Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hypoellipticity on the Heisenberg group-representation-theoretic criteria


Author: Charles Rockland
Journal: Trans. Amer. Math. Soc. 240 (1978), 1-52
MSC: Primary 22E30; Secondary 35H05, 58G05
DOI: https://doi.org/10.1090/S0002-9947-1978-0486314-0
MathSciNet review: 0486314
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group ${H_n}$; it is the precise analogue for ${H_n}$ of the statement for ${{\mathbf {R}}^n}$ that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of ${H_n}$, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely $P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}$, where ${X_i},{Y_i},i = 1, \ldots ,n$, and Z generate the Lie algebra of ${H_n}$ via the commutation relations $[{X_i},{Y_j}] = {\delta _{ij}}Z$, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on ${H_n}$ and a class of degenerate-elliptic operators on ${{\mathbf {R}}^{n + 1}}$ studied by Grušin. This connection is examined in the context of localization in enveloping algebras.


References [Enhancements On Off] (What's this?)

  • I. N. Bernšteĭn, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 26–40. MR 0320735
  • J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. II, Bull. Soc. Math. France 85 (1957), 325–388 (French). MR 95426
  • Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canadian J. Math. 10 (1958), 321–348. MR 95427, DOI https://doi.org/10.4153/CJM-1958-033-5
  • J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad. Brasil. Ci. 35 (1963), 491–519 (French). MR 182682
  • ---, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974.
  • G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI https://doi.org/10.1007/BF02386204
  • 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 367477, DOI https://doi.org/10.1002/cpa.3160270403
  • V. V. Grušin, Pseudodifferential operators in $R^{n}$ with bounded symbols, Funkcional. Anal. i Priložen 4 (1970), no. 3, 37–50 (Russian). MR 0270214
  • V. V. Grušin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.) 83 (125) (1970), 456–473 (Russian). MR 0279436
  • V. V. Grušin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.) 84 (126) (1971), 163–195 (Russian). MR 0283630
  • V. W. Guillemin, Singular symbols (to appear).
  • Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • L. Hörmander, Linear partial differential operators, Grundlehren math. Wiss., Bd. 116, Springer-Verlag, Berlin; Academic Press, New York, 1963. MR 28 #4221.
  • Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI https://doi.org/10.1007/BF02392081
  • A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
  • A. A. Kirillov, Plancherel’s measure for nilpotent Lie groups, Funkcional. Anal. i Priložen 1 (1967), no. 4, 84–85 (Russian). MR 0224748
  • Albert Messiah, Quantum mechanics. Vol. I, North-Holland Publishing Co., Amsterdam; Interscience Publishers Inc., New York, 1961. Translated from the French by G. M. Temmer. MR 0129790
  • Edward Nelson and W. Forrest Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547–560. MR 110024, DOI https://doi.org/10.2307/2372913
  • Y. Nouazé and P. Gabriel, Idéaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotente, J. Algebra 6 (1967), 77–99 (French). MR 206064, DOI https://doi.org/10.1016/0021-8693%2867%2990015-4
  • Mustapha Raïs, Solutions élémentaires des opérateurs différentiels bi-inbariants sur un groupe de Lie nilpotent, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A495–A498 (French). MR 289720
  • Charles Rockland, Hypoellipticity and eigenvalue asymptotics, Lecture Notes in Mathematics, Vol. 464, Springer-Verlag, Berlin-New York, 1975. MR 0501202
  • Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI https://doi.org/10.1007/BF02392419
  • E. M. Stein, Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 173–189. MR 0578903
  • François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
  • R. Beals, Séminaire Goulaouic-Schwartz, 1977. ---, Computes rendues du colloque de St.-Jean de Monts, June 1977.
  • Pierre Cartier, Quantum mechanical commutation relations and theta functions, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 361–383. MR 0216825
  • A. Dynin, Pseudodifferential operators on Heisenberg groups, Pseudodifferential operator with applications (Bressanone, 1977) Liguori, Naples, 1978, pp. 5–18. MR 660648
  • B. Helffer, Hypoellipticité pour des opérateurs différentiels sur des groupes de Lie nilpotents, Pseudodifferential operator with applications (Bressanone, 1977) Liguori, Naples, 1978, pp. 73–88 (French). MR 660651

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E30, 35H05, 58G05

Retrieve articles in all journals with MSC: 22E30, 35H05, 58G05


Additional Information

Keywords: Heisenberg group, nilpotent Lie group, unitary representations, Plancherel formula, dilations, homogeneous left-invariant differential operator, hypoellipticity, parametrix
Article copyright: © Copyright 1978 American Mathematical Society