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

We study linear extremal problems in the Bergman space $A^p$ of the unit disc, where $1<p<\mathrm{\infty }$. Given a functional on the dual space of $A^p$ with representing kernel $k\in A^q$, where $1\mathrm{/}p+1\mathrm{/}q=1$, 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.