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Abstract

\tolerance 6000Let $1\leq p,q \lt \infty $ and $1\leq r \leq \infty $. We show that the direct sum $(\sum_n H^p_n(H^q_n))_r$ of the mixed norm Hardy spaces and the sum $(\sum_n H^p_n(H^q_n)^*)_r$ of their dual spaces are both primary. We do so by using Bourgain’s localization method and solving the finite-dimensional factorization problem. In particular, we show that the spaces $(\sum_{n\in \mathbb N} H_n^1(H_n^s))_r$, $(\sum_{n\in \mathbb N} H_n^s(H_n^1))_r$, as well as $(\sum_{n\in \mathbb N} {\mathrm {BMO}}_n(H_n^s))_r$ and $(\sum_{n\in \mathbb N} H^s_n( {\mathrm {BMO}}_n))_r$, $1 \lt s \lt \infty $, $1\leq r \leq \infty $, are all primary.