Controlled diffeomorphic extension of homeomorphisms

Abstract

Let $\Omega$ be an internal chord-arc Jordan domain and $\phi :\mathbb{S}\to \partial \Omega$ be a homeomorphism. We show that $\phi$ has finite dyadic energy if and only if $\phi$ has a diffeomorphic extension $h:\mathbb{D}\to \Omega$ which has finite energy.

19 pages, 1 figure