On pointwise and weighted estimates for commutators of Calderón-Zygmund operators

Abstract

In recent years, it has been well understood that a Calderón-Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b, T ]$ with a locally integrable function $b$. This result is applied into two directions. If $b \in BMO$, we improve several weighted weak type bounds for $[b, T ]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b, T ]$ due to Bloom-Holmes-Lacey-Wick.