  In this paper we study the Boltzmann equation near global Maxwellians in the
$d$-dimensional whole space. A unique global-in-time mild solution to the
Cauchy problem of the equation is established in a Chemin-Lerner type space
with respect to the phase variable $(x,v)$. Both hard and soft potentials with
angular cutoff are considered. The new function space for global well-posedness
is introduced to essentially treat the case of soft potentials, and the key
point is that the velocity variable is taken in the weighted supremum norm, and
the space variable is in the $s$-order Besov space with $s\geq d/2$ including
the spatially critical regularity. The proof is based on the time-decay
properties of solutions to the linearized equation together with the bootstrap
argument. Particularly, the linear analysis in case of hard potentials is due
to the semigroup theory, where the extra time-decay plays a role in coping with
initial data in $L^2$ with respect to space variable. In case of soft
potentials, for the time-decay of linear equations we borrow the results basing
on the pure energy method and further extend them to those in $L^\infty$
framework through the technique of $L^2$--$L^\infty$ interplay. In contrast to
hard potentials, $L^1$ integrability in $x$ of initial data is necessary for
soft potentials in order to obtain global solutions to the nonlinear Cauchy
problem.
