  In this paper, we continue our study of the Boltzmann equation by use of
tools originating from the analysis of dispersive equations in quantum
dynamics. Specifically, we focus on properties of solutions to the Boltzmann
equation with collision kernel equal to a constant in the spatial domain
$\mathbb{R}^d$, $d\geq 2$, which we use as a model in this paper. Local
well-posedness for this equation has been proven using the Wigner transform
when $\left< v \right>^\beta f_0 \in L^2_v H^\alpha_x$ for $\min (\alpha,\beta)
> \frac{d-1}{2}$. We prove that if $\alpha,\beta$ are large enough, then it is
possible to propagate moments in $x$ and derivatives in $v$ (for instance,
$\left< x \right>^k \left< \nabla_v \right>^\ell f \in L^\infty_T L^2_{x,v}$ if
$f_0$ is nice enough). The mechanism is an exchange of regularity in return for
moments of the (inverse) Wigner transform of $f$. We also prove a persistence
of regularity result for the scale of Sobolev spaces $H^{\alpha,\beta}$; and,
continuity of the solution map in $H^{\alpha,\beta}$. Altogether, these results
allow us to conclude non-negativity of solutions, conservation of energy, and
the $H$-theorem for sufficiently regular solutions constructed via the Wigner
transform. Non-negativity in particular is proven to hold in $H^{\alpha,\beta}$
for any $\alpha,\beta > \frac{d-1}{2}$, without any additional regularity or
decay assumptions.
