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 $P\left(x,D\right)\sim {\sum }_{j=0}^{+\infty }{P}_{m-j}\left(x,D\right)$ 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 $\left({x}_{0},{\xi }^{0}\right)$ in the characteristic variety $L=0$ and a complex number $z$ such that ${d}_{\xi }\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)\ne 0$ at $\left({x}_{0},{\xi }^{0}\right)$ and such that the restriction of $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)$ to the bicharacteristic strip of $\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)$ vanishes of order $k<+\infty$ at $\left({x}_{0},{\xi }^{0}\right)$, changing sign there from minus to plus. It is then proved that $P\left(x,D\right)$ is not locally solvable at ${x}_{0}$, regardless of what the lower order terms ${P}_{m-j}\phantom{\rule{0.166667em}{0ex}}\left(j=1,2,...\right)$ might be.

On étudie des opérateurs pseudodifférentiels $P\left(x,D\right)\sim {\sum }_{j=0}^{+\infty }{P}_{m-j}\left(x,D\right)$ 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 $\left({x}_{0},{\xi }^{0}\right)$$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 ${d}_{\xi }\phantom{\rule{0.166667em}{0ex}}\text{Re}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)\ne 0$ en $\left({x}_{0},{\xi }^{0}\right)$ et tel que la restriction de $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)$ à la bande bicaractéristique de $\text{Re}\phantom{\rule{0.166667em}{0ex}}\left(zL\right)$, passant par ce point, a un zéro d’ordre fini $k<+\infty$ en $\left({x}_{0},{\xi }^{0}\right)$ et change de signe en ce point de moins à plus. On démontre alors que $P\left(x,D\right)$ n’est pas localement résoluble en ${x}_{0}$ quels que soient les termes d’ordre inférieur ${P}_{m-j}\phantom{\rule{0.166667em}{0ex}}\left(j=1,2,...\right)$.

@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},
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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