On the $H^1$-$L^1$ boundedness of operators

We prove that if $q$ is in $(1,\infty)$, $Y$ is a Banach space, and $T$ is a linear operator defined on the space of finite linear combinations of $(1,q)$-atoms in $\mathbb{R}^n$ with the property that

$$\sup\{\Vert{Ta}\Vert {Y}: \hbox{$a$ is a $(1,q)$-atom} \} < \infty,$$

then $T$ admits a (unique) continuous extension to a bounded linear operator from $H^1({\mathbb{R}^n})$ to $Y$. We show that the same is true if we replace $(1,q)$-atoms by continuous $(1,\infty)$-atoms. This is known to be false for $(1,\infty)$-atoms.