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Interpolation between $L_{1}$ and $L_{p}, 1 < p < \infty$


Authors: Sergei V. Astashkin and Lech Maligranda
Journal: Proc. Amer. Math. Soc. 132 (2004), 2929-2938
MSC (2000): Primary 46E30, 46B42, 46B70
DOI: https://doi.org/10.1090/S0002-9939-04-07425-8
Published electronically: May 21, 2004
MathSciNet review: 2063112
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Abstract: We show that if $X$ is a rearrangement invariant space on $[0, 1]$ that is an interpolation space between $L_{1}$ and $L_{\infty}$ and for which we have only a one-sided estimate of the Boyd index $\alpha(X) > 1/p, 1 < p < \infty$, then $X$ is an interpolation space between $L_{1}$ and $L_{p}$. This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type $(1, 1)$ and weak type $(p, p), 1 < p < \infty$.


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  • [BS] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. MR 89e:46001
  • [B67] D. W. Boyd, Spaces between a pair of reflexive Lebesgue spaces, Proc. Amer. Math. Soc. 18 (1967), 215-219. MR 35:3427
  • [B68] D. W. Boyd, The spectral radius of averaging operators, Pacific J. Math. 24 (1968), 19-28. MR 36:4360
  • [B69] D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. MR 54:909
  • [C] M. Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat. 14 (1976), 213-236. MR 56:1095
  • [G] Ju. I. Gribanov, Banach function spaces and integral operators. II., Izv. Vyss. Ucebn. Zaved. Matematika 1966, no. 6 (55), 54-63 (Russian). MR 35:3489
  • [H] T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177-199. MR 54:3440
  • [JMST] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric Structures in Banach Spaces, Memoirs Amer. Math. Soc. 19, 1979. MR 82j:46025
  • [KPS] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (Russian); English transl. in Amer. Math. Soc., Providence, RI, 1982. MR 84j:46103
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II. Function Spaces, Springer-Verlag, Berlin-New York, 1979. MR 81c:46001
  • [LS] G. G. Lorentz and T. Shimogaki, Interpolation theorems for the pairs of spaces $(L^p, L^{\infty})$ and $(L^1,L^q)$, Trans. Amer. Math. Soc. 159 (1971), 207-221. MR 52:1347
  • [M81] L. Maligranda, A generalization of the Shimogaki theorem, Studia Math. 71 (1981), 69-83. MR 83d:46034
  • [M85] L. Maligranda, Indices and interpolation, Dissertationes Math. 234 (1985), 1-54. MR 87k:46059
  • [Ms] M. Masty\lo, Interpolation of linear operators in the Köthe dual spaces, Ann. Mat. Pura Appl. 154 (1989), 231-242. MR 91f:46105
  • [R] G. I. Russu, Symmetric spaces of functions that do not have the majorization property, Mat. Issled. 4 (1969), 82-93 (Russian). MR 43:2490

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Additional Information

Sergei V. Astashkin
Affiliation: Department of Mathematics, Samara State University, Akad. Pavlova 1, 443011 Samara, Russia
Email: astashkn@ssu.samara.ru

Lech Maligranda
Affiliation: Department of Mathematics, Lulelå University of Technology, se-971 87 Luleå, Sweden
Email: lech@sm.luth.se

DOI: https://doi.org/10.1090/S0002-9939-04-07425-8
Keywords: $L_{p}$-spaces, Lorentz spaces, rearrangement invariant spaces, Boyd indices, interpolation of operators, operators of strong type, operators of weak type, $K$-functional, Marcinkiewicz spaces
Received by editor(s): October 9, 2002
Published electronically: May 21, 2004
Additional Notes: This research was supported by a grant from the Royal Swedish Academy of Sciences for cooperation between Sweden and the former Soviet Union (project 35156). The second author was also supported in part by the Swedish Natural Science Research Council (NFR)-grant M5105-20005228/2000.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society

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