Three Observations on Commutators of Singular Integral Operators with BMO Functions

Abstract

Three observations on commutators of Singular Integral Operators with BMO functions are exposed, namely

1 - The already known 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, since it cannot be better than subgaussian.

2 - It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined by $L\log L$ averages.

3 - Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if $w\in A_p\setminus A_1$ then $$\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}=\infty.$$