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Theorem 3.15. Suppose \( \left\{ {n}_{k}\right\} \) is a series of positive integers satisfying \( {n}_{k + 1} \geq \lambda {n}_{k} \) for all \( k \geq 1 \), where \( \lambda \) is a constant greater than 1 . Consider an analytic function \( f \) in \( \mathbb{D} \) whose Taylor series is of the form\n\n\[ f\left( z\r... | Proof. Recall that \( f \) is in the Bloch space of \( \mathbb{D} \) if and only if\n\n\[ \sup \left\{ {\left( {1 - {\left| z\right| }^{2}}\right) \left| {{f}^{\prime }\left( z\right) }\right| : z \in \mathbb{D}}\right\} < \infty ; \]\n\nand \( f \) is in the little Bloch space of \( \mathbb{D} \) if and only if\n\n\[ ... | Yes |
Lemma 3.18. Suppose \( \left( {X,\parallel \parallel }\right) \) is a Möbius invariant Banach space. If \( X \) contains a nonconstant function, then \( X \) contains all the polynomials. | Proof. Let \( f \) be a nonconstant function in \( X \) . If\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{m}{a}_{m}{z}^{m} \]\n\nis the Taylor expansion of \( f \), then there exists some nonzero multi-index \( m \) such that \( {a}_{m} \neq 0 \) . Fix such an \( m = \left( {{m}_{1},\cdots ,{m}_{n}}\right) \) and co... | Yes |
Lemma 3.20. Suppose \( X \) is a Banach space of holomorphic functions in \( {\mathbb{B}}_{n} \) . If \( X \) contains the constant functions and if each point evaluation is a bounded linear functional on \( X \), then every pointwise multiplier of \( X \) is in \( {H}^{\infty } \) . | Proof. Suppose \( f \) is a pointwise multiplier of \( X \) . Since \( X \) contains the constant function 1, we have \( f \in X \) . In particular, \( f \) is holomorphic. An application of the closed graph theorem then shows that \( T = {M}_{f} \), the operator of multiplication by \( f \) on \( X \), is bounded on \... | Yes |
Theorem 3.21. For a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) the following conditions are equivalent:\n\n(a) \( f\mathcal{B} \subset \mathcal{B} \) .\n\n(b) \( f{\mathcal{B}}_{0} \subset {\mathcal{B}}_{0} \) .\n\n(c) \( f \in {H}^{\infty }\left( {\mathbb{B}}_{n}\right) \) and the function\n\n\[ \left( {1 ... | Proof. Suppose \( f\mathcal{B} \subset \mathcal{B} \) . Then \( f \in {H}^{\infty }\left( {\mathbb{B}}_{n}\right) \) and there exists a positive constant \( C > 0 \) such that \( \parallel {fg}\parallel \leq C\parallel g\parallel \) for all \( g \in \mathcal{B} \) . Since\n\n\[ \nabla \left( {fg}\right) \left( z\right)... | Yes |
Lemma 3.22. There exists a constant \( C > 0 \), independent of the separation constant \( r \) for \( \left\{ {a}_{k}\right\} \) and the separation constant \( \eta \) for \( \left\{ {a}_{kj}\right\} \), such that\n\n\[ \left| {f\left( z\right) - {Sf}\left( z\right) }\right| \leq {C\sigma T}\left( \left| f\right| \rig... | Proof. Let \( p = 1 \) and \( \alpha = 0 \) in Lemma 2.29. We obtain\n\n\[ \left| {f\left( z\right) - {Sf}\left( z\right) }\right| \leq {C\sigma }\mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{{\left( 1 - {\left| {a}_{k}\right| }^{2}\right) }^{b - \left( {n + 1}\right) }}{{\left| 1 - \left\langle z,{a}_{k}\right\rangle... | Yes |
For any \( b > n \) there exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that the Bloch space \( \mathcal{B} \) consists exactly of functions of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{k}{c}_{k}\frac{{\left( 1 - {\left| {a}_{k}\right| }^{2}\right) }^{b}}{{\left( 1 - \lef... | Proof. Let \( \left\{ {a}_{k}\right\} \) be a sequence satisfying the conditions of Theorem 2.23 and let \( f \) be a function defined by (3.12).\n\nFirst observe that the series in (3.12) converges uniformly on every compact subset of \( {\mathbb{B}}_{n} \) whenver \( \left\{ {c}_{k}\right\} \) is bounded. In fact, by... | Yes |
Theorem 3.24. For any \( b > n \) there exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that \( {\mathcal{B}}_{0} \) consists exactly of functions of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }{c}_{k}\frac{{\left( 1 - {\left| {a}_{k}\right| }^{2}\right) }^{... | Proof. By the proof of Theorem 3.23, there exists a constant \( C > 0 \) such that\n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{k = 1}}^{\infty }{c}_{k}{\left( \frac{1 - {\left| {a}_{k}\right| }^{2}}{1 - \left\langle {z,{a}_{k}}\right\rangle }\right) }^{b}}\end{Vmatrix} \leq C\mathop{\sup }\limits_{{k \geq 1}}\left| {... | Yes |
Proposition 4.1. If \( f \) belongs to the ball algebra, then\n\n\[ f\left( z\right) = {\int }_{{\mathbb{S}}_{n}}C\left( {z,\zeta }\right) f\left( \zeta \right) {d\sigma }\left( \zeta \right) \]\n\nfor all \( z \in {\mathbb{B}}_{n} \) . | Proof. For any given \( z \in {\mathbb{B}}_{n} \) the Cauchy kernel \( C\left( {z,\zeta }\right) \) is bounded in \( \zeta \in {\mathbb{S}}_{n} \) . Approximating \( f \) by \( {f}_{r} \) uniformly on \( {\overline{\mathbb{B}}}_{n} \), and then approximating each \( {f}_{r} \) uniformly by its Taylor polynomials, we re... | Yes |
Proposition 4.2. If \( f \) is in the ball algebra, then\n\n\[ f\left( z\right) = {\int }_{{\mathbb{S}}_{n}}P\left( {z,\zeta }\right) f\left( \zeta \right) {d\sigma }\left( \zeta \right) \]\n\nfor all \( z \in {\mathbb{B}}_{n} \) . | Proof. Fix \( f \) in the ball algebra and \( z \) in \( {\mathbb{B}}_{n} \) . The function\n\n\[ g\left( w\right) = f\left( w\right) C\left( {w, z}\right) ,\;w \in \overline{{\mathbb{B}}_{n}}, \]\n\nalso belongs to the ball algebra. Applying Cauchy’s formula to \( g \) at the point \( z \), we\n\nobtain\n\[ \frac{f\le... | Yes |
Theorem 4.3. If \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) and \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \), then\n\n\[ P\left\lbrack {f \circ \varphi }\right\rbrack \left( z\right) = P\left\lbrack f\right\rbrack \left( {\varphi \left( z\right) }\right) \]\n\nfor all \( z \in {... | Proof. By an approximation argument, we may assume that \( f \) is continuous on the unit sphere \( {\mathbb{S}}_{n} \). \n\nWrite \( \varphi = U{\varphi }_{a} \), where \( U \) is a unitary and \( a = {\varphi }^{-1}\left( 0\right) \). By Lemmas 1.2 and 1.3,\n\n\[ P\left( {\varphi \left( z\right) ,\varphi \left( \zeta... | Yes |
Corollary 4.4. If \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \), then we have the change of variables formula\n\n\[ \n{\int }_{{\mathbb{S}}_{n}}f \circ \varphi \left( \zeta \right) {d\sigma }\left( \zeta \right) = {\int }_{{\mathbb{S}}_{n}}P\left( {a,\zeta }\right) f\left( \zeta \right) {d\sigma }\left... | Proof. Simply set \( z = 0 \) in Theorem 4.3. | Yes |
Corollary 4.5. If \( f \) is in the ball algebra, then\n\n\[{\left| f\left( z\right) \right| }^{p} \leq {\int }_{{\mathbb{S}}_{n}}P\left( {z,\zeta }\right) {\left| f\left( \zeta \right) \right| }^{p}{d\sigma }\left( \zeta \right)\]\n\nfor all \( z \in {\mathbb{B}}_{n} \) and \( 0 < p < \infty \) . | Proof. The special case \( z = 0 \) follows from the subharmonicity of \( {\left| f\right| }^{p} \) . For a general \( z \in {\mathbb{B}}_{n} \) we apply the special case to the function\n\n\[g\left( w\right) = f \circ {\varphi }_{z}\left( w\right) ,\;w \in \overline{{\mathbb{B}}_{n}},\]\n\nand make a change of variabl... | No |
