A theorem of Green’s type for functions of two complex variables

Abstract

1. Domains with a distinguished boundary surface. Functions of the extended class. The real and imaginary parts of a function of n complex variables are harmonic functions of 2n real variables. Using Green's theorem one can transform Dirichlet integrals over a 2ndimensional domain, say D, into integrals over the (2n-1)-dimensional boundary of D. Using this procedure (and exploiting some further properties of harmonic functions) one obtains generalizations of the Cauchy formula (see Bergman [1],2 Bochner-Martin [5], Martinelli [8]), certain generalizations of Nevanlinna's theory of meromorphic functions, etc. Recently, applying Green's transformations, Garabedian obtained an important generalization to the case of several variables of formulas connecting Green's and Neumann's functions with the kernel function [6]. Further, Garabedian and Spencer [7] showed that these methods can be extended to the theory of tensors defined on certain Kiihler manifolds. On the other hand, the real and imaginary parts of functions of n complex variables, n > 1, represent a very special class of harmonic functions and by the above theorems the possibilities of using theorems of Green's type are in no way exhausted. One can apply, in this special case, reduction of Green's type of k-dimensional integrals, for k>n, repeatedly. In the present paper we shall discuss an example of such a procedure in the case of functions of two complex variables. In order to explain our approach it will be useful to discuss more in detail the geometrical situation which we meet in this theory. For the sake of simplicity, we shall limit ourselves to the case n =2 and consider only a very special type of domains.3 As has been indicated in [3; 2, chap. I], the geometry of the space of functions of two complex variables differs in many respects from the geometry of four-dimensional Euclidean space. One of the reasons for this situation is the fact that analytic surfaces and segments of these surfaces take over to a certain extent the role of points in Euclidean geometry. For in-