Weighted estimates for diffeomorphic extensions of homeomorphisms

Abstract

Let $\Omega \subset {\mathbb{R}}^2$ be an internal chord-arc domain and $\phi :{\mathbb{S}}^1\to \partial \Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h:\mathbb{D}\to \Omega$ of $\phi$. We study the relationship between weighted integrability of the derivatives of $h$ and double integrals of $\phi$ and of ${\phi }^{-1}$.