Geodesic
On a criterion for hypoellipticity
Sbornik. Mathematics, Tome 14 (1971) no. 1, pp. 15-45 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper a criterion for hypoellipticity is proved which is formulated in terms of certain estimates in the $H_{(s)}$ norms, and which is a generalization of a criterion of Trèves. With the use of this criterion it is possible to prove the hypoellipticity of certain operators that do not satisfy Hörmander's criterion. It is proved, for example, that the operator $P=\partial^2/\partial x^2+\varphi^2(x)\partial^2/\partial y^2$ is hypoelliptic, where $\varphi(x)$ is an infinitely differentiable function that is not equal to zero for $x\ne0$ and has a zero of infinite order at $x=0$. Bibliography: 10 titles. 
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V. S. Fedii. On a criterion for hypoellipticity. Sbornik. Mathematics, Tome 14 (1971) no. 1, pp. 15-45. http://geodesic.mathdoc.fr/item/SM_1971_14_1_a1/

[1] F. Treves, “An invariant criterion of hypoellipticity”, Amer. J. Math., 83:4 (1961), 645–668 | DOI | MR | Zbl

[2] L. Khermander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, Moskva, 1965 | MR

[3] L. Khermander, “Psevdodifferentsialnye operatory i neellipticheskie kraevye zadachi”, Psevdodifferentsialnye operatory, Mir, Moskva, 1967, 166–296 | MR

[4] L. Khermander, “Gipoellipticheskie differentsialnye uravneniya vtorogo poryadka”, Matematika, 12:2 (1968), 88–109

[5] M. I. Vishik, V. V. Grushin, “Ob odnom klasse vyrozhdayuschikhsya ellipticheskikh uravnenii vysshikh poryadkov”, Matem. sb., 79(121) (1969), 3–36 | Zbl

[6] E. V. Radkevich, “Gipoellipticheskie operatory s kratnymi kharakteristikami”, Matem. sb., 79(121) (1969), 193–216 | Zbl

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

[8] M. S. Agranovich, “Ob uravneniyakh v chastnykh proizvodnykh s postoyannymi koeffitsientami”, Uspekhi matem. nauk, XVI:2(98) (1961), 27–94

[9] V. S. Fedii, “Otsenki v normakh $H_(s)$ i gipoelliptichnost”, DAN SSSR, 193:2 (1970), 301–303 | MR | Zbl

[10] P. P. Mosolov, “Differentsiruemost obobschennykh reshenii differentsialnykh uravnenii”, Matem. sb., 59(101) (1962), 189–194 | MR | Zbl