Improved Regularity of Harmonic Diffeomorphic Extensions on Quasihyperbolic Domains

Abstract

Let \(\mathbb{X}\) be a Jordan domain satisfying certain hyperbolic growth conditions. Assume that φ is a homeomorphism from the boundary \(\partial \mathbb{X}\) of \(\mathbb{X}\) onto the unit circle. Denote by h the harmonic diffeomorphic extension of φ 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 in [A. Koski, J. Onninen, Sobolev homeomorphic extensions, J. Eur. Math. Soc. 23 (2021) 4065–4089, Theorem 3.1].