Accessible parts of the boundary for domains in metric measure spaces

Authors

  • Ryan Gibara University of Cincinnati, Department of Mathematical Sciences
  • Riikka Korte Aalto University, Department of Mathematics and Systems Analysis

Keywords:

Visible boundary, metric measure space, John domain

Abstract

We prove in the setting of \(Q\)-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the \(s\)-dimensional Hausdorff content, then its visible boundary has large \(t\)-dimensional Hausdorff content for every \(0<t<s\leq Q-1\). The visible boundary is the set of points that can be reached by a John curve from a fixed point \(z_{0}\in \Omega\). This generalizes recent results by Koskela-Nandi-Nicolau (from \(\mathbb R^2\)) and Azzam (\(\mathbb R^n\)). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

 

Section
Articles

Published

2022-04-20

How to Cite

Gibara, R., & Korte, R. (2022). Accessible parts of the boundary for domains in metric measure spaces. Annales Fennici Mathematici, 47(2), 695–706. https://doi.org/10.54330/afm.116365