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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An interpolation theorem in symmetric function $ F$-spaces


Authors: H. Hudzik and L. Maligranda
Journal: Proc. Amer. Math. Soc. 110 (1990), 89-96
MSC: Primary 46E30; Secondary 46M35
MathSciNet review: 1000157
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Abstract: It is well known that every separable or perfect symmetric Banach function space $ X$ is an interpolation space between $ {L^1}$ and $ {L^\infty }$ (see [1] and [4]). In this paper we prove that every symmetric function $ F$-space is interpolation between $ {L^0}$ and $ {L^\infty }$, where $ {L^0}$ is the space of all measurable functions whose support has finite measure. Moreover, for any function $ f \in {L^0} + {L^\infty }$ the norm $ {\left\Vert f \right\Vert _{{L^0}}} + {L^\infty }$ is computed in the terms of the nonincreasing rearrangement function $ {f^ * }$ of $ f$ as well as in terms of its distribution function $ {d_f}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1000157-1
Keywords: Symmetric function $ F$-spaces, interpolation
Article copyright: © Copyright 1990 American Mathematical Society