Dimension dependence of factorization problems: Biparameter Hardy spaces

Lechner, Richard

doi:10.1090/proc/14364

Dimension dependence of factorization problems: Biparameter Hardy spaces
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by Richard Lechner

Proc. Amer. Math. Soc. 147 (2019), 1639-1652

DOI: https://doi.org/10.1090/proc/14364

Published electronically: January 8, 2019

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Abstract:

Given $1 \leq p,q < \infty$, and $n\in \mathbb {N}_0$, let $H_n^p(H_n^q)$ denote the finite-dimensional building blocks of the biparameter dyadic Hardy space $H^p(H^q)$. Let $(V_n : n\in \mathbb {N}_0)$ denote either $\bigl (H_n^p(H_n^q) : n\in \mathbb {N}_0\bigr )$ or $\bigl ( (H_n^p(H_n^q))^* : n\in \mathbb {N}_0\bigr )$. We show that the identity operator on $V_n$ factors through any operator $T : V_N\to V_N$ which has a large diagonal with respect to the Haar system, where $N$ depends linearly on $n$.

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