A sparse domination principle for rough singular integrals

where $Ω \in L^{q} (S^{d - 1})$ has vanishing average and $1 < q \leq \infty$ , 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 and 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.

Keywords

positive sparse operators, rough singular integrals, weighted norm inequalities

Mathematical Subject Classification 2010

Primary: 42B20

Secondary: 42B25

Milestones

Received: 19 January 2017

Accepted: 24 April 2017

Published: 1 July 2017

Authors

José M. Conde-Alonso
Departament de Matemàtiques Facultat de Ciències Universitat Autònoma de Barcelona 08193 Barcelona Spain

Amalia Culiuc
School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 United States

Francesco Di Plinio
Department of Mathematics University of Virginia Kerchof Hall Box 400137 Charlottesville, VA 22904 United States

Yumeng Ou
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 United States

Vol. 10, No. 5, 2017

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

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors