Lipschitz free p-spaces for 0 < p < 1

Abstract

This paper initiates the study of the structure of a new class of p-Banach spaces, 0 <p < 1, namely the Lipschitz free p-spaces (alternatively called Arens—Eells p-spaces) Fp(M) over p-metric spaces. We systematically develop the theory and show that some results hold as in the case of p = 1, while some new interesting phenomena appear in the case 0 <p < 1 which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free p-space over a separable ultrametric space is isomorphic to ℓp for all 0 <p ≤ 1. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces N⊂M such that the natural embedding from Fp(N) to Fp(M) is not an isometry.