Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T00:35:51.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Microlocal regularity on step two nilpotent Lie groups

Published online by Cambridge University Press:  20 January 2009

Kenneth G. Miller
Kenneth G. Miller
Affiliation:
Wichita State UniversityWichita, KS 67208
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 1 , February 1988 , pp. 25 - 39
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Beals, R., A general calculus of pseudodifferential operators, Duke Math J. 42 (1975), 1–42.CrossRefGoogle Scholar
2.Beals, R., Operateurs invariants hypoelliptiques sur un groupe de Lie nilpotent (Seminaire Goulaouic-Schwartz, 1976–1977, exposé 19), 1–8.Google Scholar
3.Grigis, A., Propagation des singularités sur des groupes de Lie nilpotents de rang 2. II, Ann. Sci. École Norm. Sup. 15 (1982), 161–171.CrossRefGoogle Scholar
4.Helffer, B. and Nourrigat, J., Caractérization des opérateurs hypoelliptiques homogènes invariants a gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4, (1979), 899–958.CrossRefGoogle Scholar
5.Hörmander, L., The Analysis of Linear Partial Differential Operators I (Springer-Verlag, Berlin and New York, 1983).Google Scholar
6.Hörmander, L., The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359–443.CrossRefGoogle Scholar
7.Kirillov, A., Unitary representations of nilpotent Lie groups, Uspekhi Mat. Nauk 17 (1962), no. 4 (106), 57–110; Russian Math. Surveys 17 (1962), 53–104.Google Scholar
8.Lascar, R., Propagation des singularités des solutions d'equations pseudo-differentielles quasihomogènes, Ann. Inst. Fourier (Grenoble) 27 (1977), 79–153.CrossRefGoogle Scholar
9.Melin, A., Parametrix constructions for some classes of right-invariant differential operators on the Heisenberg group, Comm. Partial Differential Equations 6 (1981), 1363–1405.CrossRefGoogle Scholar
10.Melin, A., Parametrix Constructions for Right-Invariant Differential Operators on Nilpotent Groups, Ann. Global Anal. Geom. 1 (1983), 79–130.Google Scholar
11.Miller, K., Parametrices for hypoelliptic operators on step two nilpotent Lie groups, Comm. Partial Differential Equations 5 (1980), 1153–1184.CrossRefGoogle Scholar
12.Miller, K., Invariant pseudodifferential operators on two step nilpotent Lie groups, Michigan Math. J. 29 (1982), 315–328; II, Michigan Math. J. 33 (1986), 395–401.CrossRefGoogle Scholar
13.Miller, K., Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups, Trans. Amer. Math. Soc. 280 (1983), 721–736.CrossRefGoogle Scholar
14.Miller, K., Microhypoellipticity on step two nilpotent Lie groups, Contemp. Math. 27 (1984), 231–235.CrossRefGoogle Scholar
15.Parenti, C. and Rodino, L., Parametrices for a class of pseudo differential operators I, II, Ann. Mat. Pura Appl. 125 (1980), 221–278.CrossRefGoogle Scholar
16.Parenti, C. and Rodino, L., Examples of hypoelliptic operators which are not microhypoelliptic, Boll. Un. Mat. Ital. B17 (1980), 390–409.Google Scholar
17.Phong, D. and Stein, E., Some further classes of pseudodifferential operators and singular-integral operators arising in boundary-value problems I: Composition of operators, Amer. J. Math 104(1982), 141–172.CrossRefGoogle Scholar
18.Rockland, C., Hypoellipticity on the Heisenberg group: representation-theoretic criteria, Trans. Amer. Math. Soc. 240 (1978), 1–52.CrossRefGoogle Scholar
19.Taylor, M., Noncommutative microlocal analysis Part I, Mem. Amer. Math. Soc. 313 (1984).Google Scholar
20.Taylor, M., Pseudodifferential Operators (Princeton University Press, Princeton, 1981).CrossRefGoogle Scholar