Smoothing effect and Cauchy problem for radially symmetric homogeneous Boltzmann equation with Debye-Yukawa potential of Shubin class initial datum

In this paper, we study the Cauchy problem for the radially symmetric homogeneous non-cutoff Boltzmann equation with Debye-Yukawa potential, the initial datum belongs to Shubin type space of the negative index which can be characterized by spectral decomposition of the harmonic oscillator, and it is a small perturbation of Maxwellian distribution. The Shubin type space of negative index contains the probability measures. Based on the spectral decomposition, we construct the weak solution with Shubin type class initial datum and prove the smoothing effect for the solution to this Cauchy problem.