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Weighted Bergman Spaces Induced by Rapidly Increasing Weights
José Ángel Peláez, Universidad de Málaga, Spain, and Jouni Rättyä, University of Eastern Finland, Joensuu, Finland

Memoirs of the American Mathematical Society
2014; 124 pp; softcover
Volume: 227
ISBN-10: 0-8218-8802-1
ISBN-13: 978-0-8218-8802-5
List Price: US$77
Individual Members: US$46.20
Institutional Members: US$61.60
Order Code: MEMO/227/1066
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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 \(\lim_{r\to 1^-}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1-r)}=\infty.\) 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 \(A^p_\omega\).

Table of Contents

  • Preface
  • Basic notation and introduction to weights
  • Description of \(q\)-Carleson measures for \(A^p_\omega\)
  • Factorization and zeros of functions in \(A^p_\omega\)
  • Integral Operators and equivalent norms
  • Non-conformally invariant space induced by \(T_g\) on \(A^p_\omega\)
  • Schatten classes of the integral operator \(T_g\) on \(A^2_\omega\)
  • Applications to differential equations
  • Further discussion
  • Bibliography
  • Index
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