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

By S. Pott

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<sup>−1</sup> both satisfy a matrix reverse Holder property introduced in [2], then the weighted Hilbert transform H :\ud L<sup>2</sup><sub>W</sub>(R,H) &#8594; L<sup>2</sup><sub>W</sub>(R,H) and also all weighted dyadic martingale transforms T<sub>&#963;</sub>: L<sup>2</sup><sub>W</sub>(R,H) &#8594; L<sup>2</sup><sub>W</sub>(R,H) are bounded.\ud We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform

Topics: QA
Publisher: Polska Akademia Nauk
Year: 2007
OAI identifier: oai:eprints.gla.ac.uk:13047
Provided by: Enlighten

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Citations

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