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–\(L^\infty \) multiplier results (in contrast to the usual \(L^\infty \)–BMO ones). Also, our approach reveals a link between the \(L^p\)-Bernstein inequality and the boundedness on \(L^p\) of the Riesz transform.

我们将经典的伯恩斯坦不等式扩展到一般设置，包括拉普拉斯-贝尔特拉米算子、薛定谔算子和黎曼流形或域上的散度椭圆算子。我们证明了\(L^p\) Bernstein 不等式以及一个“逆不等式”，即使对于紧凑流形（有边界或没有边界）也是新的。这种逆不等式可以看作是伯恩斯坦不等式的对偶。热核将是我们方法的支柱，但我们也开发了新技术。例如，一旦以半经典的方式重新表述伯恩斯坦不等式，我们证明并使用平滑函数的弱分解，如 Dixmier–Malliavin 和BMO – \(L^\infty \)乘数结果（与通常的\(L^ \infty \)– BMO的）。此外，我们的方法揭示了\(L^p\) -Bernstein 不等式与Riesz 变换的\(L^p\)上的有界性之间的联系。