I am trying to understand the motivation behind the main theorem of Keraani: See the top of page 356 and which is the motivation behind his Theorem 1.6

We consider $$i\partial_t u + \Delta u =0, u(x,0)=u_0 \in H^{1}(\mathbb R^3)$$ where $u:\mathbb R^{3+1} \to \mathbb C, $ $H^{1}(\mathbb R^3)= {f:\mathbb R^3 \to \mathbb C : \nabla f \in L^{2}(\mathbb R^3)} \}$. Put $\|f\|_{H^1}= \|\nabla f \|_{L^2}.$

By Strichartz estimate and Sobolov's inequality, we have

$$ \|e^{it \Delta}u_0\|_{L^{q}(\mathbb R_t, L^{r}(\mathbb R^3_{x}))} \leq C \|u_0\|_{H^1}$$

where a pair $(q,r)$ is $H^1-$ admissible, that is, $\frac{2}{q}+\frac{3}{r}=\frac{3}{2}$, $r\in [6, \infty).$

My Questions: When can we say the above estimte is not compact, and how to justify this? (What does this mean? How one should interprate, when authors says the estimate is not compact.) More specifically,

Let $\{x_n\}_{n\in \mathbb N}$ be a sequence of $\mathbb R^3$ and $\{t_n\}_{n\in \mathbb N}$ be a sequence of $\mathbb R$ both going to infinity. Also, let $\{h_n\}_{n\in \mathbb N}$ be a sequence of $\mathbb R^{*}_{+}$ going to 0.

Put $\tau_{x_0}\phi (x)=\phi(x-x_0), S_{h}\phi(x)=\frac{1}{\sqrt{h}} \phi (\frac{x}{h}), R_{t_0}\phi (x)= e^{it_0 \Delta} \phi (x), $ where $x, x_0 \in \mathbb R^3, t_0\in \mathbb R, h>0.$

(1) Why for every fixed function $\phi \in H^{1}$, the sequences $\tau_{x_n}\phi (x), S_{h_{n}}\phi(x), R_{t_n}\phi (x) $ converge weakly to 0 ?

(2) Why for every $H^1-$admissible pair $(q, r)$ the $L^{q}L^{r}$ norms of these sequences are equal to $\|e^{it\Delta} \phi \|_{L^{q}(\mathbb R, L^{r}(\mathbb R^3))}$ for every $n$? (These sequnces $\tau_{x_n}\phi (x)$ are functions on $\mathbb R^3,$ how one can take $L^{q}L^{r}$ for these sequnces, as my understnding is that $L^{q}L^{r}$ norm is meant for functions on $\mathbb R \times \mathbb R^3$)

(3) Why one can use (1) and (2) to conlude ``These sequences are not relatively compact in $L^{q}(\mathbb R, L^{r}(\mathbb R^3))$ "? (And what is a meaning by the sequence is not relatively compact in topological space)