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Transactions of the American Mathematical Society

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



The theory of $ p$-spaces with an application to convolution operators.

Author: Carl Herz
Journal: Trans. Amer. Math. Soc. 154 (1971), 69-82
MSC: Primary 22.65
MathSciNet review: 0272952
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Abstract: The class of p-spaces is defined to consist of those Banach spaces B such that linear transformations between spaces of numerical $ {L_p}$-functions naturally extend with the same bound to B-valued $ {L_p}$-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hilbert spaces are the same and that p-spaces are uniformly convex for $ 1 < p < \infty $. An $ {L_q}$-space is a p-space iff $ p \leqq q \leqq 2$ or $ p \geqq q \geqq 2$; this leads to the theorem that, for an amenable group, a convolution operator on $ {L_p}$ gives a convolution operator on $ {L_q}$ with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarity of notions related to p-spaces are discussed.

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Keywords: Category of Banach spaces, tensor product, $ {L_p}$-space, p-space, locally compact group, representation, representative function, regular representation on $ {L_p}$, convolution operator, measure space, Bochner integral, negative-definite function, ultraproduct
Article copyright: © Copyright 1971 American Mathematical Society

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