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


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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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).

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

Mathematics Subject Classification

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