f ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d\;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$ then there is a constant $C$ (not depending on $t \in {{{\bf}R}}_+, { \varepsilon}>0$) such that 