This is an announcement for the paper "Duality in spaces of finite linear combinations of atoms" by Fulvio Ricci and Joan Verdera. 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 $B$. Archive classification: math.FA Mathematics Subject Classification: 42B30 Remarks: 15 pages The source file(s), atoms.tex: 40423 bytes, is(are) stored in gzipped form as 0809.1719.gz with size 14kb. The corresponding postcript file has gzipped size 101kb. Submitted from: fricci@sns.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0809.1719 or http://arXiv.org/abs/0809.1719 or by email in unzipped form by transmitting an empty message with subject line uget 0809.1719 or in gzipped form by using subject line get 0809.1719 to: math@arXiv.org.