Continuity properties for linear commutators of Calderón-Zygmund operators

It is known that the linear commutator [$b, T$] of a Calderón-Zygmund operator $T$ with a $BMO$ function $b$ does not satisfy some of the continuity properties typical of a Calderón-Zygmund operator, for instance, continuity from the Hardy space $H^p$ into $L^p$ for $p\leq 1$ large enough, and weak type (1, 1). We obtain in this paper alternative results. Indeed, we prove in the first part of the paper that [$b, T$] is continuous from $H^p_b$ into $L^p$, where $H^p_b$ denotes an atomic space with atoms satisfying an extra cancellation condition involving the function $b$. In the second part of the paper we define a weak version $H^{p,\infty}_b$ of this atomic space and we prove that [$b, T$] maps continuously $H^{p,\infty}_b$ into $L^{p,\infty}$.