  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.
