Three observations on commutators of Singular Integral Operators with BMO functions

This paper contains three observations on commutators of Singular Integral Operators with BMO functions:

The subgaussian local decay for the commutator, namely $\frac{1}{∣ Q ∣} ∣ {x \in Q : ∣ [b, T] (f χ_{Q}) (x) ∣ > M^{2} f (x) t} ∣ \leq c e^{- \sqrt{c t ∥ b ∥_{B M O}}}$ is sharp, that is, it is subgaussian and not better.
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.
If $w \in A_{p} ∖ A_{1}$ then $\left| wM\left(\frac{f}{w}\right)\right|_{L^1( $R$ ^n)\rightarrow L^{1,\infty}( $R$ ^n)}=\infty$.

This paper contains three observations on commutators of Singular Integral Operators with BMO functions:

The subgaussian local decay for the commutator, namely $\frac{1}{∣ Q ∣} ∣ {x \in Q : ∣ [b, T] (f χ_{Q}) (x) ∣ > M^{2} f (x) t} ∣ \leq c e^{- \sqrt{c t ∥ b ∥_{B M O}}}$ is sharp, that is, it is subgaussian and not better.
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.
If $w \in A_{p} ∖ A_{1}$ then $\left| wM\left(\frac{f}{w}\right)\right|_{L^1( $R$ ^n)\rightarrow L^{1,\infty}( $R$ ^n)}=\infty$.

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