$\mathcal{A}$-Localization Operators

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 $$W_{\mathcal{A}}(\pi (z)f,\pi (z)g)=T_z{W}_{\mathcal{A}}(f,g),z\in {\mathbb{R}}^{2d},$$ then $$A_a^{{\phi }_1,{\phi }_2}={\mathrm{Op}}_{\mathcal{A}}(a\ast W_{\mathcal{A}}({\phi }_2,{\phi }_1)),$$ and conversely, this identity characterizes covariance. This result extends the recent representation formula of Bastianoni and Teofanov for $\tau$-operators to the full metaplectic framework. We then define the $\mathcal{A}$-localization operator $A_{a,\mathcal{A}}^{{\phi }_1,{\phi }_2}$ 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.