A necessary condition of local solvability for pseudo-differential equations with double characteristics

Cardoso, Fernando; Trèves, François

doi:10.5802/aif.499

Cardoso, Fernando ; Trèves, François

Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 225-292.

Résumé
Abstract

On étudie des opérateurs pseudodifférentiels $P (x, D) \sim \sum_{j = 0}^{+ \infty} P_{m - j} (x, D)$ du point de vue de la résolubilité locale et sous l’hypothèse que le symbole principal se factorise sous la forme $P_{m} = Q L^{2}$ au voisinage (dans le fibré cotangent) d’un point $(x_{0}, ξ^{0})$ où $L = 0$ (de plus $Q$ est elliptique en ce point, et est homogène de degré $m - 2$ ; $L$ est homogène de degré 1). On fait l’hypothèse suivante : il existe un nombre complexe $z$ tel que $d_{ξ} Re (z L) \neq 0$ en $(x_{0}, ξ^{0})$ et tel que la restriction de $Im (z L)$ à la bande bicaractéristique de $Re (z L)$ , passant par ce point, a un zéro d’ordre fini $k < + \infty$ en $(x_{0}, ξ^{0})$ et change de signe en ce point de moins à plus. On démontre alors que $P (x, D)$ n’est pas localement résoluble en $x_{0}$ quels que soient les termes d’ordre inférieur $P_{m - j} (j = 1, 2, ...)$ .

Pseudodifferential operators $P (x, D) \sim \sum_{j = 0}^{+ \infty} P_{m - j} (x, D)$ are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as $P_{m} = Q L^{2}$ with $Q$ elliptic, homogeneous of degree $m - 2$ , and $L$ homogeneous of degree one, satisfying the following condition : there is a point $(x_{0}, ξ^{0})$ in the characteristic variety $L = 0$ and a complex number $z$ such that $d_{ξ} Re (z L) \neq 0$ at $(x_{0}, ξ^{0})$ and such that the restriction of $Im (z L)$ to the bicharacteristic strip of $Re (z L)$ vanishes of order $k < + \infty$ at $(x_{0}, ξ^{0})$ , changing sign there from minus to plus. It is then proved that $P (x, D)$ is not locally solvable at $x_{0}$ , regardless of what the lower order terms $P_{m - j} (j = 1, 2, ...)$ might be.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.499

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     author = {Cardoso, Fernando and Tr\`eves, Fran\c{c}ois},
     title = {A necessary condition of local solvability for pseudo-differential equations with double characteristics},
     journal = {Annales de l'Institut Fourier},
     pages = {225--292},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {1},
     year = {1974},
     doi = {10.5802/aif.499},
     mrnumber = {50 #2726},
     zbl = {0273.35058},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.499/}
}

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Cardoso, Fernando; Trèves, François. A necessary condition of local solvability for pseudo-differential equations with double characteristics. Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 225-292. doi : 10.5802/aif.499. https://www.numdam.org/articles/10.5802/aif.499/

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