{A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform
Volume 182 / 2007
Studia Mathematica 182 (2007), 99-111
MSC: Primary 42A50, 47B37; Secondary 42A61.
DOI: 10.4064/sm182-2-1
Abstract
Let $W$ be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space ${\mathcal H}$. We show that if $W$ and its inverse $W^{-1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H:L^2_W({\mathbb R}, {\mathcal H}) \rightarrow L^2_W({\mathbb R}, {\mathcal H})$ and also all weighted dyadic martingale transforms $T_\sigma:L^2_W({\mathbb R}, {\mathcal H}) \rightarrow L^2_W({\mathbb R}, {\mathcal H})$ are bounded.
We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.
