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Factorization in mixed norm Hardy and BMO spaces (1610.01506v1)

Published 5 Oct 2016 in math.FA

Abstract: Let $1\leq p,q < \infty$ and $1\leq r \leq \infty$. We show that the direct sum of mixed norm Hardy spaces $\big(\sum_n Hp_n(Hq_n)\big)_r$ and the sum of their dual spaces $\big(\sum_n Hp_n(Hq_n)*\big)_r$ are both primary. We do so by using Bourgain's localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $\big(\sum_{n\in \mathbb N} H_n1(H_ns)\big)_r$, $\big(\sum_{n\in \mathbb N} H_ns(H_n1)\big)_r$, as well as $\big(\sum_{n\in \mathbb N} BMO_n(H_ns)\big)_r$ and $\big(\sum_{n\in \mathbb N} Hs_n(BMO_n)\big)_r$, $1 < s < \infty$, $1\leq r \leq \infty$, are all primary.

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