Propagation des singularités pour les opérateurs différentiels de type principal localement résolubles à coefficients analytiques en dimension 2
Godin, Paul
Annales de l'Institut Fourier, Tome 29 (1979), p. 223-245 / Harvested from Numdam
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Sur une variété analytique paracompacte de dimension 2, on considère un opérateur différentiel P à symbole principal p m analytique vérifiant la condition (𝒫) de Nirenberg et Treves. En ajoutant une nouvelle variable et en utilisant des estimations a priori de type Carleman, on montre qu’il y a propagation des singularités pour P, dans p m -1 (0), le long des feuilles intégrales du système différentiel engendré par les champs hamiltoniens de Rep m et Imp m .

On a paracompact analytic manifold of dimension 2, one considers a differential operator P with analytic principal symbol p m satisfying the condition (𝒫) of Nirenberg and Treves. Adding a new variable and using a priori estimates of Carleman type, one shows that there is propagation of singularities for P, in p m -1 (0), along the integral leaves of the differential system generated by the Hamiltonian vector fields of Rep m and Imp m .

@article{AIF_1979__29_2_223_0,
     author = {Godin, Paul},
     title = {Propagation des singularit\'es pour les op\'erateurs diff\'erentiels de type principal localement r\'esolubles \`a coefficients analytiques en dimension 2},
     journal = {Annales de l'Institut Fourier},
     volume = {29},
     year = {1979},
     pages = {223-245},
     doi = {10.5802/aif.748},
     mrnumber = {81m:35024},
     zbl = {0365.58019},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1979__29_2_223_0}
}

Godin, Paul. Propagation des singularités pour les opérateurs différentiels de type principal localement résolubles à coefficients analytiques en dimension 2. Annales de l'Institut Fourier, Tome 29 (1979) pp. 223-245. doi : 10.5802/aif.748. http://gdmltest.u-ga.fr/item/AIF_1979__29_2_223_0/

[1] F. Cardoso et F. Treves, Subelliptic Operators, dans Analyse fonctionnelle et applications, Hermann 1975, 161-169. | MR 58 #1565 | Zbl 0305.35087

[2] J.-J. Duistermaat et L. Hörmander, Fourier Integral Operators II, Acta Math., 128 (1972), 184-269. | MR 52 #9300 | Zbl 0232.47055

[3] B. Helffer, Addition de variables et application à la régularité (Rennes, mai 1976).

[4] L. Hörmander, Fourier Integral Operators I, Acta Math., 127 (1971), 79-183. | MR 52 #9299 | Zbl 0212.46601

[5] L. Hörmander, On the existence and the regularity of solutions of linear pseudodifferential equations, L'Ens. Math., 17 (1971), 99-163. | Zbl 0224.35084

[6] A. Menikoff, Carleman estimates for partial differential equations with real coefficients, Arch. Rational Mech. Anal., 54 (1974), 118-133. | MR 49 #5533 | Zbl 0287.35081

[7] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan, 18, 4 (1966), 398-404. | MR 33 #8005 | Zbl 0147.23502

[8] L. Nirenberg et F. Treves, On local solvability of linear partial differential equations. Part II. Sufficient conditions, Comm. Pure Appl. Math., 23 (1970), 459-510. | MR 41 #9064b | Zbl 0208.35902

[9] J. Sjöstrand, Propagation of singularities for operators with multiple involutive characteristics, Report n° 11, Institut Mittag-Leffler (1974).

[10] M. Strauss et F. Treves, First order linear partial differential equations and uniqueness in the Cauchy problem, Journal Diff. Eq., 15 (1974), 195-209. | MR 48 #9076 | Zbl 0266.35009

[11] F. Treves, On the local solvability of linear partial differential equations in two independent variables, Amer. Journ. Math., 92 (1970), 174-204. | MR 41 #3974 | Zbl 0236.35039

[12] F. Treves, A new method of proof of the subelliptic estimates, Comm. Pure Appl. Math., 24 (1971), 71-115. | MR 44 #7385 | Zbl 0206.11401

[13] F. Treves, A link between solvability of linear pseudo-differential equations and uniqueness in the Cauchy problem, Amer. J. Math., 94 (1972), 267-288. | MR 46 #7740 | Zbl 0274.35054