Bloom weighted bounds for sparse forms associated to commutators

TL;DR

This paper develops Bloom weighted bounds for bilinear sparse forms related to commutators of general operators, providing new estimates for rough singular integrals and Bochner-Riesz operators, and questioning the sharpness of existing bounds.

Contribution

It introduces Bloom weighted estimates for bilinear forms associated with commutators, extending the range of exponents and revealing potential non-sharpness of prior bounds.

Findings

Established Bloom weighted estimates for bilinear sparse forms in full exponent range.

Derived new Bloom bounds for commutators of rough singular integrals and Bochner-Riesz operators.

Identified that existing Bloom estimates may not be sharp for higher order 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 Calder\'on--Zygmund operators, the previously known quantitative Bloom weighted estimates are not sharp for the second and higher order commutators.