Failure of analytic hypoellipticity in a class of differential operators

Costin, Ovidiu; Costin, Rodica D.

Costin, Ovidiu ; Costin, Rodica D.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 21-45.

Résumé

For the hypoelliptic differential operators $P = \partial_{x}^{2} + {(x^{k} \partial_{y} - x^{l} \partial_{t})}^{2}$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k, l) = (0, 1)$ ), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

MR Zbl

Classification : 35B65

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Costin, Ovidiu; Costin, Rodica D. Failure of analytic hypoellipticity in a class of differential operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 21-45. http://www.numdam.org/item/ASNSP_2003_5_2_1_21_0/

Bibliographie
Cité par

[1] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. | MR | Zbl

[2] M. S. Baouendi - Ch. Goulaouic, Non-analytic hypoellipticity for some degenerate operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. | MR | Zbl

[3] M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 16 (1991), 1695-1707. | MR | Zbl

[4] M. Christ, A class of hypoelliptic PDE admitting nonanalytic solutions, Contemp. Math. 137 (1992), 155-167. | MR | Zbl

[5] M. Christ, Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem, Duke Math. J. 72, No. 3 (1993), 595-639. | MR | Zbl

[6] M. Christ, A necessary condition for analytic hypoellipticity, Math. Res. Lett. 1 (1994), 241-248. | MR | Zbl

[7] M. Christ, Examples of analytic nonhypoellipticity of ${\bar{\partial}}_{b}$ , Comm. Partial Differential Equations 19, no. 5-6, (1994) 911-941. | MR | Zbl

[8] M. Christ, Analytic hypoellipticity in dimension two, M.S.R.I. Preprint No. 1996-009. | MR

[9] M. Christ - D. Geller, Counterexamples to analytic hypoellipticity for domain of finite type, Ann. of Math. 135 (1992), 551-566. | MR | Zbl

[10] M. Derridj - D. S. Tartakoff, Local analyticity for the $□_{b}$ problem and the $\bar{\partial}$ -Neumann problem at certain weakly pseudoconvex domains, Comm. Partial Differential Equations 12 (1988), 1521-1600. | MR | Zbl

[11] M. Derridj - D. S. Tartakoff, Analyticité local pour le problème de $\bar{\partial}$ -Neumann en des points de faible pseudoconvexité, C.R. Acad. Sci. Paris Sér. I Math. 306 (1988), 429-432. | MR | Zbl

[12] M. Derridj - D. S. Tartakoff, Local analyticity in the $\bar{\partial}$ -Neumann problem for some model domains without maximal estimates, Duke Math. J. 64, No. 2 (1991), 377-402. | MR | Zbl

[13] M. Derridj - D. S. Tartakoff, Microlocal analyticity for the canonical solution to $\bar{\partial}$ on some rigid weakly pseudoconvex hypersurfaces in $ℂ^{2}$ , Comm. Partial Differential Equations 20 (1995), 1647-1667. | MR | Zbl

[14] M. Derridj - C. Zuily, Régularité analytique et Gevrey d'opérateurs elliptiques dégénérés, J. Math. Pures Appl. 52 (1973), 65-80. | MR | Zbl

[15] G. Francsics - N. Hanges, Analytic Regularity for the Bergman kernel, Journée Equations aux dérivées partielles, Saint-Jean de Monts, 2-5 juin 1998. | Numdam | MR | Zbl

[16] G. Francsics - N. Hanges, Treves curves and the Szegö kernel, Indiana Univ. Math. J. 47 (1998), 995-1009. | MR | Zbl

[17] G. Francsics - N. Hanges, Analytic singularities of the Bergman kernel for tubes, to appear in Duke Math. J. | MR | Zbl

[18] G. Francsics - N. Hanges, Analytic singularities, Contemp. Math. 205 (1997), 69-78. | MR | Zbl

[19] A. Grigis - J. Sjöstrand, Front d'onde analytique et somme de carres de champes de vecteurs, Duke Math. J. 52 (1985) 35-51. | MR | Zbl

[20] N. Hanges - A. A. Himonas, Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503-1511. | MR | Zbl

[21] N. Hanges - A. A. Himonas, Singular solutions for a class of Grusin type operators, Proceedings of the AMS 124 (1996), 1549-1557. | MR | Zbl

[22] N. Hanges - A. A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, Proc. of the Amer. Math. Soc. 126 (1998), 405-409. | MR | Zbl

[23] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des operateurs invariants a gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations 44 (1982), 460-481. | MR | Zbl

[24] T. Hoshiro, Failure of analytic hypoellipticity for some operators of $X^{2} + Y^{2}$ type, J. Math. Kyoto Univ. (JMKYAZ) 35 (1995) 569-581. | MR | Zbl

[25] G. Métivier, Analytic hypoellipticity for operators with multiple characteristics, Comm. Partial Differential Equations 6 (1980), 1-90. | MR | Zbl

[26] G. Métivier, Non-hypoellipticité analytique pour $D_{x}^{2} + (x^{2} + y^{2}) D_{y}^{2}$ , C.R. Acad. Sci. Paris Sér. I Math. 292 (1981), 401-404. | MR | Zbl

[27] Pham The Lai - D. Robert, Sur un problème aux valueurs propres non linèaire, Israel J. Math. 36 (1980), 169-1886. | MR | Zbl

[28] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J. 12 (1983), 392-433. | MR | Zbl

[29] D. Tartakoff, On the local real analyticity of solutions to $□_{b}$ and the $\bar{\partial}$ -problem, Acta Math.145 (1980), 117-204. | MR | Zbl

[30] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\bar{\partial}$ -problem, Comm. Partial Differential Equations 3 (1978), 475-642. | MR | Zbl

[31] F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proc. Symp. Pure Math., Vol. 65 (1999), 201-219. | MR | Zbl

[32] Y. Sibuya, “Global theory of a second order linear ordinary differential equation with a polynomial coefficient”, North-Holland Publ., 1975 | MR | Zbl

[33] V. S. Varadarajan, Meromorphic Differential Equations, Expositiones Mathematicae, 9, No. 2 (1991). | MR | Zbl

[34] W. Wasow, “Asymptotic expansions for ordinary differential equations”, New York- London-Sydney, Interscience Publishers, IX, 1965. | MR | Zbl

[35] C-C. Yu, Nonlinear eigenvalues and analytic-hypoellipticity, Mem. Amer. Math. Soc. 134, no. 636, viii + 92 pp., 1998. | MR | Zbl