Espaces de fonctions de classe C^r sur O_F

TL;DR

This paper introduces Banach spaces of C^r functions over O_F and their duals, providing bases and extension criteria, advancing the mathematical framework used in p-adic Langlands theory.

Contribution

It generalizes classical results by constructing new Banach spaces of C^r functions and establishing criteria for extending linear forms to distributions of order r.

Findings

Constructed Banach bases for C^r function spaces

Provided criteria for extending linear forms to distributions

Generalized classical results of Amice-Vélu and Vishik

Abstract

In this paper we introduce a class of Banach spaces of functions of class C^r (where r is a positive real number) and the associated dual spaces of distributions of order r, which turn out to be useful in p-adic Langlands theory. We construct a Banach basis for these spaces and we give a criterion for telling when a linear form on a space of locally Q_p-polynomial functions extends to a distribution of order r. This generalises the classical results of Amice-V\'elu and Vishik.