Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators Aaφ1,φ2 with symbols a in the wide weighted modulation space Mw∞(R2d), and windows φ1, φ2 in the Gelfand–Shilov space S(1)(Rd). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of Aaφ1,φ2 have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space S(γ)(Rd) , where the parameter γ≥ 1 is related to the growth of the considered weight. An important role is played by τ-pseudodifferential operators Opτ(σ). In that direction we show convenient continuity properties of Opτ(σ) when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Opτ(σ) when the symbol σ belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.