# Hölder continuous solutions to Monge–Ampère equations

### Jean-Pierre Demailly

Université Grenoble I, Saint-Martin-d'Hères, France### Vincent Guedj

Université Paul Sabatier, Toulouse, France### Sławomir Dinew

Jagellonian University, Krakow, Poland### Pham Hoang Hiep

Hanoi National University of Education, Vietnam### Sławomir Kołodziej

Jagellonian University, Krakow, Poland### Ahmed Zeriahi

Université Paul Sabatier, Toulouse, France

## 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, \hbox{$p>1$}. 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-Ampère operator acting on $\omega$-pluri\-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening\break 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.

## Cite this article

Jean-Pierre Demailly, Vincent Guedj, Sławomir Dinew, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi, Hölder continuous solutions to Monge–Ampère equations. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 619–647

DOI 10.4171/JEMS/442