| eBook ISBN: | 978-1-4704-1963-9 |
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| AMS Member Price: | $42.60 |
| eBook 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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 233; 2015; 91 ppMSC: 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 \(\Delta_k\), \(0\le k\le 2n+1\).
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Table of Contents
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Chapters
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Introduction
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1. Differential forms and the Hodge Laplacian on $H_n$
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2. Bargmann representations and sections of homogeneous bundles
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3. Cores, domains and self-adjoint extensions
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4. First properties of $\Delta _k$; exact and closed forms
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5. A decomposition of $L^2\Lambda _H^k$ related to the $\partial $ and $\bar \partial $ complexes
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6. Intertwining operators and different scalar forms for $\Delta _k$
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7. Unitary intertwining operators and projections
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8. Decomposition of $L^2\Lambda ^k$
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9. $L^p$-multipliers
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10. Decomposition of $L^p\Lambda ^k$ and boundedness of the Riesz transforms
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11. Applications
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12. Appendix
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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
