Accessible parts of boundary for simply connected domains

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 subdomain $\Omega _z$ 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.