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A NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN MARTINGALE INEQUALITIES IN BANACH FUNCTION SPACES

Published online by Cambridge University Press:  01 September 2007

MASATO KIKUCHI
MASATO KIKUCHI*
Affiliation:
Department of Mathematics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan e-mail: [email protected]
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Abstract

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Let X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

Keywords

60G4246E30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 3 , September 2007 , pp. 431 - 447
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1.Bennett, C. and Sharpley, R., Interpolation of operators, Pure and Applied Mathematics 129 (Academic Press, 1988).Google Scholar
2.Burkholder, D. L., Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.CrossRefGoogle Scholar
3.Burkholder, D. L., Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19–42.CrossRefGoogle Scholar
4.Chong, K. M. and Rice, N. M., Equimeasurable rearrangements of functions, Queen's Papers in Pure and Applied Mathematics, No. 28 (Queen's University, Kingston, Ontario, 1971).Google Scholar
5.Garsia, A. M., Martingale inequalities: seminar notes on recent progress (W. A. Benjamin, Inc., Massachusetts, 1973).Google Scholar
6.Kikuchi, M., Characterization of Banach function spaces that preserve the Burkholder square-function inequality, Illinois J. Math. 47 (2003), 867–882.CrossRefGoogle Scholar
7.Kikuchi, M., New martingale inequalities in rearrangement-invariant function spaces, Proc. Edinburgh Math. Soc. (2) 47 (2004), 633–657.CrossRefGoogle Scholar
8.Kikuchi, M., On the Davis inequality in Banach function spaces, preprint.Google Scholar
9.Kikuchi, M., On some mean oscillation inequalities for martingales, Publ. Mat., 50 (2006), 167–189.CrossRefGoogle Scholar
10.Shimogaki, T., Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965), 365–373.Google Scholar