Interpolation between 𝐿₁ and 𝐿_{𝑝},1<𝑝<∞

Astashkin, Sergei; Maligranda, Lech

doi:10.1090/S0002-9939-04-07425-8

Interpolation between $L_{1}$ and $L_{p}, 1 < p < \infty$
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by Sergei V. Astashkin and Lech Maligranda

Proc. Amer. Math. Soc. 132 (2004), 2929-2938

DOI: https://doi.org/10.1090/S0002-9939-04-07425-8

Published electronically: May 21, 2004

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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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