The Theory of $p$-Spaces with an Application to Convolution Operators

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 t...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1971-01, Vol.154, p.69-82
Main Author: Herz, Carl
Format: Article
Language:English
Subjects:
Banach space
Borel sets
Continuous functions
Functors
Mathematical functions
Mathematical theorems
Morphisms
Tensors
Ultraproducts
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Summary: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.
ISSN:0002-9947
DOI:10.1090/S0002-9947-1971-0272952-0