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# 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

#### 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.

F. Di Plinio was partially supported by the National Science Foundation under the grants NSF-DMS-1500449 and NSF-DMS-1650810.

Received 14 September 2016

Accepted 25 January 2017

Published 4 June 2018