Regularity of Extremal Functions in Weighted Bergman and Fock Type Spaces

TL;DR

This paper investigates the regularity and growth properties of extremal functions in weighted Bergman and Fock spaces, establishing links between the growth of the extremal functions and their defining kernels under certain conditions.

Contribution

It provides new results on the growth behavior of extremal functions in weighted Bergman and Fock spaces, especially relating kernel growth to extremal function growth under radial weights.

Findings

Slow growth of kernel implies slow growth of extremal functions

Established a relation between integrability and growth of log-convex functions

Applied results to analyze growth of integral means in Fock spaces

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 $\text{Re} \int_\Omega f(z) \overline{k(z)}\, \nu(z) \, dA(z)$ 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.