Weak type estimates associated to Burkholder's martingale inequality

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dc.creatorParcet, Javier
dc.date2005-08-24
dc.date2005-08-26
dc.date.accessioned2026-07-07T05:22:37Z
dc.date.available2026-07-07T05:22:37Z
dc.descriptionGiven a probability space $(Ω, \mathsf{A}, μ)$, let $\mathsf{A}_1, \mathsf{A}_2, ...$ be a filtration of $σ$-subalgebras of $\mathsf{A}$ and let $\mathsf{E}_1, \mathsf{E}_2, ...$ denote the corresponding family of conditional expectations. Given a martingale $f = (f_1, f_2, ...)$ adapted to this filtration and bounded in $L_p(Ω)$ for some $2 \le p < \infty$, Burkholder's inequality claims that $$\|f\|_{L_p(Ω)} \sim_{\mathrm{c}_p} \Big\| \Big(\sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^{1/2} \Big\|_{L_{p}(Ω)} + \Big(\sum_{k=1}^\infty \|df_k\|_p^p \Big)^{1/p}.$$ 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.
dc.description20 pages
dc.identifierhttps://arxiv.org/abs/math/0508447
dc.identifierhttp://arxiv.org/abs/math/0508447
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/76128
dc.subjectProbability
dc.subjectFunctional Analysis
dc.subject42B25; 60G46; 60G50
dc.titleWeak type estimates associated to Burkholder's martingale inequality
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