Analysis & PDE

• Anal. PDE
• Volume 10, Number 5 (2017), 1255-1284.

A sparse domination principle for rough singular integrals

Abstract

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

$TΩf(x) = p.v.∫ ℝdf(x − y)Ω( y |y|) ỵ |y|d,$

where $Ω ∈ Lq(Sd−1)$ has vanishing average and $1 < q ≤∞$, and to Bochner–Riesz means at the critical index in $ℝd$ 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.

Article information

Source
Anal. PDE, Volume 10, Number 5 (2017), 1255-1284.

Dates
Accepted: 24 April 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843500

Digital Object Identifier
doi:10.2140/apde.2017.10.1255

Mathematical Reviews number (MathSciNet)
MR3668591

Zentralblatt MATH identifier
06740746

Citation

Conde-Alonso, José M.; Culiuc, Amalia; Di Plinio, Francesco; Ou, Yumeng. A sparse domination principle for rough singular integrals. Anal. PDE 10 (2017), no. 5, 1255--1284. doi:10.2140/apde.2017.10.1255. https://projecteuclid.org/euclid.apde/1510843500

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