### Abstract

© 2018 American Mathematical Society. For a bounded simply connected domain Ω ⊂ R2, any point z ∈ Ω 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 ∂Ω. In fact this set in the boundary contains the intersection ∂Ωz ∩∂Ω of the boundary of a John subdomain 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.

Original language | English |
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Pages (from-to) | 3403-3412 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Keywords

- Hardy inequality
- John domain
- Simply connected

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## Cite this

Koskela, P., Nandi, D., & Nicolau, A. (2018). Accessible parts of boundary for simply connected domains.

*Proceedings of the American Mathematical Society*,*146*(8), 3403-3412. https://doi.org/10.1090/proc/13994