Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is ${}^{1,1}$ and the right hand side has a density in $L^p\left(\Omega \right)$ for some p > 1, and prove the Hölder continuity of the solution.