Lemma 4.6. There exist positive constants \( {A}_{1} \) and \( {A}_{2} \) (depending on \( n \) only) such that \[ {A}_{1} \leq \frac{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) }{{\delta }^{2n}} \leq {A}_{2} \] for all \( \zeta \in {\mathbb{S}}_{n} \) and all \( \delta \in \left( {0,\sqrt{2}}\right) \) . | Proof. The result is obvious when \( n = 1 \) . Also, it follows from symmetry that \( \sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) \) is independent of \( \zeta \) . So we may assume that \( n > 1 \) and \( \zeta = {e}_{1} \) . Applying (1.13), we obtain \[ \sigma \left( {Q\left( {\zeta ,\delta }\right) }\r... | Yes |
Lemma 4.7. Suppose \( N \) is a natural number and\n\n\[ E = \mathop{\bigcup }\limits_{{k = 1}}^{N}Q\left( {{\zeta }_{k},{\delta }_{k}}\right) \]\n\nThere exists a subsequence \( \left\{ {k}_{i}\right\} ,1 \leq i \leq M \), such that\n\n(a) The balls \( Q\left( {{\zeta }_{{k}_{i}},{\delta }_{{k}_{i}}}\right) \) are dis... | Proof. First notice that \( C \) is a finite positive constant in view of Lemma 4.6.\n\nWithout loss of generality we may assume that the finite sequence \( \left\{ {\delta }_{k}\right\} \) is nonincreasing. We let \( {k}_{1} = 1 \) and construct a subsequence \( \left\{ {k}_{i}\right\} \) inductively as follows.\n\nSu... | Yes |
Lemma 4.8. There exists a constant \( C > 0 \) such that\n\n\[ \sigma \left( {{M\mu } > t}\right) \leq \frac{C\parallel \mu \parallel }{t} \]\n\nfor every complex Borel measure \( \mu \) on \( {\mathbb{S}}_{n} \) and every \( t > 0 \) . | Proof. Fix \( \mu \) and \( t > 0 \) . If \( K \) is a compact subset of the open set \( \{ {M\mu } > t\} \) , then \( K \) is covered by a finite collection \( \Phi \) of open balls \( Q = Q\left( {\zeta ,\delta }\right) \) such that \( \left| \mu \right| \left( Q\right) > {t\sigma }\left( Q\right) \) . Let \( {\Phi }... | Yes |
Theorem 4.9. For each \( p \in \left( {1,\infty }\right) \) there exists a constant \( {C}_{p} > 0 \) such that\n\n\[{\int }_{{\mathbb{S}}_{n}}{\left| Mf\right| }^{p}{d\sigma } \leq {C}_{p}{\int }_{{\mathbb{S}}_{n}}{\left| f\right| }^{p}{d\sigma }\n\]\nfor all \( f \in {L}^{p}\left( {{\mathbb{S}}_{n},{d\sigma }}\right)... | Proof. It is obvious that the maximal operator \( M \) is sub-additive, that is,\n\n\[M\left( {f + g}\right) \leq {Mf} + {Mg}.\n\]\nIt is also obvious that \( M \) maps \( {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) into \( {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) . The desired result then follows from (4.14) ... | No |
Theorem 4.10. For every \( \alpha > 1 \) there exists a constant \( C = {C}_{\alpha } > 0 \) such that\n\n\[ \n{M}_{\alpha }P\left\lbrack \mu \right\rbrack \leq {CM\mu } \n\] \n\nfor every finite complex Borel measure \( \mu \) on \( {\mathbb{S}}_{n} \) . | Proof. Since \( M\left| \mu \right| = {M\mu } \) and \( \left| {P\left\lbrack \mu \right\rbrack }\right| \leq P\left\lbrack \left| \mu \right| \right\rbrack \), we may as well assume that \( \mu \) is positive. By Lemma 4.6, there exists a positive constant \( C \) such that \( \sigma \left( {Q\left( {\zeta ,\delta }\r... | Yes |
Corollary 4.11. For every \( 1 < p < \infty \) and \( \alpha > 1 \) there exists a constant \( C > 0 \) such that \[ {\int }_{{\mathbb{S}}_{n}}{\left| {M}_{\alpha }P\left\lbrack f\right\rbrack \right| }^{p}{d\sigma } \leq C{\int }_{{\mathbb{S}}_{n}}{\left| f\right| }^{p}{d\sigma } \] for all \( f \in {L}^{p}\left( {{\m... | Proof. This is a direct consequence of Theorems 4.9 and 4.10. | No |
Lemma 4.12. If \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \), then\n\n\[ \mathop{\lim }\limits_{{\delta \rightarrow 0}}\frac{1}{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) }{\int }_{Q\left( {\zeta ,\delta }\right) }\left| {f - f\left( \zeta \right) }\right| {d\sigma } = 0 \]\n\nfor almost e... | Proof. Define\n\n\[ {T}_{f}\left( \zeta \right) = \mathop{\limsup }\limits_{{\delta \rightarrow 0}}\frac{1}{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) }{\int }_{Q\left( {\zeta ,\delta }\right) }\left| {f - f\left( \zeta \right) }\right| {d\sigma },\;\zeta \in {\mathbb{S}}_{n}. \]\n\nFor any \( \epsilon > 0... | Yes |
Corollary 4.13. If \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \), then \( \left| {f\left( \zeta \right) }\right| \leq {Mf}\left( \zeta \right) \) for almost every \( \zeta \in {\mathbb{S}}_{n} \) . | Proof. Since\n\n\[ \left| {\frac{1}{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) }{\int }_{Q\left( {\zeta ,\delta }\right) }{fd\sigma }}\right| \leq {Mf}\left( \zeta \right) \]\n\nthe desired inequality follows from (4.18) in Lemma 4.12. | Yes |
Lemma 4.14. Suppose \( \mu \) is a finite Borel measure on \( {\mathbb{S}}_{n} \) . If \( \mu \) is singular with respect to \( \sigma \), then\n\n\[ \mathop{\limsup }\limits_{{\delta \rightarrow 0}}\frac{\mu \left( {Q\left( {\zeta ,\delta }\right) }\right) }{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) } = ... | Proof. Without loss of generality we may assume that \( \mu \) is positive. Define\n\n\[ {D\mu }\left( \zeta \right) = \mathop{\limsup }\limits_{{\delta \rightarrow 0}}\frac{\mu \left( {Q\left( {\zeta ,\delta }\right) }\right) }{\sigma \left( {Q\left( {\zeta ,\delta }\right) }\right) },\;\zeta \in {\mathbb{S}}_{n}. \]\... | Yes |
Theorem 4.17. Suppose \( 0 < p < \infty \) and \( f \in {H}^{p} \) . Then\n\n\[ \left| {f\left( z\right) }\right| \leq \frac{\parallel f{\parallel }_{p}}{{\left( 1 - {\left| z\right| }^{2}\right) }^{n/p}} \]\n\nfor all \( z \in {\mathbb{B}}_{n} \) . Furthermore, the exponent \( n/p \) is best possible. | Proof. Fix \( f \in {H}^{p} \) and \( z \in {\mathbb{B}}_{n} \) . For any \( 0 < r < 1 \) consider\n\n\[ {F}_{r}\left( w\right) = {f}_{r}\left( {{\varphi }_{z}\left( w\right) }\right) \frac{{\left( 1 - {\left| z\right| }^{2}\right) }^{n/p}}{{\left( 1-\langle w, z\rangle \right) }^{{2n}/p}},\;w \in {\mathbb{B}}_{n}, \]\... | Yes |
If \( 1 \leq p < \infty \), the Hardy space \( {H}^{p} \) is a Banach space with the norm \( {\begin{Vmatrix}\end{Vmatrix}}_{p} \) . If \( 0 < p < 1,{H}^{p} \) is a complete metric space with the distance function \[ d\left( {f, g}\right) = \parallel f - g{\parallel }_{p}^{p} \] | It suffices to show that each \( {H}^{p} \) is complete in \( {\begin{Vmatrix}\end{Vmatrix}}_{p} \) . So assume that \( \left\{ {f}_{k}\right\} \) is a Cauchy sequence in \( {H}^{p} \) . By Corollary 4.18, the sequence \( \left\{ {{f}_{k}\left( z\right) }\right\} \) is uniformly Cauchy on every compact subset of \( {\m... | Yes |
Theorem 4.20. Suppose \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) and \( f\left( 0\right) = 0 \) . Then\n\n\[ r\frac{d}{dr}{M}_{p}{\left( r, f\right) }^{p} = \frac{{p}^{2}}{2n}{\int }_{\left| z\right| < r}{\left| Rf\left( z\right) \right| }^{2}{\left| f\left( z\right) \right| }^{p - 2}{\left| z\right| }^{-{2n}}{dv... | Proof. We have just proved the case \( n = 1 \) . When \( n > 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \), we consider the slice functions\n\n\[ {f}_{\zeta }\left( w\right) = f\left( {w\zeta }\right) ,\;w \in \mathbb{D},\zeta \in {\mathbb{S}}_{n}. \]\n\nEach \( {f}_{\zeta } \) is then an analytic function... | Yes |
Corollary 4.21. If \( 0 < p < \infty \) and \( f \in {H}^{p} \), then the integral means \( {M}_{p}\left( {r, f}\right) \) are increasing in \( r \), and\n\n\[ \parallel f{\parallel }_{p} = \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{M}_{p}\left( {r, f}\right) \] | Proof. Obvious. | No |
