Vol. 10, No. 5, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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

TΩf(x) = p.v.df(x y)Ω( y |y|) |y|d,

where Ω Lq(Sd1) has vanishing average and 1 < q , and to Bochner–Riesz means at the critical index in d are dominated by sparse forms involving (1,p) averages. This domination is stronger than the weak-L1 estimates for TΩ and for Bochner–Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative Ap-weighted estimates for Bochner–Riesz means and for homogeneous singular integrals with unbounded angular part, extending previous results of Hytönen, Roncal and Tapiola for TΩ. 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