Localization operators for Wigner transform on the Chébli-Trimèche hypergroups of exponential growths

The main focus of this paper is to define the Wigner transform on Chébli-Trimèche hypergroups of exponential growth and to present several related results. Next, we introduce a new class of pseudo-differential operators \(\mathscr{L}_{\psi_1, \psi_2}(\sigma)\), called localization operators, which depend on a symbol \(\sigma\) and two admissible functions \(\psi_1\) and \(\psi_2\). We provide criteria, in terms of the symbol \(\sigma\), for their boundedness and compactness. We also show that these operators belong to the Schatten–von Neumann class \(S^p\) for all \(p \in [1, +\infty]\), and we derive a trace formula.