Tauberian Theorems for Distributions in the Lizorkin Spaces

1.1. Tauberian type theorems. As is well known, in mathematical physics there are so-called Tauberian theorems which have many applications. Tauberian theorems are usually assumed to connect asymptotical behavior of a function (distribution) at zero with asymptotical behavior of its Fourier (Laplace or other integral transforms) at infinity. The inverse theorems are usually called “Abelian” [4], [5], [10], [19] (see also the references cited therein). Multidimensional Tauberian theorems for distributions (as a rule, from the space S (R)) have been treated by V. S. Vladimirov, Yu. N. Drozzinov, B. I. Zavyalov in the fundamental book [19]. It remains to note that Tauberian theorems have many applications and are intensively used in mathematical physics (see [10], [19]). Some types of Tauberian theorems are connected with the fractional operator. In [19], as a rule, theorems of this type were proved for distributions with supports in the cone in R, n 6= 1 or in semiaxis for n = 1. This is related to the fact that such distributions constitute a convolution algebra. In this case a kernel of the fractional operator is the distribution with a support in a cone in R, n 6= 1 or in semiaxis for n = 1 [19, §2.8.]. Thus in this case, in general, the convolution of a distribution and a kernel of the fractional operator is not well defined in the sense of the space S (R)). Moreover, in general, the Schwartian test function space S(Rn) is not invariant under the fractional operators. In view of this fact Tauberian type theorems for distributions without restriction on the support have not been considered in [19].