# Three observations on commutators of Singular Integral Operators with
BMO functions

Research paper by **Carlos Pérez, Israel P. Rivera-Ríos**

Indexed on: **13 Jan '16**Published on: **13 Jan '16**Published in: **Mathematics - Classical Analysis and ODEs**

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