Vol. 10, No. 5, 2017

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A sparse domination principle for rough singular integrals

José M. Conde-Alonso, Amalia Culiuc, Francesco Di Plinio and Yumeng Ou

Vol. 10 (2017), No. 5, 1255–1284
Abstract

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

${T}_{\Omega }f\left(x\right)=p.v.\phantom{\rule{0.3em}{0ex}}{\int }_{{ℝ}^{d}}f\left(x-y\right)\Omega \left(\frac{y}{|y|}\right)\frac{ỵ}{|y{|}^{d}},$

where $\Omega \in {L}^{q}\left({S}^{d-1}\right)$ has vanishing average and $1, 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.

Keywords
positive sparse operators, rough singular integrals, weighted norm inequalities
Primary: 42B20
Secondary: 42B25