On $p$-Harmonic Measures in Half Spaces

TL;DR

This paper establishes bounds for p-harmonic measures in half spaces, showing they scale with the boundary ball radius raised to a specific exponent, and provides explicit estimates for this exponent.

Contribution

It proves the existence of a positive exponent controlling p-harmonic measure scaling in half spaces and offers explicit estimates for this exponent.

Findings

p-harmonic measure scales as δ^α(p,N) for boundary balls of radius δ

Explicit bounds are provided for the exponent α(p,N)

The results hold for all 1<p<∞ and dimensions N≥2

Abstract

For all $1<p<\infty$ and $N\ge 2$ we prove that there is a constant $\alpha(p,N)>0$ such that the $p$-harmonic measure in $\R^N_+$ of a ball of radius $0 < \delta \leq 1$ in $\R^{N-1}$ is bounded above and below by a constant times $\delta ^{\alpha (p.N)}$. We provide explicit estimates for the exponent $\alpha(p,N)$