Extremal Problems in Bergman Spaces and an Extension of Ryabykh's $H^p$ Regularity Theorem For $1<p<\infty$

Abstract

We study linear extremal problems in the Bergman space $A^p$ of the unit disc, where $1<p<\infty$. Given a functional on the dual space of $A^p$ with representing kernel $k\in A^q$, where $1/p+1/q=1$, we show that if $q\le q_1<\infty$ and $k\in H^{q_1}$, then $F\in H^{(p-1)q_1}$. This result was previously known only in the case where $p$ is an even integer. We also discuss related results.

13 pages. Extends results from arXiv:1301.7659