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A note on second order Riesz transforms in 3-dimensional Lie groups

Archiv der Mathematik volume 118, pages 291–304 (2022)Cite this article

Abstract

We prove explicit \(L^p\) bounds for second order Riesz transforms of the sub-Laplacian and of the Laplacian in the Lie groups \({\mathbb {H}}\), \(\mathbb {SU}(2)\), and \(\widetilde{\mathbb {SL}}(2)\). Our proof makes use of martingale transform techniques and specific commutation properties between the complex gradient and the sub-Laplacian in those Lie groups.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT,  06269, USA

    Fabrice Baudoin

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA,  70803, USA

    Li Chen

Authors
  1. Fabrice Baudoin
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  2. Li Chen
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Correspondence to Li Chen.

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F.B. partly supported by the NSF Grant DMS 1901315. L.C. partly supported by Simons Collaboration Grant #853249.

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Baudoin, F., Chen, L. A note on second order Riesz transforms in 3-dimensional Lie groups. Arch. Math. 118, 291–304 (2022). https://doi.org/10.1007/s00013-021-01699-6

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  • DOI: https://doi.org/10.1007/s00013-021-01699-6

Keywords

  • Second order Riesz transforms
  • Martingale transform
  • Sub-Laplacian
  • Elliptic Laplacian
  • Heisenberg groups
  • 3- dimensional Lie groups.

Mathematics Subject Classification

  • 58J65
  • 60G46
  • 22E30
  • 43A80.