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

摘要

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 both satisfy a matrix reverse Holder property introduced in [2], then the weighted Hilbert transform H :\udL(R,H) → L(R,H) and also all weighted dyadic martingale transforms T: L(R,H) → L(R,H) are bounded.\udWe also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.