Dimension dependence of factorization problems: bi-parameter Hardy spaces

Abstract

Given $1\le p,q<\infty$ and $n\in {\mathbb{N}}_0$, let $H_n^p(H_n^q)$ denote the canonical finite-dimensional bi-parameter dyadic Hardy space. Let $(V_n:n\in {\mathbb{N}}_0)$ denote either $(H_n^p(H_n^q):n\in {\mathbb{N}}_0)$ or $((H_n^p(H_n^q){)}^{\ast }:n\in {\mathbb{N}}_0)$. We show that the identity operator on $V_n$ factors through any operator $T:V_N\to V_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.

41 pages