Analysis of the Hodge Laplacian on the Heisenberg group

Müller, Detlef; Peloso, Marco; Ricci, Fulvio

doi:10.1090/memo/1095

Analysis of the Hodge Laplacian on the Heisenberg group

About this Title

Detlef Müller, Marco M. Peloso and Fulvio Ricci

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1095
ISBNs: 978-1-4704-0939-5 (print); 978-1-4704-1963-9 (online)
DOI: http://dx.doi.org/10.1090/memo/1095
Published electronically: May 19, 2014
Keywords:Hodge Laplacian, Heisenberg group

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Chapters

Introduction
Chapter 1. Differential forms and the Hodge Laplacian on $H_n$
Chapter 2. Bargmann representations and sections of homogeneous bundles
Chapter 3. Cores, domains and self-adjoint extensions
Chapter 4. First properties of $\Delta _k$; exact and closed forms
Chapter 5. A decomposition of $L^2\Lambda _H^k$ related to the $\partial $ and $\bar \partial $ complexes
Chapter 6. Intertwining operators and different scalar forms for $\Delta _k$
Chapter 7. Unitary intertwining operators and projections
Chapter 8. Decomposition of $L^2\Lambda ^k$
Chapter 9. $L^p$-multipliers
Chapter 10. Decomposition of $L^p\Lambda ^k$ and boundedness of the Riesz transforms
Chapter 11. Applications
Chapter 12. Appendix

Abstract

We 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 we address three main, related questions:

whether the $L^2$ and $L^p$ -Hodge decompositions, $1<p<\infty$ , hold on $H_n$ ;
whether the Riesz transforms $d\Delta _k^{-\frac 12}$ are $L^p$ -bounded, for $1<p<\infty$ ;
to prove a sharp Mihilin-Hörmander multiplier theorem for $\Delta _k$ , $0\le k\le 2n+1$ .

Our first main result shows that the $L^2$ -Hodge decomposition holds on $H_n$ , for $0\le k\le 2n+1$ . Moreover, we prove that $L^2\Lambda ^k(H_n)$ further decomposes into finitely many mutually orthogonal subspaces $\mathcal {V}_\nu$ with the properties:

$\operatorname {dom} \Delta _k$ splits along the $\mathcal {V}_\nu$ 's as $\sum _\nu (\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )$ ;
$\Delta _k:(\operatorname {dom}\Delta _k\cap \mathcal {V}_\nu )\mathrel {{}{}{}{}}\mathrel {\mkern -3mu}\rightarrow \mathcal {V}_\nu$ for every $\nu$ ;
for each $\nu$ , there is a Hilbert space $\mathcal {H}_\nu$ of $L^2$ -sections of a $U(n)$ -homogeneous vector bundle over $H_n$ such that the restriction of $\Delta _k$ to $\mathcal {V}_\nu$ is unitarily equivalent to an explicit scalar operator acting componentwise on $\mathcal {H}_\nu$ .

Next, we consider $L^p\Lambda ^k$ , $1<p<\infty$ . We prove that the $L^p$ -Hodge decomposition holds on $H_n$ , for the full range of $p$ and $0\le k\le 2n+1$ . Moreover, we prove that the same kind of finer decomposition as in the $L^2$ -case holds true. More precisely we show that:

the Riesz transforms $d\Delta _k^{-\frac 12 }$ are $L^p$ -bounded;
the orthogonal projection onto $\mathcal {V}_\nu$ extends from $(L^2\cap L^p)\Lambda ^k$ to a bounded operator from $L^p\Lambda ^k$ to the the $L^p$ -closure $\mathcal {V} _\nu ^p$ of $\mathcal {V} _\nu \cap L^p\Lambda ^k$ .

We then use this decomposition to prove a sharp Mihlin-Hörmander multiplier theorem for each $\Delta _k$ . We show that the operator $m(\Delta _k)$ is bounded on $L^p\Lambda ^k(H_n)$ for all $p\in (1,\infty )$ and all $k=0,\dotsc ,2n+1$ , provided $m$ satisfies a Mihlin-Hörmander condition of order $\rho >(2n+1)/2$ and prove that this restriction on $\rho$ is optimal.

Finally, we extend this multiplier theorem to the Dirac operator.

