Abstract
In this paper, the weak Orlicz space wL Φ is introduced and its applications to the martingale theory are discussed. In particular, a series of martingale inequalities including the maximal function inequality in weak Orlicz spaces are established; the relationships between these spaces are investigated. Moreover, the boundedness of several sublinear operators from one weak Orlicz space to another is proved; their vector-valued analogues are also considered.
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References
Aleksandrov A. Essays on non locally convex Hardy classes. In: Complex Analysis and Spectral Theory. LNM, 864. Berlin: Springer-Verlag, 1981
Burkholder D. Distribution function inequalities for martingales. Ann Probab, 1973, 1: 19–42
Burkholder D. A geometrical characterization of Banach space in which martingale difference sequences are unconditional. Ann Probab, 1981, 9: 997–1011
Cwikel M. The dual of weak L p. Ann Inst Fourier, 1975, 25: 85–126
Cwikel M, Fefferman C. The canonical seminorm on weak L 1. Studia Math, 1984, 78: 275–278
Fefferman R, Soria F. The space wH 1. Studia Math, 1987, 85: 1–16
Herz C. H p-spaces of martingales, 0 < p ⩽ 1. Z Wahr Geb, 1974, 28: 189–205
Kalton N. Linear operators on L p for 0 < p < 1. Trans Amer Math Soc, 1980, 259: 319–355
Kwapien S. Isomorphic characterizations of inner product spaces by orthogonal series with vector-valued coefficients. Studia Math, 1972, 44: 583–595
Lepingle D. Quleques inegalites concernant les martingales. Studia Math, 1976, 59: 63–83
Liu P. Martingales and Geometry of Banach Spaces (in Chinese). Wuhan: Wuhan Universiy Press, 1993
Long R. Martingale Spaces and Inequalities. Beijing: Peking University Press, 1993
Mei T, Liu P. On the maximal inequalities for martingales involving two functions. Proc Amer Math Soc, 2001, 130: 883–892
Pisier G. Martingales with values in uniformly convex spaces. Israel J Math, 1975, 20: 326–350
Weisz F. Martingale Hardy Spaces and Their Applications in Fourier Analysis. LNM, 1568. Berlin: Springer-Verlag, 1994
Weisz F. Weak martingale Hardy spaces. Prob Math Stat, 1998, 18: 133–148
Weisz F. Bounded operators on weak Hardy spaces and applications. Acta Math Hungarica, 1998, 80: 249–264
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Liu, P., Hou, Y. & Wang, M. Weak Orlicz space and its applications to the martingale theory. Sci. China Math. 53, 905–916 (2010). https://doi.org/10.1007/s11425-010-0065-z
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DOI: https://doi.org/10.1007/s11425-010-0065-z