On $p$-Harmonic Measures in Half Spaces

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 $\alpha(p,N)$