  It is known that the singularity in the non-cutoff cross-section of the
Boltzmann equation leads to the gain of regularity and gain of weight in the
velocity variable. By defining and analyzing a non-isotropy norm which
precisely captures the dissipation in the linearized collision operator, we
first give a new and precise coercivity estimate for the non-cutoff Boltzmann
equation for general physical cross sections. Then the Cauchy problem for the
Boltzmann equation is considered in the framework of small perturbation of an
equilibrium state. In this part, for the soft potential case in the sense that
there is no positive power gain of weight in the coercivity estimate on the
linearized operator, we derive some new functional estimates on the nonlinear
collision operator. Together with the coercivity estimates, we prove the global
existence of classical solutions for the Boltzmann equation in weighted Sobolev
spaces.
