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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
MathSciNet review: 3611328
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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 \[ T(f)(z)=\frac {\int _{|z|}^1f\left (s\frac {z}{|z|}\right ) ds}{1-|z|} \] 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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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

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