The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866914646713171968 |
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| author | Nicola, Fabio Riccardi, Federico |
| author_facet | Nicola, Fabio Riccardi, Federico |
| contents | In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators $L_Ω \colon L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$, with Gaussian window, associated with a subset $Ω\subset\mathbb{R}^{2d}$ of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of $L_Ω$ is maximized, among all subsets $Ω$ of a given finite measure, when $Ω$ is a ball and that there are no other extremizers. Actually, the main result is a quantitative version of this estimate, with sharp exponent. A similar problem is addressed for wavelet localization operators, where rearrangements are understood in the hyperbolic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators Nicola, Fabio Riccardi, Federico Classical Analysis and ODEs 42B10, 49Q20, 49R05, 81S30, 94A12 In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators $L_Ω \colon L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$, with Gaussian window, associated with a subset $Ω\subset\mathbb{R}^{2d}$ of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of $L_Ω$ is maximized, among all subsets $Ω$ of a given finite measure, when $Ω$ is a ball and that there are no other extremizers. Actually, the main result is a quantitative version of this estimate, with sharp exponent. A similar problem is addressed for wavelet localization operators, where rearrangements are understood in the hyperbolic setting. |
| title | The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators |
| topic | Classical Analysis and ODEs 42B10, 49Q20, 49R05, 81S30, 94A12 |
| url | https://arxiv.org/abs/2401.04659 |