Pseudo-differential operators on $\mathbb{Z}^n$ with applications to discrete fractional integral operators

TL;DR

This paper establishes criteria for the boundedness of discrete Fourier multipliers on 1, p) spaces and applies these results to discrete fractional integral operators, also proving discrete analogs of classical theorems.

Contribution

It provides necessary and sufficient conditions for weak(1,p) boundedness of discrete Fourier multipliers and introduces discrete versions of key theorems like Calderf3n-Vaillancourt and Gohberg Lemma.

Findings

Criteria for weak(1,p) boundedness of discrete Fourier multipliers.

Application of these criteria to discrete fractional integral operators.

Discrete Calderf3n-Vaillancourt and Gohberg Lemma proved.

Abstract

In this manuscript we provide necessary and sufficient conditions for the $\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of discrete Fourier multipliers (Fourier multipliers on $\mathbb{Z}^n$). Our main goal is to apply the results obtained to discrete fractional integral operators. Discrete versions of the Calder\'on-Vaillancourt Theorem and the Gohberg Lemma also are proved.