Extremal Problems in Bergman Spaces.

Abstract

We deal with extremal problems in Bergman spaces. If A^p denotes the Bergman space, then for any given functional phi not equal to zero in the dual space (A^p)*, an extremal function is a function F in A^p such that F has norm 1 and Re phi(F) is as large as possible.

We give a simplified proof of a theorem of Ryabykh stating that if k is in the Hardy space H^q for 1/p + 1/q = 1, and the functional phi is defined for f in A^p by phi(f) equals the integral over the unit disc of f(z) times the conjugate of k(z) d sigma, where sigma is normalized Lebesgue area measure, then the extremal function over the space A^p is actually in H^p.

We also extend Ryabykh’s theorem in the case where p is an even integer. Let p be an even integer, and let phi be defined as above. Furthermore, let p1 and q1 be a pair of numbers such that q1 is finite and greater than or equal to q and p1 = (p−1)q1. Then F is in H^(p1) if and only if k is in H^(q1) . For p an even integer, this contains the converse of Ryabykh’s theorem, which was previously unknown. We also show that F is in H^infinity if the coefficients of the Taylor expansion of k satisfy a certain growth condition.

Finally, we develop a method for finding explicit solutions to certain extremal problems in Bergman spaces. This method is applied to some particular classes of examples. Essentially the same method is used to study minimal interpolation problems, and it gives new information about canonical divisors in Bergman spaces.