  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
$f\_{t}(v)\leqCt^{-\eta}e^{-\left\vert v\right\vert ^{\lambda}}$ for some
$\eta,\lambda>0.$
