Hölder continuous solutions to Monge-Ampère equations

Abstract

Let $(X,\omega )$ 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 $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej'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.

LaTeX, 23 pages

References in corpus (3)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Exponential estimates for plurisubharmonic functions and stochastic dynamics
• Intrinsic capacities on compact Kähler manifolds

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• The Continuous Subsolution Problem for Complex Hessian Equations