Dimension estimate for the two-sided points of planar Sobolev extension domains

TL;DR

This paper estimates the Hausdorff dimension of two-sided boundary points in planar Sobolev extension domains, demonstrating sharpness with examples and establishing the equivalence of various definitions.

Contribution

It provides a precise Hausdorff dimension estimate for two-sided boundary points and proves the equivalence of different definitions in planar Sobolev extension domains.

Findings

Hausdorff dimension estimate for two-sided points

Examples demonstrating sharpness of the estimate

Equivalence of different definitions of two-sided points

Abstract

In this paper we give an estimate for the Hausdorff dimension of the set of two-sided points of the boundary of bounded simply connected Sobolev $W^{1,p}$-extension domain for $1<p<2$. Sharpness of the estimate is shown by examples. We also prove the equivalence of different definitions of two-sided points.