Dimension dependence of factorization problems: Haar system Hardy spaces
Streszczenie
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 $\Gamma ,\delta ,\varepsilon \gt 0$, and let $n,N\in \mathbb N$ be chosen such that $N \ge Cn^2$, where $C = C(\Gamma ,\delta ,\varepsilon ) \gt 0$ (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 $I_{Y_n} = ATB$ 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 $N \ge C n$ 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.
