Spherical representation of solutions of invariant differential equations on a Riemannian symmetric space of noncompact type
Izvestiya. Mathematics, Tome 27 (1986) no. 3, pp. 535-548 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An analogue of the exponential representation is obtained for solutions of invariant differential equations on spaces of the type indicated in the title. Solutions are considered which transform according to finite-dimensional representations of a maximal compact subgroup of the group of motions. The case of equations on an associated space of zero curvature is also considered. Bibliography: 34 titles.
@article{IM2_1986_27_3_a5,
     author = {P. A. Kuchment},
     title = {Spherical representation of solutions of invariant differential equations on {a~Riemannian} symmetric space of noncompact type},
     journal = {Izvestiya. Mathematics},
     pages = {535--548},
     year = {1986},
     volume = {27},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1986_27_3_a5/}
}
TY  - JOUR
AU  - P. A. Kuchment
TI  - Spherical representation of solutions of invariant differential equations on a Riemannian symmetric space of noncompact type
JO  - Izvestiya. Mathematics
PY  - 1986
SP  - 535
EP  - 548
VL  - 27
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IM2_1986_27_3_a5/
LA  - en
ID  - IM2_1986_27_3_a5
ER  - 
%0 Journal Article
%A P. A. Kuchment
%T Spherical representation of solutions of invariant differential equations on a Riemannian symmetric space of noncompact type
%J Izvestiya. Mathematics
%D 1986
%P 535-548
%V 27
%N 3
%U http://geodesic.mathdoc.fr/item/IM2_1986_27_3_a5/
%G en
%F IM2_1986_27_3_a5
P. A. Kuchment. Spherical representation of solutions of invariant differential equations on a Riemannian symmetric space of noncompact type. Izvestiya. Mathematics, Tome 27 (1986) no. 3, pp. 535-548. http://geodesic.mathdoc.fr/item/IM2_1986_27_3_a5/

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

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

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

[4] Napalkov V. V., Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR

[5] Berenstein C. A., Dostal M. A., Analytically uniform spaces and their applications to convolution equations, Lect. Not. Math., 256, Springer-Verlag, Berlin, 1972 | MR | Zbl

[6] Nikolskii N. K., “Invariantnye podprostranstva v teorii operatorov i teorii funktsii”, Itogi nauki i tekhniki. Matematicheskii analiz, 12, VINITI, 1974, 199–412

[7] Litvinov G. L., Shpiz G. B., “Primarnye razlozheniya konechnomernykh predstavlenii algebr i grupp Li”, Funkts. analiz i ego prilozh., 12:2 (1978), 86–87 | MR | Zbl

[8] Kuchment P. A., “Predstavleniya reshenii invariantnykh differentsialnykh uravnenii na nekotorykh simmetricheskikh prostranstvakh”, Dokl. AN SSSR, 259:3 (1981), 532–535 | MR | Zbl

[9] Kuchment P. A., “Teoriya Floke dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uspekhi matem. nauk, 37:4 (1982), 3–52 | MR | Zbl

[10] Kuchment P. A., “Predstavleniya reshenii periodicheskikh differentsialnykh uravnenii v chastnykh proizvodnykh”, Izv. AN SSSR. Ser. matem., 46:4 (1982), 782–809 | MR

[11] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

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

[13] Zhelobenko D. P., “Garmonicheskii analiz na reduktivnykh gruppakh Li”, Itogi nauki i tekhniki. Matematicheskii analiz, 17, VINITI, 1979, 207–269 | MR

[14] Barut A., Ronchka R., Teoriya predstavlenii grupp i ee prilozheniya, Mir, M., 1980 | Zbl

[15] Helgason S., “A duality for symmetric spaces with applications to group representations”, Adv. Math., 5:1 (1970), 1–154 | DOI | MR | Zbl

[16] Helgason S., “A duality for symmetric spaces with applications to group representations. II. Differential equations and eigenspace representations”, Adv. Math., 22 (1976), 187–219 | DOI | MR | Zbl

[17] Kashiwara M., Kowata A., Minemura K., Okamoto K., Oshima T., Tanaka M., “Eigenfunctions of invariant differential operators on a symmetric spaces”, Ann. Math., 107 (1978), 1–39 | DOI | MR | Zbl

[18] Helgason S., “Invariant differential equations on homogeneous manifolds”, Bull. Amer. Math. Soc., 83:5 (1977), 751–774 | DOI | MR | Zbl

[19] Helgason S., Analysis on Lie groups and homogeneous spaces, Conf. Board Math. Soc., AMS, 1972, 64 pp. | MR | Zbl

[20] Kostant B., “Suschestvovanie i neprivodimost nekotorykh serii predstavlenii”, Matematika, 14:2 (1970), 102–116 | Zbl

[21] Palamodov V. P., “Zamechanie ob eksponentsialnom predstavlenii reshenii differentsialnykh uravnenii s postoyannymi koeffitsientami”, Matem. sb., 76:3 (1968), 417–434 | MR | Zbl

[22] Robertson A., Robertson V., Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR | Zbl

[23] Khërmander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[24] Helgason S., “A duality for symmetric spaces with applications to group representations. III. Tangent space analysis”, Adv. Math., 36:3 (1980), 297–323 | DOI | MR | Zbl

[25] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965 | MR

[26] Kuchment P. A., “O periodicheskikh v srednem funktsiyakh na simmetricheskikh prostranstvakh”, Funkts. analiz i ego prilozh., 16:3 (1982), 68–69 | MR | Zbl

[27] Gurevich D. I., “Kontrprimery k probleme L. Shvartsa”, Funkts. analiz i ego prilozh., 9:2 (1975), 29–35 | MR | Zbl

[28] Berenstein C. A., Taylor B. A., “Interpolation problems in $\mathbf C^n$ with applications to harmonic analysis”, J. Anal. Math., 38 (1980), 188–254 | MR | Zbl

[29] Berenstein C. A., Taylor B. A., “Mean-periodic functions”, Internat. J. Math. and Math. Sci., 3:2 (1980), 199–235 | DOI | MR | Zbl

[30] Atiyah M. F., Elliptic operators and compact groups, Lect. Not. Math., 401, Springer, Berlin, 1974 | MR | Zbl

[31] Ehrenpreis L., Mautner F., “Some properties of Fourier transform on semisimple Lie groups. I”, Ann. Math., 61:3 (1955), 406–439 ; “II”, Trans. Amer. Math. Soc., 84:1 (1957), 1–55 ; “III”, Trans. Amer. Math. Soc., 90:3 (1959), 431–484 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[32] Rashevskii P. K., “Opisanie zamknutykh invariantnykh podprostranstv v nekotorykh funktsionalnykh prostranstvakh”, Tr. Mosk. matem. ob-va, 38, 1979, 139–185 | MR

[33] Platonov S. S., “Invariantnye podprostranstva v nekotorykh funktsionalnykh prostranstvakh na gruppe $SL(2,\mathbf C)$”, Trudy seminara po vektornomu i tenzornomu analizu, v. 21, MGU, M., 1982, 191–258 | MR

[34] Khinkis L. A., K zadache ob invariantnykh podprostranstvakh v prostranstve funktsii na poluprostoi gruppe Li ranga 1, dep. v VINITI No 4980-83, Voronezhskii lesotekhnich. in-t, 1983, 39 pp.