Pseudo-differential operators on ℤⁿ with applications to discrete fractional integral operators

In this manuscript we provide necessary and sufficient conditions for the weak( 1, p) boundedness, 1 < p < 8, of discrete Fourier multipliers ( Fourier multipliers on Zn). 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