Lp gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds

In this paper, we investigate the validity of first and second order Lp estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present Lp estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain Lp estimates for the second order Riesz transform (or, equivalently, the validity of Lp Calderón–Zygmund inequalities) on the whole scale 1 < p < C1 by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1 < p ≤ 2, analogous Lp bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.