Littlewood--Paley--Stein Square Functions for the Fractional Discrete Laplacian on $\mathbb{Z}$

We investigate the boundedness of ``vertical’’ Littlewood–Paley–Stein square functions for the nonlocal fractional discrete Laplacian on the lattice $\mathbb{Z}$, where the underlying graphs are not locally finite. When $q\in [2,\mathrm{\infty })$, we prove the $l^q$ boundedness of the square function by exploring the corresponding Markov jump process and applying the martingale inequality. When $q\in (1,2]$, we consider a modified version of the square function and prove its $l^q$ boundedness through a careful in on the generalized carré du champ operator. A counterexample is constructed to show that it is necessary to consider the modified version. Moreover, we extend the study to a class of nonlocal Schrödinger operators for $q\in (1,2]$.