Dimension dependence of factorization problems: Haar system Hardy spaces

TL;DR

This paper establishes quantitative factorization results for operators on Haar system Hardy spaces, revealing how the dimension dependence influences the ability to factor identity operators through bounded operators.

Contribution

It provides new factorization theorems with explicit dimension dependence for operators on Haar system Hardy spaces, including cases with positive diagonal and unconditional Haar systems.

Findings

Factorization depends on quadratic dimension growth for general operators.

Unconditional Haar systems allow linear dimension dependence.

Large positive diagonal operators admit controlled factorization with bounded norms.

Abstract

For $n\in \mathbb{N}$, let $Y_n$ 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 $H^1$). Let $I_{Y_n}$ denote the identity operator on $Y_n$. We prove the following quantitative factorization result: Fix $\Gamma,\delta,\varepsilon > 0$, and let $n,N \in \mathbb{N}$ be chosen such that $N \ge Cn^2$, where $C = C(\Gamma,\delta,\varepsilon) > 0$ (this amounts to a quasi-polynomial dependence between $\dim Y_N$ and $\dim Y_n$). Then for every linear operator $T\colon Y_N\to Y_N$ with $\|T\|\le \Gamma$, there exist operators $A,B$ with $\|A\|\|B\|\le 2(1+\varepsilon)$ such that either $I_{Y_n} = ATB$ or $I_{Y_n} = A(I_{Y_N} - T)B$. Moreover, if $T$ has $\delta$-large positive diagonal with respect to the Haar system, then we have $I_{Y_n} = ATB$ for some $A,B$ with $\|A\|\|B\|\le (1+\varepsilon)/\delta$. If the Haar system is unconditional in $Y$, then an inequality of the form $N \ge Cn$ is sufficient for the above statements to hold (hence, $\dim Y_N$ depends polynomially on $\dim Y_n$). Finally, we prove an analogous result in the case where $T$ has large but not necessarily positive diagonal entries.