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

In this manuscript we provide necessary and sufficient conditions for the $\text{weak}(1,p)$ boundedness, $1<p<\mathrm{\infty },$ 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.