Weak type estimates associated to Burkholder's martingale inequality

Abstract

Given a probability space $(\Omega ,\mathsf{A},\mu )$, let ${\mathsf{A}}_1,{\mathsf{A}}_2,...$ be a filtration of $\sigma$-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 $L_p(\Omega )$ for some $2\le p<\infty$, Burkholder's inequality claims that $$\Vert f{\Vert }_{L_p(\Omega )}{\sim }_{{\mathrm{c}}_p}\Vert (\sum _{k=1}^{\infty }{\mathsf{E}}_{k-1}(|d{f}_k{|}^2){)}^{1/2}{\Vert }_{L_p(\Omega )}+(\sum _{k=1}^{\infty }\Vert d{f}_k{\Vert }_p^p{)}^{1/p}.$$ 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.

20 pages

References in corpus (2)

• Noncommutative maximal ergodic theorems
• Gundy's decomposition for non-commutative martingales and applications