  We consider the spatially inhomogeneous non-cutoff Boltzmann equation with
moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e.
with $\gamma+2s\in(0,2]$ on the whole space $\mathbb{R}^3$. We prove that if
the initial data $f_{\text{in}}$ are close to the vacuum solution
$f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$
remains regular globally in time and approaches a solution to a linear
transport equation. Our proof uses $L^2$ estimates and we prove a multitude of
new estimates involving the Boltzmann kernel without angular cut-off. Moreover,
we rely on various previous works including those of Gressman--Strain,
Henderson--Snelson--Tarfulea and Silverstre. From the point of view of the long
time behavior we treat the Boltzmann collisional operator perturbatively. Thus
an important challenge of this problem is to exploit the dispersive properties
of the transport operator to prove integrable time decay of the collisional
operator. This requires the most care and to successfully overcome this
difficulty we draw inspiration from Luk's work [Stability of vacuum for the
Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11] and
that of Smulevici [Small data solutions of the Vlasov-Poisson system and the
vector field method, Ann. PDE, 2(2):Art. 11, 55, 2016]. In particular, to get
at least integrable time decay we need to consolidate the decay coming from the
space-time weights and the decay coming from commuting vector fields.
