A sparse domination principle for rough singular integrals

Abstract

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.

Domination of multilinear singular integrals by positive sparse forms

“…Our formulation in terms of positive sparse forms overcomes this obstacle: a similar idea, albeit not explicit, appears in the linear setting in . After the first version of this article was made public, several works based on sparse form domination have appeared within and beyond Calderón‐‐Zygmund theory, see for example and references therein.…”

Some Remarks on the Pointwise Sparse Domination

“…We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification. 42B20, 42B25.…”

Sparse domination and the strong maximal function

“…implies that T maps L log L into weak L 1 , but does not imply that T maps L 1 into weak L 1 (the proof is similar to the argument in the appendix of [5]). Our main result answers the question raised above in the negative: there can be no domination by positive sparse forms of the type (1.2) for the strong maximal function.…”

“…This example is relevant because (approximate) point masses are extremal examples for the weak-type behavior of M S and the Hardy-Littlewood maximal functions near the L 1 endpoint. Moreover, we know from the one-parameter theory that there is a connection between sparse bounds and weak-type endpoint estimates; for example, a (1, 1) sparse bound of type (1.2) implies that T maps L 1 into weak L 1 (see the appendix in [5]). Taking the above discussion into account, it is natural to ask whether or not M S admits a (1, 1) sparse bound of the type…”