Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients

V. A. Vassiliev

Izvestiya. Mathematics , Tome 28 (1987) no. 2, pp. 233-273.

Résumé

It is proved that for almost all hyperbolic operators with constant coefficients analytic sharpness of the fundamental solution everywhere is equivalent to the local Petrovskii condition. In the proximity of simple ($O$-modal) singularities of wave front sets the author finds all domains from one side of which there is sharpness. Bibliography: 24 titles.

Bibliographie

[1] Atya M. F., Bott R., Gording L., “Lakuny dlya giperbolicheskikh differentsialnykh operatorov s postoyannymi koeffitsientami. I”, Uspekhi matem. nauk, 26:2 (1971), 25–100 | MR

[2] Atya M. F., Bott R., Gording L., “Lakuny dlya giperbolicheskikh differentsialnykh operatorov s postoyannymi koeffitsientami. II”, Uspekhi matem. nauk, 39:3 (1984), 171–224 | MR

[3] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, Nauka, M., 1982 | MR

[4] Gording L., “Rezkie fronty parnykh ostsilliruyuschikh integralov”, Uspekhi matem. nauk, 38:6 (1983), 85–96 | MR

[5] Petrovskii I. G., “O diffuzii voln i lakunakh dlya giperbolicheskikh uravnenii”, Matem. sb., 17(59):3 (1945), 289–370 | MR

[6] Petrovskii I. G., “O diffuzii voln i lakunakh dlya sistem giperbolicheskikh uravnenii”, Izv. AN SSSR. Ser. matem., 8:3 (1944), 101–106 | MR | Zbl

[7] Davydova A. M., Dostatochnoe uslovie otsutstviya lakuny dlya differentsialnogo uravneniya v chastnykh proizvodnykh giperbolicheskogo tipa, Dissertatsiya na soiskanie uch. st. kand. f.-m. nauk, M., 1945, 43 pp.

[8] Borovikov V. A., “Fundamentalnye resheniya lineinykh uravnenii v chastnykh proizvodnykh s postoyannymi koeffitsientami”, Tr. Mosk. matem. o-va, 8, 1959, 199–257 | MR

[9] Vasilev V. A., “Lokalnoe uslovie Petrovskogo i teoriya Pikara–Lefshetsa”, Uspekhi matem. nauk, 39:2 (1984), 219–220 | MR

[10] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[11] Gusein-Zade S. M., “Gruppy monodromii izolirovannykh osobennostei giperpoverkhnostei”, Uspekhi matem. nauk, 32:2 (1977), 23–65 | MR | Zbl

[12] Arnold V. I., “Normalnye formy funktsii vblizi vyrozhdennykh kriticheskikh tochek”, Funkts. analiz i ego prilozh., 6:4 (1972), 3–25 | MR

[13] Briskorn E., “Monodromiya izolirovannykh osobennostei giperpoverkhnostei. Dobavlenie”, Matematika, 15:4 (1971), 156–160

[14] Gabrielov A. M., “Matritsy peresechenii dlya nekotorykh osobennostei”, Funkts. analiz i ego prilozh., 7:3 (1973), 18–32 | MR | Zbl

[15] Milnor Dzh., Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971 | MR | Zbl

[16] Varchenko A. N., “Teoremy topologicheskoi ekvisingulyarnosti semeistv algebraicheskikh mnogoobrazii i polinomialnykh otobrazhenii”, Izv. AN SSSR. Ser. matem., 36:5 (1972), 957–1019 | MR | Zbl

[17] Gabrielov A. M., “Bifurkatsii, diagrammy Dynkina i modalnost izolirovannykh osobennostei”, Funkts. analiz i ego prilozh., 8:2 (1974), 7–12 | MR | Zbl

[18] Looijenga E., “The complement of the bifurcation variety of a simple singularity”, Invent. Math., 23:2 (1974), 105–116 | DOI | MR | Zbl

[19] Gusein-Zade S. M., “Matritsy peresechenii dlya nekotorykh osobennostei funktsii dvukh peremennykh”, Funkts. analiz i ego prilozh., 8:1 (1974), 11–15 | MR | Zbl

[20] A'Campo N., “Le groupe de monodromie du déploiement des singularites isolées de courbes planes. I”, Mathematische Annalen, 213:1 (1975), 1–32 | DOI | MR

[21] A'Campo N., “Le groupe de monodromie du déploiement des singularites isolées de courbes planes. II”, Actes du Conores Internationale des Mathematiciens (Vancouver, 1974), 1, no. 1, 1975, 395–404 | MR

[22] Lyashko O. V., “Raspadeniya prostykh osobennostei funktsii”, Funkts. analiz i ego prilozh., 10:2 (1976), 49–56 | Zbl

[23] Looijenga E., “The discriminant of a real simple singularity”, Compositio Math., 37:1 (1978), 51–62 | MR | Zbl

[24] Arnold V. P., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii. II, Nauka, M., 1984 | MR