Riesz transforms of Schrödinger operators on manifolds

We consider Schrödinger operators A = − + V on Lp(M) where M is a complete Riemannian manifold of homogeneous type and V = V + − V − is a signed potential. We study boundedness of Riesz transform type operators ∇A −1 2 and |V |12 A −12 on Lp(M). When V − is strongly subcritical with constant α ∈ (0, 1) we prove that such operators are bounded on Lp(M) for p ∈ (p 0, 2] where p 0 = 1 if N ≤ 2, and p 0 = ( 2N (N−2)(1− √ 1−α) ) ∈ (1, 2) if N > 2. We also study the case p >2. With additional conditions on V and M we obtain boundedness of ∇A −1/2 and |V |1/2A −1/2 on Lp(M) for p ∈ (1, inf(q1,N)) where q1 is such that ∇(− ) −1 2 is bounded on Lr(M) for r ∈ [2, q1)