Boundedness and compactness of localization operators associated with the Laguerre-Bessel-Wigner transform

The main crux of this work is to define the Wigner transform associated with the Laguerre-Bessel operator and to give some results related to this transform. Next we introduce a new class of pseudo-differential operator \(\mathscr {L}_{u, v}(\sigma )\) ${\mathcal{L}}_{u,v}(\sigma )$ called localization operator which depend on a symbol \(\sigma\) $\sigma$ and two functions u and v. We give a criteria in terms of the symbol \(\sigma\) $\sigma$ for its boundedness and compactness, we also show that these operators belongs to the Schatten-Von Neumann class \(S^p\) $S^p$ for all \(p \in [1; +\infty ]\) $p\in [1;+\infty ]$ and we give a trace formula.