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

TL;DR

This paper demonstrates that for certain AD-regular sets of fractional dimension contained in a hyperplane or smooth manifold, the harmonic measure's Hausdorff dimension is strictly less than that of the set itself, revealing a dimension drop phenomenon.

Contribution

It establishes a new result on the Hausdorff dimension of harmonic measure for fractional-dimensional AD-regular sets within hyperplanes or smooth manifolds.

Findings

Harmonic measure dimension is strictly less than the set dimension for specified AD-regular sets.

The result applies to sets with dimension in [n-1/2, n) contained in hyperplanes or C^1 manifolds.

Shows a dimension drop phenomenon for harmonic measure in fractional dimensions.

Abstract

In this paper it is shown that if $E\subset\mathbb R^{n+1}$ is an $s$-AD regular compact set, with $s\in [n-\frac12,n)$, 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 $E$.