Besov and Triebel–Lizorkin spaces on Lie groups

Abstract

In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.

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Acknowledgements

The authors wish to thank Andrea Carbonaro for useful conversations.

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Correspondence to Marco M. Peloso.

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In memory of Elias M. Stein.

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All authors are partially supported by the grant PRIN 2015 Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis, and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Communicated by Loukas Grafakos.

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Bruno, T., Peloso, M.M. & Vallarino, M. Besov and Triebel–Lizorkin spaces on Lie groups. Math. Ann. 377, 335–377 (2020). https://doi.org/10.1007/s00208-019-01927-z

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Mathematics Subject Classification

  • 46E35
  • 22E30
  • 43A15