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Bernstein inequalities via the heat semigroup

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Abstract

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.

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Notes

  1. For which (2.2) is true but the Riesz transform is unbounded for some \(p\in (2,+\infty )\), see [12].

  2. Even in the Euclidean case, \(e^{t\Delta }g\) does not converge in \(L^1({\mathbb {R}}^d)\) as \(t\rightarrow +\infty \) for positive \(g\in L^1({\mathbb {R}})\).

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Acknowledgements

The authors would like to thank Peng Chen and Lixin Yan for their generous help for the proof of the boundedness of \(\psi (L)\) from \(L^1(M)\) into \(H^1_L(M)\) for \(\psi \in {\mathcal {C}}_c^\infty (0, \infty )\).

The research of R. Imekraz is partly supported by the ANR project ESSED ANR-18-CE40-0028. The research of E.M. Ouhabaz is partly supported by the ANR project RAGE ANR-18-CE40-0012-01.

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  1. Institut de Mathématiques (IMB), CNRS UMR 5251, University of Bordeaux, 351, Cours de la Libération, 33405, Talence, France

    Rafik Imekraz & El Maati Ouhabaz

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  1. Rafik Imekraz
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  2. El Maati Ouhabaz
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Correspondence to El Maati Ouhabaz.

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Communicated by Loukas Grafakos.

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Imekraz, R., Ouhabaz, E.M. Bernstein inequalities via the heat semigroup. Math. Ann. 382, 783–819 (2022). https://doi.org/10.1007/s00208-021-02221-7

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