Dimension dependence of factorization problems: Haar system Hardy spaces
Thomas Speckhofer
TL;DR
The paper investigates dimension-dependent factorization problems in Haar-system Hardy spaces. It develops a finite-dimensional strategy that first diagonalizes an arbitrary operator via random faithful Haar systems, then stabilizes the resulting Haar multiplier to a near-identity, and finally reduces to a positive-diagonal case using a discrete Gamlen–Gaudet construction. The main outcome is that one can guarantee a factorization of the form $I_{Y_n}=ATB$ (or through $I_{Y_N}-T$) with controlled constants, provided $N$ scales at least quadratically with $n$ (i.e., $N\gtrsim n^2$) in general, and linearly in the unconditional Haar-system setting. When the diagonal of $T$ is $\delta$-large and positive, the factorization constant improves to $(1+\varepsilon)/\delta$, and unconditionality yields polynomial dimension dependence; the paper also outlines a reduction to the positive-diagonal case and discusses extensions to large but not strictly positive diagonals. These results clarify the dimension dependence of factorization phenomena in Haar-system Hardy spaces and extend the understanding of how structural properties of the Haar basis influence quantitative operator factorizations.
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 $Γ,δ,\varepsilon > 0$, and let $n,N \in \mathbb{N}$ be chosen such that $N \ge Cn^2$, where $C = C(Γ,δ,\varepsilon) > 0$ (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 $\|T\|\le Γ$, 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 $δ$-large positive diagonal with respect to the Haar system, then we have $I_{Y_n} = ATB$ for some $A,B$ with $\|A\|\|B\|\le (1+\varepsilon)/δ$. 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.