  Let $v,~\omega_1, ~\omega_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose
that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(\omega_1, \omega_2)\in
RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in
martingale spaces, we characterize the weights for which $\mathfrak{M}$ is
bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to
$L^{p,\infty}(v)\hbox{or}L^p(v).$ If
$v=\omega_2^{\frac{p}{p_2}}\omega_2^{\frac{p}{p_2}},$ we partially give the
bilinear version of one-weight theory.
