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

英文摘要：

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