Weak [PHI]-inequalities for the Haar system and differentially subordinated martingales

For a wide class of Young functions [PHI]: [0, [infinity]) [right arrow] [0, [infinity]), we determine the best constant [C.sub.[PHI]] such that the following holds. If [([h.sub.k]).sub.k[greater than or equal to]0] is the Haar system on [0,1],then for any vectors [a.sub.k] from a separable Hilbert...

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Bibliographic Details
Published in:Proceedings of the Japan Academy. Series A. Mathematical sciences 2012-11, Vol.88A (9), p.139-144
Main Author: Osekowski, Adam
Format: Article
Language:English
Subjects:
Complement
Functional equations
Functions
Functions (mathematics)
Hilbert space
Inequalities
Inequalities (Mathematics)
Martingales
Mathematical analysis
Mathematical constants
Mathematics
Probability
Series (mathematics)
Stands
Vectors (mathematics)
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Summary:For a wide class of Young functions [PHI]: [0, [infinity]) [right arrow] [0, [infinity]), we determine the best constant [C.sub.[PHI]] such that the following holds. If [([h.sub.k]).sub.k[greater than or equal to]0] is the Haar system on [0,1],then for any vectors [a.sub.k] from a separable Hilbert space H and [[epsilon].sub.k] [member of] {-1,1}, k = 0,1,2,..., we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This is generalized to the sharp weak-[PHI] inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where X, Y stand for H-valued martingales such that Y is differentially subordinate to X. These statements complement and generalize the results of Burkholder, Suh, the author and others. Key words: Haar system; martingale; weak-[PHI] inequality; best constant.
ISSN:0386-2194
DOI:10.3792/pjaa.88.139