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 ∈ BM O, we improve several weighted weak type bounds for [b, T ]. If b belongs to the weighted BM O, we obtain a quantitative form of the two-weighted bound for [b, T ] due to Bloom-Holmes-Lacey-Wick.2010 Mathematics Subject Classification. 42B20, 42B25.

Domination of multilinear singular integrals by positive sparse forms

“…Our formulation in terms of positive sparse forms overcomes this obstacle: a similar idea, albeit not explicit, appears in the linear setting in . After the first version of this article was made public, several works based on sparse form domination have appeared within and beyond Calderón‐‐Zygmund theory, see for example and references therein.…”

Commutators of singular integrals revisited

“…It was pointed out in that similar methods can be used to obtain the corresponding estimates for Calderón–Zygmund operators.Estimate represents a quantitative version of that statement. It looks like a natural extension of the case obtained in . Note, however, that it does not seem that this estimate can be deduced via a simple inductive argument.…”

“…Recently it was extended to higher dimensions by Holmes, Lacey and Wick . Later, a quantitative form of this statement, expressed in estimate , was obtained by the authors in .As in the unweighted case, part (ii) is new in such generality. In this part was obtained, similar to , assuming that is bounded from to for every Riesz transform . Assume that .…”

“…Recently it was extended to higher dimensions by Holmes, Lacey and Wick . Later, a quantitative form of this statement, expressed in estimate , was obtained by the authors in .…”

“…We will also use the following result proved recently in [, Lemma 5.1] and closely related to : given a dyadic lattice and a sparse family , there exists a sparse family containing and such that if , then for almost every , …”