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 �B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual �⁎B⁎, the biduality result that �0⁎=�⁎B0⁎=B⁎ and �⁎⁎=�B⁎⁎=B, and a formula for the distance from an element �∈�f∈B to �0B0.