On singularity of harmonic measure in space
Jang-Mei Gloria Wu · Pacific Journal of Mathematics · 1986
We construct a topological ball D in R3, and a set E on dD lying on a 2-diniensional hyperpiane so that E has Hausdorff dimension one and has positive harmonic measure with respect to D. This shows that a theorem of Θksendal on harmonic measure in R 2 is not true in R 3. Suppose D is a bounded domain in Rm, m>2, Rm\D satisfies the corkscrew condition at each point on dD; and E is a set on dD lying also on a BMOχ surface, which is more general than a hyperpiane; then we can prove that if E has m — 1 dimensional Hausdorff measure zero then it must have harmonic measure zero with respect to D.