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memo_has_moved_text();Analysis of the Hodge Laplacian on the Heisenberg group

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)
DOI: http://dx.doi.org/10.1090/memo/1095
Published electronically: May 19, 2014
Keywords:Hodge Laplacian, Heisenberg group

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Chapters

• 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

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, hold on $H_n$;
2. whether the Riesz transforms $d\Delta _k^{-\frac 12}$ are $L^p$-bounded, for $1;
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 $\sum _\nu (\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )$;
• $\Delta _k:(\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )\mathrel {{}{}{}{}}\mathrel {\mkern -3mu}\rightarrow \mathcal {V}_\nu$ 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. 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 $k=0,\dotsc ,2n+1$, 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.