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 $A_2$-$A_{\infty }$ estimates, that in the one weight case give us the estimate $$\Vert T{\Vert }_{L^2(W)\to L^2(W)}\le C[W{]}_{A_2}^{1/2}[W{]}_{A_{\infty }}\le C[W{]}_{A_2}^{3/2}$$ where $T$ is either a Calderon--Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.

23 pages, this third version was finished on November 27, 2016

References in corpus (2)

• On pointwise estimates involving sparse operators
• The Bellman functions of the Carleson Embedding Theorem and the Doob's martingale inequality

Cited by in corpus (6)

• Sparse bounds for maximal rough singular integrals via the Fourier transform
• Uniform sparse domination of singular integrals via dyadic shifts
• Mixed $A_2$-$A_{\infty }$ estimates of the non-homogeneous vector square function with matrix weights
• On the failure of lower square function estimates in the non-homogeneous weighted setting
• Sharp bounds and T1 theorem for Calderón-Zygmund operators with matrix kernel on matrix weighted spaces
• Two weight bump conditions for matrix weights