Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including the Laplace-Beltrami operator, Schrödinger operators and divergence form elliptic operators on Riemannian manifolds or domains. We prove L p Bernstein inqualities as well as a “reverse inequality” which is new even for compact manifolds (with or without boundary). Such a reverse inequality can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques. For example, once reformulating Bernstein inequalities in a semi-classical fashion we prove and use weak factorization of smooth functions à la Dixmier–Malliavin and BMO – L ∞ multiplier results (in contrast to the usual L ∞ – BMO ones). Also, our approach reveals a link between the L p -Bernstein inequality and the boundedness on L p of the Riesz transform.