Haar systems and some results on the basis in a martingale space with mixed norm

For martingale spaces with mixed norm defined with respect to diadic flow of $\sigma$-algebras, we find a condition on summation characteristics, which implies no unconditional basis in these spaces (a generalization of the classical result of Pelczynski proved for $L_1$-spaces). In these spaces (under other conditions on characteristics) generalized Haar systems are considered; the test of the existence of an unconditional basis in terms of the Paley function is obtained and the convergence theorem for almost all choices of signs is proved.