Pseudo-Differential Operators on $${\mathbb {Z}}^n$$ Z n with Applications to Discrete Fractional Integral Operators

Abstract

In this manuscript we provide necessary and sufficient conditions for the $${\mathrm{weak}}(1,p)$$ boundedness, $$1< p<\infty ,$$ of discrete Fourier multipliers (Fourier multipliers on $${\mathbb {Z}}^n$$ ). Our main goal was to apply the results obtained to discrete fractional integral operators. Discrete versions of the Calderon–Vaillancourt theorem and the Gohberg lemma also are proved.