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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9939-04-07425-8
PII: S 0002-9939(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