) \|_{H^{s} \times H^{s-1}} & = 0
\end{array}
\label{Eqn:ScatteringRes}$$

and

$$\begin{array}{ll}
\| I u \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(\mathbb{R})} & \lesssim_{A} 1
\end{array}
\label{Eqn:DecayEstp}$$

Here

$$K(T):= \left(
\begin{array}{ll}
\cos{(T \langle D \rangle)} & \frac{\sin{(T \langle D \rangle)}}{\langle D \rangle} \\
- \langle D \rangle \, \sin{(T \langle D \rangle)} & \cos{ ( T \langle D \rangle)}
\end{array}
\right)$$

and $ \left( 2(p-1)-, 2(p-1)+ \right) \in \left( a_{p}, \, 2(p-1) \right) \times  \left( 2(p-1), \, b_{p} \right) $ (for $a_{p}$ and $b_{p}$ to be determined) that satisfy $ \frac{1}{2(p-1)-} + \frac{3}{2(p-1)+} = \frac{3}{2} -s_{c} = \frac{2}{p-1} $ [^3]

\[Thm:Scat\]

1.  On can find an an explicit value of $\tilde{s}$ if one reads carefully the proof of Theorem \[Thm:Scat\]. Nevertheless we have decided not to do this in order to make the paper easier to read.

2.  $\tilde{s}$ grows to one as $A$ goes to $\infty$.

3.  The reader can check, after reading the proof of Theorem \[Thm:Scat\] that

    -   $a_{p}$ decreases and $b_{p}$ increases as $p$ grows

    -   $\lim_{p \rightarrow 3^{+}} (a_{p},b_{p})=\left( 2(p-1),2(p-1) \right)$.

We set some notation that appear throughout the paper.\
Let $\nabla$ denote the gradient operator. If $w$ is a smooth function then

$$\begin{array}{ll}
Z_{m,s}(J,w) & := \sup_{ \substack {(q,r)-m \\ 