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

Abstract:

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