Dual construction and existence of (pluri)subharmonic minorant

Abstract

We study the existence and construction of subharmonic or plurisubharmonic function enveloping from below a function on a subset in finite--dimensional real or complex space. These problems naturally arise in theories of uniform algebras, potential and complex potential, which was reflected in works by D.A.Edwards, T.V.Gamelin, E.A.Poletsky, S.Bu and W.Schachermayer, B.J.Cole and T.J.Ransford, F.Larusson and S.Sigurdsson and many others. In works in 1990s and recently we showed that these problems play a key role in studying nontriviality of weighted spaces of holomorphic functions, in description of zero sets and subsets of functions from such spaces, in representations of meromorphic functions as a quotient of holomorphic functions with growth restrictions, in studying the approximation by exponential systems in functional spaces, etc. The main results of the paper on existence of subharmonic or plurisubharmonic function-minorant are derived from our general theoretical functional scheme, which allows us to provide a dual definition of the lower envelope with respect to a convex cone in the projective limit of vector lattices. We develop this scheme during last years and it is based on an abstract form of balayage. The ideology of the abstract balayage goes back to H.Poincare and M.V.Keldysh in the framework of balayage of measures and subharmonic functions in the potential theory. It is widely used in the probability theory, for instance, in the known monograph by P.Meyer, and it is also reflected, often implicitly, in monographs by G.P.Akilov, S.S.Kutateladze, A.M.Rubinov and others related with the theory of ordered vector spaces and lattices. Our paper is adapted for convex subcones of the cone of all subharmonic or plurisubharmonic functions. This allows us to obtain new criterion for existence of a subharmonic or plurisubharmonic minorant for functions on a domain.