Complex Monge-ampère of a Maximum

Pluri-subharmonic (psh) functions play a primary role in pluri-potential theory. They are closely related to the operator dd c = 2i∂ ¯ ∂ (with notation d = ∂ + ¯ ∂ and d c = i(¯ ∂ − ∂)), which serves as a generalization of the Laplacian from C to C dim for dim > 1. If u is smooth of class C 2 , then for 1 ≤ n ≤ dim, the coefficients of the exterior power (dd c u) n are given by the n×n sub-determinants of the matrix (∂ 2 u/∂z i ∂ ¯ z j). The top exterior power corresponds to n = dim, and in this case we have the determinant of the full matrix, which gives the complex Monge-Ampère operator. The extension of the (nonlinear) operator (dd c) n to non-smooth functions has been studied by several authors (see, for instance, [B2]). Here it will suffice to define (dd c) n on psh functions which are continuous.