Interpolation and L^p Inequalities

Abstract

In this chapter we develop a multi-component version of the Riesz-Thorin interpolation theorem and use it to derive a number of L^p inequalities which are natural relatives of inequality (4.15) and the now standard inequalities of Clarkson. These inequalities are crucial to the problem of extending Lipschitz-Hölder maps of order a between L^p spaces (see §19). In addition they are of considerable intrinsic interest, a point we here emphasize by applying certain of their number to a packing problem in L^p.