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

  $$\displaystyle{ \frac{1} {\vert Q\vert }\left \vert \left \{x \in Q\,:\, \vert [b,T](f\chi _{Q})(x)\vert> M^{2}f(x)t\right \}\right \vert \leq ce^{-\sqrt{ct\|b\|_{BMO}} }}$$

  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 LlogL averages.

• 3. Motivated by the conjugation method for commutators, it is shown the failure of the following endpoint estimate, if w ∈ A _p ∖ A ₁ then

  $$\displaystyle{ \left \|wM\left ( \frac{f} {w}\right )\right \|_{L^{1}(\mathbb{R}^{n})\rightarrow L^{1,\infty }(\mathbb{R}^{n})} = \infty. }$$

The first author was supported by Severo Ochoa Excellence Programme and the Spanish Government grant MTM2014-53850-P and the second author was supported by Grant MTM2012-30748, Spanish Government.