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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
Analysis of the Hodge Laplacian on the Heisenberg Group
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
Analysis of the Hodge Laplacian on the Heisenberg Group
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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
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
  • 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 \(\Delta_k\), \(0\le k\le 2n+1\).

  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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