$L^{p}$ gradient estimates and Calder\'on--Zygmund inequalities under Ricci lower bounds

TL;DR

This paper studies $L^{p}$ estimates for solutions of the Poisson equation on manifolds with Ricci curvature bounds, establishing new inequalities and counterexamples that depend on geometric conditions.

Contribution

It provides new $L^{p}$ gradient and Calderón--Zygmund estimates under Ricci lower bounds and constructs counterexamples showing their limitations without curvature restrictions.

Findings

Established $L^{p}$ gradient estimates under Ricci lower bounds.

Proved $L^{p}$ Calderón--Zygmund inequalities assuming positive injectivity radius.

Constructed counterexamples showing failure of estimates without curvature restrictions.

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 $L^{p}$ Calder\'on--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 $1<p \leq 2$, 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.