Uncertainty principle via variational calculus on modulation spaces

We approach uncertainty principles of Cowling-Price-Heis-enberg-type as a variational principle on modulation spaces. In our discussion we are naturally led to compact localization operators with symbols in The optimal constant these is the smallest eigenvalue inverse operator. Euler-Lagrange equations for associated functional provide eigenfunctions operators. As by-product proofs derive generalization mixed-norm spaces an inequality Wigner and Ambiguity functions due do Lieb.