Upper bounds for the function solution of the homogeneous 2D 2 D Boltzmann equation with hard potential

We deal with ft(dv), the solution of the homogeneous 2D Boltzmann equation without cutoff. The initial condition f0(dv) may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659.704): ft(dv)=ft(v)dv for every t>0. The aim of this paper is to give upper bounds for ft(v), the most significant one being of type ft(v).Ct..e.|v|. for some .,.>0.