Large amplitude solutions in $L^p_vL^\infty_TL^\infty_x$ to the Boltzmann equation for soft potentials

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 $L^{\infty}_{T}L^{\infty}_{x}L^1_v$.