We consider Schrödinger operators A = −∆ + V on L p (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 | 1 2 A − 1 2 on L p (M). When V − is strongly subcritical with constant α ∈ (0, 1) we prove that such operators are bounded on L p (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 | 1/2 A −1/2 on L p (M) for p ∈ (1, inf(q 1 , N)) where q 1 is such that ∇(−∆) − 1 2 is bounded on L r (M) for r ∈ [2, q 1).

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 is strongly sub-0 is the dual exponent of p 0 where 2 < 2N N −2 < p 0 < ∞; and we give a counterexample to the boundedness onis the reverse Sobolev exponent of p 0 . If the potential is strongly subcritical in the Kato subclassWe 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.

We prove that the mild solution of the stochastic evolution equation dX(t) = AX(t) dt + dW (t) on a Banach space E has a continuous modification if the associated Ornstein-Uhlenbeck semigroup is analytic on L 2 with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein-Uhlenbeck semigroups on domains O ⊆ E to the non-symmetric case. Denoting the generator of the Ornstein-Uhlenbeck semigroup by L O , we obtain sufficient conditions in order that the domain of √ −L O be a first order Sobolev space. 2000 Mathematics Subject Classification. Primary: 35R15 Secondary: 35J25, 42B25, 46E35, 47D05, 60H07.

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