Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Detlef Müller
Journal: Proc. Amer. Math. Soc. 121 (1994), 307-310
MSC: Primary 22E30; Secondary 22E25
DOI: https://doi.org/10.1090/S0002-9939-1994-1179590-7
MathSciNet review: 1179590
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [B] F. Battesti, Résolubilité globale d'opérateurs différentiels invariants sur certains groupes de Lie, J. Funct. Anal. 77 (1988), 261-308. MR 933970 (89m:22013)
  • [C] L. Corwin, Criteria for solvability of left-invariant operators on nilpotent Lie groups, Trans. Amer. Math. Soc. 280 (1983), 53-72. MR 712249 (85g:22013)
  • [CR] 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), 265-288. MR 639041 (83b:22010)
  • [F] G. B. Folland, Subelliptic estimates and function spaces on nilpotent groups, Ark. Mat. 13 (1975), 161-207. MR 0494315 (58:13215)
  • [FS] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Princeton Univ. Press, Princeton, NJ, 1982. MR 657581 (84h:43027)
  • [G] D. Geller, Liouville's theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), 1655-1677. MR 729197 (85f:58109)
  • [H1] L. Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555-568. MR 0124734 (23:A2044)
  • [H2] -, Linear partial differential operators, Springer, New York, 1964.
  • [HN] B. Helffer and J. Nourrigat, Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4 (1979), 899-958. MR 537467 (81i:35034)
  • [L] S. Lojasiewicz, Sur le problème de division, Studia Math. 18 (1959), 87-136. MR 0107168 (21:5893)
  • [M1] D. Müller, A new criterion for local non-solvability of homogeneous left-invariant differential operators on nilpotent Lie groups, J. Reine Angew. Math. 416 (1991), 207-219. MR 1099951 (92f:22011)
  • [M2] -, 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), 305-337. MR 1151265 (93g:35005)
  • [MR1] D. Müller and F. Ricci, Analysis of second order differential operators on Heisenberg groups. I, Invent. Math. 101 (1990), 545-582. MR 1062795 (92d:22011)
  • [MR2] -, Analysis of second order differential operators on Heisenberg groups. II, J. Funct. Anal. 108 (1992), 296-346. MR 1176678 (93j:22019)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22E30, 22E25

Retrieve articles in all journals with MSC: 22E30, 22E25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1179590-7
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society