Accessible parts of the boundary for domains in metric measure spaces

受控几何度量测度空间中区域可达边界的Hausdorff内容估计

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

本文在受控几何度量测度空间中，给出了任意区域可达边界尺寸的Hausdorff内容估计，回答了Jonas Azzam提出的问题，并将结果推广至满足余维数一界的每个倍增测度。

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

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 als…

The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property

Prime end boundaries $\partial_PΩ$ of domains $Ω$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincaré inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_PΩ$ which are Lipschitz continuous with respect to the Mazurkiewicz metr…