On symmetric spaces containing isomorphic copies of Orlicz sequence spaces

• Autorzy: Astashkin S.

Identyfikatory

DOI

10.14708/cm.v56i1.1113

• Języki publikacji: EN

Abstrakty

EN

Let an Orlicz function N be (1+ε)-convex and (2−ε)-concave at zero for some ε>0. Then the function 1/N⁻¹(t), t∈(0,1], belongs to a separable symmetric space X with the Fatou property, which is an interpolation space with respect to the couple (L₁,L₂), whenever X contains a strongly embedded subspace isomorphic to the Orlicz sequence space l_N. On the other hand, we find necessary and sufficient conditions on such an Orlicz function N under which a sequence of mean zero independent functions equimeasurable with the function 1/N⁻¹(t), 0<t<1, spans, in the Marcinkiewicz space M(φ) with φ(t) = t/N^-1(t), a strongly embedded subspace isomorphic to the Orlicz sequence l_N.

Słowa kluczowe

EN

symmetric space; Orlicz sequence space; independent functions; p-convex function; q-concave function; interpolation of operators

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2016
• Tom: Vol 56, No. 1
• Strony: 29--44
• Opis fizyczny: Bibliogr. 29 poz.

Twórcy

autor

Astashkin S.

• Department of Mathematics and mechanics, Samara State Aerospace University (SSAU), moskovskoye shosse 34, 443086, Samara, Russia

Bibliografia

• [1]. F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Springer-Verlag, New York 2006.
• [2]. S. Astashkin, F. Hernandez, and E. Semenov, Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces, Studia Math. 193 (2009), no. 3, 269-283, DOI 10.4064/sml93-3-4.
• [3]. S. Astashkin, N. Kalton, and F. Sukochev, Cesaro mean convergence of martingale differences in rearrangement invariant spaces, Positivity 12 (2008), 387-406, DOI 10.1007/sllll7-007-2146-y.
• [4]. S. Astashkin and M. Maligranda, Interpolation between Li and Lp,\ < p < ∞„ Proc. Amer. Math. Soc. 132 (2004), no. 12, 2929-2938, DOI 10.1090/S0002-9939-04-07425-8.
• [5]. S. Astashkin, E. Semenov, and F. Sukochev, The Banach-Saks p-property, Math. Ann. 332 (2005), 879-900, DOI 10.1007/s00208-005-0658-y.
• [6]. S. Astashkin and F. Sukochev, Orlicz sequence spaces spanned by identically distributed independent random variables in Lp-spaces, J. Math. Anal. Appl. 413 (2014), no. 1,1-19, DOI 10.1016/j.jmaa.2013.11.023.
• [7]. S. Astashkin, F. Sukochev, and C. Wong, Disjointification of martingale differences and conditionally independent random variables with some applications, Studia Math. 205 (2011), no. 2, 171-200, DOI 10.4064/sm205-2-3.
• [8]. S. Astashkin, F. Sukochev, and D. Zanin, On uniqueness of distribution of a random variable whose independent copies span a subspace in Lp, Studia Math. 230 (2015), no. 1, 41-57, DOI 10.4064/sm8089-l-2016.
• [9]. C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, London 1988.
• [10]. J. Bergh and J. Lofstrom, Interpolation spaces. An Introduction, Grundlehren Math. Wiss., vol. 223, Springer-Verlag, Berlin-New York 1976.
• [11]. M. Sh. Braverman, On some moment conditions for sums of independent random variables, Probab. Math. Statist. 14 (1993), no. 1, 45-56.
• [12]. M. Sh. Braverman, Independent random variables in Lorentz spaces, Bull. London Math. Soc. 28 (1996), no. 1, 79-87, DOI 10.1112/blms/28.1.79.
• [13]. M. Sh. Braverman, Independent random variables and rearrangement invariant spaces, Cambridge University Press, Cambridge 1994, DOI 10.1017/CB097805U662348.
• [14]. J. Bretagnolle and D. Dacunha-Castelle, Mesures aléatoires et espaces d’Orlicz, C. R. Acad. Sci. Paris Ser. A-B 264 (1967), A877-A880; in French.
• [15]. J. Bretagnolle and D. Dacunha-Castelle, Application de ¡’étude de certaines formes linéaires aléatoires au plongement d’espaces de Banach dans des espaces Lp, Ann. Sci. Ecole Norm. Sup. 2 (1969), no. 5, 437-480.
• [16]. D. Dacunha-Castelle, Variables aléatoires échangeables et espaces d’Orlicz, Séminaire Maurey-Schwartz 1974-1975: Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos., vol. X et XI, Centre Math., École Polytech., Paris, 1975.
• [17]. D. Dacuncha-Castelle and J.-L. Krivine, Sous espaces de L1, Israel J.Math. 26 (1977), 330-351.
• [18]. W. Johnson and G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989), 789-808.
• [19]. L. Hernandez, S. Novikov, and E. Semenov, Strictly singular embeddings between rearrangement invariant spaces, Positivity 7 (2003), 119-124, DOI 10.1023/A-.1025828302994.
• [20]. T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), no. 1,177-199.
• [21]. M. I. Kadec, Linear dimension of the spaces Lp and lq, Uspehi Mat. Nauk 84 (1958), no. 6; in Russian.
• [22]. M. Krasnoselskii and J. Rutickii, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow 1958; English transi, in Noordhoff, Groningen 1961.; in Russian.
• [23]. S. Krein, Ju. Petunin, and E. Semenov, Interpolation of linear operators, Nauka, Moscow 1978; English transl, in Translations of Math. Monographs, vol. 54, Amer. Math. Soc., Providence, RI1982.; in Russian.
• [24]. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function spaces, Springer-Verlag, Berlin- -Heidelberg-New York 1979.
• [25]. G. Ya. Lozanovskii, Transformations of ideal Banach spaces by means of concave functions, Qualitative and Approximate methods for investigation of operator equations, Yaroslavl’, 1978, 122-148; in Russian.
• [26]. M. Mastylo, Interpolation of linear operators in the Kothe dual spaces, Annali di Matematica pura ed applicata (IV) CLIV (1989), 231-242.
• [27]. E. Semenov and F. Sukochev, Banach-Saks index, Sbornik: Mathematics 195 (2004), 263-285.
• [28]. G. Sparr, Interpolation of weighted Lp-spaces, Studia Math. 62 (1978), no. 3, 229-271.
• [29]. Y. Raynaud, Complemented Hilbertian subspaces in rearrangement invariant function spaces, Illinois J. Math. 39 (1995), no. 2, 212-250.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.