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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
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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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