Global mild solution with polynomial tail for the Boltzmann equation in the whole space

We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to construct global-in-time bounded mild solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. The full range of both hard and soft potentials with cutoff can be covered. The proof is based on a decomposition of the Boltzmann equation motivated by Caflisch that can capture the time-decay property of solutions in the way that the slow velocity-decay part decays in time much faster than the fast velocity-decay part. This provides the first result on the construction of slow-decaying solutions when the spatial domain is unbounded.