Hölder continuous solutions to Monge–Ampère equations
Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi
Source 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, p>1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range \operatorname{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 \operatorname{MAH}(X,\omega) and proving that \operatorname{MAH}X,\omega) has the “ L^p -property”, p>1 . We also describe accurately the symmetric measures it contains.
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