A_{k,i,m} := \int_{y \geq 1} \frac{|\py^m Z|^2 |\py^{i - m}Z|^2 |\py^{k - i}\psi|^2}{y^{4\Bbbk - 2k - 2}} \lesssim b_1^{2L + 1 + 2(1 - \delta)(1 + \eta)},
\end{equation}

which concludes the proof of (<ref>).

To prove (<ref>), we distinguish in 3 cases:

- The initial case: $k = 0$. Since $0 \leq m \leq i \leq k$, then $k = i = m = 0$. Although this is the simplest case, it gives us a basic idea to handle other cases. From (<ref>), it is obvious to see that $|\psi|$ is uniformly bounded.  We write

\begin{align*}
A_{0,0,0}  = \int_{y \geq 1} \frac{|q|^4|\psi|^2}{y^{4\Bbbk + 2}} y^{d-1}dy  \lesssim \int_{1 \leq y \leq B_0} \frac{|q|^4}{y^{4\Bbbk + 3 - d}} dy + \int_{y \geq B_0} \frac{|q|^4}{y^{4\Bbbk + 3 - d}}dy.
\end{align*}

Using (<ref>), Definition <ref>, $b_1 \sim \frac{1}{s}$ and the fact that $d = 4\hbar + 2\gamma + 4\delta$ (see (<ref>)), we estimate

\begin{align*}
&\int_{1 \leq y \leq B_0} \frac{|q|^4}{y^{4\Bbbk + 3 - d}}dy\\
& \quad \lesssim \left\|\frac{y^{d-2}|q|^2}{y^{2(2\Bbbk - 1)}}\right\|_{L^\infty(y > 1)}\left\|\frac{y^{d-2}|q|^2}{y^{2(2\ell + 2\hbar + 3)}}\right\|_{L^\infty(y > 1)}\int_{1 \leq y \leq B_0}y^{4\ell + 5 - 4\delta - 2\gamma}dy\\
& \qquad \lesssim \Es_{2\Bbbk}\Es_{2(\ell + \hbar + 2)} B_0^{4\ell + 6 - 4\delta - 2\gamma}\\
& \quad \qquad \lesssim Kb_1^{2L + 2(1 - \delta)(1 + \eta)}b_1^{2(\ell + 1) + 2(1 - \delta) - K\eta}b_1^{-2\ell - 3 +  2\delta + \gamma}\\
&\qquad \qquad \lesssim K b_1^{2L + 2(1 - \delta)(1 + \eta)} b_1^{1 + \gamma - K\eta} \lesssim b_1^{2L + 1+ 2(1 - \delta)(1 + \eta)}.
\end{align*}

For the integral on the domain $y \geq B_0$, let us write

\begin{align*}
&\int_{y \geq B_0} \frac{|q|^4}{y^{4\Bbbk + 3 - d}}dy\\
&\quad \lesssim \left\|\frac{y^{d-2}|q|^2}{y^{2(2\Bbbk  - 2\ell - 1)}}\right\|_{L^\infty(y > 1)}\left\|\frac{y^{d-2}|q|^2}{y^{2(2\ell + 2\hbar + 1)}}\right\|_{L^\infty(y > 1)}\int_{y \geq B_0} \frac{dy}{y^{4\delta + 2\gamma - 1}}\\
& \qquad \lesssim \Es_{2(\Bbbk - \ell)}\Es_{2(\ell + \hbar + 1)} B_0^{2 - 4\delta - 2\gamma}\\
& \quad \qquad \lesssim b_1^{2(\Bbbk - \ell - \hbar -1) + 2(1 - \delta) - K\eta}b_1^{2\ell + 2(1 - \delta) - K\eta}b_1^{2\delta + \gamma -1}\\
& \qquad \qquad \lesssim b_1^{2L + 2(1 - \delta)(1 + \eta)} b_1^{1 + \gamma - (K + 2(1 - \delta))\eta} \lesssim b_1^{2L + 1+ 2(1 - \delta)(1 + \eta)}.
\end{align*}

This concludes the proof of (<ref>) when $k = i = m = 0$.

