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Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma


Authors: Fernando Pérez-González and Jouni Rättyä
Journal: Proc. Amer. Math. Soc. 145 (2017), 2155-2166
MSC (2010): Primary 30H20, 30J05
DOI: https://doi.org/10.1090/proc/13384
Published electronically: November 21, 2016
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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

$\displaystyle T(f)(z)=\frac {\int _{\vert z\vert}^1f\left (s\frac {z}{\vert z\vert}\right )\,ds}{1-\vert z\vert} $

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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Fernando Pérez-González
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain
Email: fernando.perez.gonzalez@ull.es

Jouni Rättyä
Affiliation: Department of Mathematics, University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
Email: jouni.rattya@uef.fi

DOI: https://doi.org/10.1090/proc/13384
Keywords: Inner function, Bergman space, Hardy space, Schwarz-Pick lemma, doubling weight, Muckenhoupt weight, maximal function, Carleson measure
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
Article copyright: © Copyright 2016 American Mathematical Society