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

Fernando Cardoso; François Trèves

Annales de l'institut Fourier (1974)

- Volume: 24, Issue: 1, page 225-292
- ISSN: 0373-0956

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topCardoso, 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< +\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< +\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 -

## References

top- [1] A. GILIOLI and F. TREVES, ‘An example in the solvability theory of linear PDEʹsʹ, to appear in Amer. J. of Math. Zbl0308.35022
- [2] L. HORMANDER, Linear Partial Differential Operators, Springer, Berlin, 1963. Zbl0108.09301MR28 #4221
- [3] L. HORMANDER, ‘Pseudo-differential operators and nonelliptic boundary problems, Annals of Math., Vol. 83, (1966), 129-209. Zbl0132.07402MR38 #1387
- [4] L. HORMANDER, ‘Pseudo-differential operators’, Comm. Pure Appl. Math., Vol. XVIII, (1965), 501-517. Zbl0125.33401MR31 #4970
- [5] S. MIZOHATA and Y. OHYA, ‘Sur la condition de E. E. Levi concernant des équations hyperboliques’, Publ. Res. Inst. Math. Sci. Kyoto Univ. A, 4 (1968), 511-526. Zbl0202.37401MR43 #2349b
- [6] L. NIRENBERG and F. TREVES, ‘On local solvability of linear partial differential equations. Part I : Necessary conditions’, Comm. Pure Appl. Math., Vol. XXIII, (1970), 1-38. Zbl0191.39103MR41 #9064a
- [7] J. SJOSTRAND, ‘Une classe d'opérateurs pseudodifférentiels à caractéristiques multiples’, C.R. Acad. Sc. Paris, t. 275 (1972), 817-819. Zbl0252.47052MR49 #4055
- [8] F. TREVES, Ovcyannikov theorem and hyperdifferential operators, Notas de Matematica, Rio de Janeiro (Brasil), 1968. Zbl0205.39202

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