Theorem 4.22. Suppose \( 0 < p < \infty \) . Then\n\n\[ \parallel f - f\left( 0\right) {\parallel }_{p}^{p} = \frac{{p}^{2}}{2n}{\int }_{{\mathbb{B}}_{n}}{\left| Rf\left( z\right) \right| }^{2}{\left| f\left( z\right) - f\left( 0\right) \right| }^{p - 2}{\left| z\right| }^{-{2n}}\log \frac{1}{\left| z\right| }{dv}\left... | Proof. Without loss of generality we may assume that \( f\left( 0\right) = 0 \) . In this case, we divide both sides of the equation in (4.24) by \( r \) and then integrate with respect to \( r \) from 0 to \( t \), where \( 0 < t < 1 \) . The result is\n\n\[ {M}_{p}{\left( t, f\right) }^{p} = \frac{{p}^{2}}{2n}{\int }... | Yes |
Theorem 4.24. For every \( \alpha > 1 \) there exists a constant \( C = C\left( \alpha \right) > 0 \) such that\n\n\[{\int }_{{\mathbb{S}}_{n}}{\left| {M}_{\alpha }f\right| }^{p}{d\sigma } \leq C\parallel f{\parallel }_{p}^{p}\]\n\nfor all \( p > 0 \) and \( f \in {H}^{p} \) . | Proof. Fix some \( p \in \left( {0,\infty }\right) \) and \( f \in {H}^{p} \) .\n\nFor \( g = {\left| f\right| }^{p/2} \) we can apply Corollary 4.5 to obtain\n\n\[{g}_{r}\left( z\right) \leq {\int }_{{\mathbb{S}}_{n}}P\left( {z,\zeta }\right) {g}_{r}\left( \zeta \right) {d\sigma }\left( \zeta \right) ,\;z \in {\mathbb... | Yes |
Theorem 4.25. Suppose \( f \in {H}^{p} \) with \( 0 < p < \infty \) . Then the limit\n\n\[ \n{f}^{ * }\left( \zeta \right) = \operatorname{Klim}f\left( \zeta \right) \n\]\n\nexists for almost all \( \zeta \in {\mathbb{S}}_{n} \) . Moreover,\n\n\[ \n\parallel f{\parallel }_{p}^{p} = {\int }_{{\mathbb{S}}_{n}}{\left| {f}... | Proof. We prove the case \( p \geq 1 \) here. The case \( p < 1 \) is more involved and less complex analytic; we refer the reader to [94] for a full proof.\n\nIf \( 1 < p < \infty \) and \( f \in {H}^{p} \), then the set \( \left\{ {{f}_{r} : 0 < r < 1}\right\} \) is bounded in \( {L}^{p}\left( {{\mathbb{S}}_{n},{d\si... | Yes |
Corollary 4.26. Suppose \( 0 < p < \infty \) and \( f \in {H}^{p} \) . Then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{\begin{Vmatrix}{f}_{r} - f\end{Vmatrix}}_{p} = 0 \]\n\nAlso, the set of polynomials is dense in each \( {H}^{p} \) . | Proof. The first assertion follows from Theorem 4.25. The second assertion follows from approximating each \( {f}_{r} \) uniformly by its Taylor polynomials. | No |
Corollary 4.27. Suppose \( p \geq 1 \) and \( f \in {H}^{p} \) . If \( {f}^{ * } \) is the boundary function of \( f \) , then \( f = P\left\lbrack {f}^{ * }\right\rbrack = C\left\lbrack {f}^{ * }\right\rbrack \) . | Proof. For \( r \in \left( {0,1}\right) \) consider the dilation \( {f}_{r}\left( z\right) = f\left( {rz}\right), z \in {\overline{\mathbb{B}}}_{n} \) . Then by Propositions 4.1 and 4.2, we have \( {f}_{r} = P\left\lbrack {f}_{r}\right\rbrack = C\left\lbrack {f}_{r}\right\rbrack \) for all \( r \in \left( {0,1}\right) ... | Yes |
Proposition 4.28. The Cauchy-Szegö kernel is the reproducing kernel of \( {H}^{2} \) and the Cauchy-Szegö projection \( C : {L}^{2}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \rightarrow {H}^{2} \) is simply the Cauchy transform, that is,\n\n\[ \n\left( {Cf}\right) \left( z\right) = {\int }_{{\mathbb{S}}_{n}}C\left( {z... | Proof. Recall from Proposition 4.1 that the Cauchy-Szegö kernel reproduces functions in the ball algebra. Since the ball algebra is dense in each \( {H}^{p} \), the Cauchy-Szegö kernel also reproduces functions in \( {H}^{1} \) . In particular, the Cauchy-Szegö kernel reproduces functions in \( {H}^{2} \) . By the uniq... | Yes |
Lemma 4.30. There exists a constant \( C > 0 \) such that\n\n\[ \n{\int }_{d\left( {\eta ,\zeta }\right) > {2r}}\frac{{d\sigma }\left( \eta \right) }{{\left| 1-\langle \zeta ,\eta \rangle \right| }^{n + 1/2}} \leq \frac{C}{r}\n\]\n\nfor all \( \zeta \in {\mathbb{S}}_{n} \) and \( r > 0 \) . | Proof. Let \( I\left( r\right) \) denote the concerned integral.\n\nIf \( n = 1 \), it is easy to see that\n\n\[ \nI\left( r\right) = 2{\int }_{4{r}^{2}}^{\pi }{\left| 1 - {e}^{i\theta }\right| }^{-3/2}{d\theta } < {\left( \frac{\pi }{2}\right) }^{3/2}\frac{2}{r}.\n\]\n\nIf \( n > 1 \), we can apply (1.13) to obtain\n\... | Yes |
Corollary 4.31. For \( \zeta \in {\mathbb{S}}_{n},\omega \in {\mathbb{S}}_{n},\alpha > 1 \), and \( \delta > 0 \) let\n\n\[ \n\Delta \left( {\zeta ,\omega ,\alpha ,\delta }\right) = \sup \left\{ {\left| {C\left( {z,\eta }\right) - C\left( {z,\omega }\right) }\right| : z \in {D}_{\alpha }\left( \zeta \right), d\left( {\... | Proof. This follows directly from Lemmas 4.29 and 4.30. | Yes |
For any \( \alpha > 1 \) there exists a constant \( C > 0 \) such that\n\n\[{\int }_{{\mathbb{S}}_{n}}{\left| {M}_{\alpha }C\left\lbrack f\right\rbrack \right| }^{2}{d\sigma } \leq C{\int }_{{\mathbb{S}}_{n}}{\left| f\right| }^{2}{d\sigma }\n\]\nfor all \( f \in {L}^{2}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) . | Given \( f \in {L}^{2}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \), we write \( f = g + h \), where \( g \in {H}^{2}\left( {\mathbb{S}}_{n}\right) \) and \( h \bot \) \( {H}^{2}\left( {\mathbb{S}}_{n}\right) \) . For any fixed \( z \in {\mathbb{B}}_{n} \), the function \( \zeta \mapsto \overline{C\left( {z,\zeta }\ri... | Yes |
Theorem 4.35. For any \( \alpha > 1 \) and \( 1 < p < \infty \) there exists a constant \( C > 0 \) such that \[ {\int }_{{\mathbb{S}}_{n}}{\left| {M}_{\alpha }C\left\lbrack f\right\rbrack \right| }^{p}{d\sigma } \leq C{\int }_{{\mathbb{S}}_{n}}{\left| f\right| }^{p}{d\sigma } \] for all \( f \in {L}^{p}\left( {{\mathb... | Proof. Since the mapping \( f \mapsto {M}_{\alpha }C\left\lbrack f\right\rbrack \) is subadditive, the case \( 1 < p \leq 2 \) follows from Lemma 4.33, Theorem 4.34, and the Marcinkiewicz interpolation theorem. Also, since \[ \left| {C\left\lbrack f\right\rbrack \left( {r\zeta }\right) }\right| \leq {M}_{\alpha }C\left... | Yes |
Theorem 4.36. If \( 1 < p < \infty \), the Cauchy transform \( C \) maps \( {L}^{p}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) boundedly onto \( {H}^{p} \) . | It is well known that the Cauchy transform \( C \) is unbounded on \( {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \). In fact, the boundedness of \( C \) on \( {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) would imply the boundedness of \( C \) on \( {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) by dual... | No |
Corollary 4.37. If \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \), then\n\n\[{\int }_{{\mathbb{S}}_{n}}C\left( f\right) \bar{g}{d\sigma } = {\int }_{{\mathbb{S}}_{n}}f\overline{C\left( g\right) }{d\sigma }\n\]\nholds for all \( f \in {L}^{p}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) and \( g \in {L}^{q}\left( {{\math... | Proof. The desired result clearly holds for \( f \) and \( g \) in \( \mathbb{C}\left( {\mathbb{S}}_{n}\right) \), because \( C \) is an orthogonal projection on \( {L}^{2}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \). The general case then follows from approximating \( f \) and \( g \) by functions in \( \mathbb{C}\l... | Yes |
