Failure of analytic hypoellipticity in a class of differential operators

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1. [1] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. Zbl0156.10701MR222474
2. [2] M. S. Baouendi – Ch. Goulaouic, Non-analytic hypoellipticity for some degenerate operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. Zbl0276.35023MR296507
3. [3] M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 16 (1991), 1695-1707. Zbl0762.35017MR1133746
4. [4] M. Christ, A class of hypoelliptic PDE admitting nonanalytic solutions, Contemp. Math. 137 (1992), 155-167. Zbl0804.35022MR1190978
5. [5] M. Christ, Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem, Duke Math. J. 72, No. 3 (1993), 595-639. Zbl0802.35025MR1253617
6. [6] M. Christ, A necessary condition for analytic hypoellipticity, Math. Res. Lett. 1 (1994), 241-248. Zbl0841.35026MR1266762
7. [7] M. Christ, Examples of analytic nonhypoellipticity of ${\overline{\partial }}_b$, Comm. Partial Differential Equations 19, no. 5-6, (1994) 911–941. Zbl0808.35021MR1274545
8. [8] M. Christ, Analytic hypoellipticity in dimension two, M.S.R.I. Preprint No. 1996-009. MR1453596
9. [9] M. Christ – D. Geller, Counterexamples to analytic hypoellipticity for domain of finite type, Ann. of Math. 135 (1992), 551-566. Zbl0758.35024MR1166644
10. [10] M. Derridj – D. S. Tartakoff, Local analyticity for the ${\square }_b$ problem and the $\overline{\partial }$-Neumann problem at certain weakly pseudoconvex domains, Comm. Partial Differential Equations 12 (1988), 1521-1600. Zbl0736.35070MR970155
11. [11] M. Derridj – D. S. Tartakoff, Analyticité local pour le problème de $\overline{\partial }$-Neumann en des points de faible pseudoconvexité, C.R. Acad. Sci. Paris Sér. I Math. 306 (1988), 429-432. Zbl0636.32009MR937979
12. [12] M. Derridj – D. S. Tartakoff, Local analyticity in the $\overline{\partial }$-Neumann problem for some model domains without maximal estimates, Duke Math. J. 64, No. 2 (1991), 377-402. Zbl0790.35078MR1136382
13. [13] M. Derridj – D. S. Tartakoff, Microlocal analyticity for the canonical solution to $\overline{\partial }$ on some rigid weakly pseudoconvex hypersurfaces in ${ℂ}^2$, Comm. Partial Differential Equations 20 (1995), 1647-1667. Zbl0833.35100MR1349226
14. [14] M. Derridj – C. Zuily, Régularité analytique et Gevrey d’opérateurs elliptiques dégénérés, J. Math. Pures Appl. 52 (1973), 65-80. Zbl0263.35020MR390474
15. [15] G. Francsics – N. Hanges, Analytic Regularity for the Bergman kernel, Journée Equations aux dérivées partielles, Saint-Jean de Monts, 2-5 juin 1998. Zbl1010.32003MR1640378
16. [16] G. Francsics – N. Hanges, Treves curves and the Szegö kernel, Indiana Univ. Math. J. 47 (1998), 995-1009. Zbl0939.32001MR1665733
17. [17] G. Francsics – N. Hanges, Analytic singularities of the Bergman kernel for tubes, to appear in Duke Math. J. Zbl1016.32014MR1838661
18. [18] G. Francsics – N. Hanges, Analytic singularities, Contemp. Math. 205 (1997), 69-78. Zbl0892.32019MR1447216
19. [19] A. Grigis – J. Sjöstrand, Front d’onde analytique et somme de carres de champes de vecteurs, Duke Math. J. 52 (1985) 35-51. Zbl0581.35009MR791290
20. [20] N. Hanges – A. A. Himonas, Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503-1511. Zbl0745.35011MR1132794
21. [21] N. Hanges – A. A. Himonas, Singular solutions for a class of Grusin type operators, Proceedings of the AMS 124 (1996), 1549-1557. Zbl0858.35025MR1307525
22. [22] N. Hanges – A. A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, Proc. of the Amer. Math. Soc. 126 (1998), 405-409. Zbl0906.35027MR1422872
23. [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. Zbl0458.35019MR661164
24. [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. Zbl0846.35034MR1365248
25. [25] G. Métivier, Analytic hypoellipticity for operators with multiple characteristics, Comm. Partial Differential Equations 6 (1980), 1-90. Zbl0455.35040MR597752
26. [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. Zbl0481.35033MR609762
27. [27] Pham The Lai – D. Robert, Sur un problème aux valueurs propres non linèaire, Israel J. Math. 36 (1980), 169-1886. Zbl0444.35065MR623203
28. [28] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J. 12 (1983), 392-433. Zbl0531.35022MR725588
29. [29] D. Tartakoff, On the local real analyticity of solutions to ${\square }_b$ and the $\overline{\partial }$-problem, Acta Math.145 (1980), 117-204. Zbl0456.35019MR590289
30. [30] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline{\partial }$-problem, Comm. Partial Differential Equations 3 (1978), 475-642. Zbl0384.35055MR492802
31. [31] F. Treves, Symplectic geometry and analytic hypo-ellipticity, Proc. Symp. Pure Math., Vol. 65 (1999), 201-219. Zbl0938.35038MR1662756
32. [32] Y. Sibuya, “Global theory of a second order linear ordinary differential equation with a polynomial coefficient”, North-Holland Publ., 1975 Zbl0322.34006MR486867
33. [33] V. S. Varadarajan, Meromorphic Differential Equations, Expositiones Mathematicae, 9, No. 2 (1991). Zbl0722.34006MR1101951
34. [34] W. Wasow, “Asymptotic expansions for ordinary differential equations”, New York- London-Sydney, Interscience Publishers, IX, 1965. Zbl0133.35301MR203188
35. [35] C-C. Yu, Nonlinear eigenvalues and analytic-hypoellipticity, Mem. Amer. Math. Soc. 134, no. 636, viii + 92 pp., 1998. Zbl0913.35100MR1432145

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