Some aspects of the geometry of Lipschitz free spaces [PhD thesis]

Description

Some aspects of the geometry of Lipschitz free spaces.First and foremost, we give the fundamental properties of Lipschitz free spaces. Then, we prove that the canonical image of a metric space M is weakly closed in the associated free space F(M). We prove a similar result for the set of molecules.In the second chapter, we study the circumstances in which F(M) is isometric to a dual space. In particular, we generalize a result due to Kalton on this topic. Subsequently, we focus on uniformly discrete metric spaces and on metric spaces originating from p-Banach spaces.In the next chapter, we focus on l1-like properties. Among other things, we prove that F(M) has the Schur property provided the space of little Lipschitz functions is 1-norming for F(M). Under additional assumptions, we manage to embed F(M) into an l1-sum of finite dimensional spaces.In the fourth chapter, we study the extremal structure of F(M). In particular, we show that any preserved extreme point in the unit ball of a free space is a denting point. Moreover, if F(M) admits a predual, we obtain a precise description of its extremal structure.The fifth chapter deals with vector-valued Lipschitz functions.We generalize some results obtained in the first three chapters.We finish with some considerations of norm attainment. For instance, we obtain a density result for vector-valued Lipschitz maps which attain their norm. 1. Résumé de la thèse......Page 5 2. Summary of the thesis......Page 13 3. Notation......Page 20 Definition and first properties......Page 23 Weak closure of delta(M)......Page 26 Weak closure of the set of molecules......Page 27 Perspectives......Page 29 The spaces of little Lipschitz functions......Page 31 Natural preduals......Page 34 The uniformly discrete case......Page 37 Metric spaces originating from p-Banach spaces......Page 42 Perspectives......Page 45 The Schur property......Page 47 Quantitative versions of the Schur property......Page 49 Embeddings into l1-sums......Page 51 Metric spaces originating from p-Banach spaces......Page 53 Perspectives......Page 54 Extremal structure of Lipschitz free spaces......Page 57 General results......Page 60 Extremal structure for spaces with natural preduals......Page 65 The uniformly discrete case......Page 67 Proper metric spaces......Page 69 Perspectives......Page 71 Tensor products......Page 73 Vector-valued Lipschitz free spaces......Page 75 Duality results......Page 78 Schur properties in the vector-valued case......Page 83 Norm attainment......Page 88 Perspectives......Page 91 The conjecture......Page 93 Origin of the conjecture......Page 96 Preliminary results......Page 97 Proof of the main result......Page 101 Characterization of polytopes in R1.......Page 102 Characterization of polytopes in R2.......Page 104 Construction of R3 and conclusion.......Page 106 Weakening assumption (H2)......Page 107 Introduction......Page 109 Property Q and Kalton's graphs......Page 110 The James space......Page 111 Kalton's graphs do not embed into the James space......Page 112