EXTREMAL PROBLEMS IN BERGMAN SPACES AND AN EXTENSION OF RYABYKH'S THEOREM

摘要

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 ∈ 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 ∈ H~∞. We also show that if q ≤ q1 < ∞, then F ∈ H~((p-1)q1) if and only if k ∈ H~(q1).