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Integration operators on Bergman spaces

Aleman, Alexandru LU and Siskakis, Aristomenis G (1997) In Indiana University Mathematics Journal 46(2). p.337-356
Abstract
Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable.



Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$.



In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$.... (More)
Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable.



Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$.



In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$. Thus, the question of boundedness or compactness of $T_g$ becomes the corresponding question for the operator of multiplication by $g'$ acting from $L^p_{\rm a}(w)$ to $L^p_{\rm a}(v)$.



The authors consider only $w$ which are radial: $w(re^{i\theta}) = w(r)$. In the first part of the paper it is shown that $\int |f|^pw\,dm \le C\int |f'(z)|^p v(|z|) \,dm(z)$, $p \ge 1$, where $v(r) = \int_r^1 w(u)\,du$. Under the assumption that $v(r) \le C(1 - r)w(r)$, which is valid in particular for $w(r) \equiv (1 - r)^\alpha$, $\alpha > -1$, it follows that $T_g$ is bounded when $g'(z)(1 - |z|)$ is bounded.



The converse can be proved in a more general setting. It is obtained by estimating the norm of the linear functional $D_\lambda\colon f \mapsto f'(\lambda)$ in terms of that of the evaluation functional $L_\lambda\colon f \mapsto f(\lambda)$. Rather general hypotheses on a Banach space of analytic functions are obtained in order that $\| D_\lambda \|(1 - |\lambda|) \le C\| L_\lambda \|$. This leads immediately to the converse: if the operator $T_g$ is bounded, then $g'(z)(1 - |z|)$ is bounded. Several classes of weights are shown to satisfy the hypotheses required for both the necessity and the sufficiency of the condition.



In all cases, the problem of compactness of $T_g$ is also considered, and the solution involves the little-oh versions of the same conditions for boundedness. In the special case where $p = 2$ and $w(r) = (1 - r)^\alpha$ the Schatten class of $T_g$ is determined.



The techniques are briefly applied to the weight $w(r) = \exp[ -\beta(1 - r)^{-\alpha}]$, $\alpha > 0$. In this case $v(r) \le C(1 - r)^{\alpha + 1}w(r)$, and that leads to the conclusion that $T_g$ is bounded when $g'(z)(1 - |z|)^{\alpha + 1}$ is bounded. The necessity of this condition is left open. A theorem of V. L. Oleinik [Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 120--137, 187, 192--193; MR0369705 (51 #5937) (Theorem 3.3)] shows that it is necessary at least when $\alpha > 1$. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Indiana University Mathematics Journal
volume
46
issue
2
pages
337 - 356
publisher
Indiana University
external identifiers
  • scopus:0000157297
ISSN
1943-5258
DOI
10.1512/iumj.1997.46.1373
language
English
LU publication?
yes
id
1c6f8fa0-a62a-4548-b371-9a7fdabb568a (old id 1467195)
date added to LUP
2009-09-16 12:55:49
date last changed
2017-11-12 03:20:32
@article{1c6f8fa0-a62a-4548-b371-9a7fdabb568a,
  abstract     = {Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable. <br/><br>
<br/><br>
Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$. <br/><br>
<br/><br>
In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$. Thus, the question of boundedness or compactness of $T_g$ becomes the corresponding question for the operator of multiplication by $g'$ acting from $L^p_{\rm a}(w)$ to $L^p_{\rm a}(v)$. <br/><br>
<br/><br>
The authors consider only $w$ which are radial: $w(re^{i\theta}) = w(r)$. In the first part of the paper it is shown that $\int |f|^pw\,dm \le C\int |f'(z)|^p v(|z|) \,dm(z)$, $p \ge 1$, where $v(r) = \int_r^1 w(u)\,du$. Under the assumption that $v(r) \le C(1 - r)w(r)$, which is valid in particular for $w(r) \equiv (1 - r)^\alpha$, $\alpha &gt; -1$, it follows that $T_g$ is bounded when $g'(z)(1 - |z|)$ is bounded. <br/><br>
<br/><br>
The converse can be proved in a more general setting. It is obtained by estimating the norm of the linear functional $D_\lambda\colon f \mapsto f'(\lambda)$ in terms of that of the evaluation functional $L_\lambda\colon f \mapsto f(\lambda)$. Rather general hypotheses on a Banach space of analytic functions are obtained in order that $\| D_\lambda \|(1 - |\lambda|) \le C\| L_\lambda \|$. This leads immediately to the converse: if the operator $T_g$ is bounded, then $g'(z)(1 - |z|)$ is bounded. Several classes of weights are shown to satisfy the hypotheses required for both the necessity and the sufficiency of the condition. <br/><br>
<br/><br>
In all cases, the problem of compactness of $T_g$ is also considered, and the solution involves the little-oh versions of the same conditions for boundedness. In the special case where $p = 2$ and $w(r) = (1 - r)^\alpha$ the Schatten class of $T_g$ is determined. <br/><br>
<br/><br>
The techniques are briefly applied to the weight $w(r) = \exp[ -\beta(1 - r)^{-\alpha}]$, $\alpha &gt; 0$. In this case $v(r) \le C(1 - r)^{\alpha + 1}w(r)$, and that leads to the conclusion that $T_g$ is bounded when $g'(z)(1 - |z|)^{\alpha + 1}$ is bounded. The necessity of this condition is left open. A theorem of V. L. Oleinik [Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 120--137, 187, 192--193; MR0369705 (51 #5937) (Theorem 3.3)] shows that it is necessary at least when $\alpha &gt; 1$.},
  author       = {Aleman, Alexandru and Siskakis, Aristomenis G},
  issn         = {1943-5258},
  language     = {eng},
  number       = {2},
  pages        = {337--356},
  publisher    = {Indiana University},
  series       = {Indiana University Mathematics Journal},
  title        = {Integration operators on Bergman spaces},
  url          = {http://dx.doi.org/10.1512/iumj.1997.46.1373},
  volume       = {46},
  year         = {1997},
}