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Analysis of the Hodge Laplacian on the Heisenberg Group.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence, RI : American Mathematical Society, 2015Copyright date: ©2015Edition: 1st edDescription: 1 online resource (104 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470419639
Subject(s): Genre/Form: Additional physical formats: Print version:: Analysis of the Hodge Laplacian on the Heisenberg GroupDDC classification:
  • 515/.785
LOC classification:
  • QA403 -- .M55 2014eb
Online resources:
Contents:
Cover -- Title page -- Introduction -- Chapter 1. Differential forms and the Hodge Laplacian on _{ } -- Chapter 2. Bargmann representations and sections of homogeneous bundles -- Chapter 3. Cores, domains and self-adjoint extensions -- Chapter 4. First properties of Δ_{ } -- exact and closed forms -- Chapter 5. A decomposition of ²Λ_{ }^{ } related to the ∂ and ∂ complexes -- 5.1. The subspaces -- 5.2. The action of Δ_{ } -- 5.3. Lifting by Φ -- Chapter 6. Intertwining operators and different scalar forms for Δ_{ } -- 6.1. The case of ₀^{ , } -- 6.2. The case of _{2,ℓ}^{ , } -- 6.3. The case of _{1,ℓ}^{ , } -- Chapter 7. Unitary intertwining operators and projections -- 7.1. A unitary intertwining operator for ₀^{ , } -- 7.2. Unitary intertwining operators for _{1,ℓ}^{ , ,±} -- 7.3. A unitary intertwining operator for ^{ , }_{2,ℓ} -- Chapter 8. Decomposition of ²Λ^{ } -- 8.1. The *-Hodge operator and the case &lt -- \le2 +1 -- Chapter 9. ^{ }-multipliers -- 9.1. The multiplier theorem -- 9.2. Some classes of multipliers -- Chapter 10. Decomposition of ^{ }Λ^{ } and boundedness of the Riesz transforms -- 10.1. ^{ }- boundedness of the intertwining operators _{1,ℓ}^{±} -- 10.2. ^{ }- boundedness of the intertwining operators _{2,ℓ} -- Chapter 11. Applications -- 11.1. Multipliers of Δ_{ } -- 11.2. Exact ^{ }-forms -- 11.3. The Dirac operator -- Chapter 12. Appendix -- Bibliography -- Back Cover.
Summary: 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 , hold on H_n; (2) whether the Riesz transforms d\Delta_k^{-\frac 12} are L^p-bounded, for 1.
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Cover -- Title page -- Introduction -- Chapter 1. Differential forms and the Hodge Laplacian on _{ } -- Chapter 2. Bargmann representations and sections of homogeneous bundles -- Chapter 3. Cores, domains and self-adjoint extensions -- Chapter 4. First properties of Δ_{ } -- exact and closed forms -- Chapter 5. A decomposition of ²Λ_{ }^{ } related to the ∂ and ∂ complexes -- 5.1. The subspaces -- 5.2. The action of Δ_{ } -- 5.3. Lifting by Φ -- Chapter 6. Intertwining operators and different scalar forms for Δ_{ } -- 6.1. The case of ₀^{ , } -- 6.2. The case of _{2,ℓ}^{ , } -- 6.3. The case of _{1,ℓ}^{ , } -- Chapter 7. Unitary intertwining operators and projections -- 7.1. A unitary intertwining operator for ₀^{ , } -- 7.2. Unitary intertwining operators for _{1,ℓ}^{ , ,±} -- 7.3. A unitary intertwining operator for ^{ , }_{2,ℓ} -- Chapter 8. Decomposition of ²Λ^{ } -- 8.1. The *-Hodge operator and the case &lt -- \le2 +1 -- Chapter 9. ^{ }-multipliers -- 9.1. The multiplier theorem -- 9.2. Some classes of multipliers -- Chapter 10. Decomposition of ^{ }Λ^{ } and boundedness of the Riesz transforms -- 10.1. ^{ }- boundedness of the intertwining operators _{1,ℓ}^{±} -- 10.2. ^{ }- boundedness of the intertwining operators _{2,ℓ} -- Chapter 11. Applications -- 11.1. Multipliers of Δ_{ } -- 11.2. Exact ^{ }-forms -- 11.3. The Dirac operator -- Chapter 12. Appendix -- Bibliography -- Back Cover.

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 , hold on H_n; (2) whether the Riesz transforms d\Delta_k^{-\frac 12} are L^p-bounded, for 1.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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