\end{array}
\label{Eqn:BdRi}$$

with

$$\begin{array}{ll}
R_{1}(J,w) : = \int_{J} \int_{\mathbb{R}^{3}} \frac{\nabla Iw(t,x).x}{|x|} \left( F(Iw) - I F(w) \right) \, dx dt
\end{array}$$

and

$$\begin{array}{ll}
R_{2}(J,w) := \int_{J} \int_{\mathbb{R}^{3}} \frac{Iw(t,x)}{|x|} \left( F(Iw) - I F(w) \right) \, dx dt
\end{array}$$

\[Prop:EstInt\]

[**“Almost Morawetz-Strauss Estimate”** [@triroyradnlkg]]{} Let $w$ be a solution of (\[Eqn:NlkgWdat\]). Let $x_{0} \in \mathbb{R}^{3}$ and let $T \geq 0$. Then

$$\begin{array}{ll}
\int_{0}^{T} \int_{\mathbb{R}^{3}} \frac{|Iw(t,x)|^{p+1}}{|x-x_{0}|} \, dx dt & \lesssim \sup_{t \in [0,  T]} E(Iw(t)) + R_{1}([0, \, T],w) +
R_{2}([0, \, T],w)
\end{array}
\label{Eqn:AlmMorStrEst}$$

\[Prop:AlmMor\]

Now we explain the main ideas of this paper.

In [@nakanash1; @nakanash2], the author found for all dimension $n$ an upper bound of the $L_{t}^{q} L_{x}^{r}$ norm of the solution of (\[Eqn:NlkgWdat\]) with data in the energy class (i.e $(u_{0},u_{1}) \in H^{1} \times L^{2}$) by a tower-exponential type bound of the energy and for some $(q:=q(n),r:=r(n))$ only depending on the dimension $n$: this decay estimate was the preliminary step to prove scattering. A natural question is\
*Is it possible to prove decay estimates of this form (or a modified form) in the subcritical case and for rough data (i.e for data $
(u_{0},u_{1}) \in H^{s}
\times H^{s-1}, \, s_{c} < s < 1 )$ ?*\
