Riesz transforms associated to Schrödinger operators with negative potentials

The goal of this paper is to study the Riesz transforms  ∇ A^-1/2 where A is the Schrödinger operator - ∆ - V, V ≥ 0, under different conditions on the potential V . We prove that if V i strongly subcritical,  ∇ A^-1/2 is bounded on L^p(R^N), N ≥ 3, for all p є (p´_{0 ; 2]} where p´_{0 is} the dual exponent of p₀ where 2 < 2N/N-2 < p₀ < ∞; and we give a counterexample to the boundedness on L^p (R^N) for p є (1;p´₀) ∪ (p_0*;∞) where _{p0* :=poN/N+po  is} the reverse Sobolev exponent of p₀. If the potential is strongly subcritical in the Kato subclass K∞/N, then ∇ A^-1/2 is bounded on L^p (R^N) for all p є (1;2], moreover if it is in LN/w/2 (R^N) then ∇ A^-1/2 is bounded on L^p (R^N) for all p є (1;N). We prove also boundedness of V^1/2 A^-1/2 with the same conditions on the same spaces. Finally we study these operators on manifolds. We prove that our results hold on a class of Riemannian manifolds.