Factorization in mixed norm Hardy and BMO spaces

Abstract

Let $1\le p,q<\infty$ and $1\le r\le \infty$. We show that the direct sum of mixed norm Hardy spaces $({\sum }_n{H}_n^p(H_n^q){)}_r$ and the sum of their dual spaces $({\sum }_n{H}_n^p(H_n^q{)}^{\ast }{)}_r$ 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 $({\sum }_{n\in \mathbb{N}}{H}_n^1(H_n^s){)}_r$, $({\sum }_{n\in \mathbb{N}}{H}_n^s(H_n^1){)}_r$, as well as $({\sum }_{n\in \mathbb{N}}BM{O}_n(H_n^s){)}_r$ and $({\sum }_{n\in \mathbb{N}}{H}_n^s(BM{O}_n){)}_r$, $1<s<\infty$, $1\le r\le \infty$, are all primary.

29 pages, 8 figures

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