Geodesic
Complex powers of hypoelliptic systems in $\mathbf R^n$
Sbornik. Mathematics, Tome 28 (1976) no. 3, pp. 291-300 
S. A. Smagin. Complex powers of hypoelliptic systems in $\mathbf R^n$. Sbornik. Mathematics, Tome 28 (1976) no. 3, pp. 291-300. http://geodesic.mathdoc.fr/item/SM_1976_28_3_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A system of differential operators in $\mathbf R^n$ with polynomial coefficients and whose symbol is hypoelliptic in $(x;\xi)$ is considered. The complex powers and the zeta-function of such a system are constructed. A meromorphic extension of the zeta-function is obtained, from which there follows an asymptotic result concerning the spectrum of the system. The results of Hironaka on the resolution of singularities are used in the proofs. Bibliography: 9 titles.

[1] V. V. Grushin, “Gipoellipticheskie differentsialnye uravneniya i psevdodifferentsialnye operatory s operatornoznachnymi simvolami”, Matem. sb., 88(130) (1972), 504–521 | Zbl

[2] I. N. Bernshtein, S. I. Gelfand, “O meromorfnosti funktsii $P^\lambda$”, Funkts. analiz, 3:1 (1969), 84–85 | MR

[3] M. F. Atyah, “Resolution of singularities and division of distributions”, Comm. Pure Appl. Math., 23:2 (1970), 145–150 | DOI | MR

[4] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero”, Ann. Math., 79:1–2 (1964), 104–326 | MR

[5] I. M. Gelfand, G. E. Shilov, Oboboschennye funktsii i deistviya nad nimi, Fizmatgiz, Moskva, 1959

[6] R. T. Sili, “Stepeni ellipticheskogo operatora”, Matematika, 12:1 (1968), 96–112

[7] H. Delange, “Quelques theoremes tauberieus relatifs à l'integral de Laplace et leurs applications arithmetiques”, Rend. Semin. Math. Univ. Polytechn. Torino, 14 (1954–55), 87–103 | MR

[8] S. A. Smagin, “Drobnye stepeni gipoellipticheskogo operatora v $\mathbf{R}^n$”, DAN SSSR, 209:5 (1973), 1033–1036 | MR

[9] S. A. Smagin, “O meromorfnosti $P^z$, gde $P$—matritsa”, Funkts. analiz, 9:1 (1975), 85–86 | MR | Zbl