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 \(q\in [2,\infty )\), 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 \(q\in (1,2]\).