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

Discrete Hardy Spaces and Heat Semigroup Associated with the Discrete Laplacian