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

Sandra Pot

Sandra Pot

Studia Mathematica, Tome 178 (2007), p. 99-111 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $H : L ²_{W} (ℝ,) \to L ²_{W} (ℝ,)$ and also all weighted dyadic martingale transforms $T_{σ} : L ²_{W} (ℝ,) \to L ²_{W} (ℝ,)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

Publié le : 2007-01-01

Zbl 1126.42004

EUDML-ID : urn:eudml:doc:284477

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     year = {2007},
     pages = {99-111},
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Sandra Pot. A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform. Studia Mathematica, Tome 178 (2007) pp. 99-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-2-1/