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Boundedness of vector-valued martingale transforms on extreme points and applciations

Published online by Cambridge University Press:  09 April 2009

José L. Torrea
Affiliation:
Departmento de Matemáticas, Universidad Autónoma de MadridCiudad Universitaria de Canto Blanco 28049 Madrid, Spain, e-mail: [email protected], [email protected]
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Abstract

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Let Β1, Β2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent: T is bounded from into for some p (or equivalently for every p) in the range 1 < p < ∞; T is bounded from into BMOB2; T is bounded from BMOB1 into BMOB2; T is bounded from into . Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

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