Accessible parts of the boundary for domains with lower content 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.