Uniform sparse domination of singular integrals via dyadic shifts
  Using the Calder\'on-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 Hyt\"onen'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 Journ\'e, with logarithmic-Dini type smoothness of the integral kernel. 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 $L^2(W; \mathbb R^n),$   \[   \left\|T\otimes \mathrm{Id}_{\mathbb R^n}\right\|_{L^2(W; \mathbb R^n)\rightarrow L^2(W; \mathbb R^n)} \lesssim [W]_{A_2}^{\frac32}   \] uniformly over $T\in \mathcal U$, which is the currently best known dependence. 
Amalia Culiuc, Francesco Di Plinio, Yumeng Ou
math.CA