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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
DOI: https://doi.org/10.1090/S0002-9947-04-03563-9
Published electronically: September 23, 2004
MathSciNet review: 2115376
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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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Additional Information

Jiyeon Suh
Affiliation: Department of Statistics, Purdue University, West Lafayette, Indiana 47907
Email: jsuh@stat.purdue.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03563-9
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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