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

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.