Some note on Gevrey hypoellipticity and solvability on torus

Todor GRAMCHEV; Petar POPIVANOV; Masafumi YOSHINO

doi:10.2969/jmsj/04330501

Todor GRAMCHEV, Petar POPIVANOV, Masafumi YOSHINO

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1991 年 43 巻 3 号 p. 501-514

DOI https://doi.org/10.2969/jmsj/04330501

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Published: 1991 Received: 1990/08/01 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) M.S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math., 113 (1981), 387-421.
2) L. Cattabriga, L. Rodino and L. Zanghirati, Analytic-Gevrey hypoellipticity for a class of pseudo-differential operators with multiple characteristics, Comm. Part. Diff. Eq., 15, (1990), 81-96.
3) T. Gramchev, Powers of Mizohata type operators in Gevrey classes, Bollettino U.M.I. B(7), 5 (1991), 135-156.
4) S. Greenfield and N. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31 (1972), 112-114.
5) K. Kajitani and S. Wakabayashi, Microhyperbolic operators in Gevrey class, Publ. Res. Inst. Math. Sci. Kyoto Univ., 25(1989), 169-221.
6) H. Komatsu, Ultradistributions I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. IA., 20 (1973), 25-105.
7) T. Okaji, Gevrey hypoelliptic operators which are not C^∞-hypoelliptic, J. Math. Kyoto Univ., 28 (1988), 311-322.
8) L. Rodino, On linear partial differential operators with multiple characteristics, Conf, on Part. Diff. Eq., Holzhau-DDR, 1988.
9) P. Popivanov, Doctoral thesis, Math. Inst., Bulg. Acad, of Sci. Sofia, 1986, see also Mat. Sb. (N.S.), 100 (1976), 217-241 (Russian): Mat. USSR-Sb, 29 (1976), 193-216.
10) C. Siegel, Über die Normalform analytischer Differential-gleichungen in der Nähe einer Gleichgewichtslösung, Nach. Akad. Wiss. Göttingen, (1952), 21-30.
11) K. Taira, Le principle du maximum et l'hypoellipticité globale, Seminaire Bony-Sjöstrand-Meyer 1984-1985, n°1.
12) K. Tanaka, Infinitely many periodic solutions for the equation: u_tt-u_xx±|u|^s-1u=f(x, t), Comm. Part. Diff. Eq., 10 (1985), 1317-1345.
13) W. Wasow, Linear turning point theory, Springer-Verlag, New York 1985.
14) M. Yoshino, An application of generalized implicit function theorem to Goursat problems for nonlinear Leray-Volevich systems, J. Diff. Eq., 57 (1985), 44-69.
15) M. Yoshino,The diophantine nature for the convergence of formal solutions, Tohoku Math. J., 38 (1986), 625-641.
16) M. Yoshino, A class of globally hypoelliptic operators on the torus, Math. Z., 201 (1989), 1-11.

Right : [1] M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math., 113 (1981), 387-421.
[2] L. Cattabriga, L. Rodino and L. Zanghirati, Analytic-Gevrey hypoellipticity for a class of pseudo-differential operators with multiple characteristics, Comm. Part. Diff. Eq., 15, (1990), 81-96.
[3] T. Gramchev, Powers of Mizohata type operators in Gevrey classes, Bollettino U. M. I. B (7), 5 (1991), 135-156.
[4] S. Greenfield and N. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31 (1972), 112-114.
[5] K. Kajitani and S. Wakabayashi, Microhyperbolic operators in Gevrey class, Publ. Res. Inst. Math. Sci. Kyoto Univ., 25 (1989), 169-221.
[6] H. Komatsu, Ultradistributions I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. IA., 20 (1973), 25-105.
[7] T. Okaji, Gevrey hypoelliptic operators which are not C^∞-hypoelliptic, J. Math. Kyoto Univ., 28 (1988), 311-322.
[8] L. Rodino, On linear partial differential operators with multiple characteristics, Conf. on Part. Diff. Eq., Holzhau-DDR, 1988.
[9] P. Popivanov, Doctoral thesis, Math. Inst., Bulg. Acad. of Sci. Sofia, 1986, see also Mat. Sb. (N.S.), 100 (1976), 217-241 (Russian): Mat. USSR-Sb, 29 (1976), 193-216.
[10] C. Siegel, Über die Normalform analytischer Differential-gleichungen in der Nähe einer Gleichgewichtslösung, Nach. Akad. Wiss. Göttingen, (1952), 21-30.
[11] K. Taira, Le principle du maximum et l'hypoellipticité globale, Seminaire Bony-Sjöstrand-Meyer 1984-1985, n°1.
[12] K. Tanaka, Infinitely many periodic solutions for the equation: u_tt-u_xx±|u|^s-1u=f(x,t), Comm. Part. Diff. Eq., 10 (1985), 1317-1345.
[13] W. Wasow, Linear turning point theory, Springer-Verlag, New York 1985.
[14] M. Yoshino, An application of generalized implicit function theorem to Goursat problems for nonlinear Leray-Volevich systems, J. Diff. Eq., 57 (1985), 44-69.
[15] M. Yoshino,The diophantine nature for the convergence of formal solutions, Tôhoku Math. J., 38 (1986), 625-641.
[16] M. Yoshino, A class of globally hypoelliptic operators on the torus, Math. Z., 201 (1989), 1-11.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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