Theorem 4.38. Suppose \( 1 \leq {p}_{0} < {p}_{1} < \infty \) and\n\n\[ \n\frac{1}{p} = \frac{1 - \theta }{{p}_{0}} + \frac{\theta }{{p}_{1}}\n\]\n\nwhere \( \theta \in \left( {0,1}\right) \) . Then \( {\left\lbrack {H}^{{p}_{0}},{H}^{{p}_{1}}\right\rbrack }_{\theta } = {H}^{p} \) with equivalent norms. | Proof. We prove the case \( 1 < {p}_{0} \) . When \( {p}_{0} = 1 \), the proof is much more involved and makes use of several real variable methods; we refer the reader to [36].\n\nIf we identify each \( {H}^{p} \) with a closed subspace of \( {L}^{p}\left( {\mathbb{S}}_{n}\right) \), then the inclusion \( {\left\lbrac... | No |
Theorem 4.39. Suppose \( 1 \leq k < n \) and \( 0 < p < \infty \) . Then the operator \( {R}_{k} \) maps \( {H}^{p} \) boundedly onto the weighted Bergman space \( {A}_{n - k - 1}^{p}\left( {\mathbb{B}}_{k}\right) \) of the unit ball in \( {\mathbb{C}}^{k} \) . | Proof. Let \( f \in {H}^{p} = {H}^{p}\left( {\mathbb{B}}_{n}\right) \) . For \( z \in {\mathbb{C}}^{n} \) we write \( z = \left( {w, u}\right) \), where \( w \in {\mathbb{C}}^{k} \) and \( u \in {\mathbb{C}}^{n - k} \) . By (1.15), \[ {\int }_{{\mathbb{S}}_{n}}{\left| f\right| }^{p}{d\sigma } = c{\int }_{{\mathbb{B}}_{... | Yes |
Corollary 4.40. If \( 1 \leq k < n \) and \( 0 < p < \infty \), then the operator \( {E}_{k} \) maps the weighted Bergman space \( {A}_{n - k - 1}^{p}\left( {\mathbb{B}}_{k}\right) \) into \( {H}^{p}\left( {\mathbb{B}}_{n}\right) \) . | Proof. This follows from the proof of Theorem 4.39. | No |
Theorem 4.42. Suppose \( p \in \left( {0,\infty }\right) \) and\n\n\[ \n{b}_{0} = n\max \left( {1,\frac{1}{p}}\right) \n\]\n\nThere exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that\n\n(a) If \( 2 \leq p < \infty \) and \( b > {b}_{0} \), then every function \( f \in {H}^{p} \) admits ... | Proof. First assume that \( 2 \leq p < \infty \) and \( f \in {H}^{p} \) . Fix some \( b \) with \( b > {b}_{0} \) and fix some \( t > 0 \) such that the operators \( {R}_{\alpha, t} \) and \( {R}^{\alpha, t} \) are well defined, where \( \alpha = \) \( b - \left( {n + 1}\right) \) . By Theorem 4.41, \( {R}^{\alpha, t}... | Yes |
Lemma 4.43. Suppose \( p, q \), and \( r \) are from \( (0,\infty \rbrack \) and satisfy\n\n\[ \n\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1 \]\n\nIf \( f \in {L}^{p}\left( {X,{d\mu }}\right), g \in {L}^{q}\left( {X,{d\mu }}\right) \), and \( h \in {L}^{r}\left( {X,{d\mu }}\right) \), then the product \( {fgh} \) belon... | Proof. If any of \( p, q \), and \( r \) is infinite, the result is just the classical Hölder’s inequality. So we assume that they are all finite.\n\nBy Hölder's inequality,\n\n\[ \n\left| {{\int }_{X}{fghd\mu }}\right| \leq {\left( {\int }_{X}{\left| f\right| }^{p}d\mu \right) }^{1/p}{\left( {\int }_{X}{\left| gh\righ... | Yes |
For \( f \in {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) define\n\n\[ \n{Tf}\left( z\right) = {\int }_{{\mathbb{S}}_{n}}\frac{f\left( \zeta \right) {d\sigma }\left( \zeta \right) }{{\left| 1-\langle z,\zeta \rangle \right| }^{n}},\;z \in {\mathbb{B}}_{n}.\n\]\n\nThen for any \( 1 \leq p < q < \infty \) there ... | Proof. Choose \( s > 1 \) such that\n\n\[ \n1 - \frac{1}{s} = \frac{1}{p} - \frac{1}{q}\n\]\n\nLet \( a = q, b = s/\left( {s - 1}\right) \), and \( c = p/\left( {p - 1}\right) \). Then\n\n\[ \n\frac{1}{a} + \frac{1}{b} = \frac{1}{p},\;\frac{1}{a} + \frac{1}{c} = \frac{1}{s},\;\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1... | Yes |
Theorem 4.48. Suppose \( 0 < p < q < \infty \) and\n\n\[ \alpha = {nq}\left( {\frac{1}{p} - \frac{1}{q}}\right) - 1 = \frac{nq}{p} - \left( {n + 1}\right) . \]\n\nThen \( {H}^{p} \subset {A}_{\alpha }^{q} \) and the inclusion is continuous. | Proof. For \( f \in {H}^{p} \) we let \( g = {\left| f\right| }^{p/2} \) . Then \( g \in {L}^{2}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) and \( g \leq P\left\lbrack g\right\rbrack \) on \( {\mathbb{B}}_{n} \) , because \( g \) is subharmonic. In Corollary 4.47, if we replace \( f \) by \( g, p \) by 2, and \( q \... | Yes |
Corollary 4.49. Suppose \( 0 < p < 1 \) and\n\n\[ \alpha = \frac{n}{p} - \left( {n + 1}\right) \]\n\nThen \( {H}^{p} \subset {A}_{\alpha }^{1} \) . Furthermore, there exists a constant \( C > 0 \) such that\n\n\[ {\int }_{{\mathbb{B}}_{n}}\left| {f\left( z\right) }\right| d{v}_{\alpha }\left( z\right) \leq C\parallel f... | Proof. Let \( q = 1 \) in Theorem 4.48. | No |
Theorem 4.50. Suppose \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \) . Then the dual space of \( {H}^{p} \) can be identified with \( {H}^{q} \) (with equivalent norms) under the integral pairing\n\n\[ \langle f, g\rangle = {\int }_{{\mathbb{S}}_{n}}f\left( \zeta \right) \overline{g\left( \zeta \right) }{d\sigma }\left( ... | Proof. It is obvious that every function \( g \in {H}^{q} \) induces a bounded linear functional \( F \) on \( {H}^{p} \) via the formula\n\n\[ F\left( f\right) = {\int }_{{\mathbb{S}}_{n}}f\bar{g}{d\sigma },\;f \in {H}^{p}. \]\n\nBy Hölder’s inequality, \( \parallel F\parallel \leq \parallel g{\parallel }_{q} \) .\n\n... | Yes |
Lemma 5.1. A function \( f \in {H}^{2} \) belongs to BMOA if and only if there exists a positive constant \( C \) with the following property: for any \( d \) -ball \( Q \) in \( {\mathbb{S}}_{n} \) there exists a number \( c \) such that \[ \frac{1}{\sigma \left( Q\right) }{\int }_{Q}{\left| f - c\right| }^{2}{d\sigma... | Proof. If \( f \) is in BMOA, then for any \( Q \) we can take \( c = {f}_{Q} \). Conversely, if for any \( Q \) there exists a number \( c \) such that (5.3) holds, then we apply the triangle inequality to get \[ {\left( \frac{1}{\sigma \left( Q\right) }{\int }_{Q}{\left| f - {f}_{Q}\right| }^{2}d\sigma \right) }^{1/2... | Yes |
Proposition 5.2. The space BMOA is a Banach space when equipped with the norm \( {\parallel }_{\mathrm{{BMO}}} \) . | Proof. It is clear that \( \parallel f{\parallel }_{\mathrm{{BMO}}} \) is a well-defined norm on BMOA.\n\nRecall that for \( r \geq \sqrt{2} \), we have \( Q\left( {\zeta, r}\right) = {\mathbb{S}}_{n} \) for any \( \zeta \in {\mathbb{S}}_{n} \) . It follows that \( \parallel f{\parallel }_{\mathrm{{BMO}}} \geq \paralle... | Yes |
Theorem 5.4. A positive Borel measure \( \mu \) in \( {\mathbb{B}}_{n} \) is a Carleson measure if and only if the quantity\n\n\[ \n{C}_{ * } = \mathop{\sup }\limits_{{z \in {\mathbb{B}}_{n}}}{\int }_{{\mathbb{B}}_{n}}P\left( {z, w}\right) {d\mu }\left( w\right)\n\]\n\nis finite. Here\n\n\[ \nP\left( {z, w}\right) = \f... | Proof. First assume that \( {C}_{ * } < \infty \) . For any \( \zeta \in {\mathbb{S}}_{n} \) and \( 0 < r < 1 \) we consider the point \( z = \left( {1 - {r}^{2}}\right) \zeta \) . Since\n\n\[ \n1 - \langle z, w\rangle = \left( {1 - {r}^{2}}\right) \left( {1-\langle \zeta, w\rangle }\right) + {r}^{2},\n\]\n\nwe have\n\... | Yes |
Theorem 5.5. Let \( f \in {H}^{2} \) . Then \( f \) is in BMOA if and only if the measure\n\n\[ \n{\left( 1 - {\left| z\right| }^{2}\right) }^{n}{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}{d\tau }\left( z\right) = \frac{{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}{dv}\left( z\right) }{1 - {\l... | Proof. Recall from Theorem 4.23 that\n\n\[ \n{\int }_{{\mathbb{S}}_{n}}{\left| f - f\left( 0\right) \right| }^{2}{d\sigma } = {\int }_{{\mathbb{B}}_{n}}{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}G\left( z\right) {d\tau }\left( z\right) \n\]\n\nfor every \( f \in {H}^{2} \), where\n\n\[ \nG\left( z\right) ... | Yes |
