Regularity of solutions to the Dirichlet problem for Monge-Amp\`ere equations

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 $C^{0,\alpha}(\partial \Omega)$, $0<\alpha \leq 1$.