A sparse domination principle for rough singular integrals

Abstract

We prove that bilinear forms associated to the rough homogeneous singular integrals

$$T_{\Omega }f\left(x\right)=p.v.{\int }_{{ℝ}^d}f\left(x-y\right)\Omega \left(\frac{y}{\left|y\right|}\right)\frac{ỵ}{|y{|}^d},$$

where $\Omega \in L^q\left(S^{d-1}\right)$ has vanishing average and $1<q\le \infty$, and to Bochner–Riesz means at the critical index in ${ℝ}^d$ are dominated by sparse forms involving $\left(1,p\right)$ 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 and 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.