  We study H\"older continuity of solutions to the Dirichlet problem for
measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz
measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded
strongly hyperconvex Lipschitz domain and the boundary data belongs to $
C^{0,\alpha}(\partial \Omega)$, $ 0<\alpha \leq 1$.
