Hausdorff dimension and $\sigma$ finiteness of $p-$harmonic measures in space when $p\geq n$

TL;DR

This paper investigates the properties of p-harmonic measures in Euclidean space, demonstrating their concentration on sets with finite (n-1)-dimensional Hausdorff measure under certain conditions, especially when p ≥ n.

Contribution

It establishes the Hausdorff dimension and -finiteness of p-harmonic measures for p  n, extending understanding of measure concentration in potential theory.

Findings

For p > n, p-harmonic measure is concentrated on  sets with finite H^{n-1} measure.

When p = n, the measure concentration depends on the boundary's uniform fatness in n-capacity.

The results connect geometric measure theory with p-harmonic analysis in Euclidean spaces.

Abstract

In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n we show that this p harmonic measure is concentrated on a set of \sigma- finite H^{n-1} measure while if p=n the same conclusion holds provided \Gamma is uniformly fat in the sense of n capacity.