Mixed martingale Hardy spaces

In this paper we consider the martingale Hardy spaces defined with the help of the mixed $L_{\text{pv}}$-norm. Five mixed martingale Hardy spaces will be investigated: $H_{\text{pv}}^s$, $H_{\text{pv}}^S$, $H_{\text{pv}}^M$, ${\text{cP}}_{\text{pv}}$ and ${\text{cQ}}_{\text{pv}}$. 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.