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), 6982 Request permission
Abstract:
The class of pspaces 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 Bvalued ${L_p}$functions. Some properties of pspaces are derived including norm inequalities which show that 2spaces and Hilbert spaces are the same and that pspaces are uniformly convex for $1 < p < \infty$. An ${L_q}$space is a pspace 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 pspaces are examined. Logical elementarity of notions related to pspaces are discussed.References

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Additional Information
 © Copyright 1971 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 154 (1971), 6982
 MSC: Primary 22.65
 DOI: https://doi.org/10.1090/S00029947197102729520
 MathSciNet review: 0272952