Инвариантные подпространства в функциональных пространствах медленного роста на световом конусе в $R^{3}$

S. S. Platonov

Problemy analiza, no. 18 (2011), pp. 21-60.

Résumé

We describe the structure of closed linear subspaces in tempered topological vector function spaces on the light cone $X$ in $R^{3}$ that are invariant with respect to the natural quasiregular representation of the group $R\oplus SO_{0}(1,2)$. In particular, we obtain a description of the irreducible and indecomposable invariant subspaces. The class of function spaces under consideration include, in particular, the space $S'(X)$ of all tempered distributions on $X$.

Bibliographie

[1] Berenstein C. A., “Spectral synthesis on symmetric spaces”, Contemp. Math., 63 (1987), 1–25 | DOI | MR | Zbl

[2] Berenstein C. A., Gay R., “Sur la synthèse spectrale dans les espaces symétriques”, J. Math. Pures Appl., 65 (1986), 323–333 | MR | Zbl

[3] Wawrzyńczyk A., “Spectral analysis and synthesis on symmetric spaces”, J. Math. Ann. Appl., 127 (1987), 1–17 | DOI | MR | Zbl

[4] Schvartz L., “Théorie générale des fonctions moynne-périodiques”, Ann. of Math., 48 (1947), 875–929

[5] Gurevich D. I., “Kontrprimery k gipoteze Shvartsa”, Funktsionalnyi analiz i ego prilozheniya, 9:2 (1975), 29–35 | MR | Zbl

[6] Malgrange B., “Existence et approximation des solution des équations aux dérivées partiélles et des équations de convolution”, Ann. Inst. Fourier, 6 (1956), 271–355 | DOI | MR | Zbl

[7] Ehrenpreis L., Fourier analysis in several complex variables, Wiley Interscience, New York, 1970 | MR | Zbl

[8] Palamodov V. P., Lineinye differentsialnye operatory s postoyannymi koeffitsientami, Nauka, M., 1967 | MR

[9] Platonov S. S., “Invariant subspaces in certain function spaces on Euclidean space”, Math. Scand., 76 (1995), 115–138 | MR | Zbl

[10] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na evklidovom prostranstve”, Trudy PetrGU. Ser. Matematika, 1995, no. 2, 92–112 | MR | Zbl

[11] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh polinomialnogo rosta na $\mathbb{R}^n$”, Trudy PetrGU. Ser. Matematika, 1997, no. 4, 105–124 | MR | Zbl

[12] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na $n$-mernom prostranstve Lobachevskogo”, Matem. sb., 137:4 (1988), 435–461 | MR | Zbl

[13] Platonov S. S., “O spektralnom sinteze na simmetricheskikh prostranstvakh ranga 1”, Algebra i analiz, 4:4 (1992), 174–187 | MR | Zbl

[14] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh. Ch. I”, v 3 ch., Izvestiya RAN. Ser. matematicheskaya, 59:5 (1995), 127–172 | MR | Zbl

[15] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh. Ch. II”, v 3 ch., Izvestiya RAN. Ser. matematicheskaya, 62:2 (1998), 131–168 | DOI | MR | Zbl

[16] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na simmetricheskikh prostranstvakh. Ch. III”, v 3 ch., Izvestiya RAN. Ser. matematicheskaya, 66:1 (2002), 167–202 | DOI | MR | Zbl

[17] Ehrenpreis L., Mautner F. J., “Some properties of the Fourier-transform on semisimple Lie groups, III”, Trans. Amer. Math. Soc., 90 (1959), 431–484 | MR | Zbl

[18] Rashevskii P. K., “Opisanie zamknutykh invariantnykh podprostranstv v nekotorykh funktsionalnykh prostranstvakh”, Trudy Moskovskogo matematicheskogo obschestva, 38 (1979), 139–185 | MR

[19] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na gruppe $SL(2,\mathbb{C})$”, Trudy seminara po vektornomu i tenzornomu analizu, 1983, no. 21, 191–258 | MR | Zbl

[20] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na gruppe dvizhenii evklidovoi ploskosti”, Sibirskii mat. zhurnal, 31:3 (1990), 135–146 | MR | Zbl

[21] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na prosteishei razreshimoi gruppe”, Mat. zametki, 35:1 (1984), 19–30 | MR | Zbl

[22] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, GIFML, M., 1958

[23] Wawrzyǹczyk A., “Spectral analysis on upper light cone in $\mathbb{R}^3$ and the Radon transform”, Can. J. Math., 40:6 (1988), 1458–1481 | DOI | MR | Zbl

[24] Uorner F., Osnovy teorii gladkikh mnogoobrazii i grupp Li, Mir, M., 1987 | MR

[25] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[26] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[27] Warner G., Harmonic analysis on semisimple Lie groups, v. 1, Springer-Verlag, Berlin; Heidelberg; New York, 1972

[28] Zhelobenko D. P., Garmonicheskii analiz na poluprostykh kompleksnykh gruppakh Li, Nauka, M., 1974 | MR

[29] Varadarajan V. S., Harmonic analysis on real reductive groups, Springer-Verlag, Berlin; Heidelberg; New York, 1977 | MR | Zbl

[30] Zhelobenko D. P., Shtern A. I., Predstavleniya grupp Li, Nauka, M., 1983 | MR

[31] Platonov S. S., “Spektralnyi sintez v nekotorykh funktsionalnykh topologicheskikh vektornykh prostranstvakh”, Algebra i analiz, 22:5 (2010), 154–183 | MR