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arXiv · 1907.06380

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

Luigi D'Onofrio·Luigi Greco·Karl-Mikael Perfekt·Carlo Sbordone·Roberta Schiattarella
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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 $\mathcal{B}_0$.

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Luigi D'Onofrio, Luigi Greco, Karl-Mikael Perfekt, Carlo Sbordone, Roberta Schiattarella. 2019-07-15. Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces. https://arxiv.org/abs/1907.06380

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