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arXiv · 2401.04659

The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators

Fabio Nicola·Federico Riccardi
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Abstract

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.

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Fabio Nicola, Federico Riccardi. 2024-01-21. The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators. https://arxiv.org/abs/2401.04659

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