Littlewood--Paley--Stein Square Functions for the Fractional Discrete Laplacian on $\mathbb{Z}$
Huaiqian Li, Liying Mu
TL;DR
The paper analyzes vertical Littlewood--Paley--Stein square functions for the fractional discrete Laplacian $L = (-\Delta)^s$ on the non-locally finite lattice $\mathbb{Z}$, establishing $l^q$ boundedness for different $q$-ranges: $\tilde{G}$ and $H$ on $l^q$ for $q\in(1,2]$, and $G$, $\tilde{G}$, $H$ on $l^q$ for $q\in[2,\infty)$. A counterexample shows $G$ need not be bounded when $q\in(1,2)$, motivating the modified square function, while the results extend to nonlocal Schrödinger operators $L_U = (-\Delta)^s + U$ with corresponding $l^q$ bounds for $q\in(1,2]$. The proofs combine probabilistic martingale methods for $q\ge 2$ with a pseudo-gradient framework and semigroup maximal function techniques for $q\le 2$, together with a careful comparison between gradient operators and the pseudo-gradient. Overall, the work advances harmonic analysis on nonlocally finite discrete structures and provides tools for nonlocal discrete PDEs with potential extensions to Schrödinger-type perturbations.
Abstract
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 $q\in (1,2]$.