  In this paper we consider the martingale Hardy spaces defined with the help
of the mixed $L_{\pv}$-norm. Five mixed martingale Hardy spaces will be
investigated: $H_{\pv}^{s}$, $H_{\pv}^S$, $H_{\pv}^M$, $\cP_{\pv}$ and
$\cQ_{\pv}$. Several results are proved for these spaces, like atomic
decompositions, Doob's inequality, boundedness, martingale inequalities and the
generalization of the well-known Burkholder-Davis-Gundy inequality.
