# Weighted estimates for diffeomorphic extensions of homeomorphisms

## TL;DR

This paper investigates how weighted integrability conditions on derivatives of diffeomorphic extensions relate to double integrals involving boundary homeomorphisms and their inverses in chord-arc domains.

## Contribution

It establishes new connections between weighted derivative integrability of extensions and boundary double integrals for homeomorphisms in chord-arc domains.

## Key findings

- Weighted integrability of derivatives is characterized by boundary double integrals.
- Results apply to diffeomorphic extensions in internal chord-arc domains.
- Provides new tools for analyzing boundary homeomorphisms and their extensions.

## Abstract

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