Extremal Problems in Bergman Spaces.

[摘要] 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 byphi(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 pbe an even integer, and let phi be defined as above. Furthermore, let p1 and q1 be apair 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 extremalproblems in Bergman spaces. This method is applied to some particular classesof examples. Essentially the same method is used to study minimal interpolationproblems, and it gives new information about canonical divisors in Bergman spaces.

[发布机构] University of Michigan

[效力级别] Extremal Problem

[关键词] Bergman;Extremal Problem;Hardy Space;Mathematics;Science;Mathematics