Geodesic
Smooth structure and differential operators on a~locally compact group
Izvestiya. Mathematics , Tome 59 (1995) no. 1, pp. 1-44.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe the structure of differential operators (that is, locally linear operators in the space of Bruhat smooth functions) on an arbitrary locally compact group $G$. 
@article{IM2_1995_59_1_a0,
     author = {S. S. Akbarov},
     title = {Smooth structure and differential operators on a~locally compact group},
     journal = {Izvestiya. Mathematics },
     pages = {1--44},
     publisher = {mathdoc},
     volume = {59},
     number = {1},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a0/}
}
TY  - JOUR
AU  - S. S. Akbarov
TI  - Smooth structure and differential operators on a~locally compact group
JO  - Izvestiya. Mathematics 
PY  - 1995
SP  - 1
EP  - 44
VL  - 59
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a0/
LA  - en
ID  - IM2_1995_59_1_a0
ER  - 
%0 Journal Article
%A S. S. Akbarov
%T Smooth structure and differential operators on a~locally compact group
%J Izvestiya. Mathematics 
%D 1995
%P 1-44
%V 59
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a0/
%G en
%F IM2_1995_59_1_a0
S. S. Akbarov. Smooth structure and differential operators on a~locally compact group. Izvestiya. Mathematics , Tome 59 (1995) no. 1, pp. 1-44. http://geodesic.mathdoc.fr/item/IM2_1995_59_1_a0/

[1] Glushkov V. M., “Algebry Li lokalno-bikompaktnykh grupp”, UMN, 12:2 (1957), 137–142 | MR | Zbl

[2] Lashof R. K., “Lie algebras of locally compact groups”, Pac. J. Math., 7:2 (1957), 1145–1162 | MR | Zbl

[3] Bruhat F., “Distributions sur un groupe localement compact et applications a l'etude des representations des groupes $p$-adique”, B. Soc. Math. France, 89:1 (1961), 43–75 | MR | Zbl

[4] Riss J., “Elements de calcul differentiel et theorie des distributions sur les groupes abeliens localement compacts”, Act. Math., 89:1–2 (1953), 45–105 | DOI | MR | Zbl

[5] Pytlik T., “A nuclear space of functions on a locally compact group”, B. Pol. Math. S., 17:1 (1969), 161–166 | MR | Zbl

[6] Aarnes J. F., “A large bi-invariant nuclear function space on a locally compact group”, Math. Scand., 29:2 (1971), 281–297 | MR

[7] Boseck H., Czichowski G., “Grundfunctionen und verallgemeinerte Functionen auf topologischen Gruppen, I”, Math. Nachr., 58 (1973), 215–240 | DOI | MR | Zbl

[8] Boseck H., Czichowski G., “Grundfunctionen und verallgemeinerte Functionen auf topologischen Gruppen, II”, Math. Nachr., 66 (1975), 319–332 | DOI | MR | Zbl

[9] Bauch H. F., “Grundfunctionen und verallgemeinerte Functionen auf topologischen Gruppen, IV”, Math. Nachr., 103 (1981), 167–176 | DOI | MR | Zbl

[10] Edamatsu T., “Spaces of differentiable functions on compact group”, J. Math. Kyoto Univ., 24:2 (1984), 323–360 | MR | Zbl

[11] Mackey G. W., “Infinite-dimensional group representation”, B. Am. Math. S., 69:5 (1963), 628–686 | DOI | MR | Zbl

[12] Eymard P., “Analyse fonctionnelle l'algebre de Fourier d'un groupe localement compact”, C.R. Acad. Sc. Paris, 256:7 (1963), 1429–1431 | MR | Zbl

[13] Weil A., “Sur sertains groupes d'operateurs unitaires”, Act. Math., 111:3–4 (1964), 143–211 | DOI | MR | Zbl

[14] Igusa J. I., “Harmonic analysis and theta-functions”, Act. Math., 120:3–4 (1968), 187–222 | DOI | MR | Zbl

[15] Wawrzynczyk A., “On tempered distributions and Bochner–Schwartz theorem on arbitrary locally compact abelian groups”, Colloq. Math., 19 (1968), 305–318 | MR | Zbl

