Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Abstract

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual ${\mathcal{B}}_{\ast }$, the biduality result that ${\mathcal{B}}_0^{\ast }={\mathcal{B}}_{\ast }$ and ${\mathcal{B}}_{\ast }^{\ast }=\mathcal{B}$, and a formula for the distance from an element $f\in \mathcal{B}$ to ${\mathcal{B}}_0$.