This is an announcement for the paper "Weak type estimates associated to Burkholder's martingale inequality" by Javier Parcet.
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 $$|f|_{L_p(\Omega)} \sim_{\mathrm{c}_p} \Big| \Big( \sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^{1/2} \Big|_{L_{p}(\Omega)} + \Big( \sum_{k=1}^\infty |df_k|_p^p \Big)^{1/p}.$$ 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 (as we shall explain) 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.
Archive classification: Probability; Functional Analysis
Mathematics Subject Classification: 42B25; 60G46; 60G50
Remarks: 19 pages
The source file(s), WeakBurk.tex: 66319 bytes, is(are) stored in gzipped form as 0508447.gz with size 18kb. The corresponding postcript file has gzipped size 88kb.
Submitted from: javier.parcet@uam.es
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