Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian

In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian ${\mathrm{\Delta }}_d$ in discrete Hardy spaces ${\mathcal{H}}^p(\mathbb{Z})$. We prove that the maximal operator and the Littlewood-Paley $g$ function defined by the semigroup generated by ${\mathrm{\Delta }}_d$ are bounded from ${\mathcal{H}}^p(\mathbb{Z})$ into ${\mathrm{\ell }}^p(\mathbb{Z})$, $0<p\leq 1$\mathrm{.}Also,weestablishthatevery$\Delta_d$-spectral multiplier of Laplace transform type is a bounded operator from ${\mathcal{H}}^p(\mathbb{Z})$ into itself, for every $0<p\le 1$.