玻尔兹曼方程是数学物理学中的一个基本方程，具有深刻的物理背景。 众所周知， 在初值具有有限能量的物理假设条件，玻尔兹曼方程碰撞函数的角奇异性可以导致柯西问题解的正则性。对初值具有无穷能量的情形，Cannone-karch在《玻尔兹曼方程的无穷能量解，CPAM，2014》中得到相关的结果，这一结论后来被Morimoto-Yang在《具有可测初值的玻尔兹曼方程的光滑性效应，Poincare，2015》中得以推广。对于线性化的玻尔兹曼方程，目前有Lerner, Morimoto, Pravda-Starov和Xu(徐超江教授)的作品[LMPX3]以及Glangetas，本人和Xu的工作[GLX]，只是上述结果的初值属于L^2赋范空间。对于初值属于可测函数空间的情况目前还是有待解决的热门研究课题。 在上述结果的基础上，利用谱分解技巧，我们研究具有可测初值的空间齐性玻尔兹曼方程柯西问题解的定性性质。

The Boltzmann equation is a basic equation of the mathematical physics, which has rich physical backgrouds. It is well known that the angular singularity in the cross section leads to the regularity of the solution for the Cauchy problem under the non-cutoff assumption with the initial datum of finite energy. Cannone and karch in <Infinite energy solutions to the homogeneous Boltzmann equation, CPAM,2014> prove the result that the homogeneous Boltzmann equation enjoys the global in time smoothing effect solution with the measure initial data, which have infinite energy. This idea was first intended by . This result was extended by Morimoto and Yang in <Smoothing effect of the homogeneous Boltzmann equation with measure initial datum, Poincare,2015>. Regarding the linearized Boltzmann equation, there exsits two results, Lerner, Morimoto, Pravda-Starov and Xu(Prof. Xu Chao-Jiang),<Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff,JDE,2014> and Glangetas, Li and Xu, <Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation, KRM, 2016>. However, the above result was completed with the measure initial data in L^2 spaces, the initial data in measure space is an open problem. Based on the above result, we prove that the global in time of the qualitative properties for the solutions to the Cauchy problem for the homogenous Boltzmann equation still holds for measure valued initial datum.