Nonlinear structure of some classical quasi-Banach spaces and F-spaces

We investigate the Lipschitz structure of l(p) and L-p for 0 < p < 1 as quasi-Banach spaces and as metric spaces (with the metric induced by the p-norm) and show that they are not Lipschitz isomorphic. We prove that the F-space L-0 is not uniformly homeomorphic to any other L-p space, that the L-p spaces for 0 < p < 1 embed isometrically into one another, and reduce the problem of the uniform equivalence amongst Lp spaces to their Lipschitz equivalence as metric spaces. (c) 2007 Elsevier Inc. All rights reserved.