A sparse domination principle for rough singular integrals

Abstract

We prove that bilinear forms associated to the rough homogeneous singular integrals $T_{\Omega }$ on ${\mathbb{R}}^d$, where the angular part $\Omega \in L^q(S^{d-1})$ has vanishing average and $1<q\le \infty$, and to Bochner-Riesz means at the critical index in ${\mathbb{R}}^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is stronger than the weak-$L^1$ estimates for $T_{\Omega }$ 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 $T_{\Omega }$. Our results follow from a new abstract sparse domination principle which does not rely on weak endpoint estimates for maximal truncations.

29 pages. References updated. Final version to appear on Analysis&PDE

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