On singularity of harmonic measure in space

We construct a topological ball D in R³, and a set E on ∂ D 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 Øksendal on harmonic measure in R² is not true in R³. Suppose D is a bounded domain in R^m, m ≥ 2, R^m\D satisfies the corkscrew condition at each point on ∂ D; and E is a set on ∂ D lying also on a BMO₁ 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.