Super-Hölder vectors and the field ofnorms

Let E be a field of characteristic p.In a previous paper of ours, we defined and studied super-Hlder vectors in certain E-linear representations of p .In the present paper, we define and study super-Hlder vectors in certain E-linear representations of a general p-adic Lie group.We then consider certain p-adic Lie extensions K /K of a p-adic field K , and compute the super-Hlder vectors in the tilt of K .We show that these super-Hlder vectors are the perfection of the field of norms of K /K .By specializing to the case of a Lubin-Tate extension, we are able to recover E((Y )) inside the Y -adic completion of its perfection, seen as a valued E-vector space endowed with the action of O K given by the endomorphisms of the corresponding Lubin-Tate group.Introduction 195 1. Super-Hlder functions and vectors 197 2. The field of norms 200 3.The Lubin-Tate case 203 4. Mahler expansions and super-Hlder functions 206