Bloom weighted bounds for sparse forms associated to commutators

Abstract

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner-Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón--Zygmund operators, the previously known quantitative Bloom weighted estimates are not sharp for the second and higher order commutators.

arXiv comment34 pages

References in corpus (7)

• Sharp weighted norm estimates beyond Calderón-Zygmund theory
• Sharp bounds for general commutators on weighted Lebesgue spaces
• Commutators of singular integrals revisited
• Quantitative estimates and extrapolation for multilinear weight classes
• Weighted norm inequalities for rough singular integral operators
• Improved $A_1-A_{\infty }$ and related estimates for commutators of rough singular integrals
• Fractional Bloom boundedness and compactness of commutators