Weighted estimates for diffeomorphic extensions of homeomorphisms

Abstract

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