Extremal distance and conformal mappings in Hardy and Bergman spaces

TL;DR

This paper establishes integral conditions involving extremal distance that determine when a conformal mapping belongs to Hardy or weighted Bergman spaces, and characterizes Hardy and Bergman numbers via extremal distance.

Contribution

It provides new integral criteria and characterizations linking extremal distance with membership in Hardy and Bergman spaces for conformal mappings.

Findings

Necessary and sufficient integral conditions involving extremal distance are established.

Characterizations of Hardy and Bergman numbers are given in terms of extremal distance.

The results connect geometric extremal distance with functional space membership.

Abstract

We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the Bergman number of a simply connected domain in terms of extremal distance.