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We prove the almost C regularity of the degenerate complex Monge– Ampère equation in a special case. If u is a smooth plurisubharmonic function, the Monge–Ampère operator is given by (ddcu)n = det(upq̄)dL, where upq̄ = ∂u/∂zp∂z̄q and L is a 2n-dimensional Lebesgue measure. For an arbitrary continuous plurisubharmonic function u one can define (ddcu)n to be a regular Borel measure. Let Ω be a strictly pseudoconvex domain in Cn (throughout the note we always assume n ≥ 2). Then for any nonnegative f which is continuous in Ω, and φ continuous on ∂Ω, the Dirichlet problem (1)  u ∈ PSH(Ω) ∩ C(Ω̄), (ddcu)n = fdL in Ω, u = φ on ∂Ω, has a unique continuous plurisubharmonic solution u (see [B-T]). We say that Ω is a strictly pseudoconvex domain with a C2,1 boundary if Ω = {z ∈ Cn : ρ(z) < 0} where ρ is a strictly plurisubharmonic function of class C2,1 on a neighbourhood of Ω̄ such that ∇ρ 6= 0 on ∂Ω. We say that a function u is almost C1,1 if the function ∆u is bounded. Now we can formulate the main theorem. Theorem 1. Let Ω be a strictly pseudoconvex domain with a C2,1 boundary, and f be a nonnegative function on Ω̄ such that f1/(n−1) ∈ C1,1(Ω̄) and f1/n ∈ C0,1(Ω̄). Then the Monge–Ampère equation (1) with the boundary condition φ ≡ 0 has a unique almost C1,1 solution. Remark 2. There are similar theorems for the degenerate real Monge– Ampère equation in [G] and [G-T-W] (with nonzero φ ∈ C3,1(Ω̄)). In the 2010 Mathematics Subject Classification: 32W20, 35J70.

A note on the regularity of the degenerate complex Monge–Ampère equation