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We study the Littlewood-Paley-Stein functions associated with Hodge-de Rhamand Schr{\"o}dinger operators on Riemannian manifolds. Under conditions on theRicci curvature we prove their boundedness on L p for p in some interval (p 1 ,2] and make a link to the Riesz Transform. An important fact is that we do notmake assumptions of doubling measure or estimates on the heat kernel in thiscase. For p > 2 we give a criterion to obtain the boundedness of the verticalLittlewood-Paley-Stein function associated with Schr{\"o}dinger operators on Lp .

Littlewood-Paley-Stein functions for Hodge-de Rham and Schr{\"o}dinger operators