A Sufficient Condition for the Boundedness of Operator-weighted Martingale Transforms and Hilbert Transform
نویسنده
چکیده
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 Hölder property introduced in [2], then the weighted Hilbert transform H : LW (R,H) → L 2 W (R,H) and also all weighted dyadic martingale transforms Tσ : LW (R,H)→ L 2 W (R,H) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.
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