ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES A_{mu }^1

Abstract

We consider weighted Bergman spaces A_mu ^1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of A_mu ^1. Also, as a consequence of a characterization of solid A_mu ^1-spaces, we show that, in the case of entire functions, there indeed exist solid A_mu ^1-spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of A_mu ^1, when mu equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A^1- and H^infty-spaces, the validity of which requires the assumption that -log v belongs to the class mathcal {W}_0, studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight v(z) = exp ( -1 /(1-|z|) provides an example satisfying both assumptions.