$L^p$ 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 even higher order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. In the same range of values of $p$, for manifolds with lower Ricci bounds and positive bottom of the spectrum, we show that the $L^p$ norm of the Laplacian controls the whole $W^{2,p}$-norm on compactly supported functions.

22 pages, comments welcome! arXiv:2204.04002 was merged into this paper