[16] Osborne M. S., “On the Schwartz–Bruhat space and the Paley–Wiener theorem for locally compact abelian groups”, J. Funct. Anal., 19:1 (1975), 40–49 | DOI | MR | Zbl

[17] Aarnes J. F., “Differentiable representations. 1: Induced representations and Frobenius reciprocity”, T. Am. Math. S., 220 (1976), 1–35 | DOI | MR | Zbl

[18] Reiter H., “Weil–Cartier theorem”, Monats. Math., 86:1 (1978), 13–62 | DOI | MR | Zbl

[19] Bose R. K., “Realizability theory of continuous linear operators on groups”, SIAM J. Math., 10:4 (1979), 767–777 | DOI | MR | Zbl

[20] Cartier P., “Representations of $p$-adic groups: a survey”, Proc. Symp. Pure Math., 33:1 (1979), 111–155 | MR | Zbl

[21] Roth J. P., “Closing of invariant dissipative operators”, Math. Z., 197:2 (1988), 251–258 | DOI | MR | Zbl

[22] Whyburn K., “Differentiable cohomology on locally compact groups”, Proc. Nat. Acad. Sc. USA, 67:1 (1970), 45–51 | DOI | MR | Zbl

[23] Pichaud J., “Cohomologie continue et cohomologie differentiable des groupes localement compacts”, C.R. Acad. Sc. Paris, 292:3 (1981), 171–173 | MR | Zbl

[24] Pichaud J., “Homologie continue et homologie differentiable des groupes localement compacts”, C.R. Acad. Sc. Paris, 293:3 (1981), 191–193 | MR

[25] Pichaud J., “Regularisation en homologie”, ASTERISQUE, 1985, no. 124–125, 253–268 | MR | Zbl

[26] Schreiber B. M., “Measures with bounded convolution powers”, T. Am. Math. S., 151:2 (1970), 405–432 | DOI | MR

[27] Berg C., “Potential theory on the infinite-dimentional torus”, Inv. Math., 32:1 (1976), 49–100 | DOI | MR | Zbl

[28] Bendikov A. D., “O garmonicheskikh strukturakh, porozhdennykh teplovym vinerovskim protsessom na gruppe”, UMN, 34:1 (1979), 217–218 | MR

[29] Heyer H., Probability measures on locally compact groups, Springer, Berl.–Heid.–N. Y., 1977 ; Kheier Kh., Veroyatnostnye mery na lokalno kompaktnykh gruppakh, Mir, M., 1981 | MR | Zbl

[30] Schirmeier U., “Convergence properties in harmonic spaces”, Inv. Math., 55:1 (1979), 71–95 | DOI | MR | Zbl

[31] Drisch T., Hazod W., “Analytische Vectoren von Faltungshalbgruppen, I”, Math. Z., 172:1 (1980), 1–28 | DOI | MR | Zbl

[32] Drisch T., Hazod W., “Analytische Vectoren von Faltungshalbgruppen, II”, Monats. Math., 88:2 (1979), 107–122 | DOI | MR | Zbl

[33] Drisch T., Hazod W., “Analytische Vectoren von Faltungshalbgruppen, III”, Monats. Math., 89:3 (1980), 205–218 | DOI | MR | Zbl

[34] Siebert E., “Fourier-analysis and limit theorems for convolution semigroups on a locally compact group”, Adv. Math., 39:2 (1981), 111–154 | DOI | MR | Zbl

[35] Zeuner H., “Die Existenz einer Levy–Abbildung auf einem homogenen Raum”, Math. Z., 189:4 (1985), 529–554 | DOI | MR | Zbl

[36] Hannabus K. C., “Characters and contact transformations”, Math. Proc. Camb. Phil. Soc., 90 (1981), 465–476 | DOI | MR | Zbl

[37] Hennings M. A., “The Weyl transformation and quantization for locally compact abelian groups”, Publ. Research Inst. Math. Sci., 21:6 (1985), 1223–1235 | DOI | MR | Zbl

