Harmonic analysis for a multidimensional discrete Laplacian

In this paper we analyze some classical operators in harmonic analysis associated to the multidimensional discrete Laplacian $${\mathrm{\Delta }}_{N}f(\mathbf{n})=\sum _{i=1}^N(f(\mathbf{n}+{\mathbf{e}}_i)-2f(\mathbf{n})+f(\mathbf{n}-{\mathbf{e}}_i)),\mathbf{n}\in {\mathbb{Z}}^N\mathrm{.}$$ We deal with the heat and Poisson semigroups, the fractional integrals, the Riesz transforms, the fractional powers of the Laplacian, and the $g_k$-square functions.