Abstract. We prove a pointwise estimate for positive dyadic shifts of complexity m which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.
A. We prove that bilinear forms associated to the rough homogeneous singular integralswhere Ω ∈ L q (S d −1 ) has vanishing average and 1 < q ≤ ∞, and to Bochner-Riesz means at the critical index in R d are dominated by sparse forms involving (1, p) averages. This domination is stronger than the weak-L 1 estimates for T Ω and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative A p -weighted estimates for Bochner-Riesz means and for homogeneous singular integrals with unbounded angular part, extending previous results of Hytönen-Roncal-Tapiola for T Ω . Our results follow from a new abstract sparse domination principle which does not rely on weak endpoint estimates for maximal truncations.
We study the problem of dominating the dyadic strong maximal function by (1, 1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.