Accessible parts of boundary for simply connected domains
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
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2018Copyright
© 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.
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https://converis.jyu.fi/converis/portal/detail/Publication/28098665
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The third author was partially supported by the grants 2014SGR75 of Generalitat de Catalunya and MTM2014-51824-P and MTM2017-85666-P of Ministerio de Ciencia e Innovación. The first and second authors were partially supported by the Academy of Finland grant 307333.License
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