Lemma 5.6. Suppose \( E \subset {\mathbb{B}}_{n} \) has the property that for no infinite sequence \( \left\{ {z}_{k}\right\} \) in \( E \) the \( d \) -balls \( {Q}_{{z}_{k}} \) are all disjoint. Then there exists a finite sequence \( \left\{ {z}_{k}\right\} \) in \( E \) such that the sets \( {Q}_{{z}_{k}} \) are dis... | Proof. Let\n\n\[ {T}_{1} = \sup \{ \sqrt{1 - \left| z\right| } : z \in E\} \]\n\nand choose \( {z}_{1} \in E \) such that \( 2\sqrt{1 - \left| {z}_{1}\right| } \geq {T}_{1} \) . If \( {z}_{1},\cdots ,{z}_{k - 1} \) have aready been chosen, we let \( {T}_{k} \) be the supremum of \( \sqrt{1 - \left| z\right| } \) such t... | Yes |
Theorem 5.10. Let \( \mu \) be a positive Borel measure on \( {\mathbb{B}}_{n} \) and \( p > 0 \) . Then the following conditions are equivalent:\n\n(a) \( \mu \) is a vanishing Carleson measure.\n\n(b) For every sequence \( \left\{ {f}_{k}\right\} \) that converges to 0 ultra-weakly in \( {H}^{p} \) we have\n\n\[ \mat... | Proof. That (a) implies (b) follows from Theorem 5.9 and approximating the measure \( \mu \) by the measures \( {\mu }_{r} \), where \( 0 < r < 1 \) and \( {\mu }_{r} \) is \( \mu \) times the characteristic function of \( r{\mathbb{B}}_{n} \) .\n\nChoosing\n\n\[ {f}_{k}\left( \zeta \right) = {\left\lbrack \frac{{\left... | Yes |
Theorem 5.11. For \( f \in {H}^{2} \) the following conditions are equivalent:\n\n(a) \( f \) is in VMOA.\n\n(b) \( f \) can be approximated in BMOA by functions holomorphic on the closed unit ball of \( {\mathbb{C}}^{n} \) .\n\n(c) \( f \) satisfies\n\n\[ \mathop{\lim }\limits_{{\left| a\right| \rightarrow {1}^{ - }}}... | Proof. The equivalence of (c), (d), and (e) follows from the proof of Theorems 5.3, 5.4, and 5.5. The equivalence of (a) and (b) is obvious, because a function holomorphic on the closed unit ball can be approximated uniformly on \( {\mathbb{B}}_{n} \) by polynomials.\n\nIt is obvious that every polynomial satisfies the... | Yes |
Theorem 5.12. Suppose \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) and\n\n\[ \n{d\mu }\left( z\right) = {\left( 1 - {\left| z\right| }^{2}\right) }^{n}{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}{d\tau }\left( z\right) = \frac{{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}{dv}\left( z\right)... | Proof. This follows from Theorems 5.10 and 5.11. | No |
Theorem 5.14. If \( f \) is holomorphic in \( {\mathbb{B}}_{n} \), then the following conditions are equivalent:\n\n(a) \( f \) is in BMOA.\n\n(b) \( \left\lbrack {{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}/\left( {1 - {\left| z\right| }^{2}}\right) }\right\rbrack {dv}\left( z\right) \) is a Carleson mea... | Proof. It follows from Lemma 2.14 that\n\n\[ \left( {1 - {\left| z\right| }^{2}}\right) {\left| Rf\left( z\right) \right| }^{2} \leq \left( {1 - {\left| z\right| }^{2}}\right) {\left| \nabla f\left( z\right) \right| }^{2} \leq \frac{{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}}{1 - {\left| z\right| }^{2}}.... | Yes |
Theorem 5.15. The Cauchy transform maps \( {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) boundedly onto BMOA . | Proof. By the Hahn-Banach extension theorem and the fact that the dual space of \( {L}^{1}\left( {{\mathbb{S}}_{n},{d\sigma }}\right) \) is \( {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) with respect to the integral pairing induced by \( {d\sigma } \), we see that the dual space \( {H}^{1} \) can be identified with ... | No |
Theorem 5.19. Suppose \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \) belongs to VMOA.\n\n(b) The measure \( \left( {1 - {\left| z\right| }^{2}}\right) {\left| \nabla f\left( z\right) \right| }^{2}{dv}\left( z\right) \) is a vanishing Carleson measure.\n\n... | Proof. If \( f \) is in VMOA, then by Theorem 5.12, the measure\n\n\[{\left( 1 - {\left| z\right| }^{2}\right) }^{-1}{\left| \widetilde{\nabla }f\left( z\right) \right| }^{2}{dv}\left( z\right)\]\n\nis a vanishing Carleson measure. According to Lemma 2.14,\n\n\[ \left( {1 - {\left| z\right| }^{2}}\right) {\left| \nabla... | Yes |
Theorem 5.20. Suppose \( 1 \leq k < n \) . Then the operator \( {R}_{k} \) maps BMOA and VMOA of \( {\mathbb{B}}_{n} \) onto the Bloch space and the little Bloch space of \( {\mathbb{B}}_{k} \), respectively. | Proof. If \( f \in \operatorname{BMOA}\left( {\mathbb{B}}_{n}\right) \), then there exists a function \( g \in {L}^{\infty }\left( {\mathbb{S}}_{n}\right) \) such that\n\n\[ f\left( z\right) = {\int }_{{\mathbb{S}}_{n}}\frac{g\left( \zeta \right) {d\sigma }\left( \zeta \right) }{{\left( 1-\langle z,\zeta \rangle \right... | Yes |
For each \( 1 \leq k < n \) the operator \( {E}_{k} \) maps the Bloch space and the little Bloch space of \( {\mathbb{B}}_{k} \) into BMOA and VMOA of \( {\mathbb{B}}_{n} \), respectively. | Let \( f \) be a function in the Bloch space of \( {\mathbb{B}}_{k} \) . Then there exists a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{k}\right) \) such that\n\n\[ f\left( {z}^{\prime }\right) = \left( \begin{matrix} n - 1 \\ k \end{matrix}\right) {\int }_{{\mathbb{B}}_{k}}\frac{{\left( 1 - {\left| w\right| }... | Yes |
Theorem 5.22. Suppose \( r > 0,\alpha > - 1, p \geq 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \in \mathcal{B} \) .\n\n(b) There exists a constant \( C > 0 \) such that\n\n\[ \frac{1}{{v}_{\alpha }\left( {D\left( {a, r}\right) }\right) }{\int }... | Proof. Assume that \( f \in \mathcal{B} \) . By Corollary 3.8 there exists a constant \( {C}_{1} > 0 \) such\n\nthat\n\[ {\int }_{{\mathbb{B}}_{n}}{\left| f\left( z\right) - f\left( a\right) \right| }^{p}\frac{{\left( 1 - {\left| a\right| }^{2}\right) }^{n + 1 + \alpha }}{{\left| 1-\langle a, z\rangle \right| }^{2\left... | Yes |
Lemma 5.23. Suppose \( 0 < r < 1 \) and \( R > 0 \) . There exists a constant \( \sigma \in \left( {0,1}\right) \) (depending on \( R \) but not on \( r \) ) such that\n\n\[ D\left( {a, R}\right) \subset {Q}_{r}\left( \zeta \right) \subset Q\left( {\zeta ,{r}^{\prime }}\right) \times \left( {1 - {r}^{2},1}\right) \]\n\... | Proof. First assume that \( \zeta \in {\mathbb{S}}_{n} \) and \( z \in {Q}_{r}\left( \zeta \right) \) . Then \( z \neq 0 \) and we can write \( z = \left| z\right| \eta \) for some \( \eta \in {\mathbb{S}}_{n} \) . Since\n\n\[ 1 - \langle z,\zeta \rangle = 1 - \left| z\right| + \left| z\right| \left( {1-\langle \eta ,\... | Yes |
Corollary 5.24. For any \( \alpha > - 1 \) there exist positive constants \( c \) and \( C \) such that\n\n\[ c{r}^{2\left( {n + 1 + \alpha }\right) } \leq {v}_{\alpha }\left( {{Q}_{r}\left( \zeta \right) }\right) \leq C{r}^{2\left( {n + 1 + \alpha }\right) } \]\n\nfor all \( \zeta \in {\mathbb{S}}_{n} \) and \( 0 \leq... | Proof. With notation from the lemma above, we have\n\n\[ {v}_{\alpha }\left( {D\left( {a, R}\right) }\right) \sim {\left( 1 - {\left| a\right| }^{2}\right) }^{n + 1 + \alpha } \sim {r}^{2\left( {n + 1 + \alpha }\right) } \]\n\nas \( r \rightarrow {0}^{ + } \) . Also, it follows from polar coordinates and Lemma 4.6 that... | Yes |
Theorem 5.25. Suppose \( \alpha > - 1, p \geq 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \in \mathcal{B} \) .\n\n(b) There exists a constant \( C > 0 \) with the property that for each \( r > 0 \) and \( \zeta \in {\mathbb{S}}_{n} \) there is a... | Proof. It is obvious that (c) implies (b). That (b) implies (c) follows from writing\n\n\[ f - {f}_{\alpha ,{Q}_{r}\left( \zeta \right) } = f - c - \frac{1}{{v}_{\alpha }\left( {{Q}_{r}\left( \zeta \right) }\right) }{\int }_{{Q}_{r}\left( \zeta \right) }\left( {f - c}\right) d{v}_{\alpha } \]\n\nand applying the triang... | Yes |
