On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators

TL;DR

This paper establishes pointwise estimates for commutators of Calderón-Zygmund operators using sparse operators, leading to improved weighted bounds and two-weighted estimates when the symbol function is in BMO or weighted BMO.

Contribution

It extends pointwise control techniques to commutators of Calderón-Zygmund operators and derives new weighted and two-weighted bounds based on BMO conditions.

Findings

Pointwise estimates for commutators using sparse operators.

Improved weighted weak type bounds for commutators with BMO functions.

Quantitative two-weighted bounds for commutators with weighted BMO functions.

Abstract

In recent years, it has been well understood that a Calder\'on-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.