Strong- and weak-type estimate for Littlewood–Paley operators associated with Laplace–Bessel differential operator

In this work, we consider the generalized shift operator associated with the Laplace–Bessel differential operator and define the relevant Littlewood–Paley function operators g and Sγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\gamma }$$\end{document}. We then prove the weak (1, 1) and strong (p, p) boundedness of these operators in vector-valued Calderón–Zygmund approach in some appropriate homogeneous space.