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 dimYN and dimYn). 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 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, dimYN depends polynomially on dimYn). Finally, we prove an analogous result in the case where T has large but not necessarily positive diagonal entries.