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Published online: 2014-04-02

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly^[1], Sławomir Dinew^[2], Vincent Guedj^[3], Pham Hoang Hiep^[4], Sławomir Kołodziej^[5] and Ahmed Zeriahi^[6]

(1) Université Grenoble I, Saint-Martin-d'Hères, France
(2) Jagellonian University, Krakow, Poland
(3) Université Paul Sabatier, Toulouse, France
(4) Hanoi National University of Education, Vietnam
(5) Jagellonian University, Krakow, Poland
(6) Université Paul Sabatier, Toulouse, France

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.

Keywords: Monge–Ampère operator, Kähler manifold, pluripotential theory, Hölder continuity

Demailly Jean-Pierre, Dinew Sławomir, Guedj Vincent, Hiep Pham Hoang, Kołodziej Sławomir, Zeriahi Ahmed: Hölder continuous solutions to Monge–Ampère equations. J. Eur. Math. Soc. 16 (2014), 619-647. doi: 10.4171/JEMS/442