Homeomorphic Sobolev extensions and integrability of hyperbolic metric

TL;DR

This paper investigates conditions under which boundary parametrizations of Jordan domains can be extended to Sobolev homeomorphisms, focusing on hyperbolic metrics with general integrability conditions, and provides sharpness results with counterexamples.

Contribution

It extends previous results by establishing Sobolev homeomorphic extensions under more general hyperbolic metric integrability conditions and demonstrates the sharpness of these conditions.

Findings

Extension exists under certain $oldsymbol{ ext{phi}}$-integrability conditions.

Counterexample shows the sharpness of the integrability criteria.

Generalizes previous $L^q$-integrability results to broader classes.

Abstract

Very recently, it was proved that if the hyperbolic metric of a planar Jordan domain is $L^q$-integrable for some $q\in (1,\infty)$, then every homeomorphic parametrization of the boundary Jordan curve via the unit circle can be extended to a Sobolev homeomorphism of the entire disk. This naturally raises the question of whether the extension holds under more general integrability conditions on the hyperbolic metric. In this work, we examine the case where the hyperbolic metric is $\phi$-integrable. Under appropriate conditions on the function $\phi$, we establish the existence of a Sobolev homeomorphic extension for every homeomorphic parametrization of the Jordan curve. Moreover, we demonstrate the sharpness of our result by providing an explicit counterexample.