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

We deal with $f\mathrm{\_}t(dv),$ the solution of the homogeneous $2D$ Boltzmannequation without cutoff. The initial condition $f\mathrm{\_}0(dv)$ may be anyprobability distribution (except a Dirac mass). However, for sufficiently hardpotentials, the semigroup has a regularization property (see \cite{[BF]}):$f\mathrm{\_}t(dv)=f\mathrm{\_}t(v)dv$ for every $t>0.$ The aim of this paper is to give upperbounds for $f\mathrm{\_}t(v),$ the most significant one being of type $f\mathrm{\_}t(v){\text{leqCt}}^{-\eta }{e}^{-{\mid v\mid }^{\lambda }}$ for some $\eta ,\lambda >0.$