  We prove the existence and uniqueness of global solutions to the Boltzmann
equation with non-cutoff soft potentials in the whole space when the initial
data is a small perturbation of a Maxwellian with polynomial decay in velocity.
Our method is based in the decomposition of the desired solution into two
parts: one with polynomial decay in velocity satisfying the Boltzmann equation
with only a dissipative part of the linearized operator ; the other with
Gaussian decay in velocity verifying the Boltzmann equation with a coupling
term.
