Characterization of Monge-Ampère measures with Holder continuous potentials

摘要

We show that the complex Monge-Ampère equation on a compact K?hler manifold (X, ω) of dimension n admits a Holder continuous ω-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continuous potentials or when it belongs to the Sobolev space W~(2n/p-2+?,p) (X) or to the Besov space B(∞,∞)~(?-2)_(X) for some ? > 0 and p > 1.