$U(t)$ enjoys the estimate $$\label{DispEst}
   {\left\VertU(t)f\right\Vert}_{L^r_xL^p_v} \leq \frac 1 {{\left\vertt\right\vert}^{n\left(\frac 1 p - \frac 1 r\right)}}{\left\Vertf\right\Vert}_{L^p_xL^r_v}, \quad 1 \leq p \leq r \leq \infty,$$ for all $f \in {L^{p}_xL^{r}_v}$.

Complex interpolation between the dispersive estimate (\[eq: disp est\]) and the two transport estimates (\[eq: tran est\]) with $a=1$ and $a=\infty$.

\[lem: adjoint\] The formal adjoint to $U(t)$ is the operator $U(t)^*=U(-t)$.

We denote by $\langle \cdot, \cdot \rangle$ the scalar product on $L^2({\mathbb R}^{2n})$. Thus, $$\begin{aligned}
\langle U(t)f,g\rangle &= \Int f(x-tv,v) \overline{g(x,v)}dxdv\\
          &=\Int f(x,v) \overline{g(y+tv,v)}dydv=\langle f,U(-t)g\rangle,
$$ where we have made the substitution $y=x-tv$.

\[lem: scaling\] The evolution operators $U(t)$ and $W(t)$ enjoy the following scaling properties $$\begin{aligned}
U(t)f_\lambda = f\left({x}/{\lambda} - {t v}/{\lambda} , v\right)= U(t/\lambda, x/\lambda, v)f,\\
\text{where  } f_\lambda(x,v) = f\left(x/{\lambda}, v\right),\\
U(t)f_\lambda = f\left(x / {\lambda} - {t v}/{\lambda} , {v}/{\lambda}\right)= U(t, x/\lambda, v/\lambda)f,\\
\text{where  } f_\lambda(x,v) = f\left(x/{\lambda}, v/\lambda \right),\\
W(t)F_\lambda = \lambda \int_{0}^{t / \lambda}F\left(s, x / {\lambda} - \left(t / \lambda - s \right) v, v\right)ds =  \lambda W\left(t/ \lambda,x/ \lambda,v\right)F,\\
\text{where  } F_\lambda(t,x,v) = F(t/\lambda, x/\lambda, v),\\
W(t)F_\lambda = \int_{0}^{t}F\left(s, x / {\lambda} - (t-s) v / {\lambda} , v / \lambda \right)ds= W(t,x/\lambda,v/\lambda)F,\\
\text{where  } F_\lambda(t,x,v) = F\left(t, x/ \lambda, v/ 