  In this paper we consider the Cauchy problem on the angular cutoff Boltzmann
equation near global Maxwillians for soft potentials either in the whole space
or in the torus. We establish the existence of global unique mild solutions in
the space $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ with polynomial velocity weights
for suitably large $p\leq \infty$, whenever for the initial perturbation the
weighted $L^p_vL^{\infty}_x$ norm can be arbitrarily large but the
$L^1_xL^\infty_v$ norm and the defect mass, energy and entropy are sufficiently
small. The proof is based on the local in time existence as well as the uniform
a priori estimates via an interplay in $L^p_vL^{\infty}_{T}L^{\infty}_{x}$ and
$L^{\infty}_{T}L^{\infty}_{x}L^1_v$.
