Extremal Problems in Bergman Spaces and an Extension of Ryabykh's Hardy Space Regularity Theorem for 1 < p < ∞

We study linear extremal problems in the Bergman space A(p) of the unit disc, where 1 < p < infinity. Given a functional on the dual space of Ap with representing kernel k is an element of A(q), where 1/p + 1/q = 1, we show that if q <= q(1) < infinity and k is an element of H-q1, then F is an element of H ((p-1)) (q1). This result was previously known only in the case where p is an even integer. We also discuss related results.