Lemma 5.27. Suppose \( R > 0 \) and \( \left\{ {a}_{k}\right\} \) is any sequence in \( {\mathbb{B}}_{n} \) . Then the measure\n\n\[ \n{d\mu } = \mathop{\sum }\limits_{k}{\left| {c}_{k}\right| }^{2}{\left( 1 - {\left| {a}_{k}\right| }^{2}\right) }^{n}{\delta }_{{a}_{k}}\n\]\n\nis Carleson if and only if the measure\n\n... | Proof. For any \( a \in {\mathbb{B}}_{n} \) we have\n\n\[ \n{\int }_{{\mathbb{B}}_{n}}\frac{{\left( 1 - {\left| a\right| }^{2}\right) }^{n}{d\lambda }\left( z\right) }{{\left| 1-\langle a, z\rangle \right| }^{2n}} = \mathop{\sum }\limits_{k}\frac{{\left| {c}_{k}\right| }^{2}}{{\left( 1 - {\left| {a}_{k}\right| }^{2}\ri... | Yes |
Theorem 6.1. Suppose \( 0 < p < \infty \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following two conditions are equivalent:\n\n(a) The functions\n\n\[ \n{\left( 1 - {\left| z\right| }^{2}\right) }^{N}\frac{{\partial }^{m}f}{\partial {z}^{m}}\left( z\right) ,\;\left| m\right| = N \n\]\n\nare in \(... | Proof. It suffices to show that the conditions\n\n\[ \n{\left( 1 - {\left| z\right| }^{2}\right) }^{\left| m\right| }\frac{{\partial }^{m}f}{\partial {z}^{m}}\left( z\right) \in {L}^{p}\left( {{\mathbb{B}}_{n},{d\tau }}\right) ,\;\left| m\right| = N, \n\]\n\n(6.1)\n\nare equivalent to\n\n\[ \n{\left( 1 - {\left| z\righ... | Yes |
Theorem 6.4. Suppose \( 0 < p < \infty, n + \alpha \) is not a negative integer, and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following three conditions are equivalent:\n\n(a) \( f \in {B}_{p} \) .\n\n(b) The function \( {\left( 1 - {\left| z\right| }^{2}\right) }^{t}{R}^{\alpha, t}f\left( z\right) \... | Proof. It follows from Lemma 6.3 that (c) implies (a), and that (a) implies (b).\n\nTo prove that (b) and (c) are equivalent, it suffices to show that the norms of the two functions\n\n\[{\left( 1 - {\left| z\right| }^{2}\right) }^{t}{R}^{\alpha, t}f\left( z\right) ,\;{\left( 1 - {\left| z\right| }^{2}\right) }^{s}{R}^... | Yes |
Theorem 6.6. Given any \( p \in \left( {0,\infty }\right) \) there exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that for each \( b > \max \left( {0, n\left( {p - 1}\right) /p}\right) \) the space \( {B}_{p} \) consists exactly of functions of the form\n\n\[ f\left( z\right) = \mathop{\... | Proof. Fix any \( t > n/p \) . Let \( b \) and \( {b}^{\prime } \) be two real parameters related by \( {b}^{\prime } = b + t \) .\n\nThen the condition\n\[ b > \max \left( {0,\frac{n\left( {p - 1}\right) }{p}}\right) \]\n\nis equivalent to\n\[ {b}^{\prime } > n\max \left( {1,\frac{1}{p}}\right) + \frac{{pt} - \left( {... | Yes |
Theorem 6.7. Suppose \( 1 \leq p < \infty \) and \( \alpha > - 1 \) . Then \( {B}_{p} = {P}_{\alpha }{L}^{p}\left( {{\mathbb{B}}_{n},{d\tau }}\right) \) . | Proof. Fix a parameter \( t \) such that \( t > n/p \) and such that \( {R}^{\alpha, t} \) is well defined. By Corollary 6.5, a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) belongs to \( {B}_{p} \) if and only if its fractional radial derivative \( {R}^{\alpha, t}f \) belongs to the weighted Bergman space \( ... | Yes |
Corollary 6.9. The space \( {B}_{1} \) is a Möbius invariant Banach space with the following norm: \[ \parallel f{\parallel }_{m} = \inf \left\{ {\mathop{\sum }\limits_{{k = 0}}^{\infty }\left| {c}_{k}\right| : f = {c}_{0} + \mathop{\sum }\limits_{{k = 1}}^{\infty }{c}_{k}{f}_{k}}\right\} , \] where each \( {f}_{k} \) ... | Proof. It is easy to see that \( {B}_{1} \) is a Banach space under the norm \( \parallel {\parallel }_{m} \). If \( {f}_{k} \) is the \( j \) th component of \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \), and if \( \psi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \), then \( {f}_{k} \circ... | Yes |
Theorem 6.10. If \( X \) is any Möbius invariant Banach space of holomorphic functions in \( {\mathbb{B}}_{n} \) and if \( X \) contains a nonconstant function, then \( {B}_{1} \) is continuously contained in \( X \) . | Proof. By Lemma 3.18, \( X \) contains all the polynomials. Composing the coordinate functions \( {z}_{k} \) with \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \), we see that \( X \) contains all \( n \) components of any \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\right) \) . Now if\n\n... | Yes |
Theorem 6.11. Suppose \( p > {\lambda }_{n} \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then \( f \in {B}_{p} \) if and only if \( \left| {\widetilde{\nabla }f\left( z\right) }\right| \) is in \( {L}^{p}\left( {{\mathbb{B}}_{n},{d\tau }}\right) \) . | Proof. When \( n = 1 \) and \( p > 1 \), it follows from the definition of \( {B}_{p} \) that a holomorphic function \( f \) is in \( {B}_{p} \) if and only if the function \( \left( {1 - {\left| z\right| }^{2}}\right) {f}^{\prime }\left( z\right) \) is in \( {L}^{p}\left( {\mathbb{D},{d\tau }}\right) \) . The desired ... | Yes |
Theorem 6.13. The space \( {B}_{p} \) is Möbius invariant for any \( p \in \left\lbrack {1,\infty }\right\rbrack \) . | Proof. We only need to prove the theorem for \( 1 < p < \infty \), because we already know that \( {B}_{1} \) admits a Möbius invariant norm \( \parallel f{\parallel }_{m} \) and \( {B}_{\infty } = \mathcal{B} \) admits a Möbius invariant semi-norm \( \parallel f{\parallel }_{\mathcal{B}} \) .\n\nFix \( 1 < p < \infty ... | Yes |
Theorem 6.14. Suppose \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) and\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{m}{a}_{m}{z}^{m} \]\n\nis its Taylor expansion. Then \( f \) belongs to \( {B}_{2} \) if and only if\n\n\[ \mathop{\sum }\limits_{m}\left| m\right| \frac{m!}{\left| m\right| !}{\left| {a}_{m}\right... | Proof. Let \( t = \left( {n + 1}\right) /2 \) and \( \alpha = 0 \) . By Theorem 6.4, \( f \in {B}_{2} \) if and only if the function \( {\left( 1 - {\left| z\right| }^{2}\right) }^{t}{R}^{\alpha, t}f\left( z\right) \) is in \( {L}^{2}\left( {{\mathbb{B}}_{n},{d\tau }}\right) \) if and only if \( {R}^{\alpha, t}f \) is ... | Yes |
Theorem 6.15. The semi-inner product \( \langle \) , \( \rangle \) is Möbius invariant on the space \( {B}_{2} \), that is,\n\n\[ \langle f \circ \varphi, g \circ \varphi \rangle = \langle f, g\rangle \]\n\nfor all \( f \) and \( g \) in \( {B}_{2} \) and all \( \varphi \in \operatorname{Aut}\left( {\mathbb{B}}_{n}\rig... | Proof. Let \( f = \mathop{\sum }\limits_{k}{f}_{k} \) and \( g = \mathop{\sum }\limits_{k}{g}_{k} \) be the homogeneous expansions of \( f \) and \( g \). Also, let \( f\left( z\right) = \mathop{\sum }\limits_{m}{a}_{m}{z}^{m} \) and \( g\left( z\right) = \mathop{\sum }\limits_{m}{b}_{m}{z}^{m} \) be the Taylor expansi... | Yes |
The reproducing kernel of the space \( {B}_{2} \) (when equipped with the Möbius invariant inner product) is given by\n\n\[ K\left( {z, w}\right) = \log \frac{1}{1-\langle z, w\rangle }.\] | Proof. For each multi-index \( m \) of nonnegative integers, we have\n\n\[ {\begin{Vmatrix}{z}^{m}\end{Vmatrix}}^{2} = \left| m\right| \frac{m!}{\left| m\right| !} \]\n\nIt follows that\n\n\[ K\left( {z, w}\right) = \mathop{\sum }\limits_{{\left| m\right| > 0}}\frac{{z}^{m}}{\begin{Vmatrix}{z}^{m}\end{Vmatrix}}\frac{{\... | Yes |
