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

Cardoso, Fernando, and Trèves, François. "A necessary condition of local solvability for pseudo-differential equations with double characteristics." Annales de l'institut Fourier 24.1 (1974): 225-292. <http://eudml.org/doc/74163>.

@article{Cardoso1974,
abstract = {Pseudodifferential operators $P(x,D)\sim \sum ^\{+\infty \}_\{j=0\}P_\{m-j\}(x,D)$ 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 $(x_0,\xi ^0)$ in the characteristic variety $L=0$ and a complex number $z$ such that $d_\xi \, \{\rm Re\}\, (zL)\ne 0$ at $(x_0,\xi ^0)$ and such that the restriction of $\{\rm Im\}\, (zL)$ to the bicharacteristic strip of $\{\rm Re\}\, (zL)$ vanishes of order $k&lt; +\infty $ at $(x_0,\xi ^0)$, 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 $P_\{m-j\}\, (j=1,2,\ldots )$ might be.},
author = {Cardoso, Fernando, Trèves, François},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {225-292},
publisher = {Association des Annales de l'Institut Fourier},
title = {A necessary condition of local solvability for pseudo-differential equations with double characteristics},
url = {http://eudml.org/doc/74163},
volume = {24},
year = {1974},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 1
SP - 225
EP - 292
AB - Pseudodifferential operators $P(x,D)\sim \sum ^{+\infty }_{j=0}P_{m-j}(x,D)$ 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 $(x_0,\xi ^0)$ in the characteristic variety $L=0$ and a complex number $z$ such that $d_\xi \, {\rm Re}\, (zL)\ne 0$ at $(x_0,\xi ^0)$ and such that the restriction of ${\rm Im}\, (zL)$ to the bicharacteristic strip of ${\rm Re}\, (zL)$ vanishes of order $k&lt; +\infty $ at $(x_0,\xi ^0)$, 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 $P_{m-j}\, (j=1,2,\ldots )$ might be.
LA - eng
UR - http://eudml.org/doc/74163
ER -