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 |$s\in [n-\frac 12,n)$|⁠ , 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 |$s$|⁠ , that is, than the Hausdorff dimension of |$E$|⁠. [ABSTRACT FROM AUTHOR]