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Subspaces of $ L\sb {p,q}$


Authors: N. L. Carothers and S. J. Dilworth
Journal: Proc. Amer. Math. Soc. 104 (1988), 537-545
MSC: Primary 46E30; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1988-0962825-8
MathSciNet review: 962825
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Abstract: We examine the subspace structure of the Lorentz function space $ {L_{p,q}}[0,\infty )$. Our main result is that a subspace of $ {L_{p,q}}[0,\infty ),p \ne 2,q < \infty $, must either strongly embed in $ {L_p}[0,1]$ or contain a complemented copy of $ {l_q}$.


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  • [1] Z. Altshuler, P. G. Casazza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 144-155. MR 0328553 (48:6895)
  • [2] J. Bergh and J. Löfstrom, Interpolation spaces, Grundlehren der mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, Heidelberg, and New York, 1976.
  • [3] N. L. Carothers, Rearrangement invariant subspaces of Lorentz function spaces. II, Rocky Mountain J. Math. 17 (1987), 607-616. MR 908267 (89a:46065)
  • [4] N. L. Carothers and S. J. Dilworth, Equidistributed random variables in $ {L_{p,q}}$, J. Funct. Anal. (to appear).
  • [5] P. G. Casazza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces. II, Israel J. Math. 17 (1974), 191-218. MR 0348443 (50:941)
  • [6] J. Creekmore, Type and cotype in Lorentz $ {L_{p,q}}$ spaces, Indag. Math. 43 (1981), 145-152. MR 707247 (84i:46032)
  • [7] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math., vol. 92, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1984. MR 737004 (85i:46020)
  • [8] P. Enflo and H. P. Rosenthal, Some results concerning $ {L_p}(\mu )$-spaces, J. Funct. Anal. 14 (1973), 325-348. MR 0350402 (50:2895)
  • [9] T. Figiel, W. B. Johnson and L. Tzafriri, On Banach lattices and spaces having local unconditional structure with applications to Lorentz function spaces, J. Approx. Theory 13 (1975), 297-312. MR 0367624 (51:3866)
  • [10] S. Guerre and M. Levy, Espaces $ {l_p}$ dans les sous-espaces de $ {L_1}$, Trans. Amer. Math. Soc. 279 (1983), 611-616. MR 709571 (85d:46033)
  • [11] R. A. Hunt, On $ L(p,q)$ spaces, Enseign. Math. 12 (1966), 249-274. MR 0223874 (36:6921)
  • [12] W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 217 (1979). MR 527010 (82j:46025)
  • [13] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $ {L_p}$, Studia Math. 21 (1962), 161-176.
  • [14] N. J. Kalton, Linear operators on $ {L_p}$ for $ 0 < p < 1$, Trans. Amer. Math. Soc. 259 (1980), 319-355. MR 567084 (81d:47022)
  • [15] -, Compact and strictly singular operators on certain function spaces, Arch. Math. 43 (1984), 66-78. MR 758342 (85m:47027)
  • [16] -, Banach spaces embedding into $ {L_0}$, Israel J. Math. 52 (1985), 305-319. MR 829361 (87k:46045)
  • [17] M. Levy, Thesis, Université Paris VI, 1980 (= C.R. Acad. Sci. Paris 289 (1979), 675-677).
  • [18] J. Lindenstrauss and L. Tzafriri, On Orlicz sequences spaces. II, Israel J. Math. 11 (1970), 355-379. MR 0310592 (46:9690)
  • [19] -, Classical Borel Spaces. I. Sequence Spaces, Ergebnisse Math. Grenzgebiete, Bd. 92, Springer-Verlag, Berlin, Heidelberg, and New York, 1977. MR 0500056 (58:17766)
  • [20] -, Classical Banach Spaces. II. Function Spaces, Ergebnisse Math. Grenzgebiete, Bd. 97, Springer-Verlag, Berlin, Heidelberg, and New York, 1979. MR 540367 (81c:46001)
  • [21] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5-68. MR 0165343 (29:2627)
  • [22] I. P. Natanson, Theory of functions of a real variable, Vol. 1, Ungar, New York, 1961.
  • [23] R. O'Neil, Integral transforms and tensor products on Orlicz spaces and $ L(p,q)$ spaces, J. Analyse Math. 21 (1968), 1-276. MR 0626853 (58:30125)
  • [24] Y. Raynaud, Thesis, Université Paris VII, 1981.
  • [25] H. P. Rosenthal, On subspaces of $ {L_p}$, Ann. of Math. 97 (1973), 344-373. MR 0312222 (47:784)
  • [26] E. M. Stein and G. Weiss, Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0962825-8
Keywords: Lorentz function spaces, subspaces
Article copyright: © Copyright 1988 American Mathematical Society

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