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arXiv · 1601.03193

Three observations on commutators of Singular Integral Operators with BMO functions

Carlos Pérez·Israel P. Rivera-Ríos
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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 \[\frac{1}{|Q|}\left|\left\{x\in Q\, : \, |[b,T](fχ_Q)(x)|>M^2f(x)t\right\}\right|\leq c e^{-\sqrt{ct\|b\|_{BMO}}} \] 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 $\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}=\infty$.

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Carlos Pérez, Israel P. Rivera-Ríos. 2017-01-25. Three observations on commutators of Singular Integral Operators with BMO functions. https://doi.org/10.1007/978-3-319-51593-9_12

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