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Title

Calderón couples of rearrangement invariant spaces

Authors 1

Affiliations

  1. Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.

Abstract

We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with L∞. We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair (LF,L∞) forms a Calderón pair.

Bibliography

  1. J. Arazy and M. Cwikel, A new characterization of the interpolation spaces between Lp and Lq, Math. Scand. 55 (1984), 253-270.
  2. S. F. Bellenot, The Banach spaces of Maurey and Rosenthal and totally incomparable bases, J. Funct. Anal. 95 (1991), 96-105.
  3. S. F. Bellenot, R. Haydon and E. Odell, Quasi-reflexive and tree spaces constructed in the spirit of R. C. James, in: Contemp. Math. 85, Amer. Math. Soc., 1987, 19-43.
  4. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York 1988.
  5. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin 1976.
  6. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge Univ. Press, 1987.
  7. Yu. Brudnyĭ and N. Kruglyak, Real interpolation functors, Soviet Math. Dokl. 23 (1981), 5-8.
  8. Yu. Brudnyĭ and N. Kruglyak, Interpolation Functors and Interpolation Spaces, North-Holland, 1991.
  9. A. P. Calderón, Spaces between L1 and L∞ and the theorems of Marcinkiewicz, Studia Math. 26 (1966), 273-299.
  10. P. G. Casazza, W. B. Johnson and L. Tzafriri, On Tsirelson's space, Israel J. Math. 47 (1984), 81-98.
  11. P. G. Casazza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces, II, ibid. 17 (1974), 191-218.
  12. P. G. Casazza and T. J. Shura, Tsirelson's space, Lecture Notes in Math. 1363, Springer, Berlin 1989.
  13. M. Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat. 14 (1976), 213-236.
  14. M. Cwikel, Monotonicity properties of interpolation spaces II, ibid. 19 (1981), 123-136.
  15. M. Cwikel, K-divisibility of the K-functional and Calderón couples, ibid. 22 (1984), 39-62.
  16. M. Cwikel and P. Nilsson, On Calderón-Mityagin couples of Banach lattices, in: Proc. Conf. Constructive Theory of Functions, Varna 1984, Bulgarian Acad. Sci., 1984, 232-236.
  17. M. Cwikel and P. Nilsson, Interpolation of Marcinkiewicz spaces, Math. Scand. 56 (1985), 29-42.
  18. M. Cwikel and P. Nilsson, Interpolation of weighted Banach lattices, Mem. Amer. Math. Soc., to appear.
  19. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1934.
  20. F. L. Hernandez and B. Rodriguez-Salinas, On ℓp-complemented copies in Orlicz spaces II, Israel J. Math. 68 (1989), 27-55.
  21. P. W. Jones, On interpolation between H1 and H∞, in: Interpolation Spaces and Allied Topics in Analysis, Proc. Lund Conference 1983, M. Cwikel and J. Peetre (eds.), Lecture Notes in Math. 1070, Springer, 1984, 143-151.
  22. J. L. Krivine, Sous espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. 104 (1976), 1-29.
  23. J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263-269.
  24. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin 1977.
  25. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin 1979.
  26. G. G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127-132.
  27. G. G. Lorentz and T. Shimogaki, Interpolation theorems for the pairs of spaces (L1,Lp) and (Lp,L∞), Trans. Amer. Math. Soc. 159 (1971), 207-222.
  28. L. Maligranda, On Orlicz results in interpolation theory, in: Proc. Orlicz Memorial Conference, Univ. of Mississippi, 1990.
  29. L. Maligranda and V. I. Ovchinnikov, On interpolation between L1+L∞ and L1∩L∞, J. Funct. Anal. 107 (1992), 342-351.
  30. C. Merucci, Interpolation réelle avec fonction paramètre: réitération et applications aux espaces Λp(ϕ)(0<pℓ+∞), C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 427-430.
  31. C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in: Interpolation Spaces and Allied Topics in Analysis, Proc. Lund Conference 1983, M. Cwikel and J. Peetre (eds.), Lecture Notes in Math. 1070, Springer, 1984, 183-201.
  32. B. S. Mityagin, An interpolation theorem for modular spaces, Mat. Sb. 66 (1965), 473-482 (in Russian).
  33. S. J. Montgomery-Smith, Comparison of Orlicz-Lorentz spaces, Studia Math. 103 (1992), 161-189.
  34. V. I. Ovchinnikov, On the estimates of interpolation orbits, Mat. Sb. 115 (1981), 642-652 (in Russian) (= Math. USSR-Sb. 43 (1982), 573-583).
  35. A. A. Sedaev and E. M. Semenov, On the possibility of describing interpolation spaces in terms of Peetre's K-method, Optimizatsiya 4 (1971), 98-114 (in Russian).
  36. G. Sparr, Interpolation of weighted Lp-spaces, Studia Math. 62 (1978), 229-271.
  37. B. S. Tsirelson, Not every Banach space contains an embedding of ℓp or c0, Functional Anal. Appl. 8 (1974), 138-141.
  38. Q. Xu, Notes on interpolation of Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), 877-889.
  39. M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265-272.
Studia Mathematica
1993/106/3
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Pages:
233-277
Main language of publication
English
Received
1992-10-27
Published
1993
Exact and natural sciences