Extremal Problems in Bergman Spaces and an Extension of 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.

arXiv comment16 pages. To appear in the Illinois Journal of Mathematics