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

Consider the discrete Laplacian ${\mathrm{\Delta }}_d$ defined on the set of integers $\mathbb{Z}$ by $${\mathrm{\Delta }}_{d}f(n)=-f(n+1)+2f(n)-f(n-1),n\in \mathbb{Z},$$ where $f$ is a function defined on $\mathbb{Z}$. In this paper, we define Hardy spaces, Besov spaces and Triebel–Lizorkin spaces associated with ${\mathrm{\Delta }}_d$ 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 ${\mathrm{\Delta }}_d$ on these function spaces.