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

Abstract

We show that if $0<t<s\le n-1$, $\Omega \subseteq {\mathbb{R}}^n$ 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 ${\mathcal{H}}_{\infty }^t(\partial {\Omega }_z\cap \partial \Omega ){\gtrsim }_t\text{dist}(z,{\Omega }^c{)}^t$. 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 Lehrbäck; Lehrbäck 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.

Corrected typos, added some more detail. To appear in Annales Academiæ Scientiarum Fennicæ