A sparse domination principle for rough singular integrals

TL;DR

This paper establishes a new sparse domination principle for rough singular integrals and Bochner-Riesz means, leading to sharper weighted estimates and extending previous results in harmonic analysis.

Contribution

It introduces an abstract sparse domination principle applicable to rough singular integrals and Bochner-Riesz means, improving weighted estimates without relying on weak endpoint bounds.

Findings

Sparse domination yields stronger bounds than weak-L^1 estimates.

New sharp quantitative A_p-weighted estimates are derived.

Extends previous results to unbounded angular parts in 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 $1<q\leq \infty$, 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 Hyt\"onen-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.