Improved regularity of harmonic diffeomorphic extensions on quasihyperbolic domains

TL;DR

This paper proves optimal regularity and weighted Sobolev estimates for harmonic diffeomorphic extensions of boundary homeomorphisms on quasihyperbolic domains, generalizing previous Sobolev regularity results.

Contribution

It establishes the optimal Orlicz-Sobolev regularity and weighted Sobolev estimates for harmonic extensions on quasihyperbolic domains, extending prior Sobolev regularity results.

Findings

Proved optimal Orlicz-Sobolev regularity of harmonic extensions.

Established weighted Sobolev estimates for these extensions.

Generalized previous Sobolev regularity results by Koski-Onninen.

Abstract

Let $\mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions. Assume that $\varphi$ is a homeomorphism from the boundary $\partial \mathbb{X}$ of $\mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic diffeomorphic extension of $\varphi $ from $\mathbb{X}$ onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of $h.$ These generalize the Sobolev regularity of $h$ by Koski-Onninen [21, Theorem 3.1].