  Let $f:=(f_n)_{n\in \mathbb{Z}_+}$ and $g:=(g_n)_{n\in \mathbb{Z}_+}$ be two
martingales related to the probability space $(\Omega,\mathcal F,\mathbb P)$
equipped with the filtration $(\mathcal F_n)_{n\in \mathbb{Z}_+}.$ Assume that
$f$ is in the martingale Hardy space $H_1$ and $g$ is in its dual space, namely
the martingale $\rm BMO.$ Then the semi-martingale $f\cdot g:=(f_ng_n)_{n\in
\mathbb{Z}_+}$ may be written as the sum $$f\cdot g=G(f, g)+L( f,g).$$ Here $L(
f,g):=(L( f,g)_n)_{n\in\mathbb{Z}_+}$ with $L(
f,g)_n:=\sum_{k=0}^n(f_k-f_{k-1})(g_k-g_{k-1)})$ for any $n\in\mathbb{Z}_+$,
where $f_{-1}:=0=:g_{-1}$. The authors prove that $L( f,g)$ is a process with
bounded variation and limit in $L^1,$ while $G(f,g)$ belongs to the martingale
Hardy-Orlicz space $H_{\log}$ associated with the Orlicz function
$$\Phi(t):=\frac{t}{\log(e+t)},\quad \forall\, t\in[0,\infty).$$ The above
bilinear decomposition $L^1+H_{\log}$ is sharp in the sense that, for
particular martingales, the space $L^1+H_{\log}$ cannot be replaced by a
smaller space having a larger dual. As an application, the authors characterize
the largest subspace of $H_1$, denoted by $H^b_1$ with $b\in {\rm BMO}$, such
that the commutators $[T, b]$ with classical sublinear operators $T$ are
bounded from $H^b_1$ to $L^1$. This endpoint boundedness of commutators allow
the authors to give more applications. On the one hand, in the martingale
setting, the authors obtain the endpoint estimates of commutators for both
martingale transforms and martingale fractional integrals. On the other hand,
in harmonic analysis, the authors establish the endpoint estimates of
commutators both for the dyadic Hilbert transform beyond doubling measures and
for the maximal operator of Ces\`{a}ro means of Walsh--Fourier series.
