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

Fabio Nicola, Federico Riccardi

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.

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

Abstract

In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators , with Gaussian window, associated with a subset of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of is maximized, among all subsets of a given finite measure, when 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.
Paper Structure (12 sections, 9 theorems, 93 equations, 3 figures)

This paper contains 12 sections, 9 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Hilbert-Schmidt norm of localization operators
  4. A general setting for localization operators
  5. Time-frequency localization operators
  6. Quantitative estimate
  7. Some remarks on the sharpness of Proposition \ref{['prop : quantitative estimate for HS norm']}
  8. Sharpness of the power $\alpha[\Omega]^2$
  9. Sharpness of the power $t^{2+1/d}$ for $0<t<1$
  10. The behavior of $\beta(t)$ as $t \rightarrow +\infty$
  11. The Hilbert-Schmidt norm of wavelet localization operators
  12. Remarks on the uniqueness of the extremizers in Riesz' rearrangement inequality

Key Result

Theorem 2.1

(liebloss) Let $f,g$ and $h$ be three nonnegative measurable functions on $\mathbb{R}^d$. Then we have with the understanding that if the left-hand side is $+\infty$ then also the right-hand side is. If, in addition, $g$ is strictly symmetric decreasing and $f$ and $h$ are not zero and the above integrals are finite, equality occurs if and only $f(x) = f^*(x-y)$ and $h(x)=h^*(x-y)$ for almost eve

Figures (3)

  • Figure 1: Representation of $\Omega_r$ with $d=1$ and $v = e_1$.
  • Figure 2: The area of the darker gray region is $(\chi_{\Omega_r^1} * \chi_{B_{r/3}})(z)$
  • Figure 3: Example of $g$ that is symmetric decreasing but not strictly (left, where $g(x_1,0)$ is represented) and the corresponding set $\Omega$ in dimension $d=2$ (right).

Theorems & Definitions (19)

  • Theorem 2.1: Riesz' rearrangement inequality
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 9 more