Gelfand–Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation

We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand–Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.