  For the Maxwellian molecules or hard potentials case, we verify the smoothing
effect for the spatially inhomogeneous Boltzmann equation without angular
cutoff. Given initial data with low regularity, we prove its solutions at any
positive time are analytic for strong angular singularity, and in Gevrey class
with optimal index for mild angular singularity. To overcome the degeneracy in
the spatial variable, a family of well-chosen vector fields with time-dependent
coefficients will play a crucial role, and the sharp regularization effect of
weak solutions relies on a quantitative estimate on directional derivatives in
these vector fields.
