On solid cores and hulls of weighted Bergman spaces $A_{\mu}^1$

TL;DR

This paper characterizes the solid core and hull of weighted Bergman spaces $A_m^1$ on the unit disc and entire functions, using duality and specific weight conditions, revealing the existence of solid spaces.

Contribution

It extends techniques to characterize solid cores and hulls of $A_m^1$, including for entire functions, under new weight conditions and duality relations.

Findings

Characterization of the solid core of $A_m^1$ spaces.

Existence of solid $A_m^1$-spaces among entire functions.

Explicit description of the solid hull for certain weights.

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 paper 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 weight $v$. The results are based on a duality relation of 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 condition $(b)$, introduced by the authors. The exponentially decreasing weight $v(z) = \exp( -1 /(1-|z|)$ provides an example satisfying both assumptions.