Interpolation on families of characteristic functions

Michael Cwikel; Archil Gulisashvili

Studia Mathematica (2000)

  • Volume: 138, Issue: 3, page 209-224
  • ISSN: 0039-3223

Abstract

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We study a problem of interpolating a linear operator which is bounded on some family of characteristic functions. A new example is given of a Banach couple of function spaces for which such interpolation is possible. This couple is of the form Φ ¯ = ( B , L ∞ ) where B is an arbitrary Banach lattice of measurable functions on a σ-finite nonatomic measure space (Ω,Σ,μ). We also give an equivalent expression for the norm of a function ⨍ in the real interpolation space ( B , L ∞ ) θ , p in terms of the characteristic functions of the level sets of ⨍.

How to cite

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Cwikel, Michael, and Gulisashvili, Archil. "Interpolation on families of characteristic functions." Studia Mathematica 138.3 (2000): 209-224. <http://eudml.org/doc/216700>.

@article{Cwikel2000,
abstract = {We study a problem of interpolating a linear operator which is bounded on some family of characteristic functions. A new example is given of a Banach couple of function spaces for which such interpolation is possible. This couple is of the form $\overline\{Φ\} =(B,L^∞)$ where B is an arbitrary Banach lattice of measurable functions on a σ-finite nonatomic measure space (Ω,Σ,μ). We also give an equivalent expression for the norm of a function ⨍ in the real interpolation space $(B,L^∞)_\{θ,p\}$ in terms of the characteristic functions of the level sets of ⨍.},
author = {Cwikel, Michael, Gulisashvili, Archil},
journal = {Studia Mathematica},
keywords = {real interpolation method; -functional; couple of Banach lattices},
language = {eng},
number = {3},
pages = {209-224},
title = {Interpolation on families of characteristic functions},
url = {http://eudml.org/doc/216700},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Cwikel, Michael
AU - Gulisashvili, Archil
TI - Interpolation on families of characteristic functions
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 3
SP - 209
EP - 224
AB - We study a problem of interpolating a linear operator which is bounded on some family of characteristic functions. A new example is given of a Banach couple of function spaces for which such interpolation is possible. This couple is of the form $\overline{Φ} =(B,L^∞)$ where B is an arbitrary Banach lattice of measurable functions on a σ-finite nonatomic measure space (Ω,Σ,μ). We also give an equivalent expression for the norm of a function ⨍ in the real interpolation space $(B,L^∞)_{θ,p}$ in terms of the characteristic functions of the level sets of ⨍.
LA - eng
KW - real interpolation method; -functional; couple of Banach lattices
UR - http://eudml.org/doc/216700
ER -

References

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  1. [B] B. Beauzamy, Espaces d'interpolation réels: topologie et géométrie, Lecture Notes in Math. 666, Springer, Berlin, 1978. Zbl0382.46021
  2. [BL] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. Zbl0344.46071
  3. [CJM] M. Cwikel, B. Jawerth, and M. Milman, The couple ( B , L ∞ ) and commutator estimates, unpublished manuscript. 
  4. [CP] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl. 60 (1981), 1-50. 
  5. [DU] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977. 
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  8. [F] K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer, Berlin, 1980. Zbl0437.46006
  9. [G1] A. Gulisashvili, An interpolation theorem of weak type and the behavior of the Fourier transform of a function having prescribed Lebesgue sets, Dokl. Akad. Nauk SSSR 218 (1974), 1268-1271 (in Russian); English transl.: Soviet Math. Dokl. 15 (1974), 1481-1485. 
  10. [G2] A. Gulisashvili, The interpolation theorem on subsets, Bull. Georgian Acad. Sci. 88 (1977), 545-548. 
  11. [G3] A. Gulisashvili, The individual interpolation theorem, ibid. 94 (1979), 33-36. 
  12. [G4] A. Gulisashvili, Estimates for the Pettis integral in interpolation spaces and some inverse embedding theorems, Dokl. Akad. Nauk SSSR 263 (1982), 793-798 (in Russian); English transl.: Soviet Math. Dokl. 25 (1982), 428-432. 
  13. [G5] A. Gulisashvili, Estimates for the Pettis integral in interpolation spaces with some applications, in: Banach Space Theory and its Applications ( Bucharest, 1981), Lecture Notes in Math. 991, Springer, Berlin, 1983, 55-76. 
  14. [G6] A. Gulisashvili, Rearrangements of functions on a locally compact abelian group and integrability of the Fourier transform, J. Funct. Anal. 146 (1997), 62-115. Zbl0870.43001
  15. [KPS] S. G. Krein, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, RI, 1982. 
  16. [P] J. D. Pryce, A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. 23 (1971), 532-546. Zbl0221.46012
  17. [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971. 
  18. [T] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 110 (1979), 407-438. Zbl0393.46019

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