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 $\epsilon > 0$, $$\| \nabla f \|_p^2 \le C_\epsilon \| \Delta^{\frac{1}{2} + \epsilon} f \|_{p}\| \Delta^{\frac{1}{2} - \epsilon} f \|_{p}.$$The estimate is dimension free. This inequality is even proved in the abstract setting of generators of sub-Markov semigroups.