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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.03962 |
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| _version_ | 1866908352307527680 |
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| author | Edmunds, David E. Gurka, Petr Lang, Jan |
| author_facet | Edmunds, David E. Gurka, Petr Lang, Jan |
| contents | We establish that the Fourier transform $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p',p}(\mathbb{R}^d)$, for $d\in\mathbb{N}$ and $1<p<2$, is not strictly singular, thereby confirming the optimality of the source and target spaces. A~similar result is obtained for Fourier series on $L^p(\mathbb{T}^n)$, with sequence Lorentz spaces as the target. These findings complement known results, which state that $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p'}(\mathbb{R}^d)$ is finitely strictly singular and then also strictly singular, and provide further insight into the degrees of non-compactness of~$\mathcal{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Non-Compactness Properties of the Fourier Transform on Optimal Spaces Edmunds, David E. Gurka, Petr Lang, Jan Functional Analysis 42B25, 47B06 We establish that the Fourier transform $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p',p}(\mathbb{R}^d)$, for $d\in\mathbb{N}$ and $1<p<2$, is not strictly singular, thereby confirming the optimality of the source and target spaces. A~similar result is obtained for Fourier series on $L^p(\mathbb{T}^n)$, with sequence Lorentz spaces as the target. These findings complement known results, which state that $\mathcal{F}: L^p(\mathbb{R}^d)\to L^{p'}(\mathbb{R}^d)$ is finitely strictly singular and then also strictly singular, and provide further insight into the degrees of non-compactness of~$\mathcal{F}$. |
| title | Quantitative Non-Compactness Properties of the Fourier Transform on Optimal Spaces |
| topic | Functional Analysis 42B25, 47B06 |
| url | https://arxiv.org/abs/2505.03962 |