Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators $A_a^{\varphi_1,\varphi_2}$ with symbols $a$ in the wide weighted modulation space $M^\infty_{w}(\mathbb{R}^{2d})$, and windows $\varphi_1, \varphi_2$ 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 $A_a^{\varphi_1,\varphi_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $\mathcal{S}^{(\gamma)} (\mathbb{R}^{d})$, where the parameter $\gamma \geq 1$ is related to the growth of the considered weight. An important role is played by $\tau$-pseudodifferential operators $\mathrm{Op}_\tau(\sigma)$. In that direction we show convenient continuity properties of $\mathrm{Op}_\tau(\sigma)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\mathrm{Op}_\tau(\sigma)$ when the symbol $\sigma$ 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.