Littlewood–Paley–Stein functionals: an ℛ-boundedness approach

Let L=Δ+V be a Schrödinger operator with a non-negative potential V on a complete Riemannian manifold M. We prove that the vertical Littlewood-Paley-Stein functional associated with L is bounded on L p (M) if and only if the set {t∇e -tL ,t>0} is ℛ-bounded on L p (M). We also introduce and study more general functionals. For a sequence of functions m k :[0,∞)→ℂ, we define