  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.