- Case II: $k \geq 1$ and $k = i$. We first use the Leibniz rule to write

\begin{equation}\label{eq:pylZexp}
\forall l \in \mathbb{N}, \quad |\py^l Z|^2 \lesssim \sum_{j = 0}^l \frac{|\py^j q|^2}{y^{2 + 2l - 2j}},
\end{equation}

from which,

\begin{align*}
A_{k,k,m} &\lesssim \sum_{j = 0}^m \sum_{l = 0}^{k-m} \int_{y \geq 1} \frac{|\py^j q|^2 |\py^l q|^2}{y^{4 \Bbbk - 2j - 2l + 2}}y^{d-1}dy.
\end{align*}

We claim that for all $(j,l) \in \mathbb{N}^2$ and $1 \leq j + l \leq 2\Bbbk -1$,

\begin{equation}\label{est:Bjl}
B_{j,l,0}:= \int_{y \geq 1} \frac{|\py^j q|^2 |\py^l q|^2}{y^{4 \Bbbk - 2j - 2l + 2}}y^{d-1}dy \lesssim b_1^{2L +1 + 2(1 - \delta)(1 + \eta) + \frac{(\gamma - 1)}{2}},
\end{equation}

which immediately follows (<ref>) for the case when $k = i$.

To prove (<ref>), we proceed as for the case $k = 0$ by splitting the integral in two parts as follows:

\begin{align*}
B_{j,l,0} &= \int_{1 \leq y \leq B_0} \frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{4\Bbbk - 2j - 2l + 4\hbar + 6}}y^{7 - 4\delta - 2\gamma}dy\\
&\quad  + \int_{y \geq B_0} \frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{4\Bbbk - 2j - 2l + 4\hbar}}\frac{dy}{y^{4\delta + 2\gamma - 1}}\\
&\qquad \lesssim \left\|\frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{4\Bbbk - 2j - 2l + 4\hbar + 6}} \right\|_{L^\infty(y \geq 1)}b_1^{2\delta + \gamma - 4}\\
&\qquad \quad + \left\|\frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{4\Bbbk - 2j - 2l + 4\hbar}} \right\|_{L^\infty(y \geq 1)}b_1^{2\delta + \gamma - 1}\\
& \qquad \qquad = \left\|\frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{2J_1 - 2j + 2J_2 - 2l}} \right\|_{L^\infty(y \geq 1)}b_1^{2\delta + \gamma - 4}\\
&\qquad \qquad \quad + \left\|\frac{\big(y^{d-2}|\py^j q|^2\big)\big(y^{d-2}|\py^l q|^2\big)}{y^{2J_3 - 2j + 2J_4 - 2l}} \right\|_{L^\infty(y \geq 1)}b_1^{2\delta + \gamma - 1}\\
& \qquad \qquad \qquad := B_{j,l,0,J_1, J_2} b_1^{2\delta + \gamma - 4} + B_{j,l,0, J_3, J_4}b_1^{2\delta + \gamma - 1},
\end{align*}

where $J_n (n = 1,2,3,4)$ satisfy

$$J_1 + J_2 = 2\Bbbk + 2\hbar + 3, \quad J_3 + J_4 = 2\Bbbk + 2\hbar.$$

We now estimate $B_{j,l,0,J_1, J_2}$.

- If $l$ is even, we take

$$J_2 = \left\{ \begin{array}{ll}
l + 2 &\quad \text{if}\quad l \leq 2\Bbbk - 4,\\
l &\quad \text{if}\quad l = 2\Bbbk - 2.
\end{array} \right.$$

This gives

$$2\hbar + 4 \leq J_2 \leq 2\Bbbk - 2, \quad 2\hbar + 5 \leq J_1 = 2\Bbbk + 2\hbar + 3 - J_2 \leq 2\Bbbk - 1.$$

Using (<ref>), we have the estimate

\begin{align*}
B_{j,l,0,J_1, J_2}& \lesssim \left\|\frac{y^{d-2}|\py^j q|^2}{y^{2J_1 - 2j}}\right\|_{L^\infty(y \geq 1)} \left\|\frac{y^{d-2}|\py^l q|^2}{y^{2J_2 - 2l}} \right\|_{L^\infty(y \geq 1)}\\
&\quad  \lesssim \Es_{J_1 + 1} \sqrt{\Es_{J_2} \Es_{J_2 + 2}}.
\end{align*}

