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

Studia Mathematica (2007)

• Volume: 182, Issue: 2, page 99-111
• ISSN: 0039-3223

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## Abstract

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

## How to cite

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Sandra Pot. "A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform." Studia Mathematica 182.2 (2007): 99-111. <http://eudml.org/doc/284477>.

@article{SandraPot2007,
abstract = {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 $H:L²_\{W\}(ℝ, ) → L²_\{W\}(ℝ, )$ and also all weighted dyadic martingale transforms $T_\{σ\}: L²_\{W\}(ℝ, ) → L²_\{W\}(ℝ, )$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.},
author = {Sandra Pot},
journal = {Studia Mathematica},
keywords = {operator weight; dyadic martingale transform; weighted Hilbert transform; reverse Hölder property; weighted square function},
language = {eng},
number = {2},
pages = {99-111},
title = {A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform},
url = {http://eudml.org/doc/284477},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Sandra Pot
TI - A sufficient condition for the boundedness of operator-weighted martingale transforms and Hilbert transform
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 2
SP - 99
EP - 111
AB - 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 $H:L²_{W}(ℝ, ) → L²_{W}(ℝ, )$ and also all weighted dyadic martingale transforms $T_{σ}: L²_{W}(ℝ, ) → L²_{W}(ℝ, )$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.
LA - eng
KW - operator weight; dyadic martingale transform; weighted Hilbert transform; reverse Hölder property; weighted square function
UR - http://eudml.org/doc/284477
ER -

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