- journal article metadata
European Mathematical Society Publishing House
2018-02-09 23:30:02
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
34
2018
1
Tangent measures and absolute continuity of harmonic measure
Jonas
Azzam
Universitat Autònoma de Barcelona, Bellaterra, Spain
Mihalis
Mourgoglou
Universitat Autònoma de Barcelona, Bellaterra, Spain and Centre de Recerca Matemàtica, Barcelona, Spain
Harmonic measure, Wolff snowflakes, non-tangentially accessible (NTA) domains, uniform domains, capacity density condition, tangent measures, absolute continuity
We show that for uniform domains $\Omega\subseteq \mathbb R^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $\alpha$-dimensional Hausdorff measure unless $\alpha\leq d$. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.
Potential theory
Measure and integration
305
330
10.4171/RMI/986
http://www.ems-ph.org/doi/10.4171/RMI/986
2
6
2018