Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups

Miller, Kenneth G.

doi:10.1090/S0002-9947-1983-0716847-2

Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
HTML articles powered by AMS MathViewer

by Kenneth G. Miller

Trans. Amer. Math. Soc. 280 (1983), 721-736

DOI: https://doi.org/10.1090/S0002-9947-1983-0716847-2

PDF | Request permission

Abstract:

Let $P$ be a right-invariant pseudodifferential operator with principal part ${P_0}$ on a simply connected two-step nilpotent Lie group $G$ of type $H$. It will be shown that if $\pi (P_0)$ is injective in ${\mathcal {S}_\pi }$ for every nontrivial irreducible unitary representation $\pi$ of $G$, then $P$ has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on $G$. If, in addition, $\pi (P)$ is injective in ${\mathcal {S}_\pi }$ for every irreducible unitary representation of $G$, it will be shown that $P$ has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.

References

Richard Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1–42. MR 367730
Richard Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), no. 1, 45–57. MR 435933
R. Beals, Opérateurs invariants hypoelliptiques sur un groupe de Lie nilpotent, Séminaire Goulaouic-Schwartz 1976–1977: Équations aux dérivées partielles et analyse fonctionnelle, Centre Math., École Polytech., Palaiseau, 1977, pp. Exp. No. 19, 8 (French). MR 0494313
A. S. Dynin, An algebra of pseudodifferential operators on the Heisenberg groups. Symbolic calculus, Dokl. Akad. Nauk SSSR 227 (1976), no. 4, 792–795 (Russian). MR 0423432
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
Daryl Geller, Local solvability and homogeneous distributions on the Heisenberg group, Comm. Partial Differential Equations 5 (1980), no. 5, 475–560. MR 571049, DOI 10.1080/0360530800882146
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
B. Helffer and F. Nourrigat, Hypoellipticité pour des groupes nilpotents de rang de nilpotence $3$, Comm. Partial Differential Equations 3 (1978), no. 8, 643–743 (French). MR 499352, DOI 10.1080/03605307808820075
B. Helffer and J. Nourrigat, Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4 (1979), no. 8, 899–958 (French). MR 537467, DOI 10.1080/03605307908820115
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081

The Weyl calculus of pseudo-differential operators

32

Pierre Lévy-Bruhl, Résolubilité locale et globale d’opérateurs invariants du second ordre sur des groupes de Lie nilpotents, Bull. Sci. Math. (2) 104 (1980), no. 4, 369–391 (French, with English summary). MR 602406
Anders Melin, Parametrix constructions for some classes of right-invariant differential operators on the Heisenberg group, Comm. Partial Differential Equations 6 (1981), no. 12, 1363–1405. MR 640161, DOI 10.1080/03605308108820214
Guy Métivier, Hypoellipticité analytique sur des groupes nilpotents de rang $2$, Duke Math. J. 47 (1980), no. 1, 195–221 (French). MR 563376
Kenneth G. Miller, Hypoellipticity on the Heisenberg group, J. Functional Analysis 31 (1979), no. 3, 306–320. MR 531133, DOI 10.1016/0022-1236(79)90005-3
Kenneth G. Miller, Parametrices for hypoelliptic operators on step two nilpotent Lie groups, Comm. Partial Differential Equations 5 (1980), no. 11, 1153–1184. MR 589618, DOI 10.1080/03605308008820166
Kenneth G. Miller, Invariant pseudodifferential operators on two-step nilpotent Lie groups, Michigan Math. J. 29 (1982), no. 3, 315–328. MR 674285
Charles Rockland, Hypoellipticity on the Heisenberg group-representation-theoretic criteria, Trans. Amer. Math. Soc. 240 (1978), 1–52. MR 486314, DOI 10.1090/S0002-9947-1978-0486314-0
Linda Preiss Rothschild and David S. Tartakoff, Inversion of analytic matrices and local solvability of some invariant differential operators on nilpotent Lie groups, Comm. Partial Differential Equations 6 (1981), no. 6, 625–650. MR 617784, DOI 10.1080/03605308108820187
K\B{o}ichi Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. J. (2) 31 (1979), no. 4, 383–437. MR 558675, DOI 10.2748/tmj/1178229728