[A] G. Alexopoulos, $L^p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Proc. Amer. math. Soc. 120 (1994), 973-979.
[An] J.-Ph. Anker, Spectral multipliers on Lie groups of polynomial growth, Ann. Math. 132 (1990), 597-628.
[AnLoh] Jean-Philippe Anker and Noël Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), no. 6, 1303–1353 (French). MR 868894, 10.2307/2374528
[ACDH] Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957 (English, with English and French summaries). MR 2119242, 10.1016/j.ansens.2004.10.003
[Ch] Michael Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), no. 1, 73–81. MR 1104196, 10.1090/S0002-9947-1991-1104196-7
[ChM] M. Christ and D. Müller, On $L^p$ spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996), no. 5, 860–876. MR 1415763, 10.1007/BF02246787
[ClS] J. L. Clerc and E. M. Stein, $L^{p}$-multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912. MR 0367561
[CD] Thierry Coulhon and Xuan Thinh Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56 (2003), no. 12, 1728–1751. MR 2001444, 10.1002/cpa.3040
[CMZ] Thierry Coulhon, Detlef Müller, and Jacek Zienkiewicz, About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (1996), no. 2, 369–379. MR 1391221, 10.1007/BF01444227
[CoKSi] Michael G. Cowling, Oldrich Klima, and Adam Sikora, Spectral multipliers for the Kohn sublaplacian on the sphere in $\Bbb C^n$, Trans. Amer. Math. Soc. 363 (2011), no. 2, 611–631. MR 2728580, 10.1090/S0002-9947-2010-04920-7
[D] Harold Donnelly, The differential form spectrum of hyperbolic space, Manuscripta Math. 33 (1980/81), no. 3-4, 365–385. MR 612619, 10.1007/BF01798234
[F] Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366
[FS] G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 0367477
[GSj] Garth Gaudry and Peter Sjögren, Singular integrals on the complex affine group, Colloq. Math. 75 (1998), no. 1, 133–148. MR 1487974
[He] Waldemar Hebisch, Multiplier theorem on generalized Heisenberg groups, Colloq. Math. 65 (1993), no. 2, 231–239. MR 1240169
[HeZ] Waldemar Hebisch and Jacek Zienkiewicz, Multiplier theorem on generalized Heisenberg groups. II, Colloq. Math. 69 (1995), no. 1, 29–36. MR 1341678
[Hu] Andrzej Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), no. 3, 253–266. MR 782662
[Ka] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
[KST] C. E. Kenig, R. J. Stanton, and P. A. Tomas, Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982), no. 1, 28–44. MR 654463, 10.1016/0022-1236(82)90042-8
[Li1] Xiang Dong Li, Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds, Rev. Mat. Iberoam. 22 (2006), no. 2, 591–648, loose erratum. MR 2294791, 10.4171/RMI/467
[Li2] Xiang-Dong Li, On the strong $L^p$-Hodge decomposition over complete Riemannian manifolds, J. Funct. Anal. 257 (2009), no. 11, 3617–3646. MR 2572263, 10.1016/j.jfa.2009.08.002
[Li3] Xiang-Dong Li, Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds, Rev. Mat. Iberoam. 26 (2010), no. 2, 481–528. MR 2677005, 10.4171/RMI/607
[Loh] Noël Lohoué, Estimation des projecteurs de de Rham Hodge de certaines variétés riemaniennes non compactes, Math. Nachr. 279 (2006), no. 3, 272–298 (French, with English and French summaries). MR 2200666, 10.1002/mana.200310361
[Loh2] Noël Lohoué, Transformées de Riesz et fonctions sommables, Amer. J. Math. 114 (1992), no. 4, 875–922 (French). MR 1175694, 10.2307/2374800
[LohMu] Noël Lohoué and Sami Mustapha, Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes, Canad. J. Math. 50 (1998), no. 5, 1090–1104 (French, with French summary). MR 1650934, 10.4153/CJM-1998-052-9
[Lot] John Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), no. 2, 471–510. MR 1158345
[LuM] J. Ludwig and D. Müller, Sub-Laplacians of holomorphic $L^p$-type on rank one $AN$-groups and related solvable groups, J. Funct. Anal. 170 (2000), no. 2, 366–427. MR 1740657, 10.1006/jfan.1999.3517
[LuMS] Jean Ludwig, Detlef Müller, and Sofiane Souaifi, Holomorphic $L^p$-type for sub-Laplacians on connected Lie groups, J. Funct. Anal. 255 (2008), no. 6, 1297–1338. MR 2565710, 10.1016/j.jfa.2008.06.014
[L\"u] Wolfgang Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649
[Mar] Alessio Martini, Analysis of joint spectral multipliers on Lie groups of polynomial growth, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1215–1263 (English, with English and French summaries). MR 3025742
[MauMe] Giancarlo Mauceri and Stefano Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 141–154. MR 1125759, 10.4171/RMI/100
[MPR1] Detlef Müller, Marco M. Peloso, and Fulvio Ricci, $L^p$-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group, Geom. Funct. Anal. 17 (2007), no. 3, 852–886. MR 2346278, 10.1007/s00039-007-0612-0
[MPR2] Detlef Müller, Marco M. Peloso, and Fulvio Ricci, Eigenvalues of the Hodge Laplacian on a quotient of the Heisenberg group, Collect. Math. Vol Extra (2006), 327-342.
[MRS1] Detlef Müller, Fulvio Ricci, and Elias M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math. 119 (1995), no. 2, 199–233. MR 1312498, 10.1007/BF01245180
[MRS2] Detlef Müller, Fulvio Ricci, and Elias M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II, Math. Z. 221 (1996), no. 2, 267–291. MR 1376298, 10.1007/BF02622116
[MS] D. Müller and E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9) 73 (1994), no. 4, 413–440. MR 1290494
[Ra] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923
[RS] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
[Ru1] Michel Rumin, Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994), no. 2, 281–330 (French). MR 1267892
[R2] Michel Rumin, Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal. 10 (2000), no. 2, 407–452. MR 1771424, 10.1007/s000390050013
[Si] A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation, J. Funct. Anal. 52 (1983), 48-79.
[St] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Math. Z. 247 (2004), 643-662.
[Ta] Michael E. Taylor, $L^p$-estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), no. 3, 773–793. MR 1016445, 10.1215/S0012-7094-89-05836-5
[Th] Sundaram Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Mathematics, vol. 159, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1633042
[W] André Weil, Introduction à l’étude des variétés kählériennes, Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056

