Atomic decomposition of finite signed measures on compacts of ${\mathbb{R}}^n$

Abstract

Recently there has been interest in pairs of Banach spaces $(E_0,E)$ in an $o-O$ relation and with $E_0^{\ast \ast }=E$. It is known that this can be done for Lipschitz spaces on suitable metric spaces. In this paper we consider the case of a compact subset $K$ of ${\mathbb{R}}^n$ with the euclidean metric, which does not give an $o-O$ structure, but we use part of the theory concerning these pairs to find an atomic decomposition of the predual of $Lip(K)$. In particular, since the space $\mathfrak{M}(K)$ of finite signed measures on $K$, when endowed with the Kantorovich-Rubinstein norm, has as dual space $Lip(K)$, we can give an atomic decomposition for this space.