\end{array}$$

and $N>>1$ is a dyadic number playing the role of a parameter to be chosen. We shall abuse the notation and write $m (|\xi|)$ for $m(\xi)$, thus for instance $m(N)=1$. If $w$ is a function and $B(x_{0},R):= \{ x \in \mathbb{R}, \, |x-x_{0}| \leq R  \}$ then

$$\begin{array}{ll}
E(Iw) & := \frac{1}{2} \int_{\mathbb{R}^{3}} |\partial_{t} I w (x)|^{2} \, dx + \frac{1}{2} \int_{\mathbb{R}^{3}} |I w(x)|^{2} \, dx \\
& + \frac{1}{2} \int_{\mathbb{R}^{3}} |\nabla I w (x)|^{2} \, dx + \frac{1}{p+1} \int_{\mathbb{R}^{3}} | I w(x)|^{p+1} \, dx
\end{array}
\label{Eqn:DefMolNrj}$$

$$\begin{array}{ll}
E(Iw,B(x_{0},R)) & := \frac{1}{2} \int_{B(x_{0},R)} |\partial_{t} Iw (x)|^{2} \, dx + \frac{1}{2} \int_{B(x_{0},R)} |I w(x)|^{2} \, dx    \\
& + \frac{1}{2} \int_{B(x_{0},R)} |\nabla I w (x)|^{2} \, dx + \frac{1}{p+1} \int_{B(x_{0},R)} |Iw(x)|^{p+1} \, dx
\end{array}$$

and

$$\begin{array}{l}
E_{c}(Iw) := \frac{1}{2} \int_{\mathbb{R}^{3}} |\partial_{t} I w (x)|^{2} \, dx + \frac{1}{2} \int_{\mathbb{R}^{2}} |Iw(x)|^{2} \, dx +
\frac{1}{2} \int_{\mathbb{R}^{3}} |\nabla I w (x)|^{2} \, dx
\end{array}$$

Some estimates that we establish throughout the paper require a Paley-Littlewood decomposition. We set it up now. Let $\phi(\xi)$ be a real, radial, nonincreasing function that is equal to $1$ on the unit ball $\left\{ \xi \in \mathbb{R}^{3}: \, |\xi| \leq 1 \right\}$ and that that is supported 