Continuity properties for linear commutators of Calderon–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 Hp into Lp for p ≤ 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 fromHpb intoL p, whereHpb 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 Hp,∞b of this atomic space and we prove that [b, T] maps continuously Hp,∞b into L p,∞