A perturbation result for the Riesz transform

Let (M, g) be a Riemannian manifold. The Riesz transform problem, namely giving conditions on p and on the manifold such that the operator d∆−1/2 – the so-called Riesz transform – is bounded on L, has recently undergone certain progresses. A pioneering result which goes back to 1985 is a theorem of D. Bakry [2] which asserts that if the Ricci curvature of M is non-negative, then the Riesz transform on M is bounded on L for every 1 < p < ∞. However, it is only recently that some progresses have been made to understand the behaviour of the Riesz transform if some amount of negative Ricci curvature is allowed. A general question is the following: