Subharmonic envelopes for functions on domains

Abstract

One of the most common problems in various fields of real and complex analysis is the questions of the existence and construction for a given function of an envelope from below or from above of a function from a special class H. We consider a case when H is the convex cone of all subharmonic functions on the domain D of a finite-dimensional Euclidean space over the field of real numbers. For a pair of subharmonic functions u and M from this convex cone H, dual necessary and sufficient conditions are established under which there is a subharmonic function h ̸≡ −∞, “dampening the growth” of the function u in the sense that the values of the sum of u + h at each point of D is not greater than the value of the function M at the same point. These results are supposed to be applied in the future to questions of non-triviality of weight classes of holomorphic functions, to the description of zero sets and uniqueness sets for such classes, to approximation problems of the function theory, etc.