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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma
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by Fernando Pérez-González and Jouni Rättyä PDF
Proc. Amer. Math. Soc. 145 (2017), 2155-2166 Request permission

Abstract:

We characterize those radial doubling weights $\omega$ for which the Schwarz-Pick lemma applied to the derivative of any inner function in the norm of the Bergman space $A^p_\omega$ does not cause any essential loss of information. The approach we employ is based on operator theory and leads to a characterization of when the linear average operator \[ T(f)(z)=\frac {\int _{|z|}^1f\left (s\frac {z}{|z|}\right ) ds}{1-|z|} \] is bounded from $A^p_\omega$ to $L^p_\omega$. The characterizing integral condition is self-improving and therefore $T:A^p_\omega \to L^p_\omega$ is bounded if and only if $T:A^{p-\varepsilon }_\omega \to L^{p-\varepsilon }_\omega$ is bounded for all sufficiently small $\varepsilon >0$. This study also reveals the fact that, under appropriate hypothesis on $\omega$, the average operator $T:A^p_\omega \to L^p_\omega$ is bounded if and only if the Bergman projection $P:L^p_\omega \to L^p_\omega$ is bounded if and only if the classical Hilbert operator $\mathcal {H}:L^{p+1}_{\widehat {\omega }}([0,1))\to A^{p+1}_\omega$ is bounded, where $\widehat {\omega }(r)=\int _r^1\omega (s) ds$.
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Additional Information
  • Fernando Pérez-González
  • Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain
  • MR Author ID: 137985
  • Email: fernando.perez.gonzalez@ull.es
  • Jouni Rättyä
  • Affiliation: Department of Mathematics, University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
  • MR Author ID: 686390
  • Email: jouni.rattya@uef.fi
  • Received by editor(s): June 8, 2016
  • Received by editor(s) in revised form: July 15, 2016
  • Published electronically: November 21, 2016
  • Additional Notes: This research was supported in part by Ministerio de Economia y Competitividad, Spain, projects MTM2011-26538 and MTM2014-52685-P and Academy of Finland project no. 268009, and the Faculty of Science and Forestry of University of Eastern Finland project no. 930349
  • Communicated by: Stephan Ramon Garcia
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2155-2166
  • MSC (2010): Primary 30H20, 30J05
  • DOI: https://doi.org/10.1090/proc/13384
  • MathSciNet review: 3611328