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A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales

Author: Jiyeon Suh
Journal: Trans. Amer. Math. Soc. 357 (2005), 1545-1564
MSC (2000): Primary 60G44, 60G42; Secondary 60G46
Published electronically: September 23, 2004
MathSciNet review: 2115376
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Abstract | References | Similar Articles | Additional Information

Abstract: If $(d_{n})_{n\geq 0}$ is a martingale difference sequence, $(\varepsilon_{n})_{n\geq 0}$ a sequence of numbers in $\{ 1,-1\}$, and $n$ a positive integer, then

\begin{displaymath}P(\max _{0\leq m\leq n}\vert \sum_{k=0}^{m} \varepsilon_{k}d_... ...geq 1) \leq \alpha_{p}\Vert\sum_{k=0}^{n} d_{k}\Vert_{p}^{p}. \end{displaymath}

Here $\alpha_{p}$ denotes the best constant. If $1\leq p\leq 2$, then $\alpha_{p}= 2/\Gamma(p+1)$ as was shown by Burkholder. We show here that $\alpha_p=p^{p-1}/2$ for the case $p > 2$, and that $p^{p-1}/2$ is also the best constant in the analogous inequality for two martingales $M$ and $N$ indexed by $[0,\infty)$, right continuous with limits from the left, adapted to the same filtration, and such that $[M,M]_t-[N,N]_t$ is nonnegative and nondecreasing in $t$. In Section 7, we prove a similar inequality for harmonic functions.

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  • 1. M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with formulas, graphs, and mathematical tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992. MR 94b:00012
  • 2. A. Baernstein, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), no. 5, 833-852. MR 80g:30022
  • 3. R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no 3, 575-600. MR 96k:60108
  • 4. R. Bañuelos and G. Wang, Orthogonal martingales under differentiable subordination and applications to Riesz transforms, Illinois J. Math. 40 (1996), 678-691. MR 99a:60047
  • 5. R. Bañuelos and G. Wang, Davis's inequality for orthogonal martingales under differential subordination, Michigan Math. J. 47 (2000), 109-124. MR 2001g:60100
  • 6. D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. MR 34:8456
  • 7. D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702. MR 86b:60080
  • 8. D. L. Burkholder, A sharp and strict $L^p$-inequality for stochastic integrals, Ann. Probab. 15 (1987), 268-273. MR 88d:60156
  • 9. D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158 (1988), 75-94. MR 90b:60051
  • 10. D. L. Burkholder, Differential subordination of harmonic functions and martingales, Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Math. 1384 (1989), 1-23. MR 90k:31004
  • 11. D. L. Burkholder, Explorations in martingale theory and its applications, Lecture Notes in Math. 1464 (1991), 1-66. MR 92m:60037
  • 12. D. L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), 995-1025. MR 95h:60085
  • 13. D. L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress (East Lansing, MI, 1994), 343-358, Proc. Sympos. Appl. Math. 52, Amer. Math. Soc., Providence, RI (1997). MR 98f:60103
  • 14. B. Davis, On the weak type $(1,1)$ inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307-311. MR 50:879
  • 15. C. Dellacherie and P. A. Meyer, Probability and Potential: The Theory of Martingales, North-Holland, Amsterdam, 1982. MR 85e:60001
  • 16. J. L. Doob, Stochastic Processes, Wiley, New York, 1953. MR 15:445b
  • 17. M. Essén, On sharp constants in the weak type $(p,p)$-inequalities, $2<p<\infty$, Report No. 43 1999/2000, Institut Mittag-Leffler.
  • 18. H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209-245. MR 36:945
  • 19. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995. MR 98c:34001
  • 20. P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer-Verlag, Berlin, 1990. MR 91i:60148
  • 21. B. Tomaszewski, Some sharp weak-type inequalities for holomorphic functions on the unit ball of $\mathbf{C}^n$, Proc. Amer. Math. Soc. 95 (1985), 271-274. MR 87a:32005
  • 22. G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551. MR 96b:60120

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Additional Information

Jiyeon Suh
Affiliation: Department of Statistics, Purdue University, West Lafayette, Indiana 47907

Keywords: Martingale transform, differential subordination, biconcave majorant
Received by editor(s): February 19, 2003
Received by editor(s) in revised form: November 4, 2003
Published electronically: September 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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