  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.
