Sobolev homeomorphisms and Jordan curves with positive area

Abstract

Each homeomorphic parametrization of a Jordan curve via the unit circle $\mathbb{S}\subset \mathbb{S}^2$ extends to a homeomorphism of the entire $\mathbb{S}^2.$ We show that there exists a Jordan curve of positive area whose every parametrization via $\mathbb{S}$ admits a homeomorphic extension that has $p$-Sobolev regularity on $\mathbb{S}^2$ for all exponents $1\le p<2 .$ This is obtained via a new sharp extension result for the case of $p<2.$