On Pseudo-Difference Operators on the Lattice \(\mathbb Z^n\)

This article is an extract of the paper (Botchway et al., J Funct Anal 278:11–108473, 2020). Here, we introduce some of the properties related to the pseudo-differential operators on the lattice \(\mathbb {Z}^n\) . The frequency space in this case is simply the n-dimensional torus \(\mathbb {T}^n\) . We present the composition formulae and the conditions for the \(\ell ^p\) -boundedness and \(\ell ^p\) -compactness of these operators. The results are applied to estimate the solutions of the difference equations on the lattice \(\mathbb {Z}^n\) .