Hardy spaces, Besov spaces and Triebel--Lizorkin spaces associated with a discrete Laplacian and applications

Abstract

Consider the discrete Laplacian ${\Delta }_d$ defined on the set of integers $\mathbb{Z}$ by \[ Δ_d f(n) = -f(n+1) + 2f(n) -f(n-1), \ \ \ \ n\in \mathbb Z, \] where $f$ is a function defined on $\mathbb{Z}$. In this paper, we define Hardy spaces, Besov spaces and Triebel--Lizorkin spaces associated with ${\Delta }_d$ and then show that these function spaces coincide with the classical function spaces defined on $\mathbb{Z}$. As applications, we prove the boundedness of the spectral multipliers and the Riesz transforms associated with ${\Delta }_d$ on these function spaces.

49 pages. Accepted to publish in Annali Scuola Normale Speriore - Classe de Scienze in 2023