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Weighted Bergman spaces induced by rapidly increasing weights

About this Title

José Ángel Peláez, Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain and Jouni Rättyä, University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 227, Number 1066
ISBNs: 978-0-8218-8802-5 (print); 978-1-4704-1427-6 (online)
DOI: https://doi.org/10.1090/memo/1066
Published electronically: June 24, 2013
Keywords: Bergman space, Hardy space, regular weight, rapidly increasing weight, normal weight, Bekollé-Bonami weight, Carleson measure, maximal function, integral operator, Schatten class, factorization, zero distribution, linear differential equation, growth of solutions, oscillation of solutions, zero set.
MSC: Primary 30H20; Secondary 47G10

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Table of Contents

Chapters

• Preface
• 1. Basic Notation and Introduction to Weights
• 2. Description of $q$-Carleson Measures for $A^p_\omega$
• 3. Factorization and Zeros of Functions in $A^p_\omega$
• 4. Integral Operators and Equivalent Norms
• 5. Non-conformally Invariant Space Induced by $T_g$ on $A^p_\omega$
• 6. Schatten Classes of the Integral Operator $T_g$ on $A^2_\omega$
• 7. Applications to Differential Equations
• 8. Further Discussion

Abstract

This monograph is devoted to the study of the weighted Bergman space $A^p_\omega$ of the unit disc $\mathbb {D}$ that is induced by a radial continuous weight $\omega$ satisfying \begin{equation} \lim _{r\to 1^-}\frac {\int _r^1\omega (s)\,ds}{\omega (r)(1-r)}=\infty .\tag {\dag } \end{equation} Every such $A^p_\omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\alpha$. Even if it is well known that $H^p$ is the limit of $A^p_\alpha$, as $\alpha \to -1$, in many respects, it is shown that $A^p_\omega$ lies “closer” to $H^p$ than any $A^p_\alpha$, and that several finer function-theoretic properties of $A^p_\alpha$ do not carry over to $A^p_\omega$.

As to concrete objects to be studied, positive Borel measures $\mu$ on $\mathbb {D}$ such that $A^p_\omega \subset L^q(\mu )$, $0<p\le q<\infty$, are characterized in terms of a neat geometric condition involving Carleson squares. These measures are shown to coincide with those for which a Hörmander-type maximal function from $L^p_\omega$ to $L^q(\mu )$ is bounded. It is also proved that each $f\in A^p_\omega$ can be represented in the form $f=f_1\cdot f_2$, where $f_1\in A^{p_1}_\omega$, $f_2\in A^{p_2}_\omega$ and $\frac {1}{p_1}+ \frac {1}{p_2}=\frac {1}{p}$. Because of the tricky nature of $A^p_\omega$ several new concepts are introduced. In particular, the use of a certain equivalent norm involving a square area function and a non-tangential maximal function related to lens type regions with vertexes at points in $\mathbb {D}$, gives raise to a some what new approach to the study of the integral operator \[ T_g(f)(z)=\int _{0}^{z}f(\zeta )\,g’(\zeta )\,d\zeta . \] This study reveals the fact that $T_g:A^p_\omega \to A^p_\omega$ is bounded if and only if $g$ belongs to a certain space of analytic functions that is not conformally invariant. The lack of this invariance is one of the things that cause difficulties in the proof, leading the above-mentioned new concepts, and thus further illustrates the significant difference between $A^p_\omega$ and the standard weighted Bergman space $A^p_\alpha$. The symbols $g$ for which $T_g$ belongs to the Schatten $p$-class $\mathcal {S}_p(A^2_\omega )$ are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.