Singular Sets for Harmonic Measure on Locally Flat Domains with Locally Finite Surface Measure
Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa · International Mathematics Research Notices · 2016
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in nontangentially accessible domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each |$d>1$|, there exists a Reifenberg flat domain |$\Omega\subset \mathbb{R}^{d+1}$| with |$\mathcal{H}^{d}(\partial\Omega)<\infty$| and a subset |$E\subset \partial\Omega$| with positive harmonic measure yet zero |$\mathcal{H}^{d}$|-measure. In particular, this implies that a classical theorem of F. and M.~Riesz fails in higher dimensions for this type of domains.