Uniform sparse domination of singular integrals via dyadic shifts

Using the Caldeŕon-Zygmund decomposition, we give a novel and simple proof that L2 bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with Hyẗonen's dyadic representation theorem, upgrades to a positive sparse domination of the class U of singular integrals satisfying the assumptions of the classical T(1)-theorem of David and Jourńe. Furthermore, our proof extends rather easily to the Rn-valued case, yielding as a corollary the operator norm bound on the matrix weighted space L2(W; Rn) uniformly over T E μ, which is the currently best known dependence.