Bi-Sobolev extensions

TL;DR

This paper characterizes circle homeomorphisms that extend to the unit disk with finite bi-Sobolev norm, introduces a bi-conformal extension variant, and discusses limitations of existing extension methods.

Contribution

It provides a full characterization of bi-Sobolev extendable circle homeomorphisms and introduces a bi-conformal extension theorem, addressing limitations of harmonic and Beurling-Ahlfors techniques.

Findings

Characterization of circle homeomorphisms with bi-Sobolev extensions

A bi-conformal variant of Beurling-Ahlfors extension theorem

Existing extension methods are ineffective in degenerated cases

Abstract

We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling-Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling-Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.