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 "-functions naturally extend with the same bound to -valued "-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hubert spaces are the same and that ^-spaces are uniformly convex for 1 <p< . An ,-space is a p-space iff/><7=i2 or p^q^2; this leads to the theorem that, for an amenable group, a convolution operator on Lp gives a convolution operator on Lq with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarily of notions related to /-spaces are discussed. 0. Introduction. Let R designate the field of real or complex numbers. We denote by 38 the category whose objects are complete normed linear spaces over R and whose morphisms are the bounded i?-linear transformations of norm f 1.