Electronic ISBN:  9781470419639 
Product Code:  MEMO/233/1095.E 
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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 leftinvariant 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 \(\Delta_k\), \(0\le k\le 2n+1\). 
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 selfadjoint 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


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The authors consider the Hodge Laplacian \(\Delta\) on the Heisenberg group \(H_n\), endowed with a leftinvariant 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 selfadjoint 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