Title: The cauchy problem for radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand-shilov smoothing effect,
Abstract: In this paper, we study the Cauchy problem for radially symmetric homogeneous
non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum
belongs to Shubin space of the negative index which can be characterized by
spectral decomposition of the harmonic oscillators. The Shubin space of the
negative index contains the measure functions. Based on this spectral
decomposition, we construct the weak solution with Shubin class initial datum,
we also prove that the Cauchy problem enjoys Gelfand-Shilov smoothing effect,
meaning that the smoothing properties are the same as the Cauchy problem
defined by the evolution equation associated to a fractional harmonic
oscillator.