[38] Volovich I. V., “$p$-Adicheskoe prostranstvo-vremya i teoriya strun”, TMF, 71:3 (1987), 337–340 | MR | Zbl

[39] Vladimirov V. S., Volovich I. V., “$p$-Adic quantum mechanics”, Comm. Math. Phys., 123 (1989), 569–576 | DOI | MR

[40] Vladimirov V. S., “Obobschennye funktsii nad polem $p$-adicheskikh chisel”, UMN, 43:5 (1988), 19–64 | MR | Zbl

[41] Vladimirov V. S., “O spektre nekotorykh psevdodifferentsialnykh operatorov nad polem $p$-adicheskikh chisel”, Algebra i analiz, 2:6 (1990), 107–124

[42] Khrennikov A. Yu., “Obobschennye funktsii i gaussovskie kontinualnye integraly po nearkhimedovym funktsionalnym prostranstvam”, Izv. AN SSSR. Ser. matem., 55:4 (1991), 780–814 | MR

[43] Khrennikov A. Yu., “Fundamentalnye resheniya nad polem $p$-adicheskikh chisel”, Algebra i analiz, 4:3 (1992), 248–266 | MR | Zbl

[44] Boseck H., Czichowski G., Rudolph K. P., Analysis on topological groups – general Lie theory, V. 37, Teubner, Leipzig, 1985 | MR | Zbl

[45] Antonets M. A., Shereshevskii I. A., “Kvantovanie Veilya na kompaktnykh abelevykh gruppakh i kvantovaya mekhanika pochti-periodicheskikh sistem”, TMF, 48:1 (1981), 49–59 | MR | Zbl

[46] Shereshevskii I. A., Geometriya i kvantovanie na kompaktnykh abelevykh gruppakh i sferakh, Preprint No 35, IPF, Gorkii, 1981

[47] Shereshevski I. A., “Quantization and commutative superalgebras”, Lett. Math. Phys., 5:5 (1981), 423–435 | DOI | MR

[48] Akbarov S. S., Gamiltonova mekhanika i kvantovanie na lokalno kompaktnykh abelevykh gruppakh, Diss. ...kand. fiz.-mat. nauk, M., 1989

[49] Peetre J., “Un caracterization abstraite des operateurs differentiels”, Math. Scand., 7:1 (1959), 211–218 | MR | Zbl

[50] Akbarov S. S., “Stroenie differentsialnykh operatorov na lokalno kompaktnoi abelevoi gruppe”, Mat. zametki, 52:4 (1992), 3–14 | MR | Zbl

[51] Edvards R., Funktsionalnyi analiz, Mir, M., 1969

[52] Martineau A., “Sur une properiete caracteristique d'un produit de droites”, Arch. Math., 1960, no. 11, 423–426 | DOI | MR | Zbl

[53] Shefer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR

[54] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959

[55] Bukur I., Delyanu A., Vvedenie v teoriyu kategorii i funktorov, Mir, M., 1972 | MR

[56] Burbaki N., Obschaya topologiya. Topologicheskie gruppy. Chisla i svyazannye s nimi gruppy i prostranstva, Nauka, M., 1969 | MR

[57] Glushkov V. M., “Stroenie lokalno-bikompaktnykh grupp i pyataya problema Gilberta”, UMN, 12:2 (1957), 3–41 | MR | Zbl

[58] Montgomery D., Zippin L., Topological transformation groups, Interscience, N.Y., 1966 | MR | Zbl

[59] Born E., “Projective Lie algebra bases of a locally compact groupe and uniform differentiability”, Math. Z., 200:2 (1989), 279–292 | DOI | MR | Zbl

[60] Narasimkhan R., Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, Mir, M., 1971 | Zbl

[61] Edamatsu T., “Differential operators on locally compact groups”, J. Math. Kyoto Univ., 28:3 (1988), 445–503 | MR | Zbl

[62] Vinogradov A. M., Krasilschik I. S., “Chto takoe gamiltonov formalizm?”, UMN, 30:1(181) (1975), 172–198

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

[64] Petrovskii I. G., Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1970 | MR