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Decembar, 2007 Weak type estimates associated to Burkholder's martingale inequality
Javier Parcet
Rev. Mat. Iberoamericana 23(3): 1011-1037 (Decembar, 2007).

Abstract

Given a probability space $(\Omega, \mathsf{A}, \mu)$, let $\mathsf{A}_1, \mathsf{A}_2, \ldots$ be a filtration of $\sigma$-subalgebras of $\mathsf{A}$ and let $\mathsf{E}_1, \mathsf{E}_2, \ldots$ denote the corresponding family of conditional expectations. Given a martingale $f = (f_1, f_2, \ldots)$ adapted to this filtration and bounded in $L_p(\Omega)$ for some $2 \le p < \infty$, Burkholder's inequality claims that $$ \|f\|_p \sim_{\mathrm{c}_p} \Big\| \Big( \sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^\frac12 \Big\|_p + \Big( \sum_{k=1}^\infty \|df_k\|_p^p \Big)^\frac1p. $$ Motivated by quantum probability, Junge and Xu recently extended this result to the range $1 < p < 2$. In this paper we study Burkholder's inequality for $p=1$, for which the techniques must be different. Quite surprisingly, we obtain two non-equivalent estimates which play the role of the weak type $(1,1)$ analog of Burkholder's inequality. As application we obtain new properties of Davis decomposition for martingales.

Citation

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Javier Parcet. "Weak type estimates associated to Burkholder's martingale inequality." Rev. Mat. Iberoamericana 23 (3) 1011 - 1037, Decembar, 2007.

Information

Published: Decembar, 2007
First available in Project Euclid: 27 February 2008

zbMATH: 1155.46033
MathSciNet: MR2414501

Subjects:
Primary: 42B25, 60G46, 60G50

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.23 • No. 3 • Decembar, 2007
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