Duality in spaces of finite linear combinations of atoms

Abstract:

In this paper we describe the dual and the completion of the space of finite linear combinations of $(p,\infty )$-atoms, $0<p\leq 1$. 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 $B$.