Accessible Parts of Boundary for Simply Connected Domains

For a bounded simply connected domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, any point $z\in \mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$. In fact this set in the boundary contains the intersection $\mathrm{\partial }{\mathrm{\Omega }}_z\cap \mathrm{\partial }\mathrm{\Omega }$ of the boundary of a John sub-domain ${\mathrm{\Omega }}_z$ of $\mathrm{\Omega }$, centered at $z$, with the boundary of $\mathrm{\Omega }$. 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.