A quantum harmonic analysis approach to nonlinear time-frequency concentration

We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.