  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 $\alpha(p,N)$
