Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators $A_a^{φ_1,φ_2}$ with symbols $a$ in the wide weighted modulation space $M^\infty_{w}(\mathbb{R}^{2d})$, and windows $φ_1, φ_2$ in the Gelfand-Shilov space $\mathcal{S}^{\left(1\right)}(\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{φ_1,φ_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $\mathcal{S}^{(γ)} (\mathbb{R}^{d})$, where the parameter $γ\geq 1$ is related to the growth of the considered weight. An important role is played by $τ$-pseudodifferential operators $\mathrm{Op}_τ(σ)$. In that direction we show convenient continuity properties of $\mathrm{Op}_τ(σ)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\mathrm{Op}_τ(σ)$ when the symbol $σ$ belongs to a modulation space with appropriately chosen weight functions. As a tool we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

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