Hölder continuous solutions to Monge–Ampère equations

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 -pluri-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening Ko odziej's result that measures with L^p -density belong to MAH(X,ω) and proving that MAHX,ω) has the “ L^p -property”, p>1 . We also describe accurately the symmetric measures it contains.