Regularity of Extremal Functions in Weighted Bergman and Fock Type 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.