Let L = $\Delta$ + V be a Schr{\"o}dinger operator with a non-negative potential V on a complete Riemannian manifold M. We prove that the vertical Littlewwod-Paley-Stein functional associated with L is bounded on L p (M) if and only if the set { $\sqrt$ t $\nabla$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, $\infty$) $\rightarrow$ C, we define H((f k)) := ( \sum\_k \int\_0^\infty |$\nabla$m k (tL)f \_k |^2 dt )^1/2 + (\sum\_k \int\_0^\infty | $\sqrt$ V m k (tL)f \_k | 2 dt )^1/2. Under fairly reasonable assumptions on M we prove for certain functions m k the boundedness of H on L p (M) in the sense \| H((f \_k)) \|\_p $\le$ C \| (\sum\_k |f \_k | 2 )^1/2 \|\_p for some constant C independent of (f \_k) \_k. A lower estimate is also proved on the dual space L p. We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper. read more

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Analysis of PDEs
Functional Analysis