Geodesic
Spaces defined by the Paley function
Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 937-957 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is concerned with Haar and Rademacher series in symmetric spaces, and also with the properties of spaces defined by the Paley function. In particular, the symmetric hull of the space of functions with uniformly bounded Paley function is found. Bibliography: 27 titles.
Keywords: symmetric space, Haar functions, Rademacher functions, Paley function, real interpolation method. 
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S. V. Astashkin; E. M. Semenov. Spaces defined by the Paley function. Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 937-957. http://geodesic.mathdoc.fr/item/SM_2013_204_7_a0/

[1] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces. II. Function spaces, Ergeb. Math. Grenzgeb., 97, Springer-Verlag, Berlin–New York, 1979 | MR | Zbl

[2] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, RI, 1982 | MR | MR | Zbl | Zbl

[3] Ya. B. Rutitskii, “O nekotorykh klassakh izmerimykh funktsii”, UMN, 20:4 (1965), 205–208

[4] S. V. Astashkin, “Rademacher functions in symmetric spaces”, J. Math. Sci. (N. Y.), 169:6 (2010), 725–886 | DOI | MR | Zbl

[5] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Nauka, M., 1984 ; B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989 | MR | Zbl | MR | Zbl

[6] P. F. Müller, Isomorphisms between $H^1$ spaces, IMPAN Monogr. Mat. (N. S.), 66, Birkhäuser, Basel, 2005 | MR | Zbl

[7] B. I. Golubov, “Series with respect to the Haar system”, J. Soviet Math., 1:6 (1973), 704–726 | DOI | MR | Zbl | Zbl

[8] I. Novikov, E. Semenov, Haar series and linear operators, Math. Appl., 367, Kluwer Acad. Publ., Dordrecht, 1996 | MR | Zbl

[9] S. V. Astashkin, E. M. Semenov, “Haar series and spaces determined by the Paley function”, Dokl. Math., 86:1 (2012), 539–541 | DOI | Zbl

[10] C. Franchetti, E. M. Semenov, “A Hilbert space characterization among function spaces”, Anal. Math., 21:2 (1995), 85–93 | DOI | MR | Zbl

[11] W. B. Johnson, G. Schechtman, “Martingale inequalities in rearrangement invariant function spaces”, Israel J. Math., 64:3 (1988), 267–275 | DOI | MR | Zbl

[12] E. M. Semenov, “$\mathrm{RUC}$-basis properties of the Haar system”, Dokl. Math., 53:3 (2008), 397–398 | MR | Zbl

[13] P. Billard, S. Kwapień, A. Pelczyński, Ch. Samuel, “Biorthogonal systems of random unconditional convergence in Banach spaces”, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986), Longhorn Notes, Univ. Texas, Austin, TX, 1986, 13–35 | MR | Zbl

[14] S. J. Szarek, “On the best constants in the Khinchin inequality”, Studia Math., 58:2 (1976), 197–208 | MR | Zbl

[15] M. Kikuchi, “On the Davis inequality in Banach function spaces”, Math. Nachr., 281:5 (2008), 697–709 | DOI | MR | Zbl

[16] V. A. Rodin, E. M. Semenov, “Complementability of the subspace generated by the Rademacher system in a symmetric space”, Funct. Anal. Appl., 13:2 (1979), 150–151 | DOI | MR | Zbl | Zbl

[17] Y. Raynaud, “Complemented Hilbertian subspaces in rearrangement invariant function spaces”, Illinois J. Math., 39:2 (1995), 212–250 | MR | Zbl

[18] E. V. Tokarev, “Complemented subspaces of symmetric function spaces”, Math. Notes, 32:4 (1982), 726–728 | DOI | MR | Zbl

[19] C. Bennett, R. A. DeVore, R. Sharpley, “Weak–$L^\infty$ and $\mathrm{BMO}$”, Ann. of Math. (2), 113:3 (1981), 601–611 | DOI | MR | Zbl

[20] C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Boston, MA, 1988 | MR | Zbl

[21] A. Korenovskii, Mean oscillations and equimeasurable rearrangements of functions, Lect. Notes Unione Mat. Ital., 4, Springer-Verlag, Berlin, 2007 | MR | Zbl

[22] S. V. Astashkin, M. Leibov, L. Maligranda, “Rademacher functions in $\mathrm{BMO}$”, Studia Math., 205:1 (2011), 83–100 | DOI | MR | Zbl

[23] A. Zygmund, Trigonometric series, v. I, Cambridge Univ. Press, New York, 1959 | MR | MR | Zbl | Zbl

[24] V. A. Rodin, E. M. Semyonov, “Rademacher series in symmetric spaces”, Anal. Math., 1:3 (1975), 207–222 | DOI | MR | Zbl

[25] Yu. A. Brudnyi, N. Ya. Kruglyak, Interpolation functors and interpolation spaces, v. 1, North-Holland Math. Library, 47, Amsterdam, North-Holland, 1991 | MR | Zbl

[26] J. Bergh, J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | MR | MR | Zbl

[27] E. M. Nikishin, “On a property of sums of independent variables”, Math. Notes, 16:5 (1974), 1015–1017 | DOI | MR | Zbl