Dimension dependence of factorization problems: bi-parameter Hardy spaces

Given $1 leq p,q < infty$ and $ninmathbb{N}_0$, let $H_n^p(H_n^q)$ denote the canonical finite-dimensional bi-parameter dyadic Hardy space. Let $(V_n : ninmathbb{N}_0)$ denote either $bigl(H_n^p(H_n^q) : ninmathbb{N}_0bigr)$ or $bigl( (H_n^p(H_n^q))^* : ninmathbb{N}_0bigr)$. We show that the identity operator on $V_n$ factors through any operator $T : V_Nto V_N$ which has large diagonal with respect to the Haar system, where $N$ depends emph{linearly} on $n$.