A sparse domination principle for rough singular integrals

By José M. Conde-Alonso, Amalia Culiuc, Francesco Di Plinio, Yumeng Ou. 2016

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2fa575c4f0f03512e1b111b1d3ce18539a26aed3

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Sparse bounds for spherical maximal functions (2017, background): “Some of these arguments are rather short and elegant, using familiar TT ∗ style arguments [7] to provide remarkably sharp control of rough singular integrals.”