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Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

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Abstract

The \(L^1\)-Sobolev inequality states that for compactly supported functions u on the Euclidean n-space, the \(L^{n/(n-1)}\)-norm of a compactly supported function is controlled by the \(L^1\)-norm of its gradient. The generalization to differential forms (due to Lanzani and Stein and Bourgain and Brezis) is recent, and states that a the \(L^{n/(n-1)}\)-norm of a compactly supported differential h-form is controlled by the \(L^1\)-norm of its exterior differential du and its exterior codifferential \(\delta u\) (in special cases the \(L^1\)-norm must be replaced by the \(\mathcal H^1\)-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Annalisa Baldi & Bruno Franchi

  2. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Pierre Pansu

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  1. Annalisa Baldi
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  2. Bruno Franchi
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  3. Pierre Pansu
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Correspondence to Annalisa Baldi.

Additional information

A.B. and B.F. are supported by University of Bologna, funds for selected research topics, by GNAMPA of INdAM, Italy and by MAnET Marie Curie Initial Training Network. P.P. is supported by Agence Nationale de la Recherche, ANR-10-BLAN 0116.

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Baldi, A., Franchi, B. & Pansu, P. Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups. Math. Ann. 365, 1633–1667 (2016). https://doi.org/10.1007/s00208-015-1337-2

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  • DOI: https://doi.org/10.1007/s00208-015-1337-2

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