A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform

Abstract

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space H. We show that if W and its inverse W−1 both satisfy a matrix reverse Holder property introduced in [2], then the weighted Hilbert transform H : L2W(R,H) → L2W(R,H) and also all weighted dyadic martingale transforms Tσ: L2W(R,H) → L2W(R,H) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform