On solid cores and hulls of weighted Bergman spaces A^1_mu

Abstract

[EN] We consider weighted Bergman spaces A^1_mu 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^1_mu. Also, as a consequence of a characterization of solid A^1_mu-spaces, we show that, in the case of entire functions, there indeed exist solid A^1_mu-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^1_mu, when mu equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A1- and H infinite-spaces, the validity of which requires the assumption that -log v belongs to the class W0, 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.