Extremal Problems in Bergman Spaces and an Extension of Ryabykh's Theorem

TL;DR

This paper investigates extremal problems in Bergman spaces for even integer p, establishing conditions under which extremal functions belong to certain Hardy spaces and extending Ryabykh's theorem.

Contribution

It extends Ryabykh's theorem by linking the smoothness of the kernel to the regularity of extremal functions in Bergman spaces.

Findings

Small Taylor coefficients of the kernel imply the extremal function is in H-infinity.

If q ≤ q1, then the extremal function's membership in H^{(p-1)q1} is equivalent to the kernel being in H^{q1}.

Results provide a partial converse to Ryabykh's theorem.

Abstract

We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. 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 the Taylor coefficients of $k$ are sufficiently small, then the extremal function $F \in H^{\infty}$. We also show that if $q \le q_1 < \infty$, then $F \in H^{(p-1)q_1}$ if and only if $k \in H^{q_1}$. These results extend and provide a partial converse to a theorem of Ryabykh.