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

Abstract

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 \(\MAH(X,\omega)\) 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 \(\MAH(X,\omega)\) and proving that \(\MAH(X,\omega)\) has the "\(L^p\)-property", \(p>1\). We also describe accurately the symmetric measures it contains.