Lemma 6.18. If \( f \) and \( g \) are bounded holomorphic functions in \( {\mathbb{B}}_{n} \), then\n\n\[ \n{\int }_{{\mathbb{B}}_{n}}f\left( z\right) \overline{g\left( z\right) }{dv}\left( z\right) = {\int }_{{\mathbb{B}}_{n}}f\left( z\right) \overline{{Vg}\left( z\right) }{dv}\left( z\right) .\n\] | Proof. Represent \( {R}^{0, n + 1}g \) by Corollary 2.3, use Fubini’s theorem, and then apply the reproducing formula in Theorem 2.2. The integral on the right-hand side is then reduced to the one on the left-hand side. | No |
Theorem 6.19. Suppose \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \) . Then the dual space of \( {B}_{p} \) can be identified with \( {B}_{q} \) (with equivalent norms) under the pairing\n\n\[ \langle f, g\rangle = {\int }_{{\mathbb{B}}_{n}}{Vf}\left( z\right) \overline{{Vg}\left( z\right) }{d\tau }\left( z\right) \] | Proof. By Theorem 6.4 and Hölder’s inequality, every \( g \in {B}_{q} \) induces a bounded linear functional on \( {B}_{p} \) via the integral pairing (6.20).\n\nTo show that every bounded linear functional on \( {B}_{p} \) arises this way, recall from Corollay 6.5 that \( {R}^{0, n + 1} \) is a bounded invertible oper... | Yes |
Theorem 6.20. Under the integral pairing in (6.20) we can identify (with equivalent norms) the dual space of \( {B}_{1} \) as \( \mathcal{B} \), and the dual space of \( {\mathcal{B}}_{0} \) with \( {B}_{1} \) . | Proof. It follows from Theorem 6.4 that, via the integral pairing in (6.20), every function \( g \in \mathcal{B} \) induces a bounded linear functional on \( {B}_{1} \), and every function \( g \in \) \( {B}_{1} \) induces a bounded linear functional on \( {\mathcal{B}}_{0} \) .\n\nIf \( F \) is a bounded linear functi... | Yes |
Theorem 6.21. Suppose \( 0 < p \leq 1 \) and \( t > n/p \) . Then the dual space of \( {B}_{p} \) can be identified with \( \mathcal{B} \) under the following duality pairing:\n\n\[ \langle f, g\rangle = {\int }_{{\mathbb{B}}_{n}}{\left( 1 - {\left| z\right| }^{2}\right) }^{t}{R}^{\alpha, t}f\left( z\right) \overline{g... | Proof. Recall from Corollary 6.5 that \( {R}^{\alpha, t} \) is a bounded invertible operator from \( {B}_{p} \) onto \( {A}_{{pt} - \left( {n + 1}\right) }^{p} \) . So \( F \) is a bounded linear functional on \( {B}_{p} \) if and only if \( F \circ {R}_{\alpha, t} \) is a bounded linear functional on \( {A}_{{pt} - \l... | Yes |
Lemma 6.22. We have\n\n\[ D = R{R}^{-n, n - 1} = n\left( {{R}^{-n, n} - {R}^{-n, n - 1}}\right) ,\]\n\nand \( D \) acts on \( H\left( {\mathbb{B}}_{n}\right) \) as a linear partial differential operator of order \( n \) with polynomial coefficients. | Proof. For any fixed \( w \in {\mathbb{B}}_{n} \) we have\n\n\[ D\frac{1}{1-\langle z, w\rangle } = \mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{\Gamma \left( {n + k}\right) }{\Gamma \left( n\right) \Gamma \left( k\right) }\langle z, w{\rangle }^{k} = R\mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{\Gamma \left( {n +... | Yes |
Proposition 6.23. For every holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) we have\n\n\[ \n{\int }_{{\mathbb{B}}_{n}}{\left| Df\left( z\right) \right| }^{2}\frac{{\left( 1 - {\left| z\right| }^{2}\right) }^{n - 1}}{{\left| z\right| }^{2n}}{dv}\left( z\right) = n\mathop{\sum }\limits_{m}\left| m\right| \frac{m!}... | Proof. It suffices to prove the result for polynomials (to avoid issues of convergence); the general case then follows from an approximation argument.\n\nIf \( f \) is a polynomial, then the integral\n\n\[ \nI\left( f\right) = {\int }_{{\mathbb{B}}_{n}}{\left| Df\left( z\right) \right| }^{2}\frac{{\left( 1 - {\left| z\... | Yes |
Lemma 6.24. Suppose \( p > 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then \( f \in {B}_{p} \) if and only if \[ {\int }_{{\mathbb{B}}_{n}}{\left( 1 - {\left| z\right| }^{2}\right) }^{pn}{\left| Df\left( z\right) \right| }^{p}{\left| z\right| }^{-{2n}}{d\tau }\left( z\right) < \infty . \] | Proof. Since \( {Df} \) vanishes at the origin, the integral in (6.23) is always convergent near \( z = 0 \), and so (6.23) holds if and only if \[ {\int }_{{\mathbb{B}}_{n}}{\left( 1 - {\left| z\right| }^{2}\right) }^{pn}{\left| Df\left( z\right) \right| }^{p}{d\tau }\left( z\right) < \infty , \] which is equivalent t... | Yes |
Theorem 6.25. Suppose \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \) . Then the dual space of \( {B}_{p} \) can be identified with \( {B}_{q} \) (with equivalent norms) under the Möbius invariant pairing\n\n\[ \langle f, g\rangle = \mathop{\sum }\limits_{m}\left| m\right| \frac{m!}{\left| m\right| !}{a}_{m}\overline{{b}_... | Proof. By polarizing the identity in Proposition 6.23 we can write the Möbius invariant semi-inner product as\n\n\[ \langle f, g\rangle = \frac{1}{n}{\int }_{{\mathbb{B}}_{n}}{\left( 1 - {\left| z\right| }^{2}\right) }^{n}{Df}\left( z\right) \overline{{\left( 1 - {\left| z\right| }^{2}\right) }^{n}{Dg}\left( z\right) }... | Yes |
Lemma 6.26. Suppose \( 1 < p < \infty \) and \( f\left( z\right) = \mathop{\sum }\limits_{m}{a}_{m}{z}^{m} \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then \( f \) is in \( {B}_{p} \) if and only if the function\n\n\[ F\left( z\right) = \mathop{\sum }\limits_{{\left| m\right| > 0}}\frac{\left| m\right| !\Gamma \left(... | Proof. It is easy to see that there exists a constant \( c > 0 \) such that\n\n\[ {DF} = c{R}_{-1,1}{R}^{0, n + 1}f \]\n\nprovided that \( f\left( 0\right) = 0 \) . By Lemma 6.24, \( F \in {B}_{p} \) if and only if \( {DF} \in {A}_{{pn} - \left( {n + 1}\right) }^{p} \) , which, according to Theorem 2.19, is equivalent ... | Yes |
Theorem 6.28. Suppose \( p > {\lambda }_{n},\alpha > - 1 \), and \( f \) is holomorphic in \( {B}_{n} \) . Then \( f \in \) \( {B}_{p} \) if and only if\n\n\[{\int }_{{\mathbb{B}}_{n}}{\int }_{{\mathbb{B}}_{n}}\frac{{\left| f\left( z\right) - f\left( w\right) \right| }^{p}d{v}_{\alpha }\left( z\right) d{v}_{\alpha }\le... | Proof. For a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) let\n\n\[{I}_{\alpha }\left( f\right) = {\int }_{{\mathbb{B}}_{n}}{\int }_{{\mathbb{B}}_{n}}\frac{{\left| f\left( z\right) - f\left( w\right) \right| }^{p}d{v}_{\alpha }\left( z\right) d{v}_{\alpha }\left( w\right) }{{\left| 1-\langle z, w\rangle \righ... | Yes |
Theorem 7.1. Suppose \( \alpha > 0,\beta > - 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \in {\mathcal{B}}_{\alpha } \) .\n\n(b) The function \( {\left( 1 - {\left| z\right| }^{2}\right) }^{\alpha }\left| {{Rf}\left( z\right) }\right| \) is boun... | Proof. It is obvious that (a) implies (b).\n\nIf (b) holds, then the function\n\n\[ g\left( z\right) = \frac{{c}_{\alpha + \beta }}{{c}_{\beta }}{\left( 1 - {\left| z\right| }^{2}\right) }^{\alpha }\left( {f\left( z\right) + \frac{{Rf}\left( z\right) }{n + \alpha + \beta }}\right) \]\n\nis bounded in \( {\mathbb{B}}_{n... | Yes |
Theorem 7.2. Suppose \( n > 1 \) and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \). (a) If \( \alpha > \frac{1}{2} \), then \( f \in {\mathcal{B}}_{\alpha } \) if and only if the function \[ {\left( 1 - {\left| z\right| }^{2}\right) }^{\alpha - 1}\left| {\widetilde{\nabla }f\left( z\right) }\right| \] is bounded in ... | Proof. Recall from Lemma 2.14 that \[ \left( {1 - {\left| z\right| }^{2}}\right) \left| {\nabla f\left( z\right) }\right| \leq \left| {\widetilde{\nabla }f\left( z\right) }\right| ,\;z \in {\mathbb{B}}_{n}. \] So the boundedness of \( {\left( 1 - {\left| z\right| }^{2}\right) }^{\alpha - 1}\left| {\widetilde{\nabla }f\... | Yes |
