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In this paper, we study the Cauchy problem for radially symmetric homogeneousnon-cutoff Boltzmann equation with Maxwellian molecules, the initial datumbelongs to Shubin space of the negative index which can be characterized byspectral decomposition of the harmonic oscillators. The Shubin space of thenegative index contains the measure functions. Based on this spectraldecomposition, 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 problemdefined by the evolution equation associated to a fractional harmonicoscillator.

The cauchy problem for radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand-shilov smoothing effect