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 $$\|f\|_{ L_p(\Omega)} \sim_{ \mathrm{ c}_p} \Big\| \Big(\sum_{ k=1}^\infty \mathsf{ E}_{ k-1}(|df_k|^2) \Big)^{ 1/2} \Big\|_{ L_{ p}(\Omega)} + \Big(\sum_{ k=1}^\infty \|df_k\|_p^p \Big)^{ 1/p}.$$ 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.
Categories: math.PR math.FA