  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.
