On $p$-Harmonic Measures in Half Spaces

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