Subharmonic envelopes for functions on domains

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 $h\not\equiv -\infty$, “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.