Dimension dependence of factorization problems: Biparameter Hardy spaces

Given 1 \leq p,q < \infty , and n\in \mathbb{N}_0, let H_n^p(H_n^q) denote the finite-dimensional building blocks of the biparameter dyadic Hardy space H^p(H^q). Let (V_n : n\in \mathbb{N}_0) denote either \bigl (H_n^p(H_n^q) : n\in \mathbb{N}_0\bigr ) or \bigl ( (H_n^p(H_n^q))^* : n\in \mathbb{N}_0\bigr ). We show that the identity operator on V_n factors through any operator T : V_N\to V_N which has a large diagonal with respect to the Haar system, where N depends linearly on n.