Operator valued BMO and commutators

Abstract

If $E$ is a Banach space, $b\in \mathit{BMO}({\mathbb R}^n,\mathcal{L}(E))$ and $T$ is a $\mathcal{L}(E)$-valued Calderón-Zygmund type operator with operator-valued kernel $k$, we show the boundedness of the commutator $T_b(f)= b T(f)- T(bf)$ on $L^p({\mathbb R}^n,E)$ for $1<p<\infty$ whenever $b$ and $k$ verify some commuting properties. Some endpoint estimates are also provided.