Geodesic
Nondegenerate subelliptic pseudodifferential operators
Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 291-309 
Yu. V. Egorov. Nondegenerate subelliptic pseudodifferential operators. Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 291-309. http://geodesic.mathdoc.fr/item/SM_1970_11_3_a0/
@article{SM_1970_11_3_a0,
     author = {Yu. V. Egorov},
     title = {Nondegenerate subelliptic pseudodifferential operators},
     journal = {Sbornik. Mathematics},
     pages = {291--309},
     year = {1970},
     volume = {11},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_11_3_a0/}
}
TY  - JOUR
AU  - Yu. V. Egorov
TI  - Nondegenerate subelliptic pseudodifferential operators
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 291
EP  - 309
VL  - 11
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1970_11_3_a0/
LA  - en
ID  - SM_1970_11_3_a0
ER  - 
%0 Journal Article
%A Yu. V. Egorov
%T Nondegenerate subelliptic pseudodifferential operators
%J Sbornik. Mathematics
%D 1970
%P 291-309
%V 11
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1970_11_3_a0/
%G en
%F SM_1970_11_3_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we study scalar pseudodifferential operators for which the gradient $\operatorname{grad}_{x,\xi}p^0(x,\xi)$ of the principal part of the symbol does not vanish and is not proportional to a real vector at any characteristic point $(x,\xi)\in\Omega\times\{\mathbf R^n\setminus0\}$. Such operators are called nondegenerate. It is assumed in addition that for each point of $\Omega\times\{\mathbf R^n\setminus0\}$ there exists an operator in the Lie algebra generated by the operators $P$ and $P^*$ the principal part of the symbol of which does not vanish at this point. For these operators we present here hypoellipticity conditions, conditions for the local solvability of the equation $Pu=f$, a theorem on the smoothness of the solutions of this equation, and so on. All of the conditions obtained have a simple algebraic character and are exact, necessary and sufficient. Bibliography: 13 titles.

[1] L. Hörmander, “Pseudo-differential operators”, Comm. Pure Appl. Math., 18:3 (1965), 501–517 | DOI | MR | Zbl

[2] L. Hörmander, “Pseudo-differential operators and nonelliptic boundary problems”, Ann. of Math. (2), 83:6 (1966), 129–209 | DOI | MR | Zbl

[3] L. Hörmander, “Pseudo-differential operators and hypoelliptic equations”, Proc. of Symp. in Pure Math., X, 1967, 129 | MR

[4] L. Hormander, “Hypoelliptic second order differential equations”, Acta Math., 119 (1968), 147–171 | DOI | MR

[5] K. Moren, Metody gilbertova prostranstva, Mir, Moskva, 1965 | MR

[6] E. V. Radkevich, “Ob odnoi teoreme L. Khërmandera”, Uspekhi matem. nauk, XXIV:2(146) (1969), 233–234

[7] Yu. V. Egorov, “Psevdodifferentsialnye operatory glavnogo tipa”, Matem. sb., 73(115) (1967), 356–374 | Zbl

[8] Yu. V. Egorov, “Ob odnom klasse psevdodifferentsialnykh operatorov”, Matem. sb., 79(121) (1969), 59–77 | MR

[9] Yu. V. Egorov, “Ob apriornykh otsenkakh dlya differentsialnykh operatorov pervogo poryadka”, DAN SSSR, 185:3 (1969), 507–508 | MR | Zbl

[10] Yu. V. Egorov, “Apriornye otsenki dlya differentsialnykh operatorov pervogo poryadka”, Funk. analiz, 3:3 (1969), 53–60 | MR | Zbl

[11] Yu. V. Egorov, “O kanonicheskikh preobrazovaniyakh i psevdodifferentsialnykh operatorakh”, Uspekhi matem. nauk, XXIV:5(149) (1969), 235–236 | MR | Zbl

[12] Yu. V. Egorov, “Ob usloviyakh razreshimosti psevdodifferentsialnykh uravnenii”, DAN SSSR, 187:6 (1969), 1232–1234 | Zbl

[13] Yu. V. Egorov, “O nevyrozhdennykh subellipticheskikh psevdodifferentsialnykh operatorakh”, DAN SSSR, 186:5 (1969), 1006–1008