L_{x}^{r}$ spaces” [@triroyradnlkg]**]{} Assume that $w$ satisfies the following Klein-Gordon equation on $\mathbb{R}^{3}$

$$\left\{
\begin{array}{ccl}
\partial_{tt} w  - \Delta w + w & = & Q \\
w(0,x)& = & w_{0}(x)    \\
\partial_{t} w(0,x) & = & w_{1}(x)
\end{array}
\right. \label{Eqn:KlGH}$$

Let $T \geq 0$. Then

$$\begin{array}{l}
\| w \|_{L_{t}^{q} L_{x}^{r} ([0,T])} + \| \partial_{t} \langle D \rangle^{-1} w \|_{L_{t}^{q} L_{x}^{q} ([0,T]) } +  \| w \|_{L_{t}^{\infty}
H^{m} [0,T] } +  \| \partial_{t} w \|_{L_{t}^{\infty} H^{m-1} [0, T]} \\ \\
\lesssim \| w_{0} \|_{H^{m}} + \| w_{1} \|_{H^{m-1}} + \| Q \|_{L_{t}^{\tilde{q}} L_{x}^{\tilde{r}} ([0,T])}
\end{array}
\label{Eqn:StrNlkg}$$

under the following assumptions

-   $(q,r)$ is $m$- wave admissible, i.e $ (q,r) \in (2, \infty] \times [2, \infty) $ and $ \frac{1}{q} + \frac{1}{r} \leq \frac{1}{2} $ and it obeys the following constraint $\frac{1}{q} + \frac{3}{r}  =\frac{3}{2} -m $

-   $(q,r)  \neq \left( 2, \infty \right) $

-   $(\tilde{q},\tilde{r})$ satisfies $ \frac{1}{\tilde{q}} + \frac{1}{q}=1 $, $\frac{1}{\tilde{r}} + \frac{1}{r}=1$ and $\frac{1}{\tilde{q}} + \frac{3}{\tilde{r}} -2    = \frac{1}{q} + \frac{3}{r} $.

\[prop:StrEstLtqLxr\]

Next we recall some propositions

[“**Mollified energy at time $0$ is bounded by $N^{2(1-s)}$**” [@triroyradnlkg]]{} Let $w$ be a solution of (\[Eqn:NlkgWdat\]). Then there exists a constant $C_{E} >0$ such that

$$\begin{array}{ll}
E(Iw_{0}) & 