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2019 Regularity of extremal functions in weighted Bergman and Fock type spaces
Timothy Ferguson
Rocky Mountain J. Math. 49(1): 47-71 (2019). DOI: 10.1216/RMJ-2019-49-1-47

Abstract

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $Re \int _\Omega f(z) \overline {k(z)}\, \nu (z) \, dA(z)$ over all functions $f$ of unit norm, where $\nu $ is the weight function and $\Omega $ is the domain of the functions in the space. We consider the case where $\nu (z)$ is a decreasing radial function satisfying some additional assumptions, and where $\Omega $ is either a disc centered at the origin or the entire complex plane. We show that, if $k$ grows slowly in a certain sense, then $f$ must grow slowly in a related sense. We also discuss a relation between the integrability and growth of certain log-convex functions and apply the result to obtain information about the growth of integral means of extremal functions in Fock type spaces.

Citation

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Timothy Ferguson. "Regularity of extremal functions in weighted Bergman and Fock type spaces." Rocky Mountain J. Math. 49 (1) 47 - 71, 2019. https://doi.org/10.1216/RMJ-2019-49-1-47

Information

Published: 2019
First available in Project Euclid: 10 March 2019

zbMATH: 07036618
MathSciNet: MR3921866
Digital Object Identifier: 10.1216/RMJ-2019-49-1-47

Subjects:
Primary: 30H20
Secondary: ‎46E15

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 1 • 2019
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