Hypoellipticity on the Heisenberg group-representation-theoretic criteria

Rockland, Charles

doi:10.1090/S0002-9947-1978-0486314-0

Hypoellipticity on the Heisenberg group-representation-theoretic criteria
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Trans. Amer. Math. Soc. 240 (1978), 1-52 Request permission

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

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, DOI 10.24033/bsmf.1492
Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canadian J. Math. 10 (1958), 321–348. MR 95427, DOI 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

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 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 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

Singular symbols

Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455

Linear partial differential operators

28

Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 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 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 10.1016/0021-8693(67)90015-4
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, DOI 10.1007/BFb0096096
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 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
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