$\mathcal{A}$-Localization Operators

Abstract

Time-frequency localization operators, originally introduced by Daubechies (1988), provide a framework for localizing signals in the phase space and have become a central tool in time-frequency analysis. In this paper we introduce and study a broad generalization of these operators, called $\mathcal{A}$-localization operators, associated with a metaplectic Wigner distribution $W_{\mathcal{A}}$ and the corresponding $\mathcal{A}$-pseudodifferential calculus. We first show that the classical relation between localization operators and Weyl quantization extends to any \emph{covariant metaplectic Wigner distribution}. Specifically, if $W_{\mathcal{A}}$ satisfies the covariance property \[ W_\mathcal{A}(π(z)f,π(z)g)=T_zW_\mathcal{A}(f,g), \qquad z\in\mathbb{R}^{2d}, \] then \[ A_{a}^{φ_1,φ_2} = \operatorname{Op}_\mathcal{A}\big(a * W_\mathcal{A}(φ_2,φ_1)\big), \] and conversely, this identity characterizes covariance. This result extends the recent representation formula of Bastianoni and Teofanov for $\ddot{I}„$-operators to the full metaplectic framework. We then define the $\mathcal{A}$-localization operator $A_{a,\mathcal{A}}^{\ddot{I}{†}_1,\ddot{I}{†}_2}$ and investigate its analytical properties. We establish boundedness results on modulation spaces and provide sufficient conditions for Schatten-von Neumann class membership. These findings connect the structure of metaplectic representations with time-frequency localization theory, offering a unified approach to quantization and signal analysis.

24 pages