Weak type estimates associated to Burkholder's martingale inequality

Given a probability space $(\mathrm{\Omega },\mathsf{A},\mu )$, let $$\mathsf{A}$_1, $\mathsf{A}$_2, …$beafiltrationof$\sigma$-subalgebrasof$$\mathsf{A}$$ and let ${\mathsf{E}}_1,{\mathsf{E}}_2,\mathrm{.}\mathrm{.}\mathrm{.}$ denote the corresponding family of conditional expectations. Given a martingale $f=(f_1,f_2,\mathrm{.}\mathrm{.}\mathrm{.})$ adapted to this filtration and bounded in $L_p(\mathrm{\Omega })$ for some $2\le p<\mathrm{\infty }$, Burkholder’s inequality claims that $$\mathrm{\parallel }f{\mathrm{\parallel }}_{L_p(\mathrm{\Omega })}{\sim }_{{\mathrm{c}}_p}\parallel (\sum _{k=1}^{\mathrm{\infty }}{\mathsf{E}}_{k-1}(\mathrm{\mid }d{f}_k{\mathrm{\mid }}^2){)}^{1\mathrm{/}2}{\parallel }_{L_p(\mathrm{\Omega })}+(\sum _{k=1}^{\mathrm{\infty }}\mathrm{\parallel }d{f}_k{\mathrm{\parallel }}_p^p{)}^{1\mathrm{/}p}\mathrm{.}$$ 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.