Geodesic
Smirnov–Sobolev spaces and their embeddings
Sbornik. Mathematics, Tome 194 (2003) no. 4, pp. 541-550 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $G$ be a bounded simply connected domain with rectifiable boundary $\partial G$ and assume that $0. Let $E_p(G)$ be the Smirnov space of analytic functions $f$ in $G$. The Smirnov–Sobolev space $E_p^s(G)$, $s\in\mathbb N$, consists of analytic functions $f$ in $G$ such that $f^{(s)}\in E_p(G)$. If $G$ is a disc, then $E_p(G)$ is the Hardy space and $E_p^s(G)$ is the Hardy–Sobolev space. In that case the following Hardy–Littlewood embedding is known: $$ E_\sigma^s(G)\subset E_p(G), \qquad 1/\sigma=s+1/p. $$ The author has recently extended this embedding to Smirnov–Sobolev spaces in Lavrent'ev domains. A further generalization of the Hardy–Littlewood embedding is obtained in the present paper. Namely, it is shown that such an embedding holds if the domain $G$ satisfies the following condition: for all points $\xi$ and $\eta$ in $\partial G$, $$ |\Gamma(\xi,\eta)|\leqslant \chi\rho^+(\xi,\eta), \qquad \chi=\chi(G)\geqslant 1. $$ Here $|\Gamma(\xi,\eta)|$ is the length of the smallest of the two arcs of $\partial G$ connecting $\xi$ and $\eta$; $\rho^+(\xi,\eta)$ is the interior distance (with respect to $G$) between the points $\xi$ and $\eta$. 
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     author = {A. A. Pekarskii},
     title = {Smirnov{\textendash}Sobolev spaces and their embeddings},
     journal = {Sbornik. Mathematics},
     pages = {541--550},
     year = {2003},
     volume = {194},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_4_a3/}
}
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A. A. Pekarskii. Smirnov–Sobolev spaces and their embeddings. Sbornik. Mathematics, Tome 194 (2003) no. 4, pp. 541-550. http://geodesic.mathdoc.fr/item/SM_2003_194_4_a3/

[1] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M.–L., 1950

[2] Duren P., Theory of $H^p$ spaces, Academic Press, New York, 1970 | MR

[3] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[4] Alexandrov A. B., “Essay on non locally convex Hardy classes”, Lecture Notes in Math., 864, 1981, 1–89

[5] Lorentz G. G., Golitschek M., Makovoz Yu., Constructive approximation. Advanced problems, Springer-Verlag, Berlin, 1996 | MR

[6] Pekarskii A. A., “Ratsionalnye priblizheniya funktsii s proizvodnymi iz prostranstva V. I. Smirnova”, Algebra i analiz., 13:2 (2001), 165–190 | MR

[7] Andrievskii V. V., “Geometricheskie svoistva oblastei V. K. Dzyadyka”, Ukr. matem. zhurn., 33:6 (1981), 723–727 | MR | Zbl

[8] Dynkin E. M., “Metody teorii singulyarnykh integralov – II. Teoriya Littlvuda–Peli i ee prilozheniya”, Sovr. problemy matem. Fundament. napravleniya, 42, VINITI, M., 1989, 105–198 | MR

[9] Dynkin E. M., “Otsenki analiticheskikh funktsii v zhordanovykh oblastyakh”, Zapiski nauchn. sem. LOMI, 73, 1977, 70–90 | MR | Zbl

[10] Privalov I. I., Vvedenie v teoriyu funktsii kompleksnogo peremennogo, Nauka, M., 1977 | MR