Factorization in Haar system Hardy spaces

TL;DR

This work establishes a unified factorization framework for operators on Haar-system Hardy spaces $Y$, which include all separable rearrangement-invariant spaces and the dyadic Hardy space $H^1$. By combining strategically reproducible bases with randomized faithful Haar systems, the authors reduce arbitrary bounded operators to Haar multipliers and ultimately to scalar multiples of the identity, yielding that the identity factors through either $T$ or $I_Y-T$ and that the maximal ideal $\mathcal{M}_Y$ is unique. They further extend these factorization results to infinite direct sums $\ell^p(Y)$, proving primariness for these spaces, and develop a stabilization technique showing that bounded Haar multipliers can be approximated by stable multipliers near $cI_Y$. Collectively, the paper generalizes classical primariness/factorization results for $L^p$ and $H^1$ to the broad class of Haar-system Hardy spaces and their sums, with explicit bounds and a robust operator-ideal framework.

Abstract

A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I)_I$, either under a rearrangement-invariant norm $\|\cdot \|$ or under the associated square function norm \begin{equation*} \Bigl\| \sum_Ia_Ih_I \Bigr\|_{*} = \Bigl\| \Bigl( \sum_I a_I^2 h_I^2 \Bigr)^{1/2} \Bigr\|. \end{equation*} Apart from $L^p$, $1\le p<\infty$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\ne C(Δ)$ ($C(Δ)$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y - T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \begin{equation*} \mathcal{M}_Y = \{ T\in \mathcal{B}(Y) : I_Y \ne ATB\text{ for all } A, B\in \mathcal{B}(Y) \} \end{equation*} is the unique maximal ideal of the algebra $\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces $\ell^p(Y)$, $1\le p \leq \infty$, and use them to show that they are primary.

Figures (2)

• Figure 1: The collections $\mathcal{G}_n(\mathcal{A})$
• Figure 2: The first three functions of a $(\varkappa_I)_{I\in \mathcal{D}}$-faithful Haar system $(\tilde{h}_I)_{I\in \mathcal{D}}$ which is extended to a faithful Haar system $(\hat{h}_I)_{I\in \mathcal{D}}$ by adding the dashed Haar functions with the light blue shading

Theorems & Definitions (94)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Definition 2.8