We prove a Liouville type theorem for entire maximal m-subharmonic functions in C n with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the program, initiated in [HMW], of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some L p , with a sharp bound on p.

We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain Ω⊂ℂn. We show, under a mild technical assumption, that the unique solution u to this problem is Hölder continuous if the boundary data ϕ is Hölder continuous and the density f belongs to Lp(Ω) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.

We prove some L ∞ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of C n and on compact Kähler manifolds. We also show optimal L p integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of B locki. Finally we obtain a local regularity result for W 2,p solutions of the real and complex Hessian equations under suitable regularity assumptions on the right hand side. In the real case the method of this proof improves a result of Urbas.

Abstract. Let (X, ω) 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 MAH(X, ω) of the complex Monge-Ampère operator acting on ω-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Ko lodziej's result that measures with L p -density belong to MAH(X, ω) and proving that MAH(X, ω) has the "L p -property", p > 1. We also describe accurately the symmetric measures it contains.