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.

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In this paper we study relations between the weighted energy class E χ introduced by S. Benelkourchi, V. Guedj and A. Zeriahi recently with Cegrell's classes E and N . Next we establish a generalized comparison principle for the operator M χ . As an application, we prove a version of existence of solutions of Monge-Ampère type equations in the class E χ (H, ).

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In this article we will first prove a result about the convergence in capacity. Next we will obtain a general decomposition theorem for complex Monge-Ampère measures which will be used to prove a comparison principle for the complex Monge-Ampère operator.

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