Regularity of Extremal Functions in Weighted Bergman and Fock Type Spaces

2014/11/07 by Timothy Ferguson, Ferguson, Timothy
Mathematics · #30H10 #30H20 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:30H10 #msc:30H20

paper · pdf · doi:10.48550/arxiv.1411.1979

17 pages

arxiv created 2014/11/07 · openalex publication_date 2014/11/07 · arxiv updated 2014/11/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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 ∫_Ωf(z) k(z) ν(z) dA(z) over all functions f of unit norm, where ν is the weight function and Ω is the domain of the functions in the space. We consider the case where ν(z) is a decreasing radial function satisfying some additional assumptions, and where Ω 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.

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