Non-analytic hypoellipticity in the presence of symplecticity

Hanges, Nicholas; Himonas, A.

doi:10.1090/S0002-9939-98-04115-X

Non-analytic hypoellipticity in the presence of symplecticity
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by Nicholas Hanges and A. Alexandrou Himonas PDF

Proc. Amer. Math. Soc. 126 (1998), 405-409 Request permission

Abstract:

Here we construct non-analytic solutions to a class of hypoelliptic operators with symplectic characteristic set and in the form of a sum of squares of real analytic vector fields.

References

M. S. Baouendi and C. Goulaouic, Nonanalytic-hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483–486. MR 296507, DOI 10.1090/S0002-9904-1972-12955-0
Michael Christ, A class of hypoelliptic PDE admitting nonanalytic solutions, The Madison Symposium on Complex Analysis (Madison, WI, 1991) Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 155–167. MR 1190978, DOI 10.1090/conm/137/1190978
Michael Christ, A necessary condition for analytic hypoellipticity, Math. Res. Lett. 1 (1994), no. 2, 241–248. MR 1266762, DOI 10.4310/MRL.1994.v1.n2.a11
Michael Christ, On local and global analytic and Gevrey hypoellipticity, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1995) École Polytech., Palaiseau, 1995, pp. Exp. No. IX, 7. MR 1360478
M. Derridj and C. Zuily, Régularité analytique et Gevrey d’opérateurs elliptiques dégénérés, J. Math. Pures Appl. (9) 52 (1973), 65–80 (French). MR 390474
A. Grigis and J. Sjöstrand, Front d’onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J. 52 (1985), no. 1, 35–51 (French). MR 791290, DOI 10.1215/S0012-7094-85-05203-2
Nicholas Hanges and A. Alexandrou Himonas, Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1503–1511. MR 1132794, DOI 10.1080/03605309108820808
Nicholas Hanges and A. Alexandrou Himonas, Singular solutions for a class of Grusin type operators, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1549–1557. MR 1307525, DOI 10.1090/S0002-9939-96-03180-2
Bernard Helffer, Conditions nécessaires d’hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations 44 (1982), no. 3, 460–481 (French). MR 661164, DOI 10.1016/0022-0396(82)90008-0
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
J. J. Kohn, Pseudo-differential operators and hypoellipticity, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 61–69. MR 0338592
Guy Métivier, Une classe d’opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980), no. 6, 823–860 (French). MR 589650, DOI 10.1512/iumj.1980.29.29059
O. A. Oleĭnik and E. V. Radkevič, The analyticity of the solutions of linear differential equations and systems, Dokl. Akad. Nauk SSSR 207 (1972), 785–788 (Russian). MR 0324192
O. A. Oleĭnik and E. V. Radkevič, Second order equations with nonnegative characteristic form, Plenum Press, New York-London, 1973. Translated from the Russian by Paul C. Fife. MR 0457908, DOI 10.1007/978-1-4684-8965-1
Phạm The Lại and D. Robert, Sur un problème aux valeurs propres non linéaire, Israel J. Math. 36 (1980), no. 2, 169–186 (French, with English summary). MR 623203, DOI 10.1007/BF02937354
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
David S. Tartakoff, The local real analyticity of solutions to $\square _{b}$ and the $\bar \partial$-Neumann problem, Acta Math. 145 (1980), no. 3-4, 177–204. MR 590289, DOI 10.1007/BF02414189
François Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline \partial$-Neumann problem, Comm. Partial Differential Equations 3 (1978), no. 6-7, 475–642. MR 492802, DOI 10.1080/03605307808820074