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


Let (X,ω)(X,\omega) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on XX with LpL^p right hand side, \hbox{p>1p>1}. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range \MAH(X,ω)\MAH(X,\omega) of the complex Monge-Ampère operator acting on ω\omega-pluri\-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening\break Ko\l odziej's result that measures with LpL^p-density belong to \MAH(X,ω)\MAH(X,\omega) and proving that \MAH(X,ω)\MAH(X,\omega) has the "LpL^p-property'', p>1p>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