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

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