Hölder regularity of the solution to the complex Monge-Ampère equation with L p density

On a smooth domain ⊂⊂ Cn,we consider the Dirichlet problem for the complex Monge-Ampère equation ((ddcu)n = f dV, u|b ≡ φ). We state the Hölder regularity of the solution u when the boundary value φ is Hölder continuous and the density f is only L p, p > 1. Note that in former literature (Guedj-Kolodziej-Zeriahi) the weakness of the assumption f ∈ L p was balanced by taking φ ∈ C1,1 (in addition to assuming strongly pseudoconvex).