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