Lorentz norms in some of the estimates above is not a great obstacle to applications. For example, if we restrict to finite time intervals $[0, T]$, we have the continuous embeddings $$\begin{aligned}
    L^{q,r}([0, T]) \hookrightarrow L^{p}([0, T]),   \quad q > p, \; 1\leq q,p,r\leq \infty,\\
    L^{p}([0, T])   \hookrightarrow L^{q,r}([0, T]), \quad p > q, \; 1\leq q,p,r\leq \infty,\end{aligned}$$ see [@BS p. 217]. Let us recall also the continuous embedding $L^{q,c}({\mathbb R}^n) \hookrightarrow L^q({\mathbb R}^n)$ whenever $q\geq c$. For example, let $(\infty, r,p)$ and $\tript$ be such that estimate holds and let $1\leq \widetilde Q < \tilde q$. Then we have the local inhomogeneous estimate $$\begin{aligned}
      \normQRPloc{W(t)F}{\infty}{q}{r}{[0,T]} \lesssim_T \normQRPloc{F}{\widetilde Q'}{\tilde r'}{\tilde p'}{[0,T]}.\end{aligned}$$

Preliminaries
=============

The kinetic transport propagator $U(t)$ enjoys the estimate $$\label{eq: disp est 2}
  {\left\VertU(t)f\right\Vert}_{L^{\infty}_xL^{1}_v} \leq \frac 1 {{\left\vertt\right\vert}^{n}}{\left\Vertf\right\Vert}_{L^{1}_xL^{\infty}_v},$$ for all $f \in {L^{1}_xL^{\infty}_v}$.

$$\begin{split}
  \int_{{\mathbb R}^n} {\left\vertU(t)f\right\vert} dv = \int_{{\mathbb R}^n} {\left\vertf(x-tv,v)\right\vert}dv \leq \int_{{\mathbb R}^n} \sup_{y \in {\mathbb R}^n} {\left\vertf(x-tv,y)\right\vert} dv \\
  \leq \rp {{\left\vertt\right\vert}^n} \int_{{\mathbb R}^n} \sup_{y \in {\mathbb R}^n} {\left\vertf(z,y)\right\vert} dz = \rp {{\left\vertt\right\vert}^n} \normBC{f}{1}{\infty}.
 \end{split}$$

The kinetic transport propagator $U(t)$ enjoys the estimate $$\label{eq: tran est}
  \normQRP{U(t)f}{\infty}{a}{a} \leq \normA{f}{a}, \quad 0<a\leq \infty,$$ for all $f \in L^{a}_{x,v}$.

Trivial.

\[cor: dec est\] The kinetic transport propagator 