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

TL;DR

This paper extends the understanding of extremal problems in Bergman spaces, showing that certain regularity results hold for a broader range of p and q values, generalizing previous results limited to even integers.

Contribution

The paper generalizes Ryabykh's $H^p$ regularity theorem for extremal problems in Bergman spaces to include non-even integer p, under specific conditions on the kernel's regularity.

Findings

If $k \

The regularity of the extremal function $F$ is linked to the regularity of the kernel $k$ in Hardy spaces.

The results extend known theorems to a wider class of p and q values, beyond even integers.

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.