On subspaces of Orlicz spaces spanned by independent copies

S. V. Astashkin

Izvestiya. Mathematics , Tome 88 (2024) no. 4, pp. 601-625.

Résumé

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska–Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

Détail

Keywords: independent functions, symmetric space, strongly embedded subspace, Orlicz function, Orlicz space, Matuszewska–Orlicz indices.

Bibliographie

[1] A. Zygmund, Trigonometric series, v. 1, 2nd ed., Cambridge Univ. Press, New York, 1959 | MR | Zbl

[2] F. Albiac and N. J. Kalton, Topics in Banach space theory, Grad. Texts in Math., 233, Springer, New York, 2006 | DOI | MR | Zbl

[3] W. Rudin, “Trigonometric series with gaps”, J. Math. Mech., 9 (1960), 203–227 | DOI | MR | Zbl

[4] J. Bourgain, “Bounded orthogonal systems and the $\Lambda(p)$-set problem”, Acta Math., 162:3-4 (1989), 227–245 | DOI | MR | Zbl

[5] G. F. Bachelis and S. E. Ebenstein, “On $\Lambda(p)$ sets”, Pacific J. Math., 54:1 (1974), 35–38 | DOI | MR | Zbl

[6] J. Bourgain, “$\Lambda_p$-sets in analysis: results, problems and related aspects”, Handbook of the geometry of Banach spaces, v. 1, North-Holland Publishing Co., Amsterdam, 2001, 195–232 | DOI | MR | Zbl

[7] H. P. Rosenthal, “On subspaces of $L^p$”, Ann. of Math. (2), 97:2 (1973), 344–373 | DOI | MR | Zbl

[8] S. V. Astashkin, “The structure of subspaces in Orlicz spaces lying between $L^1$ and $L^2$”, Math. Z., 303:4 (2023), 91 | DOI | MR | Zbl

[9] M. I. Kadets, “Linear dimension of the spaces $L_p$ and $l_q$”, Uspekhi Mat. Nauk, 13:6(84) (1958), 95–98 | MR | Zbl

[10] J. Bretagnolle and D. Dacunha-Castelle, “Mesures aléatoires et espaces d'Orlicz”, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A877–A880 | MR | Zbl

[11] J. Bretagnolle and D. Dacunha-Castelle, “Application de l'étude de certaines formes linéaires aléatoires au plongement d'espaces de Banach dans des espaces $L^p$”, Ann. Sci. École Norm. Sup. (4), 2:4 (1969), 437–480 | DOI | MR | Zbl

[12] D. Dacunha-Castelle, “Variables aléatoires échangeables et espaces d'Orlicz”, Séminaire Maurey–Schwartz 1974–1975. Espaces $L^p$, applications radonifiantes et géométrie des espaces de Banach, École Polytech., Centre Math., Paris, 1975, Exp. X, XI | MR | Zbl

[13] M. Sh. Braverman, “On some moment conditions for sums of independent random variables”, Probab. Math. Statist., 14:1 (1993), 45–56 | MR | Zbl

[14] M. Braverman, “Independent random variables in Lorentz spaces”, Bull. London Math. Soc., 28:1 (1996), 79–87 | DOI | MR | Zbl

[15] S. V. Astashkin and F. A. Sukochev, “Orlicz sequence spaces spanned by identically distributed independent random variables in $L_p$-spaces”, J. Math. Anal. Appl., 413:1 (2014), 1–19 | DOI | MR | Zbl

[16] S. Astashkin, F. Sukochev, and D. Zanin, “On uniqueness of distribution of a random variable whose independent copies span a subspace in $L^p$”, Studia Math., 230:1 (2015), 41–57 | DOI | MR | Zbl

[17] S. Astashkin, F. Sukochev, and D. Zanin, “The distribution of a random variable whose independent copies span $\ell_M$ is unique”, Rev. Mat. Complut., 35:3 (2022), 815–834 | DOI | MR | Zbl

[18] S. Astashkin, “On symmetric spaces containing isomorphic copies of Orlicz sequence spaces”, Comment. Math., 56:1 (2016), 29–44 | DOI | MR | Zbl

[19] S. V. Astashkin, “On subspaces of an Orlicz space spanned by independent identically distributed functions”, Dokl. Math., 108:1 (2023), 297–299 | DOI

[20] S. G. Kreĭn, Ju. I. Petunin, and E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, RI, 1982 | MR | Zbl

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

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

[23] M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961 | MR | Zbl

[24] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, Inc., New York, 1991 | MR | Zbl

[25] L. Maligranda, Orlicz spaces and interpolation, Sem. Mat., 5, Univ. Estad. Campinas, Dep. de Matemática, Campinas, SP, 1989 | MR | Zbl

[26] J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces. III”, Israel J. Math., 14 (1973), 368–389 | DOI | MR | Zbl

[27] A. Kamińska and Y. Raynaud, “Isomorphic copies in the lattice $E$ and its symmetrization $E^{*}$ with applications to Orlicz–Lorentz spaces”, J. Funct. Anal., 257:1 (2009), 271–331 | DOI | MR | Zbl

[28] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, v. I, Ergeb. Math. Grenzgeb., 92, Sequence spaces, Springer-Verlag, Berlin–New York, 1977 | MR | Zbl

[29] S. V. Astashkin, “$\Lambda(p)$-spaces”, J. Funct. Anal., 266:8 (2014), 5174–5198 | DOI | MR | Zbl

[30] S. Montgomery-Smith and E. Semenov, “Random rearrangements and operators”, Voronezh winter mathematical schools, Amer. Math. Soc. Transl. Ser. 2, 184, Adv. Math. Sci., 37, Amer. Math. Soc., Providence, RI, 1998, 157–183 | DOI | MR | Zbl

[31] L. V. Kantorovich and G. P. Akilov, Functional analysis, Pergamon Press, Oxford–Elmsford, NY, 1982 | MR | Zbl

[32] J. Alexopoulos, “De La Vallée Poussin's theorem and weakly compact sets in Orlicz spaces”, Quaest. Math., 17:2 (1994), 231–248 | DOI | MR | Zbl

[33] R. del Campo, A. Fernández, F. Mayoral, and F. Naranjo, “Compactness in quasi-Banach function spaces with applications to $L^1$ of the semivariation of a vector measure”, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114:3 (2020), 112 | DOI | MR | Zbl

[34] K. Leśnik, L. Maligranda, and J. Tomaszewski, “Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices”, Math. Nachr., 295:3 (2022), 574–592 | DOI | MR | Zbl

[35] B. S. Kashin and A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989 | DOI | MR | Zbl

[36] W. B. Johnson and G. Schechtman, “Sums of independent random variables in rearrangement invariant function spaces”, Ann. Probab., 17:2 (1989), 789–808 | DOI | MR | Zbl

[37] S. V. Astashkin, “Independent functions in rearrangement invariant spaces and the Kruglov property”, Sb. Math., 199:7 (2008), 945–963 | DOI

[38] S. Montgomery-Smith, “Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables”, Israel J. Math., 131 (2002), 51–60 | DOI | MR | Zbl

[39] S. V. Astashkin and E. M. Semenov, “Some properties of embeddings of rearrangement invariant spaces”, Sb. Math., 210:10 (2019), 1361–1379 | DOI