A note on p-harmonic measure

Abstract

Let $1<p<\infty$. We identify the optimal decay exponent of the $p$-harmonic measure ${\omega }_p(B(\xi ,t)\cap \partial \mathbb{B},0,\mathbb{B})$ for small $t$, where $\mathbb{B}$ is the unit ball in ${\mathbb{R}}^n$, $\xi \in \partial \mathbb{B}$ and $B(\xi ,t)$ denotes the euclidean ball centered at $\xi$ of radius $t$. As a consequence and, under suitable geometrical restrictions on a domain $\Omega \subset {\mathbb{R}}^n$, we identify the optimal decay exponent of ${\omega }_p(B(\xi ,t)\cap \partial \Omega ,a,\Omega )$ for small $t$, where $a\in \Omega$ and $\xi \in \partial \Omega$. When restricted to the range $1<p<n$ we provide new estimates complementing previous results of different authors in the unit ball, the unit disc or the upper half-space.