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The $ {\rm T}1$ theorem for martingales


Author: Andrew G. Bennett
Journal: Proc. Amer. Math. Soc. 107 (1989), 493-502
MSC: Primary 60G46; Secondary 42B20
DOI: https://doi.org/10.1090/S0002-9939-1989-0979217-9
MathSciNet review: 979217
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Abstract: The $ T1$ theorem of David and Journé gives necessary and sufficient conditions that a singular integral operator be bounded from $ {L^2}({R^n})$ to $ {L^2}({R^n})$. In this paper, the definition of singular integral operator is extended to the setting of operators on $ {L^2}(\Omega )$ where $ \Omega $ denotes Wiener space. The main theorem is that the $ T1$ theorem holds in this new setting.


References [Enhancements On Off] (What's this?)

  • [1] R. Bañuelos, Martingale transforms and related singular integrals, Trans. Amer. Math. Soc. 293 (1986), 547-563. MR 816309 (87h:60095)
  • [2] R. Bañuelos and A. Bennett, Paraproducts and commutators of martingale transforms, Proc. Amer. Math. Soc. 103 (1988), 1226-1234. MR 955015 (89m:60108)
  • [3] D. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. MR 0208647 (34:8456)
  • [4] -, Distribution function inequalities for martingales, Ann. Probab. 1 (1973) 19-42. MR 0365692 (51:1944)
  • [5] D. Burkholder, R. Gundy, and M. Silverstein, A maximal function characterization of the class $ {H^p}$, Trans. Amer. Math. Soc. 157 (1971), 137-153. MR 0274767 (43:527)
  • [6] G David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397. MR 763911 (85k:42041)
  • [7] R. Durrett, Brownian motion and martingales in analysis, Wadsworth, 1984. MR 750829 (87a:60054)
  • [8] R. Gundy and N. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sci. Paris Sér. A 289 (1979) 13-16. MR 545671 (82e:60089)
  • [9] N. Varopoulos, Aspects of probabilistic Littlewood-Paley theory J. Funct. Anal. 38 (1980), 25-60. MR 583240 (82d:42016)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0979217-9
Article copyright: © Copyright 1989 American Mathematical Society

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