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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 of the unit disc that is induced by a radial continuous weight satisfying

Every such lies between the Hardy space and every classical weighted Bergman space . Even if it is well known that is the limit of , as , in many respects, it is shown that lies “closer” to than any , and that several finer function-theoretic properties of do not carry over to . As to concrete objects to be studied, positive Borel measures on such that , , 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 to is bounded. It is also proved that each can be represented in the form , where , and . Because of the tricky nature of 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 , gives raise to a some what new approach to the study of the integral operator

This study reveals the fact that is bounded if and only if 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 and the standard weighted Bergman space . The symbols for which belongs to the Schatten -class are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.

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