Duality in spaces of finite linear combinations of atoms

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