Geodesic
Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order
Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 395-405 
Yu. A. Dubinskii. Limits of Banach spaces. Imbedding theorems. Applications to Sobolev spaces of infinite order. Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 395-405. http://geodesic.mathdoc.fr/item/SM_1981_38_3_a4/
@article{SM_1981_38_3_a4,
     author = {Yu. A. Dubinskii},
     title = {Limits of {Banach} spaces. {Imbedding} theorems. {Applications} to {Sobolev} spaces of infinite order},
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     volume = {38},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_3_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For a sequence of Banach spaces ${X_1}\supset{X_2}\supset\dotsb$, a concept of limit $X_\infty=\lim_{r\to\infty}X_r$ is introduced that is a natural generalization of the concept of the limit of a monotonically decreasing numerical sequence. Necessary and sufficient conditions are obtained for an imbedding $X_\infty\subset Y_\infty$ and for a compact imbedding. Applications are given to the Sobolev spaces of infinite order $W^\infty\{a_\alpha,p\}$. Necessary and sufficient conditions bearing an algebraic character are established for the imbedding $W^\infty\{a_\alpha,2\}(\mathbf R^\nu)\subset W^\infty\{b_\alpha,2\}(\mathbf R^\nu)$. Sufficient algebraic imbedding conditions are obtained for the spaces $W^\infty\{a_\alpha,p\}(\mathbf R^1)$ for any $p>1$. Bibliography: 8 titles.

[1] Yu. A. Dubinskii, “Prostranstva Soboleva beskonechnogo poryadka i povedenie reshenii nekotorykh kraevykh zadach pri neogranichennom vozrastanii poryadka uravneniya”, Matem. sb., 98(140) (1975), 163–184 | MR | Zbl

[2] Yu. A. Dubinskii, “Netrivialnost prostranstv Soboleva beskonechnogo poryadka v sluchae polnogo evklidova prostranstva i tora”, Matem. sb., 100(142) (1976), 436–446 | MR | Zbl

[3] Yu. A. Dubinskii, “O sledakh funktsii v prostranstvakh Soboleva beskonechnogo poryadka i neodnorodnykh kraevykh zadachakh”, DAN SSSR, 235:6 (1977), 1248–1252 | MR

[4] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo SO AN SSSR, Novosibirsk, 1962

[5] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, izd-vo “Nauka”, Moskva, 1975 | MR | Zbl

[6] B. V. Shabat, Vvedenie v kompleksnyi analiz, ch. 1, izd-vo “Nauka”, Moskva, 1975 | MR

[7] L. R. Volevich, B. P. Paneyakh, “Nekotorye prostranstva obobschennykh funktsii i teoremy vlozheniya”, Uspekhi matem. nauk, XX:1(121) (1965), 3–74

[8] S. Mandelbroit, Primykayuschie ryady. Regulyarizatsiya posledovatelnostei. Primeneniya, Gostekhizdat, Moskva, 1955