Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff

In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^{\mathrm{\infty }}$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto’s recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $\gamma$ satisfying $\gamma + 2s \in (-1,1)$,$s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $\mathrm{\Phi }(v)=\mathrm{\mid }v{\mathrm{\mid }}^{\gamma }$ instead of $\text{la}v{\text{ra}}^{\gamma }$ as Morimoto did before.