  For the spatially homogeneous Boltzmann equation with hard po- tentials and
Grad's cutoff (e.g. hard spheres), we give quantitative estimates of
exponential convergence to equilibrium, and we show that the rate of
exponential decay is governed by the spectral gap for the linearized equation,
on which we provide a lower bound. Our approach is based on establishing
spectral gap-like estimates valid near the equilibrium, and then connecting the
latter to the quantitative nonlinear theory. This leads us to an explicit study
of the linearized Boltzmann collision operator in functional spaces larger than
the usual linearization setting.
