A version of Burkholder’s theorem for operator-weighted spaces

Abstract

Let W W be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space H \mathcal {H} . We prove that if the dyadic martingale transforms are uniformly bounded on L R 2 ( W ) L^2_{\mathbb {R}}(W) for each dyadic grid in R \mathbb {R} , then the Hilbert transform is bounded on L R 2 ( W ) L^2_{\mathbb {R}}(W) as well, thus providing an analogue of Burkholder’s theorem for operator-weighted L 2 L^2 -spaces. We also give a short new proof of Burkholder’s theorem itself. Our proof is based on the decomposition of the Hilbert transform into “dyadic shifts”.