Free-transport semigroup
= Free-transport semigroup
{title2=$U_tg(x,v)=g(x-tv,v)$}
For $1\leq p<\infty$, these operators are isometries on the phase-space <Lp space>, with $U_tU_s=U_{t+s}$. They form a <strongly continuous semigroup> for $t\geq0$ and extend to a group for all real times. Strong continuity follows by smooth compactly supported approximation and measure preservation. They preserve the <essential supremum> as well, but strong continuity is not claimed on the whole $L^\infty$ space.