- If $l$ is odd, we simply take $J_2 = l + 1$ which gives

$$2\hbar + 4 \leq J_2 \leq 2\Bbbk - 2, \quad 2\hbar + 5 \leq J_1 \leq 2\Bbbk - 1.$$

\begin{align*}
B_{j,l,0,J_1, J_2}&\lesssim \Es_{J_1 + 1} \sqrt{\Es_{J_2} \Es_{J_2 + 2}}.
\end{align*}

Recall from Definition (<ref>) that for all even integer $m$ in the range $2\hbar + 4 \leq m \leq 2\Bbbk$,

\begin{equation}\label{est:Esmeven}
\Es_{m} \leq \left\{
\begin{array}{ll}
b_1^{\frac{\ell}{2\ell - \gamma}(2m - d)}&\quad \text{for}\quad 2\hbar + 4 \leq m \leq 2\hbar +2\ell.\\
b_1^{m - 2\hbar -2 + 2(1 -\delta) - K\eta}&\quad \text{for}\quad 2\hbar + 2\ell + 2 \leq m \leq 2\Bbbk.
\end{array}
\right.
\end{equation}

- If $J_1 + 1 \geq 2\hbar + 2\ell + 2$ and  $J_2 \geq 2\hbar + 2\ell + 2$, then

$$B_{j,l,0,J_1, J_2} \lesssim b_1^{J_1 + J_2 - 4\hbar -2 + 4(1 - \delta) - 2K\eta} \lesssim b_1^{2L + 2 + 4(1 - \delta) - K\eta}.$$

- If $J_1 + 1 \leq 2\hbar + 2\ell$, then $J_2 = 2\Bbbk + 2\hbar + 3 - J_1 \geq 2\Bbbk - 2\ell + 4 \geq 2\hbar + 2\ell + 2$ because $\Bbbk \gg \ell$. This follows

$$B_{j,l,0,J_1, J_2} \lesssim b_1^{\frac{\ell}{2\ell - \gamma}(2J_1 + 2 - d) + J_2 + 1 - 2(\hbar +1) + 2(1 - \delta) - K\eta} \lesssim b_1^{2L + 2 + 4(1 - \delta)  - K\eta}.$$

Hence, we obtain

$$B_{j,l,0,J_1, J_2} \lesssim b_1^{2L + 2 + 4(1 - \delta)  - K\eta} \quad \text{for}\quad J_1 + J_2 = 2\Bbbk + 2\hbar +3.$$

Similarly, one prove that

$$B_{j,l,0,J_3, J_4} \lesssim b_1^{2L - 1 + 4(1 - \delta) - K\eta} \quad \text{for} \quad J_3 + J_4 = 2\Bbbk + 2\hbar.$$

\begin{align*}
B_{j,l,0} & \lesssim b_1^{2L + 2 + 4(1 - \delta) - K\eta}b_1^{2\delta + \gamma - 4} + b_1^{2L -1 + 4(1 - \delta) - K\eta}b_1^{2\delta + \gamma - 1}\\
& \quad \lesssim b_1^{2L + 1 + 2(1 - \delta)(1 + \eta) + (\gamma-1) - (K + 2 -2\delta)\eta} \lesssim b_1^{2L + 1 + 2(1 - \delta)(1 + \eta) + \frac{(\gamma - 1)}{2}},
\end{align*}

for $\eta \leq \frac{\gamma - 1}{2(K + 2 - 2\delta)}$. This concludes the proof of (<ref>) as well as (<ref>) when $k = i$.

- Case III: $k \geq 1$ and $k - i \geq 1$. Let us write from (<ref>) and (<ref>),

\begin{equation}\label{eq:Akmi11}
A_{k,m,i} \lesssim \sum_{j = 0}^{m}\sum_{l = 0}^{i - m} \int_{y \geq 1} \frac{|\py^j q|^2 |\py^l q|^2}{y^{4\Bbbk - 2j - 2l + 2}} \frac{|\py^{k - i} \psi|^2}{y^{-2(k - i)}}.
\end{equation}

