Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian

TL;DR

This paper investigates harmonic analysis operators related to the discrete Laplacian on Hardy spaces over integers, establishing boundedness of maximal functions, Littlewood-Paley functions, and spectral multipliers in this setting.

Contribution

It proves boundedness of key harmonic analysis operators associated with the discrete Laplacian on Hardy spaces, extending classical results to the discrete setting.

Findings

Maximal operator is bounded from ^p( Z) to ^p( Z) for 0<p

Littlewood-Paley g-function is bounded from ^p( Z) to ^p( Z) for 0<p

Spectral multipliers of Laplace transform type are bounded on ^p( Z) for 0<p

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 $\ell^p(\mathbb Z)$, $0<p\leq 1$. 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 $0<p\leq 1$.