Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian

Abstract

In this paper we study the behavior of some harmonic analysis operators associated with the discrete Laplacian ${\Delta }_d$ 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 ${\Delta }_d$ are bounded from ${\mathcal{H}}^p(\mathbb{Z})$ into ${\ell }^p(\mathbb{Z})$, $0<p\le 1$. Also, we establish that every ${\Delta }_d$-spectral multiplier of Laplace transform type is a bounded operator from ${\mathcal{H}}^p(\mathbb{Z})$ into itself, for every $0<p\le 1$.