Three observations on commutators of Singular Integral Operators with BMO functions

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

Proof of (12) and Proof 2 of observation 2 have been corrected. To appear in AWM-Springer Series, Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory. Celebrating Cora Sadosky's Life. Vol.2. Editors: S. Marcantognini, C. Pereyra, A. Stokolos y W. Urbina

References in corpus (1)

• On pointwise estimates involving sparse operators

Cited by in corpus (4)

• Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates
• A revisit on the compactness of commutators
• Sparse domination of sharp variational truncations
• On the compactness of oscillation and variation of commutators