  The Boltzmann equation without an angular cutoff is considered when the
initial data is a small perturbation of a global Maxwellian with an algebraic
decay in the velocity variable. A well-posedness theory in the perturbative
framework is obtained for both mild and strong angular singularities by
combining three ingredients: the moment propagation, the spectral gap of the
linearized operator, and the regularizing effect of the linearized operator
when the initial data is in a Sobolev space with a negative index. A carefully
designed pseudo-differential operator plays an central role in capturing the
regularizing effect. Moreover, some intrinsic symmetry with respect to the
collision operator and an intrinsic functional in the coercivity estimate are
essentially used in the commutator estimates for the collision operator with
velocity weights.
