Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Some nonanalytic-hypoelliptic sums of squares of vector fields


Author: Michael Christ
Journal: Bull. Amer. Math. Soc. 26 (1992), 137-140
MSC (2000): Primary 35H05
DOI: https://doi.org/10.1090/S0273-0979-1992-00258-6
MathSciNet review: 1110438
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Certain second-order partial differential operators, which are expressed as sums of squares of real-analytic vector fields in $ {\mathbb{R}^3}$ and which are well known to be $ {C^\infty }$ hypoelliptic, fail to be analytic hypoelliptic.


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

  • [BG] M. S. Baouendi and C. Goulaouic, Nonanalytic-hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. MR 0296507 (45:5567)
  • [C1] M. Christ, Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains, Internat. Math. Res. Notices 1 (1991), 31-40. MR 1115151 (92g:32031)
  • [C2] -, Remarks on the breakdown of analyticity for $ {\bar \partial _b}$, and Szegõ kernels, Proc. 1990 Sendai Conf. on Harmonic Analysis, Lecture Notes in Math. (to appear).
  • [C3] -, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations (to appear). MR 1133746 (92k:35056)
  • [C4] -, On the $ \bar \partial $ equation in weighted $ {L^2}$ norms in $ {\mathbb{C}^1}$, J. Geom. Anal. (to appear).
  • [CG] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. (to appear). MR 1166644 (93i:35034)
  • [CL] E. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • [DZ] M. Derridj and C. Zuily, Régularité analytique et Gevrey pour des classes d'opérateurs elliptiques paraboliques dégénérés du second ordre, Astérisque 2, 3 (1973), 371-381. MR 0393788 (52:14597)
  • [GS] A. Grigis and J. Sjöstrand, Front d'onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J. 52 (1985), 35-51. MR 791290 (86h:58136)
  • [HH] N. Hanges and A. A. Himonas, Singular solutions for sums of squares of vector fields, preprint. MR 1132794 (92i:35031)
  • [He] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations 44 (1982), 460-481. MR 661164 (84c:35026)
  • [Hl] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474 (36:5526)
  • [H2] -, The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, 1983. MR 717035 (85g:35002a)
  • [K] J. J. Kohn, Boundary behavior of $ \bar \partial $ on weakly pseudo-convex manifolds of dimension two, J. Differential Geom. 6 (1971), 523-542. MR 0322365 (48:727)
  • [M1] G. Métivier, Une class d'opérateurs non hypoélliptiques analytiques, Indiana Univ. Math. J. 29(1980), 823-860.
  • [M2] -, Analytic hypoellipticity for operators with multiple characteristics, Comm. Partial Differential Equations 6 (1981), 1-90. MR 597752 (82g:35030)
  • [N] A. Nagel, Vector fields and nonisotropic metrics, Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., no. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 241-306. MR 864374 (88f:42045)
  • [PR] Pham The Lai and D. Robert, Sur un problème aux valeurs propres non linéaire, Israel J. Math. 36 (1980), 169-186. MR 623203 (83b:35132)
  • [S] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J. 12 (1983), 392-433. MR 725588 (85e:35022)
  • [Ta] D. S. Tartakoff, On the local real analyticity of solutions to $ {\square _b}$ and the $ \bar \partial $-Neumann problem, Acta Math. 145 (1980), 117-204. MR 590289 (81k:35033)
  • [Tp] J.-M. Trepeau, Sur l'hypoellipticité analytique microlocale des opérateurs de type principal, Comm. Partial Differential Equations 9 (1984), 1119-1146. MR 759240 (86m:58144)
  • [Tv1] F. Trèves, Analytic-hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math. 24 (1971), 537-570. MR 0296509 (45:5569)
  • [Tv2] -, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $ \bar \partial $-Neumann problem, Comm. Partial Differential Equations 3 (1978), 475-642. MR 0492802 (58:11867)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 35H05

Retrieve articles in all journals with MSC (2000): 35H05


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1992-00258-6
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society