Hölder continuity of solutions to the Monge - Ampère equations on compact Kähler manifolds

X ω = 1. An upper semicontinuous function φ : X → [−∞,+∞) is called ω-plurisubharmonic (ω-psh) if φ ∈ L(X) and ωφ := ω + dd φ ≥ 0. By PSH(X,ω) (resp. PSH(X,ω)) we denote the set of ω-psh (resp. negative ω-psh) functions on X . The complex Monge-Ampère equation ω u = fω n was solved for smooth positive f in the fundamental work of S. T. Yau (see [Yau]). Later S. Kolodziej showed that there exists a continuous solution if f ∈ L(ω), f ≥ 0, p > 1 (see [Ko2]). Recently in [Ko5] he proved that this solution is Hölder continuous in this case (see also [EGZ] for the case X = CP). In Corollary 1.2 in [DNS] the authors have shown that the measure ω u is moderate if u is Hölder continuous. The main result is the following theorem which give a partial answer to the converse problem: