Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including the Laplace-Beltrami operator, Schrodinger 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 neweven 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 a la Dixmier-Malliavin and BMO-L-infinity multiplier results (in contrast to the usual L-infinity-BMO ones). Also, our approach reveals a link between the L-p-Bernsteinin equality and the boundedness on L-p of the Riesz transform.

Automated summary

In this work, the classical Bernstein inequality is extended to a more general setting involving various types of operators on Riemannian manifolds or domains. The L-p Bernstein inequalities are proved, along with a novel reverse inequality that is applicable even for compact manifolds. The relationship between L-p Bernstein inequality and the boundedness of the Riesz transform on L-p is highlighted, with the development of new techniques for reformulating the Bernstein inequalities.