Dimension dependence of factorization problems: Haar system Hardy spaces
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916703254872064 |
|---|---|
| author | Speckhofer, Thomas |
| author_facet | Speckhofer, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05187 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension dependence of factorization problems: Haar system Hardy spaces Speckhofer, Thomas Functional Analysis 47A68, 46B07, 46B25, 46E30, 30H10 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. |
| title | Dimension dependence of factorization problems: Haar system Hardy spaces |
| topic | Functional Analysis 47A68, 46B07, 46B25, 46E30, 30H10 |
| url | https://arxiv.org/abs/2407.05187 |