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# memo_has_moved_text();Weighted Bergman spaces induced by rapidly increasing weights

### About this Title

José Ángel Peláez and Jouni Rättyä

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: http://dx.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

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### Table of Contents

Chapters

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

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, 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

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.