We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in C n . 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 C 1,1 and the right hand side has a density in L p (Ω) for some p > 1 and prove the Hölder continuity of the solution.
We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order m (1 ≤ m ≤ n) with a continuous right hand side and a continuous boundary data in a bounded strongly m-pseudoconvex domain Ω ⋐ C n . Moreover when the right hand side is in L p (Ω), for some p > n/m and the boundary value function is C 1,1 we prove that the solution is Hölder continuous.
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