Gevrey and Gelfand–Shilov smoothing effect for non-cutoff Boltzmann equation with hard potentials

In this paper, we study the Gevrey and Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff in hard potentials case or Maxwellian molecules case. More explicitly, for the initial data sufficiently small in Hx2Lv2, we prove that the solutions at any positive time are analytic and Gelfand-Shilov regularity G1(Rx3)×S11(Rv3) for strong angular singularity and in Gevrey and Gelfand-Shilov class G1/2s(Rx3)×S1/2s1/2s(Rv3) for mild angular singularity.