Theorem 7.7. Suppose \( \alpha > 0 \) and\n\n\[ b > \max \left( {n, n + \alpha - 1}\right) \text{.} \]\n\nThere exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that \( {\mathcal{B}}_{\alpha } \) consists exactly of functions of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k =... | Proof. The proof is similar to that of Theorem 3.23. We leave the details to the interested reader. | No |
For each \( \alpha \in \left( {0,1}\right) \) the holomorphic Lipschitz space \( {\Lambda }_{\alpha } \) is a Banach space with the norm\n\n\[ \parallel f\parallel = \left| {f\left( 0\right) }\right| + \parallel f{\parallel }_{\alpha } \] | Proof. If \( \left\{ {f}_{k}\right\} \) is a Cauchy sequence in \( {\Lambda }_{\alpha } \), then \( \left\{ {f}_{k}\right\} \) is uniformly Cauchy in \( {\mathbb{B}}_{n} \) , so \( \left\{ {f}_{k}\right\} \) converges uniformly to some holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) .\n\nGiven any \( \epsilon > ... | Yes |
Theorem 7.9. Suppose \( 0 < \alpha < 1,\beta > - 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \) is in \( {\Lambda }_{\alpha } \) .\n\n(b) \( f \) is in the ball algebra and its boundary values satisfy\n\n\[ \sup \left\{ {\frac{\left| f\left( \ze... | Proof. The equivalence of (c), (d), and (e) follows from Theorem 7.1.\n\nIt is obvious that (a) implies (b).\n\nSuppose \( f \) is in the ball algebra and its boundary function satisfies\n\n\[ \left| {f\left( {\zeta }_{1}\right) - f\left( {\zeta }_{2}\right) }\right| \leq C{\left| {\zeta }_{1} - {\zeta }_{2}\right| }^{... | Yes |
Theorem 7.10. Suppose \( 0 < \alpha < 1,\beta > - 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . Then the following conditions are equivalent:\n\n(a) \( f \in {\Lambda }_{\alpha ,0} \) .\n\n(b) The restriction of \( f \) on \( {\mathbb{S}}_{n} \) can be approximated in the Lipschitz \( \alpha \) -norm of ... | Proof. The proof is similar to that of Theorem 7.9. We omit the details. | No |
Theorem 7.11. Let \( \beta > - 1 \) and \( f \) be holomorphic in \( {\mathbb{B}}_{n} \) . The following conditions are equivalent:\n\n(a) \( f \in {\Lambda }_{1} \), that is, \( \partial f/\partial {z}_{k} \) is in the Bloch space for each \( 1 \leq k \leq n \) .\n\n(b) \( {Rf} \) is in the Bloch space.\n\n(c) There e... | Proof. Since each coordinate function \( {z}_{k},1 \leq k \leq n \), is a pointwise multiplier of the Bloch space \( \mathcal{B} \), it is clear that (a) implies (b).\n\nIf \( {Rf} \) is in the Bloch space, then so is \( f \) . By Theorem 3.4, there exists a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{n}\right)... | Yes |
Theorem 7.12. For \( \beta > - 1 \) and \( f \) holomorphic in \( {\mathbb{B}}_{n} \) the following conditions are equivalent:\n\n(a) \( f \in {\Lambda }_{1,0} \) .\n\n(b) Each \( \partial f/\partial {z}_{k},1 \leq k \leq n \), is in the little Bloch space \( {\mathcal{B}}_{0} \) .\n\n(c) \( R \) fis in \( {\mathcal{B}... | Proof. The proof is similar to that of Theorem 7.11. We leave the details to the interested reader. | No |
Theorem 7.14. Suppose \( \alpha > 1,\beta > - 1 \), and \( n + \beta - \alpha \) is not a negative integer. Then a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) belongs to \( {\Lambda }_{\alpha } \) if and only if there exists a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{n}\right) \) such that\n\n\[ f... | Proof. First assume that \( f \) admits the integral representation. Differentiating under the integral sign and applying Theorem 1.12, we see that condition (c) in Theorem 7.13 holds, so \( f \in {\Lambda }_{\alpha } \) .\n\nNext we assume that \( f \in {\Lambda }_{\alpha } \) . Let \( k \) be the positive integer sat... | Yes |
Theorem 7.15. Suppose \( \alpha > 1,\beta > - 1 \), and \( k \) is the positive integer such that \( k \leq \alpha < k + 1 \) . If \( n + \beta - \alpha \) is not a negative integer, then the following conditions are equivalent for a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) :\n\n(a) The function \( f \) b... | Proof. The proof is similar to those of Theorems 7.13 and 7.14. We omit the details. | No |
Theorem 7.16. Suppose \( \alpha > 0,\beta > - 1 \), and \( f \) is holomorphic in \( {\mathbb{B}}_{n} \) . If \( n + \beta - \alpha \) is not a negative integer, then \( f \in {\Lambda }_{\alpha } \) if and only if there exists a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{n}\right) \) such that \[ f\left( z\ri... | Proof. This follows from Theorems 7.9, 7.11, and 7.14. | No |
Theorem 7.17. Suppose \( t > \alpha > 0 \) . If \( s \) is a real parameter such that neither \( n + s \) nor \( n + s + t \) is a negative integer, then a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) belongs to \( {\Lambda }_{\alpha } \) if and only if the function \[ \varphi \left( z\right) = {\left( 1 - {\... | Proof. Let \( \beta = s + \alpha + N \), where \( N \) is a sufficiently large positive integer. If \( f \in {\Lambda }_{\alpha } \), then by Theorem 7.16, we can then find a function \( g \in {L}^{\infty }\left( {\mathbb{B}}_{n}\right) \) such that \[ f\left( z\right) = {\int }_{{\mathbb{B}}_{n}}\frac{g\left( w\right)... | Yes |
Theorem 7.19. Suppose \( \alpha > 0 \) and \( s \) is a real parameter such that neither \( n + s \) nor \( n + s + \alpha \) is a negative integer. Then the fractional differential operator \( {R}^{s,\alpha } \) maps the Lipschitz space \( {\Lambda }_{\alpha } \) onto the Bloch space \( \mathcal{B} \) . Equivalently, ... | Proof. Let \( t \) be a positive number large enough so that \( t > \alpha \) and \( n + s - t + \alpha \) is not a negative integer. Then the operators\n\n\[ \n{R}^{s - t + \alpha, t},\;{R}^{s - t + \alpha, t - \alpha },\;{R}^{s,\alpha },\n\]\n\nare all well defined. By Theorem 7.17, the assumption \( f \in {\Lambda }... | Yes |
Theorem 7.21. Suppose \( \alpha > 0, b > n \), and \( b - \alpha - 1 \) is not a negative integer. Then there exists a sequence \( \left\{ {a}_{k}\right\} \) in \( {\mathbb{B}}_{n} \) such that \( {\Lambda }_{\alpha } \) consists exactly of functions of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k = 1}}^... | Proof. Write \( b - \alpha = n + 1 + s \) . The assumptions on \( b \) imply that neither \( n + s \) nor \( n + s + \alpha \) is a negative integer, so that the operator \( {R}^{s,\alpha } \) is well defined. Now a holomorphic function \( f \) in \( {\mathbb{B}}_{n} \) admits a representation as in (7.8) if and only i... | Yes |
Theorem 7.23. Suppose \( n > 1 \) and \( f \in {\Lambda }_{\alpha } \). (a) If \( 0 < \alpha < \frac{1}{2} \), then there exists a constant \( C > 0 \) such that \[ {\left( 1 - {\left| z\right| }^{2}\right) }^{-\alpha }\left| {\widetilde{\nabla }f\left( z\right) }\right| \leq C,\;{\left( 1 - {\left| z\right| }^{2}\righ... | Proof. Since \( {\Lambda }_{\alpha } = {\mathcal{B}}_{1 - \alpha } \) for \( 0 < \alpha < 1 \), the estimates for \( \left| {\widetilde{\nabla }f\left( z\right) }\right| \) follow from Theorem 7.2. The estimates for \( \left| {{\nabla }_{T}f\left( z\right) }\right| \) then follow from the corresponding ones for \( \lef... | Yes |
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