Metaplectic Wigner distributions

Metaplectic Wigner distributions were recently investigated as natural generalizations of the classical Wigner distribution, and provide a wide class of time-frequency representations that exploits the structure of the symplectic group. This work serves as a survey on metaplectic Wigner distributions and their applications to the time-frequency analysis of modulation spaces and pseudodifferential operators, topics that are all still poorly understood. We also give some new results, generalizing Lieb’s uncertainty principle to the so-called matrix Wigner distributions and proving the continuity on $M_{v_s}^p({\mathbb{R}}^d)$ spaces of metaplectic pseudodifferential operators with symbols in $M_{v_{-s}}^{p^{\mathrm{\prime }}}({\mathbb{R}}^d)$.