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The singular integral characterization of $ H\sp{p}$ on simple martingales

Author: Akihito Uchiyama
Journal: Proc. Amer. Math. Soc. 88 (1983), 617-621
MSC: Primary 42B20; Secondary 60G46
Corrigendum: Proc. Amer. Math. Soc. 111 (1991), null.
MathSciNet review: 702286
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Abstract: The singular integral characterization of $ {H^1}$ on simple martingales was given by S. Janson. We show that his result cannot be extended to $ {H^p}$ if $ p( > 0)$ is very small.

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  • [1] J. A. Chao, Conjugate characterizations of 𝐻¹ dyadic martingales, Math. Ann. 240 (1979), no. 1, 63–67. MR 524002, 10.1007/BF01428300
  • [2] J. A. Chao, Hardy spaces on regular martingales, Martingale theory in harmonic analysis and Banach spaces (Cleveland, Ohio, 1981) Lecture Notes in Math., vol. 939, Springer, Berlin-New York, 1982, pp. 18–28. MR 668534
  • [3] J. A. Chao and Svante Janson, A note on 𝐻¹ 𝑞-martingales, Pacific J. Math. 97 (1981), no. 2, 307–317. MR 641160
  • [4] Jia Arng Chao and Mitchell H. Taibleson, A sub-regularity inequality for conjugate systems on local fields, Studia Math. 46 (1973), 249–257. MR 0338695
  • [5] R. R. Coifman and Guido Weiss, On subharmonicity inequalities involving solutions of generalized Cauchy-Riemann equations, Studia Math. 36 (1970), 77–83. MR 0271382
  • [6] C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953
  • [7] R. F. Gundy, Inégalités pour martingales à un et deux indices: l’espace 𝐻^{𝑝}, Eighth Saint Flour Probability Summer School—1978 (Saint Flour, 1978), Lecture Notes in Math., vol. 774, Springer, Berlin, 1980, pp. 251–334 (French). MR 590628
  • [8] R. F. Gundy and N. Th. Varopoulos, A martingale that occurs in harmonic analysis, Ark. Mat. 14 (1976), no. 2, 179–187. MR 0448539
  • [9] Svante Janson, Characterizations of 𝐻¹ by singular integral transforms on martingales and 𝑅ⁿ, Math. Scand. 41 (1977), no. 1, 140–152. MR 0463789
  • [10] M. H. Taibleson, Fourier analysis on local fields, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0487295
  • [11] Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO on simple martingales, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 495–505. MR 730087
  • [12] Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO (𝑅ⁿ), Acta Math. 148 (1982), 215–241. MR 666111, 10.1007/BF02392729

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