Mathematical Research Letters

Volume 25 (2018)

Number 1

Uniform sparse domination of singular integrals via dyadic shifts

Pages: 21 – 42

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n1.a2

Authors

Amalia Culiuc (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Francesco Di Plinio (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

Yumeng Ou (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Abstract

Using the Calderón–Zygmund decomposition, we give a novel and simple proof that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with Hytönen’s dyadic representation theorem, upgrades to a positive sparse domination of the class $\mathcal{U}$ of singular integrals satisfying the assumptions of the classical $T(1)$-theorem of David and Journé. Furthermore, our proof extends rather easily to the $\mathbb{R}^n$-valued case, yielding as a corollary the operator norm bound on the matrix weighted space $L^2 (W ; \mathbb{R}^n)$\[{ \lVert T \otimes \mathrm{Id}_{\mathbb{R}^n} \rVert }_{L^2(W ; \mathbb{R}^n) \to L^2(W; \mathbb{R}^n)} \lesssim {[W]}^{\frac{3}{2}}_{A_2}\]uniformly over $T \in \mathcal{U}$, which is the currently best known dependence.

Keywords

positive sparse operators, $T(1)$ theorems, weighted norm inequalities, matrix weights

2010 Mathematics Subject Classification

Primary 42B20. Secondary 42B25.

Received 14 September 2016

Accepted 25 January 2017

Published 4 June 2018