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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Interpolation theorems for the pairs of spaces $ (L\sp{p},\,L\sp{\infty })$ and $ (L\sp{1},\,L\sp{q})$

Authors: George G. Lorentz and Tetsuya Shimogaki
Journal: Trans. Amer. Math. Soc. 159 (1971), 207-221
MSC: Primary 46M35; Secondary 46E30
MathSciNet review: 0380447
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Abstract: A Banach space $ Z$ has the interpolation property with respect to the pair $ (X,Y)$ if each $ T$, which is a bounded linear operator from $ X$ to $ X$ and from $ Y$ to $ Y$, can be extended to a bounded linear operator from $ Z$ to $ Z$. If $ X = {L^p},Y = {L^\infty }$ we give a necessary and sufficient condition for a Banach function space $ Z$ on $ (0,l),0 < l \leqq + \infty $, to have this property. The condition is that $ g \prec {}^pf$ and $ f \in Z$ should imply $ g \in Z$; here $ g \prec {}^pf$ means that $ {g^{ \ast p}} \prec {f^{ \ast p}}$ in the Hardy-Littlewood-Pólya sense, while $ {h^ \ast }$ denotes the decreasing rearrangement of the function $ \vert h\vert$.

If the norms $ \vert\vert T\vert{\vert _X},\vert\vert T\vert{\vert _Y}$ are given, we can estimate $ \vert\vert T\vert{\vert _Z}$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding $ {\lambda _p},{\lambda _p} \leqq {2^{1/q}},1/p + 1/q = 1$.

Similar results hold if $ X = {L^1},Y = {L^q}$. For all these theorems, the complete continuity of $ T$ on $ Z$ is assured if $ T$ has this property on $ X$ or on $ Y$, and if $ Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $ \vert\vert{\sigma _a}\vert{\vert _Z},a > 0$, where $ {\sigma _a}$ is the compression operator $ {\sigma _a}f(t) = f(at),0 \leqq t < l$.

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Keywords: Interpolation property, interpolation theorem, quasi-order, rearrangement invariant Banach function space, space monotone with respect to a quasi-order, completely continuous operator, compression operator, Orlicz space
Article copyright: © Copyright 1971 American Mathematical Society