = \frac{1}{q} + \frac{1}{r} \] then every function \( f \in {A}_{\alpha }^{p} \) admits a decomposition \[ f\left( z\right) = \mathop{\sum }\limits_{k}{g}_{k}\left( z\right) {h}_{k}\left( z\... | Proof. If \( f \) is nonvanishing in \( {\mathbb{B}}_{n} \), then we have the factorization \( f = {gh} \), where \( g = \) \( {f}^{p/q} \) is in \( {A}_{\alpha }^{q} \) and \( h = {f}^{p/r} \) is in \( {A}_{\alpha }^{r} \) . In general, we use the atomic decomposition of \( f \) , \[ f\left( z\right) = \mathop{\sum }\... | Yes |
Theorem 2.34. Suppose \( \alpha > - 1 \) and \( 1 \leq {p}_{0} < {p}_{1} < \infty \) . If\n\n\[ \n\frac{1}{p} = \frac{1 - \theta }{{p}_{0}} + \frac{\theta }{{p}_{1}}\n\]\n\nfor some \( \theta \in \left( {0,1}\right) \), then\n\n\[ \n{\left\lbrack {A}_{\alpha }^{{p}_{0}},{A}_{\alpha }^{{p}_{1}}\right\rbrack }_{\theta } ... | Proof. First assume that \( f \in {A}_{\alpha }^{p} \) . For any complex parameter \( \zeta \) with \( \operatorname{Re}\zeta \in \left\lbrack {0,1}\right\rbrack \) we consider the function\n\n\[ \n{h}_{\zeta }\left( z\right) = \frac{f\left( z\right) }{\left| f\left( z\right) \right| }{\left| f\left( z\right) \right| }... | Yes |
Theorem 3.1. For \( z \in {\mathbb{B}}_{n} \) and \( f \) holomorphic in \( {\mathbb{B}}_{n} \) the following quantities are all equal:\n\n(a) \( {Q}_{f}\left( z\right) \).\n\n(b) \( {\left\langle \overline{B{\left( z\right) }^{-1}}\nabla f\left( z\right) ,\nabla f\left( z\right) \right\rangle }^{1/2} \).\n\n(c) \( \fr... | Proof. It follows from Lemma 2.13 and (2.12) that the quantities in (c), (d), and (e) are the same.\n\nWe can replace \( w \) by \( B{\left( z\right) }^{-1/2}w \) in the definition \( {Q}_{f}\left( z\right) \) to obtain\n\n\[ {Q}_{f}\left( z\right) = \sup \left\{ {\frac{\left| \left\langle \nabla f\left( z\right) ,\ove... | Yes |
Proposition 3.2. The semi-norm \( \parallel {\parallel }_{\mathcal{B}} \) is complete and invariant under the action of \( \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \), that is,\n\n\[ \parallel f \circ \varphi {\parallel }_{\mathcal{B}} = \parallel f{\parallel }_{\mathcal{B}} \]\n\nfor all \( f \in \mathcal{B} \... | Proof. The Möbius invariance of the semi-norm \( \parallel f{\parallel }_{\mathcal{B}} \) follows from that of the invariant gradient; see (2.13).\n\nTo show that \( \parallel {\parallel }_{\mathcal{B}} \) is complete, assume that \( \left\{ {f}_{k}\right\} \) is a sequence of functions in \( \mathcal{B} \) with the pr... | Yes |
Theorem 3.4. Suppose \( \alpha > - 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \) is in \( \mathcal{B} \) .\n\n(b) \( \left( {1 - {\left| z\right| }^{2}}\right) \left| {\nabla f\left( z\right) }\right| \) is bounded in \( {\mathbb{B}}_{n} \) .\n\... | Proof. By Lemma 2.14, condition (a) implies (b), and condition (b) implies (c).\n\nTo show that (c) implies (d), suppose \( \left( {1 - {\left| z\right| }^{2}}\right) {Rf}\left( z\right) \) is bounded in \( {\mathbb{B}}_{n} \) . Consider the function\n\n\[ g\left( z\right) = \frac{{c}_{\alpha + 1}}{{c}_{\alpha }}\left(... | Yes |
Theorem 3.5. Suppose \( N \) is a positive integer, \( t > 0 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . If \( \alpha \) is a real parameter such that neither \( n + \alpha \) nor \( n + \alpha + t \) is a negative integer; then the following conditions are equivalent:\n\n(1) \( f \in \mathcal{B} \) .\n\... | Proof. If \( f \in \mathcal{B} \), then by Theorem 3.4 there exists a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{n}\right) \) such that\n\n\[ \nf\left( z\right) = {\int }_{{\mathbb{B}}_{n}}\frac{g\left( w\right) d{v}_{\beta }\left( w\right) }{{\left( 1-\langle z, w\rangle \right) }^{n + 1 + \beta }}.\n\]\n\nHe... | Yes |
