Characterization of Monge-Ampère measures with Hölder continuous potentials

We show that the complex Monge-Ampère equation on a compact Kähler manifold (X,ω) of dimension n admits a Hölder continuous ω-psh solution if and only if its right-hand side is a positive measure with Hölder continuous super-potential. This property is true in particular when the measure has locally Hölder continuous potentials or when it belongs to the Sobolev space W2n/p−2+ϵ,p(X) or to the Besov space B∞,∞ϵ−2(X) for some ϵ>0 and p>1.