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.
Categories: math.CA