Tauberian theorems for generalized functions with values in Banach spaces

We state and prove Tauberian theorems of a new type. In these theorems we give sufficient conditions under which the values of a generalized function (distribution) that are assumed to lie in a locally convex topological space actually belong to some narrower (Banach) space. These conditions are stated in terms of “general class estimates” for the standard average of this generalized function with a fixed kernel belonging to a space of test functions.
The applications of these theorems are based, in particular, on the fact that asymptotical (and some other) properties of the generalized functions under investigation can be described in terms of membership of certain Banach spaces. We apply these theorems to the study of asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of generalized functions of small growth (tempered distributions), and to the study of Banach spaces of Besov–Nikol'skii type.