Journal of the European Mathematical Society
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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