Open AccessHölder continuous solutions to Monge–Ampère equations
Author(s) -
Jean-Pierre Demailly,
Sławomir Dinew,
Vincent Guedj,
Phạm Hoàng Hiệp,
Sławomir Kołodziej,
Ahmed Zériahi
Publication year - 2014
Publication title -
journal of the european mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.549
H-Index - 64
eISSN - 1435-9863
pISSN - 1435-9855
DOI - 10.4171/jems/442
Subject(s) - mathematics , omega , cohomology , hölder condition , operator (biology) , pure mathematics , combinatorics , kähler manifold , subharmonic , mathematical analysis , physics , nonlinear system , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
LaTeX, 23 pagesInternational audienceLet $(X,\omega)$ 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 $\MAH(X,\omega)$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder 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