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On the duality of some martingale spaces

Published online by Cambridge University Press:  17 April 2009

N.L. Bassily  and
A.M. Abdel-Fattah
N.L. Bassily
Affiliation:
Department of Mathematics, American University of Cairo, P.O. Box 2511 Cairo, Egypt
A.M. Abdel-Fattah
Affiliation:
The Institute of Statistical Studies, Cairo University, Cairo, Egypt
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Abstract

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Fefferman has proved that the dual space of the martingale Hardy space H1 is the BMO1-space. Garsia went further and proved that the dual of Hp is the so-called martingale Kp-space, where p and q are two conjugate numbers and 1 ≤ p < 2.

The martingale Hardy spaces HΦ with general Young function Φ, were investigated by Bassily and Mogyoródi. In this paper we show that the dual of the martingale Hardy space HΦ is the martingale Hardy space HΦ where (Φ, Ψ) is a pair of conjugate Young functions such that both Φ and Ψ have finite power. Moreover, two other remarkable dualities are presented.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 1 , February 1992 , pp. 43 - 52
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Fefferman, C., ‘Characterizations of bounded mean oscillations’, Bull. Amer. Math. Soc. 77 (1971), 587–588.CrossRefGoogle Scholar
[2]Garsia, A.M., Martingale Inequalities (Benjamin Readings, Massachusetts, 1973).Google Scholar
[3]Bassily, N.L. and Mogyoródi, J., ‘On the K Φspaces with general Young function Φ’, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. XXVII (1985), 205–214.Google Scholar
[4]Neveu, J., Discrete parameter martingales (North-Holland, Amsterdam, 1975).Google Scholar
[5]Kranoselskii, M.A. and Rutickii, Ya.B., Convex functions and Orlicz spaces (Noordhoff, Gröningen, 1961).Google Scholar
[6]Burkholder, D.L., Davis, B. and Gundy, R.F., ‘Integral inequalities for convex functions of operators on martingales’, in Proceedings 6th Berkeley symposium on mathematical statistics and probability, pp. 223–240 (University of California Press, 1972).Google Scholar
[7]Mogyoródi, J. and Mori, P., ‘Necessary and sufficient condition for the maximal inequality of convex Young functions’, Acta Sci. Math. 45 (1983), 325–332.Google Scholar
[8]Ishak, S. and Mogyoródi, J., ‘On the generalization of the Fefferman-Garsia inequality’, Stochastic Differential Systems, in Lecture notes in control and information sciences, pp. 85–97 (Springer-Verlag, Berlin, 1981).Google Scholar
[9]Ishak, S. and Mogyoródi, J., ‘On the P Φ-spaces and the generalization of Hertz' and Fefferman's inequalities I, II and III’, Studia Sci. Math. Hungar. 17 (1982), 229–234. 18, pp. 205–210 and 18 (1983), 211–219.Google Scholar
[10]Schipp, F., ‘The dual space of the martingale VMO-space’, in Proceedings of the 3rd Pannonian symposium on mathematical statistics, pp. 305–311 (Visagrád, Hungary, 1982).Google Scholar
[11]Burkholder, D.L., ‘Distribution function inequalities for martingales’, Ann. Prob. 1 (1973), 19–42.CrossRefGoogle Scholar