Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866911423091703808 |
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| author | Bampouras, Konstantinos |
| author_facet | Bampouras, Konstantinos |
| contents | Recently it was proven that for a convex subset of $\mathbb{R}^{n}$ that has infinitely many extreme vectors, the Nehari theorem fails, that is, there exists a bounded Hankel operator $\Ha_ϕ$ on the Paley--Wiener space $\PW(Ω)$ that does not admit a bounded symbol. In this paper we examine whether Nehari's theorem can hold under the stronger assumption that the Hankel operator $\Ha_ϕ$ is in the Schatten class $S^{p}(\PW(Ω))$. We prove that this fails for $p>4$ for any convex subset of $\mathbb{R}^{n}$, $n\geq2$, of boundary with a $C^{2}$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in $\mathbb{R}^{n}$, the inequality $$\int_{2P}\dfrac{|\widehat{f}(x)|}{m(P\cap (x-P))}dx\leq C(P)\|f\|_{L^{1}},$$ holds for all $f\in \PW^{1}(2P)$, and consequently any Hilbert--Schmidt Hankel operator on a Paley--Wiener space of a polytope is generated by a bounded function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces Bampouras, Konstantinos Functional Analysis 47B35, 47B10 Recently it was proven that for a convex subset of $\mathbb{R}^{n}$ that has infinitely many extreme vectors, the Nehari theorem fails, that is, there exists a bounded Hankel operator $\Ha_ϕ$ on the Paley--Wiener space $\PW(Ω)$ that does not admit a bounded symbol. In this paper we examine whether Nehari's theorem can hold under the stronger assumption that the Hankel operator $\Ha_ϕ$ is in the Schatten class $S^{p}(\PW(Ω))$. We prove that this fails for $p>4$ for any convex subset of $\mathbb{R}^{n}$, $n\geq2$, of boundary with a $C^{2}$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in $\mathbb{R}^{n}$, the inequality $$\int_{2P}\dfrac{|\widehat{f}(x)|}{m(P\cap (x-P))}dx\leq C(P)\|f\|_{L^{1}},$$ holds for all $f\in \PW^{1}(2P)$, and consequently any Hilbert--Schmidt Hankel operator on a Paley--Wiener space of a polytope is generated by a bounded function. |
| title | Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces |
| topic | Functional Analysis 47B35, 47B10 |
| url | https://arxiv.org/abs/2504.05986 |