Accessible Parts of Boundary for Simply Connected Domains

TL;DR

This paper provides a quantitative lower bound on the Hausdorff content of boundary parts accessible via John curves in simply connected domains, and applies this to weighted Hardy inequalities.

Contribution

It introduces a new lower bound for boundary accessibility in simply connected domains using John curves, extending Makarov's results with quantitative estimates.

Findings

Lower bound for Hausdorff content of accessible boundary parts

Identification of boundary intersection with John sub-domain boundary

Application to weighted Hardy inequalities

Abstract

For a bounded simply connected domain $\Omega\subset\mathbb{R}^2$, any point $z\in\Omega$ and any $0<\alpha<1$, we give a lower bound for the $\alpha$-dimensional Hausdorff content of the set of points in the boundary of $\Omega$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $\alpha$, in terms of the distance of $z$ to $\partial\Omega$. In fact this set in the boundary contains the intersection $\partial\Omega_z\cap\partial\Omega$ of the boundary of a John sub-domain $\Omega_z$ of $\Omega$, centered at $z$, with the boundary of $\Omega$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.