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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The theory of $p$-spaces with an application to convolution operators.
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by Carl Herz PDF
Trans. Amer. Math. Soc. 154 (1971), 69-82 Request permission

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.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 154 (1971), 69-82
  • MSC: Primary 22.65
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0272952-0
  • MathSciNet review: 0272952