Weak type estimates associated to Burkholder's martingale inequality

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.