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2000 | 138 | 3 | 209-224
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Interpolation on families of characteristic functions

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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 $\overline Φ =(B,L^∞)$ 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 ⨍.
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Bibliografia
  • [B] B. Beauzamy, Espaces d'interpolation réels: topologie et géométrie, Lecture Notes in Math. 666, Springer, Berlin, 1978.
  • [BL] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
  • [CJM] M. Cwikel, B. Jawerth, and M. Milman, The couple $(B,L^∞)$ and commutator estimates, unpublished manuscript.
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  • [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.
  • [G2] A. Gulisashvili, The interpolation theorem on subsets, Bull. Georgian Acad. Sci. 88 (1977), 545-548.
  • [G3] A. Gulisashvili, The individual interpolation theorem, ibid. 94 (1979), 33-36.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
  • [T] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 110 (1979), 407-438.
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bwmeta1.element.bwnjournal-article-smv138i3p209bwm
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