Multidimensional Tauberian theorems for Banach-space valued generalized functions

Multidimensional Tauberian theorems for the standard averages of tempered Banach-space valued distributions are stated and proved. These results enable one to determine from the asymptotic behaviour of the averages the asymptotic behaviour of the generalized function itself. The role of the asymptotic scale in these results is performed by the class of regularly varying functions. Special attention is paid to averaging kernels such that several of their moments or linear combinations of moments vanish. Important in these results is the structure of the zero set of the Fourier transformations of the kernels in question.
The results so established are applied to the study of the asymptotic properties of solutions of the Cauchy problem for the heat equation in the class of tempered distributions, to the problem of the diffusion of a many-component gas, and to the problem of the absence of the phenomenon of compensation of singularities for holomorphic functions in tube domains over acute cones.