The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One

Let Omega subset of Rn+1 be open and let E subset of partial derivative Omega with 0 < H-s(E) < infinity, for some s is an element of(n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure H-s on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that Omega is a uniform domain.