  In this paper the p -adic Lizorkin spaces of test functions and distributions
are introduced, and multidimensional Vladimirov's and Taibleson's fractional
operators are studied on these spaces. Since the p -adic Lizorkin spaces are
invariant under the Vladimirov and Taibleson operators, they can play a key
role in considerations related to fractional operator problems. A class of p
-adic pseudo-differential operators in the Lizorkin spaces is also introduced
and solutions of pseudo-differential equations are constructed. p -Adic
multidimensional Tauberian theorems connected with fractional operators and
pseudo-differential operators for the Lizorkin distributions are also proved.
