A homogeneous, globally solvable differential operator on a nilpotent Lie group which has no tempered fundamental solution

Müller, Detlef

doi:10.1090/S0002-9939-1994-1179590-7

A homogeneous, globally solvable differential operator on a nilpotent Lie group which has no tempered fundamental solution
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Proc. Amer. Math. Soc. 121 (1994), 307-310 Request permission

Abstract:

We present an example of a homogeneous, left-invariant differential operator on the Heisenberg group ${H_3}$ which admits fundamental solutions but no tempered ones. This answers a question raised by Corwin in the negative.

References

Françoise Battesti, Résolubilité globale d’opérateurs différentiels invariants sur certains groupes de Lie, J. Funct. Anal. 77 (1988), no. 2, 261–308 (French, with English summary). MR 933970, DOI 10.1016/0022-1236(88)90088-2
Lawrence Corwin, Criteria for solvability of left invariant operators on nilpotent Lie groups, Trans. Amer. Math. Soc. 280 (1983), no. 1, 53–72. MR 712249, DOI 10.1090/S0002-9947-1983-0712249-3
L. Corwin and L. P. Rothschild, Necessary conditions for local solvability of homogeneous left invariant differential operators on nilpotent Lie groups, Acta Math. 147 (1981), no. 3-4, 265–288. MR 639041, DOI 10.1007/BF02392875
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 Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Daryl Geller, Liouville’s theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), no. 15, 1665–1677. MR 729197, DOI 10.1080/03605308308820319
Lars Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555–568. MR 124734, DOI 10.1007/BF02589517

Linear partial differential operators

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
S. Łojasiewicz, Sur le problème de la division, Studia Math. 18 (1959), 87–136 (French). MR 107168, DOI 10.4064/sm-18-1-87-136
D. Müller, A new criterion for local nonsolvability of homogeneous left invariant differential operators on nilpotent Lie groups, J. Reine Angew. Math. 416 (1991), 207–219. MR 1099951, DOI 10.1515/crll.1991.416.207
Detlef Müller, Local solvability of first order differential operators near a critical point, operators with quadratic symbols and the Heisenberg group, Comm. Partial Differential Equations 17 (1992), no. 1-2, 305–337. MR 1151265, DOI 10.1080/03605309208820843
D. Müller and F. Ricci, Analysis of second order differential operators on Heisenberg groups. I, Invent. Math. 101 (1990), no. 3, 545–582. MR 1062795, DOI 10.1007/BF01231515
D. Müller and F. Ricci, Analysis of second order differential operators on Heisenberg groups. II, J. Funct. Anal. 108 (1992), no. 2, 296–346. MR 1176678, DOI 10.1016/0022-1236(92)90027-G