  We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian
molecules, under the perturbation frame work to global equilibrium. We show
that if $H^r_x(L^2_v), r >3/2$ norm of the initial perturbation is small
enough, then the Cauchy problem of the nonlinear Landau equation admits a
unique global solution which becomes analytic with respect to both position $x$
and velocity $v$ variables for any time $t>0$. This is the first result of
analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic
equation. The method used here is microlocal analysis and energy estimates. The
key point is adopting a time integral weight associated with the kinetic
transport operator.