Corollary 3.8. Suppose \( \alpha > - 1, p > 0 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then \( f \in \mathcal{B} \) if and only if there exists a constant \( C > 0 \) such that\n\n\[{\int }_{{\mathbb{B}}_{n}}{\left| f \circ {\varphi }_{a}\left( z\right) - f\left( a\right) \right| }^{p}d{v}_{\alpha }\l... | Proof. That the two integrals are equal follows from a change of variables; see Proposition 1.13.\n\nIf \( f \) is in the Bloch space, then there exists a constant \( C > 0 \) such that\n\n\[ \left| {f\left( z\right) - f\left( w\right) }\right| \leq {C\beta }\left( {z, w}\right) \]\n\nfor all \( z \) and \( w \) in \( ... | Yes |
Theorem 3.9. For any \( z \) and \( w \) in \( {\mathbb{B}}_{n} \) we have\n\n\[ \beta \left( {z, w}\right) = \sup \left\{ {\left| {f\left( z\right) - f\left( w\right) }\right| : \parallel f{\parallel }_{\mathcal{B}} \leq 1}\right\} . \] | Proof. According to Theorem 3.6,\n\n\[ \left| {f\left( z\right) - f\left( w\right) }\right| \leq \parallel f{\parallel }_{\mathcal{B}}\beta \left( {z, w}\right) \]\n\nfor all \( f \in \mathcal{B} \) and all points \( z \) and \( w \) in \( {\mathbb{B}}_{n} \) . It follows that\n\n\[ \sup \left\{ {\left| {f\left( z\righ... | Yes |
Proposition 3.10. \( {\mathcal{B}}_{0} \) is a closed subspace of \( \mathcal{B} \) and the set of polynomials is dense in \( {\mathcal{B}}_{0} \) . | Proof. It is obvious that \( {\mathcal{B}}_{0} \) is closed in \( \mathcal{B} \) . Given \( f \in {\mathcal{B}}_{0} \), we have \( {\begin{Vmatrix}f - {f}_{r}\end{Vmatrix}}_{\mathcal{B}} \rightarrow 0 \) as \( r \rightarrow {1}^{ - } \), where \( {f}_{r}\left( z\right) = f\left( {rz}\right) \) . Since each \( {f}_{r} \... | No |
Theorem 3.11. Suppose \( \alpha > - 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \in {\mathcal{B}}_{0} \) .\n\n(b) The function \( \left( {1 - {\left| z\right| }^{2}}\right) \left| {\nabla f\left( z\right) }\right| \) belongs to \( {\mathbb{C}}_{0... | Proof. By Lemma 2.14 we have that (a) implies (b), and (b) implies (c).\n\nRecall from the proof of Theorem 3.4 that \( f = {P}_{\alpha }g \) whenever \( f \in \mathcal{B} \), where\n\n\[ g\left( z\right) = \frac{{c}_{\alpha + 1}}{{c}_{\alpha }}\left\lbrack {\left( {1 - {\left| z\right| }^{2}}\right) f\left( z\right) +... | Yes |
Theorem 3.12. Suppose \( \alpha > - 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \) belongs to the little Bloch space.\n\n(b) \( \left| {\widetilde{\nabla }f\left( z\right) }\right| \) belongs to \( \mathbb{C}\left( {\overline{\mathbb{B}}}_{n}\rig... | Proof. It is trivial that (a) implies (b). It follows from Lemma 2.14 that (b) implies (c), and (c) implies (d).\n\nThat (d) implies (e) follows from the same construction used in the proof of Theorems 3.4 and 3.11.\n\nThe last part of the proof of Theorem 3.11 actually shows that (e) implies (a). | Yes |
Theorem 3.13. Suppose \( N \) is a positive integer; \( \alpha \) is real, and \( t \) is positive. If neither \( n + \alpha \) nor \( n + \alpha + t \) is a negative integer, then the following conditions are equivalent for a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) :\n\n(a) \( f \in {\mathcal{B}}_{0} \)... | Proof. The proof is similar to that of Theorem 3.5. We omit the details. | No |
Theorem 3.14. For any \( z \) and \( w \) in \( {\mathbb{B}}_{n} \) we have\n\n\[ \beta \left( {z, w}\right) = \sup \left\{ {\left| {f\left( z\right) - f\left( w\right) }\right| : \parallel f{\parallel }_{\mathcal{B}} \leq 1, f \in {\mathcal{B}}_{0}}\right\} . \] | Proof. The proof is the same as that of Theorem 3.9, except that we use the functions\n\n\[ {h}_{r}\left( w\right) = \frac{1}{2}\log \frac{\left| z\right| + r\langle w, z\rangle }{\left| z\right| - r\langle w, z\rangle },\;w \in {\mathbb{B}}_{n}, \]\n\nwhere \( z \in {\mathbb{B}}_{n} - \{ 0\} \) is fixed and \( r \in \... | Yes |
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