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ISSN 1088-9485(e) ISSN 0273-0979(p)

Some nonanalytic-hypoelliptic sums of squares of vector fields

Author(s): Michael Christ
Journal: Bull. Amer. Math. Soc. 26 (1992), 137-140.
MSC (2000): Primary 35H05
MathSciNet review: 1110438
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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:

[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)

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