Extremal problems on Bergman spaces A¹α and Besov spaces

Abstract

Extremal problems in different function spaces have long been investigated. Ferguson provides a method, using Bergman projections, to solve certain types of extremal problems in Bergman spaces for 1 < p < ∞ in his work [3]. Later the method is extended to weighted Bergman spaces for 1 < p < ∞ in [13]. Now, we extend this method to the p = 1 case. The two cases differ in the structure of Bergman projections and dual spaces. First, we define some function spaces, namely weighted Bergman spaces, the Bloch space, and Besov spaces, and show the usage of Bergman projection on these spaces. Then, we find some conditions to ensure the existence of unique solutions for extremal problems. Later, we use Bergman projection to find a candidate function for the solution in the p = 1 case, and we prove that the candidate function is the solution if it never attains the value 0. Finally, under special conditions, we solve a similar problem in Besov spaces.