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

Abstract

AbstractWe consider weighted Bergman spaces $$A_\mu ^1$$ A μ 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 $$A_\mu ^1$$ A μ 1 . Also, as a consequence of a characterization of solid $$A_\mu ^1$$ A μ 1 -spaces, we show that, in the case of entire functions, there indeed exist solid $$A_\mu ^1$$ A μ 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 $$A_\mu ^1$$ A μ 1 , when $$\mu$$ μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted $$A^1$$ A 1 - and $$H^\infty$$ H ∞ -spaces, the validity of which requires the assumption that $$-\log v$$ - log v belongs to the class $$\mathcal {W}_0$$ W 0 , studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight $$v(z) = \exp ( -1 /(1-|z|)$$ v ( z ) = exp ( - 1 / ( 1 - | z | ) provides an example satisfying both assumptions.