Accessible parts of boundary for simply connected domains

Abstract

For a bounded simply connected domain Ω ⊂ R 2 , any point z ∈ Ω and any 0 < α < 1, we give a lower bound for the αdimensional Hausdorff content of the set of points in the boundary of Ω which can be joined to z by a John curve with a suitable John constant depending only on α, in terms of the distance of z to ∂Ω. In fact this set in the boundary contains the intersection ∂Ωz ∩∂Ω of the boundary of a John sub-domain Ωz of Ω, centered at z, with the boundary of Ω. 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.