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arXiv · 1709.06802

Accessible Parts of Boundary for Simply Connected Domains

Pekka Koskela·Debanjan Nandi·Artur Nicolau
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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.

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Pekka Koskela, Debanjan Nandi, Artur Nicolau. 2017-09-20. Accessible Parts of Boundary for Simply Connected Domains. https://arxiv.org/abs/1709.06802

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