$L^p$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds

Abstract

In this paper, we investigate the validity of first and second order $L^p$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^p$ 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 $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^p$ Calderón–Zygmund inequalities) on the whole scale $1<p<+\infty$ 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 $1<p\le 2$, analogous $L^p$ bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.