Three observations on commutators of Singular Integral Operators with BMO functions

TL;DR

This paper presents three key insights into commutators of singular integral operators with BMO functions, including the sharpness of subgaussian decay, limitations of sparse domination, and unboundedness of certain weighted maximal functions.

Contribution

It provides new results on decay rates, demonstrates the impossibility of sparse domination in certain cases, and explores boundedness properties of weighted maximal functions.

Findings

Subgaussian local decay for commutators is sharp.

Pointwise control by finite sparse operators with $L ext{log}L$ averages is impossible.

Certain weighted maximal functions are unbounded on $L^1$.

Abstract

This paper contains three observations on commutators of Singular Integral Operators with BMO functions: 1) The subgaussian local decay for the commutator, namely \[\frac{1}{|Q|}\left|\left\{x\in Q\, : \, |[b,T](f\chi_Q)(x)|>M^2f(x)t\right\}\right|\leq c e^{-\sqrt{ct\|b\|_{BMO}}} \] is sharp, that is, it is subgaussian and not better. 2) It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined with $L\log L$ averages. 3) If $w\in A_p\setminus A_1$ then $\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}=\infty$.