Bloom weighted bounds for sparse forms associated to commutators

Abstract

In this paper we consider bilinear sparse forms intimately related to iterated commutators ofa rather general class of operators. We establish Bloom weighted estimates for these formsin 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 singularintegrals and the Bochner–Riesz operator at the critical index. We also raise the questionabout the sharpness of our estimates. In particular we obtain the surprising fact that even inthe case of Calderón–Zygmund operators, the previously known quantitative Bloom boundsare not sharp for the second and higher order commutators.