Pseudodifferential operators are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as with elliptic, homogeneous of degree , and homogeneous of degree one, satisfying the following condition : there is a point in the characteristic variety and a complex number such that at and such that the restriction of to the bicharacteristic strip of vanishes of order at , changing sign there from minus to plus. It is then proved that is not locally solvable at , regardless of what the lower order terms might be.
On étudie des opérateurs pseudodifférentiels du point de vue de la résolubilité locale et sous l’hypothèse que le symbole principal se factorise sous la forme au voisinage (dans le fibré cotangent) d’un point où (de plus est elliptique en ce point, et est homogène de degré ; est homogène de degré 1). On fait l’hypothèse suivante : il existe un nombre complexe tel que en et tel que la restriction de à la bande bicaractéristique de , passant par ce point, a un zéro d’ordre fini en et change de signe en ce point de moins à plus. On démontre alors que n’est pas localement résoluble en quels que soient les termes d’ordre inférieur .
@article{AIF_1974__24_1_225_0, 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 = {http://www.numdam.org/articles/10.5802/aif.499/} }
TY - JOUR AU - Cardoso, Fernando AU - Trèves, François TI - A necessary condition of local solvability for pseudo-differential equations with double characteristics JO - Annales de l'Institut Fourier PY - 1974 SP - 225 EP - 292 VL - 24 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.499/ DO - 10.5802/aif.499 LA - en ID - AIF_1974__24_1_225_0 ER -
%0 Journal Article %A Cardoso, Fernando %A Trèves, François %T A necessary condition of local solvability for pseudo-differential equations with double characteristics %J Annales de l'Institut Fourier %D 1974 %P 225-292 %V 24 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.499/ %R 10.5802/aif.499 %G en %F 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/articles/10.5802/aif.499/
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