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Accessible parts of boundary for simply connected domains

Authors: Pekka Koskela, Debanjan Nandi and Artur Nicolau
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 26D15, 30C35
Published electronically: February 28, 2018
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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 subdomain $ \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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Additional Information

Pekka Koskela
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland

Debanjan Nandi
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland

Artur Nicolau
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra. Barcelona, Spain

Keywords: Simply connected, John domain, Hardy inequality
Received by editor(s): June 19, 2017
Received by editor(s) in revised form: October 19, 2017
Published electronically: February 28, 2018
Additional Notes: 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.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2018 American Mathematical Society

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