  We extend the classical Bernstein inequality to a general setting including
Schr{\"o}dinger operators and divergence form elliptic operators on Riemannian
manifolds or domains. Moreover , we prove a new reverse inequality that 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 such as
semi-classical Bernstein inequalities, weak factorization of smooth functions
{\`a} la Dixmier-Malliavin and BM O -- L $\infty$ multiplier results (in
contrast to the usual L $\infty$ -- BM O ones). Also, our approach reveals a
link between the L p-Bernstein inequality and the boundedness on L p of the
Riesz transform. The later being an important subject in harmonic analysis.
2010 Mathematics Subject Classifications: 35P20, 58J50, 42B37 and 47F05.
