H\"older continuous solutions to Monge-Amp\`ere equations

Let $(X,\omega )$ be a compact K"ahler manifold. We obtain uniform H"older regularity for solutions to the complex Monge-Amp`ere 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 $\text{MAH}(X,\omega )$ of the complex Monge-Amp`ere operator acting on $\omega$-plurisubharmonic H"older continuous functions. We show that this set is convex, by sharpening Ko{\l}odziej’s result that measures with $L^p$-density belong to $\text{MAH}(X,\omega )$ and proving that $\text{MAH}(X,\omega )$ has the “$L^p$-property”, $p>1$. We also describe accurately the symmetric measures it contains.