Accessible Parts of Boundary for Simply Connected Domains
英文摘要：
For a bounded simply connected domain $Ω\subset\mathbb{R}^2$, any point $z\inΩ$ and any $0<α<1$, we give a lower bound for the $α$-dimensional 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 $\partialΩ$. In fact this set in the boundary contains the intersection $\partialΩ_z\cap\partialΩ$ of the boundary of a John sub-domain $Ω_z$ 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.
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