p-Adic analysis in the Lizorkin type spaces: fractional operators, pseudo-differential equations and Tauberian theorems

TL;DR

This paper introduces p-adic Lizorkin spaces, studies fractional and pseudo-differential operators within them, and proves related Tauberian theorems, advancing the mathematical understanding of p-adic analysis and operator equations.

Contribution

It develops the theory of p-adic Lizorkin spaces, analyzes fractional and pseudo-differential operators on these spaces, and establishes new Tauberian theorems in this context.

Findings

Invariance of Lizorkin spaces under fractional operators

Construction of solutions to p-adic pseudo-differential equations

Proof of p-adic Tauberian theorems related to fractional operators

Abstract

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.