Littlewood–Paley–Stein functions for non-local Schrödinger operators

Abstract

Motivated by recent works on vertical Littlewood–Paley–Stein functions for symmetric non-local Dirichlet forms and local Schrodinger type operators respectively, we study Littlewood–Paley–Stein functions for non-local Schrodinger type operators with non-negative potential in metric measure spaces. We prove the $$L^p$$ -boundedness of Littlewood–Paley–Stein functions for all $$p\in (1,2]$$ , and further find that the $$L^p$$ -boundedness for some $$p\in (2,\infty )$$ implies the vanishment of a particular class of potentials.