Factorization in mixed norm Hardy and BMO spaces

TL;DR

This paper proves that certain mixed norm Hardy and BMO spaces, along with their duals, are primary, using Bourgain's localization method and finite dimensional factorization techniques.

Contribution

It establishes the primarity of a broad class of mixed norm Hardy and BMO spaces and their duals, extending previous results in the field.

Findings

Mixed norm Hardy spaces are primary.

Dual spaces of mixed norm Hardy spaces are primary.

Several specific mixed norm Hardy and BMO spaces are shown to be primary.

Abstract

Let $1\leq p,q < \infty$ and $1\leq r \leq \infty$. We show that the direct sum of mixed norm Hardy spaces $\big(\sum_n H^p_n(H^q_n)\big)_r$ 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 Bourgain's localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $\big(\sum_{n\in \mathbb N} H_n^1(H_n^s)\big)_r$, $\big(\sum_{n\in \mathbb N} H_n^s(H_n^1)\big)_r$, as well as $\big(\sum_{n\in \mathbb N} BMO_n(H_n^s)\big)_r$ and $\big(\sum_{n\in \mathbb N} H^s_n(BMO_n)\big)_r$, $1 < s < \infty$, $1\leq r \leq \infty$, are all primary.