We construct a topological ball
D in R3 , and a set E on ∂D lying on a 2-diniensional hyperplane 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 Rm, m ≥ 2, Rm∖ 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 hyperplane; 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.
