An integral operator on $H\sp p$
(1995) In Complex Variables, Theory & Application 28(2). p.149-158- Abstract
- To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator.
The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation.
... (More) - To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator.
The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation.
The main results of the paper are as follows. Theorem 1: Let $g$ be an analytic function in $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is bounded on $H^p$ if and only if $g\in{\rm BMOA}$. Theorem 2: Let $g$ be an analytic function in $\bold D$. (i) If $1<p<\infty$ then $T_g\in S_p(H^2)$ if and only if $g\in B_p$. (ii) If $0<p\leq 1$ then $T_g\in S_p(H^2)$ if and only if $g$ is constant. Theorem 1 generalizes an earlier result of Ch. Pommerenke [Comment. Math. Helv. 52 (1977), no. 4, 591--602; MR0454017 (56 #12268)] in the context of $H^2$. It also implies the following corollary: Let $g$ be an analytic function on $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is compact on $H^p$ if and only if $g\in{\rm VMOA}$. The main tool in the proof of Theorem 2 is Luecking's results [D. H. Luecking, J. Funct. Anal. 73 (1987), no. 2, 345--368; MR0899655 (88m:47046)] on Cauchy transforms of Borel measures $\mu$ on $\bold D$. $$Q_\mu(f)(w)\coloneq\int_{\bf D}\frac{f(z)}{1-w\overline z}d\mu(z).$$ Let $\{R_j\}^\infty_{j=1}$ be disjoint ``Carleson cubes'' which cover $\bold D$, and let $l(R_j)$ be the distance from the center of $R_j$ to $\partial\bold D$. Then Luecking's theorem [op. cit.] says that $Q_µ\in S_p(H^2)$ if and only if $\sum^\infty_{j=1}(\mu(R_j)/l(R_j))^p<\infty$. The connection to Luecking's theorem is via the relation $T^\ast_gT_g=Q_\mu$ with $d\mu(z)=2|g'(z)|^2\log(1/|z|)dm(z)$, where $m$ is Lebesgue measure. (Less)
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https://lup.lub.lu.se/record/1467247
- author
- Aleman, Alexandru LU and Siskakis, Aristomenis G
- publishing date
- 1995
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Complex Variables, Theory & Application
- volume
- 28
- issue
- 2
- pages
- 149 - 158
- publisher
- New York ; Gordon and Breach, 1982-
- ISSN
- 1563-5066
- DOI
- 10.1080/17476939508814844
- language
- English
- LU publication?
- no
- additional info
- Complex Variables, Theory and Application: An International Journal forsättes av (2006) Complex Variables and Elliptic Equations An International Journal ISSN: 1747-6941 (electronic) 1747-6933 (paper) Taylor&Frances
- id
- 1ea4416d-da8c-488f-94d1-51a298c9169d (old id 1467247)
- date added to LUP
- 2016-04-01 12:17:36
- date last changed
- 2018-11-21 20:05:56
@article{1ea4416d-da8c-488f-94d1-51a298c9169d, abstract = {{To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator. <br/><br> <br/><br> The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation. <br/><br> <br/><br> The main results of the paper are as follows. Theorem 1: Let $g$ be an analytic function in $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is bounded on $H^p$ if and only if $g\in{\rm BMOA}$. Theorem 2: Let $g$ be an analytic function in $\bold D$. (i) If $1<p<\infty$ then $T_g\in S_p(H^2)$ if and only if $g\in B_p$. (ii) If $0<p\leq 1$ then $T_g\in S_p(H^2)$ if and only if $g$ is constant. Theorem 1 generalizes an earlier result of Ch. Pommerenke [Comment. Math. Helv. 52 (1977), no. 4, 591--602; MR0454017 (56 #12268)] in the context of $H^2$. It also implies the following corollary: Let $g$ be an analytic function on $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is compact on $H^p$ if and only if $g\in{\rm VMOA}$. The main tool in the proof of Theorem 2 is Luecking's results [D. H. Luecking, J. Funct. Anal. 73 (1987), no. 2, 345--368; MR0899655 (88m:47046)] on Cauchy transforms of Borel measures $\mu$ on $\bold D$. $$Q_\mu(f)(w)\coloneq\int_{\bf D}\frac{f(z)}{1-w\overline z}d\mu(z).$$ Let $\{R_j\}^\infty_{j=1}$ be disjoint ``Carleson cubes'' which cover $\bold D$, and let $l(R_j)$ be the distance from the center of $R_j$ to $\partial\bold D$. Then Luecking's theorem [op. cit.] says that $Q_µ\in S_p(H^2)$ if and only if $\sum^\infty_{j=1}(\mu(R_j)/l(R_j))^p<\infty$. The connection to Luecking's theorem is via the relation $T^\ast_gT_g=Q_\mu$ with $d\mu(z)=2|g'(z)|^2\log(1/|z|)dm(z)$, where $m$ is Lebesgue measure.}}, author = {{Aleman, Alexandru and Siskakis, Aristomenis G}}, issn = {{1563-5066}}, language = {{eng}}, number = {{2}}, pages = {{149--158}}, publisher = {{New York ; Gordon and Breach, 1982-}}, series = {{Complex Variables, Theory & Application}}, title = {{An integral operator on $H\sp p$}}, url = {{http://dx.doi.org/10.1080/17476939508814844}}, doi = {{10.1080/17476939508814844}}, volume = {{28}}, year = {{1995}}, }