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arXiv · 0809.1719

Duality in spaces of finite linear combinations of atoms

Fulvio Ricci·Joan Verdera
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Abstract

In this note we describe the dual and the completion of the space of finite linear combinations of $(p,\infty)$-atoms, $0<p\leq 1$ on ${\mathbb R}^n$. As an application, we show an extension result for operators uniformly bounded on $(p,\infty)$-atoms, $0<p < 1$, whose analogue for $p=1$ is known to be false. Let $0 < p <1$ and let $T$ be a linear operator defined on the space of finite linear combinations of $(p,\infty)$-atoms, $0<p < 1 $, which takes values in a Banach space $B$. If $T$ is uniformly bounded on $(p,\infty)$-atoms, then $T$ extends to a bounded operator from $H^p({\mathbb R}^n)$ into $B$.

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Fulvio Ricci, Joan Verdera. 2008-09-10. Duality in spaces of finite linear combinations of atoms. https://arxiv.org/abs/0809.1719

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