The dimension of harmonic measure on some AD-regular flat sets of fractional dimension

In this paper it is shown that if $E\subset {\mathbb{R}}^{n+1}$ is an $s$-AD regular compact set, with $s\in [n-\frac{1}{2},n)$, and $E$ is contained in a hyperplane or, more generally, in an $n$-dimensional $C^1$ manifold, then the Hausdorff dimension of the harmonic measure for the domain ${\mathbb{R}}^{n+1}\setminus E$ is strictly smaller than $s$, i.e., than the Hausdorff dimension of $E$.