References

[A] G. Alexopoulos, $L^p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Proc. Amer. math. Soc. 120 (1994), 973-979.
[An] J.-Ph. Anker, Spectral multipliers on Lie groups of polynomial growth, Ann. Math. 132 (1990), 597-628.
[AnLoh] Jean-Philippe Anker and Noël Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), no. 6, 1303–1353 (French). MR 868894 (88c:43008), http://dx.doi.org/10.2307/2374528
[ACDH] Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957 (English, with English and French summaries). MR 2119242 (2005k:58043), http://dx.doi.org/10.1016/j.ansens.2004.10.003
[Ch] Michael Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), no. 1, 73–81. MR 1104196 (92k:42017), http://dx.doi.org/10.2307/2001877
[ChM] M. Christ and D. Müller, On $L^p$ spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996), no. 5, 860–876. MR 1415763 (98b:22017), http://dx.doi.org/10.1007/BF02246787
[ClS] J. L. Clerc and E. M. Stein, $L^{p}$ -multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912. MR 0367561 (51 \#3803)
[CD] Thierry Coulhon and Xuan Thinh Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56 (2003), no. 12, 1728–1751. MR 2001444 (2004j:58020), http://dx.doi.org/10.1002/cpa.3040
[CMZ] Thierry Coulhon, Detlef Müller, and Jacek Zienkiewicz, About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (1996), no. 2, 369–379. MR 1391221 (97f:22015), http://dx.doi.org/10.1007/BF01444227
[CoKSi] Michael G. Cowling, Oldrich Klima, and Adam Sikora, Spectral multipliers for the Kohn sublaplacian on the sphere in $\mathbb {C}^n$ , Trans. Amer. Math. Soc. 363 (2011), no. 2, 611–631. MR 2728580 (2011i:42018), http://dx.doi.org/10.1090/S0002-9947-2010-04920-7
[D] Harold Donnelly, The differential form spectrum of hyperbolic space, Manuscripta Math. 33 (1980/81), no. 3-4, 365–385. MR 612619 (82f:58085), http://dx.doi.org/10.1007/BF01798234
[F] Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
[FS] G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 0367477 (51 \#3719)
[GSj] Garth Gaudry and Peter Sjögren, Singular integrals on the complex affine group, Colloq. Math. 75 (1998), no. 1, 133–148. MR 1487974 (99k:43003)
[He] Waldemar Hebisch, Multiplier theorem on generalized Heisenberg groups, Colloq. Math. 65 (1993), no. 2, 231–239. MR 1240169 (94m:43013)
[HeZ] Waldemar Hebisch and Jacek Zienkiewicz, Multiplier theorem on generalized Heisenberg groups. II, Colloq. Math. 69 (1995), no. 1, 29–36. MR 1341678 (96h:43007)
[Hu] Andrzej Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), no. 3, 253–266. MR 782662 (86g:22009)
[Ka] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 \#3324)
[KST] C. E. Kenig, R. J. Stanton, and P. A. Tomas, Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982), no. 1, 28–44. MR 654463 (83k:22025), http://dx.doi.org/10.1016/0022-1236(82)90042-8
[Li1] Xiang Dong Li, Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds, Rev. Mat. Iberoam. 22 (2006), no. 2, 591–648, loose erratum. MR 2294791 (2008d:58027), http://dx.doi.org/10.4171/RMI/467
[Li2] Xiang-Dong Li, On the strong $L^p$ -Hodge decomposition over complete Riemannian manifolds, J. Funct. Anal. 257 (2009), no. 11, 3617–3646. MR 2572263 (2011b:58005), http://dx.doi.org/10.1016/j.jfa.2009.08.002
[Li3] Xiang-Dong Li, Riesz transforms on forms and $L^p$ -Hodge decomposition on complete Riemannian manifolds, Rev. Mat. Iberoam. 26 (2010), no. 2, 481–528. MR 2677005 (2011f:58064), http://dx.doi.org/10.4171/RMI/607
[Loh] Noël Lohoué, Estimation des projecteurs de de Rham Hodge de certaines variétés riemaniennes non compactes, Math. Nachr. 279 (2006), no. 3, 272–298 (French, with English and French summaries). MR 2200666 (2006j:58034), http://dx.doi.org/10.1002/mana.200310361
