Geometric characterizations for conformal mappings in weighted Bergman spaces

TL;DR

This paper characterizes conformal mappings in weighted Bergman spaces using geometric and integral conditions, extending known Hardy space results and introducing a new Hardy number analogue.

Contribution

It provides a geometric characterization of conformal mappings in weighted Bergman spaces and introduces an analogue of the Hardy number for these spaces.

Findings

Conformal mappings belong to weighted Bergman spaces iff certain harmonic measure integrals converge.

Geometric conditions involving Euclidean areas characterize these mappings.

New relations between Hardy and weighted Bergman spaces are established.

Abstract

We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions for this number and hence we establish new results for the Hardy number and the relation between Hardy and weighted Bergman spaces.