Generalized quasidisks and conformality II

TL;DR

This paper introduces a weaker condition related to local connectivity that ensures the extendability of conformal maps from the unit disk to the entire plane as homeomorphisms with controlled distortion, broadening understanding of quasidisks.

Contribution

It presents a new, weaker variant of the three point property that guarantees conformal map extendability with locally p-integrable distortion.

Findings

Weaker local connectivity condition suffices for conformal map extension.

Conditions for extendability with exponentially integrable distortion are established.

New criteria for quasidisks and conformality are proposed.

Abstract

We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [12], sufficient to guarantee the extendability of a conformal map f from the unit disk onto a domain to the entire plane as a homeomorphism of locally exponentially integrable distortion. Sufficient conditions for extendability to a homeomorphism of locally p-integrable distortion are also given.