The Quantitative Isoperimetric Inequality for the Hilbert–Schmidt Norm of Localization Operators

• DOI: 10.1007/s00041-025-10145-y
• Publication Date: 2025-02-04T21:15:51Z
• Authors: Fabio Nicola; Federico Riccardi
• Keywords: Poincaré inequality

Abstract In this paper we study the Hilbert–Schmidt norm of time-frequency localization operators

$$L_{\Omega } :L^2(\mathbb {R}^d) \rightarrow L^2(\mathbb {R}^d)$$

L Ω

:

L 2

(

R

d

)

→

L 2

(

R

d

)

, with Gaussian window, associated with a subset

$$\Omega \subset \mathbb {R}^{2d}$$

Ω ⊂

R

2 d

of finite measure. We prove, in particular, that the Hilbert–Schmidt norm of

$$L_\Omega $$

L Ω

is maximized, among all subsets

$$\Omega $$

Ω

of a given finite measure, when

$$\Omega $$

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.