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Strong Differential Subordination and Sharp Inequalities for Orthogonal Processes

  • Adam Osękowski
Article

Abstract

We introduce a strong differential α-subordination for continuous-time processes, which generalizes this notion from the discrete-time setting, due to Burkholder and Choi. Then we determine the best constants in the L p estimates for a nonnegative submartingale and its strong α-subordinate under an additional assumption on the orthogonality of these two processes.

Keywords

Martingale Submartingale Orthogonal processes Differential subordination Strong differential subordination 

Mathematics Subject Classification (2000)

60G44 60G48 

References

  1. 1.
    Bañuelos, R., Wang, G.: Sharp inequalities for martingales with applications to the Beurling–Ahlfors and Riesz transformations. Duke Math. J. 80, 575–600 (1995) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bañuelos, R., Wang, G.: Orthogonal martingales under differential subordination and application to Riesz transforms. Ill. J. Math. 40, 678–691 (1996) MATHGoogle Scholar
  3. 3.
    Burkholder, D.L.: Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math. 26, 182–205 (1977) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burkholder, D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, 647–702 (1984) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burkholder, D.L.: Explorations in martingale theory and its applications. In: École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math., vol. 1464, pp. 1–66. Springer, Berlin (1991) Google Scholar
  6. 6.
    Burkholder, D.L.: Strong differential subordination and stochastic integration. Ann. Probab. 22, 995–1025 (1994) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Choi, C.: A submartingale inequality. Proc. Am. Math. Soc. 124, 2549–2553 (1996) MATHCrossRefGoogle Scholar
  8. 8.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential B. North-Holland, Amsterdam (1982) MATHGoogle Scholar
  9. 9.
    Iwaniec, T., Martin, G.: The Beurling–Ahlfors transform in ℝn and related singular integrals. J. Reine Angew. Math. 473, 25–57 (1996) MATHMathSciNetGoogle Scholar
  10. 10.
    Osękowski, A.: Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions. Ill. J. Math. (to appear) Google Scholar
  11. 11.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) MATHGoogle Scholar
  12. 12.
    Suh, Y.: A sharp weak type (p,p) inequality (p>2) for martingale transforms and other subordinate martingales. Trans. Am. Math. Soc. 357(4), 1545–1564 (2005) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Wang, G.: Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities. Ann. Probab. 23, 522–551 (1995) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland

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