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where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) "= 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic functions in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of functions of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of "inner functions" on the domain O. If h is a bounded analytic function

The dirichlet problem for a complex Monge-Amp�re equation