Accessible parts of the boundary for domains with lower content regular complements

TL;DR

This paper proves that for domains with lower content regular complements, there exist boundary parts supporting Hardy inequalities, providing a characterization of such domains in terms of lower content regularity.

Contribution

It extends previous results by establishing the existence of chord-arc boundary parts supporting Hardy inequalities in higher dimensions with lower content regular complements.

Findings

Existence of boundary parts supporting Hardy inequalities in domains with lower content regular complements.

Characterization of domains supporting pointwise $(p,\beta)$-Hardy inequalities based on lower content regularity.

Generalization of planar domain results to higher dimensions with regular complements.

Abstract

We show that if $0<t<s\leq n-1$, $\Omega\subseteq \mathbb{R}^{n}$ with lower $s$-content regular complement, and $z\in \Omega$, there is a chord-arc domain $\Omega_{z}\subseteq \Omega $ with center $z$ so that $\mathscr{H}^{t}_{\infty}(\partial\Omega_{z}\cap \partial\Omega)\gtrsim_{t} \textrm{dist}(z,\Omega^{c})^{t}$. This was originally shown by Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when $n=2$, $s=1$, and $\Omega$ is a simply connected planar domain. Domains satisfying the conclusion of this result support $(p,\beta)$-Hardy inequalities for $\beta<p-n+t$ by a result of Koskela and Lehrb\"{a}ck; Lehrb\"{a}ck also showed that $s$-content regularity of the complement for some $s>n-p+\beta$ was necessary. Thus, the combination of these results gives a characterization of which domains support pointwise $(p,\beta)$-Hardy inequalities for $\beta<p-1$ in terms of lower content regularity.