On singularity of harmonic measure in space (Q1076853)

M. A. Lavrentiev (1936) found a simply-connected domain D in \(R^ 2\) and a set \(E\subset \partial D\) which has zero linear measure and positive harmonic measure with respect to D. B. Øksendal (1981) has proved that if D is a simply-connected domain in \(R^ 2\) and \(E\subset \partial D\) is situated on a line with vanishing linear measure then E has also vanishing harmonic measure with respect to D (this result was generalized by R. Kaufman and J.-M. Wu (1982)). In the present paper the author constructs a topological ball D in \(R^ 3\) and a set \(E\subset \partial D\) lying on a 2-dimensional hyperplane so that E has Hausdorff dimension one and has positive harmonic measure with respect to D. Further it is proved that if D is a bounded domain in \(R^ m\) (m\(\geq 2)\), \(R^ m-D\) satisfies the corkscrew condition at each point on \(\partial D\), E is a set on \(\partial D\) lying on a \(BMO_ 1\) surface and E has m-1 dimensional Hausdorff measure zero then it must have harmonic measure zero with respect to D.