wave \, adm} } \| \partial_{t} \langle D \rangle^{-m} I w \|_{L_{t}^{q}(J) L_{x}^{r}} + \|
\langle D \rangle^{1-m} I w \|_{L_{t}^{q}(J) L_{x}^{r}}
\end{array}$$

and

$$\begin{array}{ll}
Z(J,w) & := \sup_{m \in [0,1]} Z_{m,s}(J,w)
\end{array}$$

We write $F(w)$ for the following function

$$\begin{array}{ll}
F(w) & := |w|^{p-1} w
\end{array}$$

If $w$ is a solution of (\[Eqn:NlkgWdat\]) with data $(w_{0},w_{1})$ then $w(t)=w^{lin}(t)+ w^{nlin}(t)$ with

$$\begin{array}{ll}
w^{lin}(t) & := \cos{(t \langle D \rangle)} w_{0} + \frac{ \sin{(t \langle D \rangle)}}{ \langle D \rangle} w_{1}
\end{array}$$

and

$$\begin{array}{ll}
w^{nlin}(t) & := - \int_{0}^{t} \frac{ \sin{(t-s) \langle D \rangle}} {\langle D \rangle} F(w) \, ds
\end{array}$$

If $(C,Q) \in (\mathbb{R}^{+})^{2}$ then let $X_{C,Q}:=Y_{0}$ where $(Y_{i})_{ 0 \leq i \leq m}$ is an element of the class of sequences $(Z_{i})_{0 \leq i \leq m}$ satisfying

$$\left\{
\begin{array}{l}
Z_{i} := e^{C^{C} C^{C Z_{i+1}} A^{C} C^{A ^{C}}}   \\
Z_{m} << 1 \\
m \sim Q
\end{array}
\right.$$

Let $I$ be the following multiplier

$$\begin{array}{ll}
\widehat{If}(\xi) & := m(\xi) \hat{f}(\xi)
\end{array}$$

where $m(\xi): =  \eta \left( \frac{\xi}{N} \right)$, $\eta$ is a smooth, radial, nonincreasing in $|\xi|$ such that

$$\begin{array}{ll}
\eta (\xi) & :=  \left\{
\begin{array}{l}
1, \, |\xi| \leq 1 \\
\left( \frac{1}{|\xi|} \right)^{1-s}, \, |\xi| \geq 2
\end{array}
\right.
