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Transactions of the American Mathematical Society

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



A pair of indices for function spaces on the circle

Author: Colin Bennett
Journal: Trans. Amer. Math. Soc. 174 (1972), 289-304
MSC: Primary 46E30
MathSciNet review: 0333699
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Abstract: We give here some of the basic properties of the classes $ \{ {\Phi _r}\} $, $ \{ {\Psi _r}\} , - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces $ \mathfrak{X}$ on the circle It is shown that to each space $ \mathfrak{X}$ there correspond two numbers $ \xi ,\eta $, called indices, which satisfy $ 0 \leq \eta \leq \xi \leq 1$; these numbers represent the rate of growth or decay of $ \left\Vert {{\Psi _r}} \right\Vert$ as $ r \to \pm 1$.

By using the operators $ {\Psi _r}$ to obtain estimates for certain averaging operators $ {A_\gamma }$, we are able to show that the indices $ (\xi ,\eta )$ coincide with the Boyd indices $ (\alpha ,\beta )$. As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.

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Keywords: Rearrangement-invariant space, interpolation theorem, dilation operator, indices
Article copyright: © Copyright 1972 American Mathematical Society

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