Factorization in mixed norm Hardy and BMO spaces

Let $1leq p,q < infty$ and $1leq r leq infty$. We show that the direct sum of mixed norm Hardy spaces $big(sum_n H^p_n(H^q_n)big)_r$ and the sum of their dual spaces $big(sum_n H^p_n(H^q_n)^*big)_r$ are both primary. We do so by using Bourgains localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $big(sum_{nin mathbb N} H_n^1(H_n^s)big)_r$, $big(sum_{nin mathbb N} H_n^s(H_n^1)big)_r$, as well as $big(sum_{nin mathbb N} BMO_n(H_n^s)big)_r$ and $big(sum_{nin mathbb N} H^s_n(BMO_n)big)_r$, $1 < s < infty$, $1leq r leq infty$, are all primary.