Convex body domination and weighted estimates with matrix weights

Abstract

We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions.We prove that Calderón–Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight A2–A∞ estimates, that in the one weight case give us the estimate‖T‖L2(W)→L2(W)≤C[W]A21/2[W]A∞≤C[W]A23/2 where T is either Calderón–Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.