On planar Sobolev $L^m_p$-extension domains

TL;DR

This paper characterizes bounded simply connected Sobolev extension domains in the plane using intrinsic subhyperbolic metrics, introducing a new Square Separation Theorem crucial for understanding domain geometry.

Contribution

It provides a novel geometric criterion for Sobolev extension domains in R^2 based on subhyperbolic metrics and a new Square Separation Theorem.

Findings

Characterization of Sobolev $L^m_p$-extension domains in the plane.

Introduction of a new Square Separation Theorem.

Development of chains of squares connecting points in the domain.

Abstract

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a series of results related to the existence of special chains of squares joining given points $x$ and $y$ in $\Omega$. An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point $x$ and a square $S\subset\Omega$ there exists a similar square $Q\subset\Omega$ which touches $S$ and has the property that $x$ and $S$ belong to distinct connected components of $\Omega\setminus Q$.