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 for 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 \(H_b^p\) into \(L^p\), where \(H_b^p\) 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 \(H_b^{p,\infty}\) of this atomic space and we prove that \([b,T]\) maps continuously \(H_b^{p,\infty}\) into \(L^{p,\infty}\).
scientific article; zbMATH DE number 1186310
scientific article