A multiplicative inequality of Riesz transform type on general Riemannian manifolds

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