Abstract
Our goal in this paper is to continue the study initiated by the authors in [4] of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0 < p <= 1, denoted F-p(M). Here we develop new techniques to show that, by analogy with the case p = 1, the space l(p) embeds isomorphically in F-p(M) for 0 < p < 1.
Going further we see that despite the fact that, unlike the case p = 1, this embedding need not be complemented in general, complementability of l(p) in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in F-p([0, 1]), we obtain examples of p-Banach spaces for p < 1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis. (C) 2019 Elsevier Inc.
All rights reserved.