Analytic smoothing effect of the time variable for the spatially homogeneous Boltzmann equations

In this article, we study the analytic smoothing effect of Cauchy problem for the spatially homogeneous Boltzmann equation without angular cutoff in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable at positive time. Our proof relies on the perturbed Boltzmann equation to reduce the derivative order with respect to the time and the elementary $ L^2 $ weighted estimate.