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Analysis of the Hodge Laplacian on the Heisenberg Group

Detlef Müller Universität Kiel, Germany
Marco M. Peloso Universita Degli Studi Di Mila, Milano, Italy
Fulvio Ricci Scuola Normale Superiore, Pisa, Italy
Available Formats:
Electronic ISBN: 978-1-4704-1963-9
Product Code: MEMO/233/1095.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view Analysis of the Hodge Laplacian on the Heisenberg Group Detlef Müller Universität Kiel, Germany Marco M. Peloso Universita Degli Studi Di Mila, Milano, Italy Fulvio Ricci Scuola Normale Superiore, Pisa, Italy Available Formats:  Electronic ISBN: 978-1-4704-1963-9 Product Code: MEMO/233/1095.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2332015; 91 pp
MSC: Primary 43; 42;

The authors 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 they 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 $1<<\infty$;

(3) how to prove a sharp Mihilin–Hörmander multiplier theorem for $\Delta_k$, $0\le k\le 2n+1$.

• 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
• Requests

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Volume: 2332015; 91 pp
MSC: Primary 43; 42;

The authors 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 they 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 $1<<\infty$;

(3) how to prove a sharp Mihilin–Hörmander multiplier theorem for $\Delta_k$, $0\le k\le 2n+1$.

• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
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