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 B0⁎=B⁎ and B⁎⁎=B, and a formula for the distance from an element f∈B to B0.
Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces
D'Onofrio L.;
2020
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 B0⁎=B⁎ and B⁎⁎=B, and a formula for the distance from an element f∈B to B0.File in questo prodotto:
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