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Analysis of the Hodge Laplacian on the Heisenberg group

About this Title

Detlef Müller, Marco M. Peloso and Fulvio Ricci

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1095
ISBNs: 978-1-4704-0939-5 (print); 978-1-4704-1963-9 (online)
Published electronically: May 19, 2014
Keywords:Hodge Laplacian, Heisenberg group

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Table of Contents


  • Introduction
  • Chapter 1. Differential forms and the Hodge Laplacian on $H_n$
  • Chapter 2. Bargmann representations and sections of homogeneous bundles
  • Chapter 3. Cores, domains and self-adjoint extensions
  • Chapter 4. First properties of $\Delta _k$; exact and closed forms
  • Chapter 5. A decomposition of $L^2\Lambda _H^k$ related to the $\partial $ and $\bar \partial $ complexes
  • Chapter 6. Intertwining operators and different scalar forms for $\Delta _k$
  • Chapter 7. Unitary intertwining operators and projections
  • Chapter 8. Decomposition of $L^2\Lambda ^k$
  • Chapter 9. $L^p$-multipliers
  • Chapter 10. Decomposition of $L^p\Lambda ^k$ and boundedness of the Riesz transforms
  • Chapter 11. Applications
  • Chapter 12. Appendix


We consider the Hodge Laplacian on the Heisenberg group , endowed with a left-invariant and -invariant Riemannian metric. For , let denote the Hodge Laplacian restricted to -forms.

In this paper we address three main, related questions:

  1. whether the and -Hodge decompositions, , hold on ;
  2. whether the Riesz transforms are -bounded, for ;
  3. to prove a sharp Mihilin-Hörmander multiplier theorem for , .

Our first main result shows that the -Hodge decomposition holds on , for . Moreover, we prove that further decomposes into finitely many mutually orthogonal subspaces with the properties:

  • splits along the 's as ;
  • for every ;
  • for each , there is a Hilbert space of -sections of a -homogeneous vector bundle over such that the restriction of to is unitarily equivalent to an explicit scalar operator acting componentwise on .

Next, we consider , . We prove that the -Hodge decomposition holds on , for the full range of and . Moreover, we prove that the same kind of finer decomposition as in the -case holds true. More precisely we show that:

  • the Riesz transforms are -bounded;
  • the orthogonal projection onto extends from to a bounded operator from to the the -closure of .

We then use this decomposition to prove a sharp Mihlin-Hörmander multiplier theorem for each . We show that the operator is bounded on for all and all , provided satisfies a Mihlin-Hörmander condition of order and prove that this restriction on is optimal.

Finally, we extend this multiplier theorem to the Dirac operator.

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