A multiplicative inequality of Riesz transform type on general Riemannian manifolds

Abstract

Given any complete Riemannian manifold $M$, we prove that for every $p\in (1,2]$ and every $\epsilon >0$, $$\Vert \nabla f{\Vert }_p^2\le C_{\epsilon }\Vert {\Delta }^{\frac{1}{2}+\epsilon }f{\Vert }_p\Vert {\Delta }^{\frac{1}{2}-\epsilon }f{\Vert }_p.$$The estimate is dimension free. This inequality is even proved in the abstract setting of generators of sub-Markov semigroups.

Expanded version, three new section are added