Accessible parts of boundary for simply connected domains
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- by Pekka Koskela, Debanjan Nandi and Artur Nicolau PDF
- Proc. Amer. Math. Soc. 146 (2018), 3403-3412 Request permission
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.References
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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
- MR Author ID: 289254
- Email: pekka.j.koskela@jyu.fi
- Debanjan Nandi
- Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
- Email: debanjan.s.nandi@jyu.fi
- Artur Nicolau
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra. Barcelona, Spain
- MR Author ID: 254630
- Email: artur@mat.uab.cat
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3403-3412
- MSC (2010): Primary 26D15, 30C35
- DOI: https://doi.org/10.1090/proc/13994
- MathSciNet review: 3803665