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

Sandra Pot — 2007

Studia Mathematica

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . 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 ( , ) L ² W ( , ) and also all weighted dyadic martingale transforms T σ : L ² W ( , ) L ² W ( , ) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

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