Hölder continuous solutions to Monge–Ampère equations

Let (X, ) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with Lᵖ 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, ) of the complex Monge-Ampère operator acting on -pluri-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening Ko odziej's result that measures with Lᵖ -density belong to MAH (X, ) and proving that MAHX, ) has the “ Lᵖ -property”, p>1. We also describe accurately the symmetric measures it contains.