Geodesic
Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus
Sbornik. Mathematics, Tome 29 (1976) no. 3, pp. 393-401
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In this paper we obtain necessary and sufficient conditions for nontriviality of the spaces $$ W^\infty\{a_\alpha,p_\alpha\}\equiv\biggl\{u(x):\rho(u)\equiv\sum_{|\alpha|=0}^\infty a_\alpha\|D^\alpha u\|^{p_\alpha}_{L_{r_\alpha}}<\infty\biggr\} $$ in the cases where the functions $u(x)$ are defined on $\mathbf R^n$ or are periodic with period $2\pi$ with respect to $n$ variables. Bibliography: 5 titles. 
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Yu. A. Dubinskii. Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus. Sbornik. Mathematics, Tome 29 (1976) no. 3, pp. 393-401. http://geodesic.mathdoc.fr/item/SM_1976_29_3_a6/

[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] I. Stein, G. Veiss, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, izd-vo «Mir», Moskva, 1974 | MR

[3] A. Zigmund, Trigonometricheskie ryady, t. 1, 2, izd-vo «Mir», Moskva, 1965 | MR

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

[5] Yu. A. Dubinskii, “Prostranstva Soboleva beskonechnogo poryadka na tore i nekotorye voprosy teorii periodicheskikh reshenii differentsialnykh uravnenii”, DAN SSSR, 222:2 (1975), 269–272 | MR | Zbl