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

Abstract:

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 \[ \sup \left \{ \Vert Ta \Vert Y: \text {$a$ is a $(1,q)$-atom} \right \} < \infty , \] 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.