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Analysis of the Hodge Laplacian on the Heisenberg group

About this Title

Detlef Müller, Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Straße 4, D-24098 Kiel, Germany, Marco M. Peloso, Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy and Fulvio Ricci, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

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)
DOI: https://doi.org/10.1090/memo/1095
Published electronically: May 19, 2014
Keywords: Hodge Laplacian, Heisenberg group, spectral multiplier
MSC: Primary 43A80, 42B15

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Table of Contents

Chapters

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

Abstract

We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and $U(n)$-invariant Riemannian metric. For $0\le k\le 2n+1$, let $\Delta _k$ denote the Hodge Laplacian restricted to $k$-forms.

In this paper we address three main, related questions:

  1. whether the $L^2$ and $L^p$-Hodge decompositions, $1<p<\infty$, hold on $H_n$;

  2. whether the Riesz transforms $d\Delta _k^{-\frac 12}$ are $L^p$-bounded, for $1<p<\infty$;

  3. to prove a sharp Mihilin–Hörmander multiplier theorem for $\Delta _k$, $0\le k\le 2n+1$.

Our first main result shows that the $L^2$-Hodge decomposition holds on $H_n$, for $0\le k\le 2n+1$. Moreover, we prove that $L^2\Lambda ^k(H_n)$ further decomposes into finitely many mutually orthogonal subspaces $\mathcal {V}_\nu$ with the properties:

  • $\operatorname {dom} \Delta _k$ splits along the $\mathcal {V}_\nu$’s as $\sum _\nu (\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )$;

  • $\Delta _k:(\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )\longrightarrow \mathcal {V}_\nu$ for every $\nu$;

  • for each $\nu$, there is a Hilbert space $\mathcal {H}_\nu$ of $L^2$-sections of a $U(n)$-homogeneous vector bundle over $H_n$ such that the restriction of $\Delta _k$ to $\mathcal {V}_\nu$ is unitarily equivalent to an explicit scalar operator acting componentwise on $\mathcal {H}_\nu$.

    • Next, we consider $L^p\Lambda ^k$, $1<p<\infty$. We prove that the $L^p$-Hodge decomposition holds on $H_n$, for the full range of $p$ and $0\le k\le 2n+1$. Moreover, we prove that the same kind of finer decomposition as in the $L^2$-case holds true. More precisely we show that:

  • the Riesz transforms $d\Delta _k^{-\frac 12 }$ are $L^p$-bounded;

  • the orthogonal projection onto $\mathcal {V}_\nu$ extends from $(L^2\cap L^p)\Lambda ^k$ to a bounded operator from $L^p\Lambda ^k$ to the the $L^p$-closure $\mathcal {V} _\nu ^p$ of $\mathcal {V} _\nu \cap L^p\Lambda ^k$.

    • We then use this decomposition to prove a sharp Mihlin–Hörmander multiplier theorem for each $\Delta _k$. We show that the operator $m(\Delta _k)$ is bounded on $L^p\Lambda ^k(H_n)$ for all $p\in (1,\infty )$ and all $k=0,\dots ,2n+1$, provided $m$ satisfies a Mihlin–Hörmander

    condition of order $\rho >(2n+1)/2$ and prove that this restriction on $\rho$ is optimal.

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

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