Accessible parts of boundary for simply connected domains

Pekka Koskela, Debanjan Nandi, Artur Nicolau

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1 Citation (Scopus)

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 languageEnglish
Pages (from-to)3403-3412
JournalProceedings of the American Mathematical Society
Volume146
Issue number8
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Hardy inequality
  • John domain
  • Simply connected

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    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