JournalsrmiVol. 23, No. 3pp. 1011–1037

Weak type estimates associated to Burkholder’s martingale inequality

  • Javier Parcet

    Consejo Superior de Investigaciones Científicas, Madrid, Spain
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Abstract

Given a probability space (Ω,A,μ)(\Omega, \mathsf{A}, \mu), let A1,A2,\mathsf{A}_1, \mathsf{A}_2, \ldots be a filtration of σ\sigma-subalgebras of A\mathsf{A} and let E1,E2,\mathsf{E}_1, \mathsf{E}_2, \ldots denote the corresponding family of conditional expectations. Given a martingale f=(f1,f2,)f = (f_1, f_2, \ldots) adapted to this filtration and bounded in Lp(Ω)L_p(\Omega) for some 2p<2 \le p < \infty, Burkholder's inequality claims that

fpcp(k=1Ek1(dfk2))12p+(k=1dfkpp)1p.\|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<21 < p < 2. In this paper we study Burkholder's inequality for p=1p=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)(1,1) analog of Burkholder's inequality. As application we obtain new properties of Davis decomposition for martingales.

Cite this article

Javier Parcet, Weak type estimates associated to Burkholder’s martingale inequality. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 1011–1037

DOI 10.4171/RMI/522