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St. Petersburg Mathematical Journal

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Finite representability of $ \ell_p$-spaces in symmetric spaces


Author: S. V. Astashkin
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 257-273
MSC (2010): Primary 46E30
DOI: https://doi.org/10.1090/S1061-0022-2012-01196-9
Published electronically: January 23, 2012
MathSciNet review: 2841673
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Abstract: For a separable rearrangement invariant space $ X$ on the semiaxis, $ {\mathcal F}(X)$ is defined to be the set of all $ p\in [1,\infty ]$ such that $ \ell _p$ is finitely representable in $ X$ in such a way that the standard basis vectors of $ \ell _p$ correspond to equimeasurable mutually disjoint functions. In the paper, a characterization of the set $ {\mathcal F}(X)$ is obtained. As a consequence, a version of Krivine's well-known theorem is proved for rearrangement invariant spaces. Next, a description of the sets $ {\mathcal F}(X)$ for certain Lorentz spaces is found.


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Additional Information

S. V. Astashkin
Affiliation: Samara State University, ul. Akademika Pavlova 1, Samara 443011, Russia
Email: astashkn@ssu.samara.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01196-9
Keywords: Finite representability of $ℓ_{p}$-spaces, symmetric spaces, Boyd indices, Lorentz space, spectrum, weighted spaces
Received by editor(s): October 7, 2009
Published electronically: January 23, 2012
Additional Notes: Supported in part by RFBR, grant no. 07-01-96603
Article copyright: © Copyright 2012 American Mathematical Society

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