[Loh2] Noël Lohoué, Transformées de Riesz et fonctions sommables, Amer. J. Math. 114 (1992), no. 4, 875–922 (French). MR 1175694 (93k:58214), http://dx.doi.org/10.2307/2374800
[LohMu] Noël Lohoué and Sami Mustapha, Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes, Canad. J. Math. 50 (1998), no. 5, 1090–1104 (French, with French summary). MR 1650934 (2000a:43008), http://dx.doi.org/10.4153/CJM-1998-052-9
[Lot] John Lott, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (1992), no. 2, 471–510. MR 1158345 (93b:58140)
[LuM] J. Ludwig and D. Müller, Sub-Laplacians of holomorphic $L^p$ -type on rank one $AN$ -groups and related solvable groups, J. Funct. Anal. 170 (2000), no. 2, 366–427. MR 1740657 (2001g:43010), http://dx.doi.org/10.1006/jfan.1999.3517
[LuMS] Jean Ludwig, Detlef Müller, and Sofiane Souaifi, Holomorphic $L^p$ -type for sub-Laplacians on connected Lie groups, J. Funct. Anal. 255 (2008), no. 6, 1297–1338. MR 2565710 (2011g:43013), http://dx.doi.org/10.1016/j.jfa.2008.06.014
[L\"u] Wolfgang Lück, $L^2$ -invariants: theory and applications to geometry and $K$ -theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649 (2003m:58033)
[Mar] Alessio Martini, Analysis of joint spectral multipliers on Lie groups of polynomial growth, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1215–1263 (English, with English and French summaries). MR 3025742
[MauMe] Giancarlo Mauceri and Stefano Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 141–154. MR 1125759 (92j:42014), http://dx.doi.org/10.4171/RMI/100
[MPR1] Detlef Müller, Marco M. Peloso, and Fulvio Ricci, $L^p$ -spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group, Geom. Funct. Anal. 17 (2007), no. 3, 852–886. MR 2346278 (2008k:43019), http://dx.doi.org/10.1007/s00039-007-0612-0
[MPR2] Detlef Müller, Marco M. Peloso, and Fulvio Ricci, Eigenvalues of the Hodge Laplacian on a quotient of the Heisenberg group, Collect. Math. Vol Extra (2006), 327-342.
[MRS1] Detlef Müller, Fulvio Ricci, and Elias M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math. 119 (1995), no. 2, 199–233. MR 1312498 (96b:43005), http://dx.doi.org/10.1007/BF01245180
[MRS2] Detlef Müller, Fulvio Ricci, and Elias M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II, Math. Z. 221 (1996), no. 2, 267–291. MR 1376298 (97c:43007), http://dx.doi.org/10.1007/BF02622116
[MS] D. Müller and E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9) 73 (1994), no. 4, 413–440. MR 1290494 (96m:43004)
[Ra] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923 (87i:32001)
[RS] Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. Functional analysis. MR 751959 (85e:46002)
[Ru1] Michel Rumin, Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994), no. 2, 281–330 (French). MR 1267892 (95g:58221)
[R2] Michel Rumin, Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal. 10 (2000), no. 2, 407–452. MR 1771424 (2002f:53044), http://dx.doi.org/10.1007/s000390050013
[Si] A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation, J. Funct. Anal. 52 (1983), 48-79.
[St] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Math. Z. 247 (2004), 643-662.
[Ta] Michael E. Taylor, $L^p$ -estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), no. 3, 773–793. MR 1016445 (91d:58253), http://dx.doi.org/10.1215/S0012-7094-89-05836-5
[Th] Sundaram Thangavelu, Harmonic analysis on the Heisenberg group, Progress in Mathematics, vol. 159, Birkhäuser Boston Inc., Boston, MA, 1998. MR 1633042 (99h:43001)
[W] André Weil, Introduction à l'étude des variétés kählériennes, Publications de l'Institut de Mathématique de l'Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056 (22 \#1921)

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Analysis of the Hodge Laplacian on the Heisenberg group