A necessary condition of local solvability for pseudo-differential equations with double characteristics
Annales de l'Institut Fourier, Volume 24 (1974) no. 1, p. 225-292

Pseudodifferential operators P(x,D) j=0 + 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 =QL 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 (zL)0 at (x 0 ,ξ 0 ) and such that the restriction of Im (zL) to the bicharacteristic strip of Re (zL) vanishes of order k<+ 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.

On étudie des opérateurs pseudodifférentiels P(x,D) j=0 + 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 =QL 2 au voisinage (dans le fibré cotangent) d’un point (x 0 ,ξ 0 )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(zL)0 en (x 0 ,ξ 0 ) et tel que la restriction de Im (zL) à la bande bicaractéristique de Re(zL), passant par ce point, a un zéro d’ordre fini k<+ 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,...).

@article{AIF_1974__24_1_225_0,
     author = {Cardoso, Fernando and Tr\`eves, Fran\c cois},
     title = {A necessary condition of local solvability for pseudo-differential equations with double characteristics},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {24},
     number = {1},
     year = {1974},
     pages = {225-292},
     doi = {10.5802/aif.499},
     zbl = {0273.35058},
     mrnumber = {50 \#2726},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1974__24_1_225_0}
}
Cardoso, Fernando; Trèves, François. A necessary condition of local solvability for pseudo-differential equations with double characteristics. Annales de l'Institut Fourier, Volume 24 (1974) no. 1, pp. 225-292. doi : 10.5802/aif.499. http://www.numdam.org/item/AIF_1974__24_1_225_0/

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