  In this paper, we investigate the problems of Cauchy theory and exponential
stability for the inhomogeneous Boltzmann equation without angular cut-off. We
only deal with the physical case of hard potentials type interactions (with a
moderate angular singularity). We prove a result of existence and uniqueness of
solutions in a close-to-equilibrium regime for this equation in weighted
Sobolev spaces with a polynomial weight, contrary to previous works on the
subject, all developed with a weight prescribed by the equilibrium. It is the
first result in this more physically relevant frameworkfor this equation.
Moreover, we prove an exponential stability for such a solution, with a rate as
close as we want to the optimal rate given by the semigroup decay of the
linearized equation. Let us highlight the fact that a key point of the
development of our Cauchy theory is the proof of new regularization estimates
in short time for the linearized operator thanks to pseudo-differential tools.
