Discrete Triebel-Lizorkin spaces and expansive matrices
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| Format: | Preprint |
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2024
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| _version_ | 1866910020303585280 |
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| author | van Velthoven, Jordy Timo Voigtlaender, Felix |
| author_facet | van Velthoven, Jordy Timo Voigtlaender, Felix |
| contents | We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that $\dot{\mathbf{f}}^α_{p,q}(A) = \dot{\mathbf{f}}^α_{p,q}(B)$ for all $α\in \mathbb{R}$ and $p, q \in (0, \infty]$ if and only if the set $\{A^j B^{-j} : j \in \mathbb{Z}\}$ is finite, or in the trivial case when $p = q$ and $|\det(A)|^{α+ 1/2 - 1/p} = |\det(B)|^{α+ 1/2 - 1/p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Triebel-Lizorkin spaces and expansive matrices van Velthoven, Jordy Timo Voigtlaender, Felix Classical Analysis and ODEs Functional Analysis We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that $\dot{\mathbf{f}}^α_{p,q}(A) = \dot{\mathbf{f}}^α_{p,q}(B)$ for all $α\in \mathbb{R}$ and $p, q \in (0, \infty]$ if and only if the set $\{A^j B^{-j} : j \in \mathbb{Z}\}$ is finite, or in the trivial case when $p = q$ and $|\det(A)|^{α+ 1/2 - 1/p} = |\det(B)|^{α+ 1/2 - 1/p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces. |
| title | Discrete Triebel-Lizorkin spaces and expansive matrices |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2409.01849 |