Dimension dependence of factorization problems: Haar system Hardy spaces

For n∈ N, let Yₙ denote the linear span of the first $n+1$ levels of the Haar system in a Haar system Hardy space Y (this class contains all separable rearrangement-invariant function spaces and also related spaces such as dyadic H¹). Let IYₙ denote the identity operator on Yₙ. We prove the following quantitative factorization result: Fix Γ,δ,ε > 0, and let n,N ∈ N be chosen such that N ≥ Cn², where C = C(Γ,δ,ε) > 0 (this amounts to a quasi-polynomial dependence between YN and Yₙ). Then for every linear operator T YN→ YN with \|T\|≤ Γ, there exist operators $A,B$ with \|A\|\|B\|≤ 2(1+ε) such that either IYₙ = ATB or IYₙ = A(IYN - T)B. Moreover, if T has δ-large positive diagonal with respect to the Haar system, then we have IYₙ = ATB for some $A,B$ with \|A\|\|B\|≤ (1+ε)/δ. If the Haar system is unconditional in Y, then an inequality of the form N ≥ Cn is sufficient for the above statements to hold (hence, YN depends polynomially on Yₙ). Finally, we prove an analogous result in the case where T has large but not necessarily positive diagonal entries.