At this stage, we need to precise the decay of $|\py^n \psi|$ to archive the bound (<ref>). To do so, let us recall that $T_i$ is admissible of degree $(i,i)$ (see Lemma <ref>) and $S_i$ is homogeneous of degree $(i,i-1,i)$ (see Proposition <ref>). Together with (<ref>), we estimate

$$ \forall j \geq 1,\quad  |\py^j \tilde{Q}_b | \lesssim \frac{1}{y^{\gamma + j}} + \sum_{l = 1}^{2L + 2}\frac{b_1^l y^{2l}}{y^{\gamma + j}} \mathbf{1}_{\{y \leq 2B_1\}} \lesssim \frac{b_1^{-(2L+2)\eta}}{y^{\gamma + j}}.$$

Let $\tau \in [0,1]$ and $v_\tau = \tilde{Q}_b + \tau q$. We use the Faa di Bruno formula to write

\begin{align*}
\forall n \in \mathbb{N}, \quad |\py^n \psi|^2 &\lesssim \int_0^1 \sum_{m^* = n} |\partial_{v_\tau}^{m_1 + \cdots+m_n}\sin(v_\tau)|^2  \prod_{i = 1}^n |\py^i \tilde{Q}_b + \py^iq|^{2m_i} d\tau \nonumber\\
&\quad \lesssim \sum_{m^* = n}  \prod_{i = 1}^n \left(\frac{b_1^{-C(L)\eta}}{y^{2\gamma + 2i}} +  |\py^iq|^2\right)^{m_i}, \quad m^* = \sum_{i = 1}^n im_i.
\end{align*}

For $1 \leq y \leq B_0$, we use (<ref>) to estimate

\begin{equation*}
|\py^iq|^2 = y^{4\Bbbk - 2i -2}\left|\frac{\py^i q}{y^{2\Bbbk - i - 1}}\right|^2 \leq B_0^{4\Bbbk - 2i - d}\Es_{2\Bbbk} \leq b_1^{-C(K)\eta + i + \gamma} \leq \frac{b_1^{-C(K)\eta}}{y^{2\gamma + 2i}},
\end{equation*}

from which, we have

\begin{equation}\label{est:psin1B0}
|\py^n \psi|^2 \lesssim \sum_{m^* = n}  \prod_{i = 1}^n \left(\frac{b_1^{-C(L)\eta}}{y^{2\gamma + 2i}} +  \frac{b_1^{-C(K)\eta}}{y^{2\gamma + 2i}}\right)^{m_i} \lesssim \frac{b_1^{-C(K, L)\eta}}{y^{2\gamma + 2n}}, \quad \forall 1 \leq y \leq B_0.
\end{equation}

For $y \geq B_0$, we use again (<ref>) to write for all $1 \leq n \leq 2\Bbbk - 1$,

\begin{align}
|\py^n \psi|^2 &\lesssim \sum_{m^* = n}  \prod_{i = 1}^{2\hbar + 2\ell + 1}\left(\frac{b_1^{-C(L)\eta}}{y^{2\gamma + 2i}} +  y^{4\hbar + 4\ell + 2 - 2i}\left|\frac{\py^i q}{y^{2\hbar + 2\ell + 1 - i}}\right|^2\right)^{m_i}\nonumber\\
&\quad \times \prod^{n}_{i = 2\hbar + 2\ell + 1}\left(\frac{b_1^{-C(L)\eta}}{y^{2\gamma + 2i}} +  \left|\py^i q\right|^2\right)^{m_i}\nonumber\\
& \lesssim \sum_{m^* = n}  \prod_{i = 1}^{2\hbar + 2\ell + 1} \left(b_1^{-C(L)\eta + \gamma + i} + b_1^{-K\eta + i+ \gamma}b_1^{2\ell + 2(1 - \delta)}y^{4\ell + 4(1 - \delta)} \right)^{m_i}\nonumber\\
& \quad \times \prod^{n}_{i = 2\hbar + 2\ell + 1}\left(b_1^{-C(L)\eta + \gamma + i} + b_1^{-K\eta + \gamma + i}\right)^{m_i}\nonumber\\
&\lesssim b_1^{-C(L,K)\eta + n + \gamma \sum_{i = 1}^n m_i}\left(b_1y^2\right)^{(2\ell + 2(1 - \delta))\sum_{i = 1}^{2\hbar +2\ell + 1}m_i}, \quad \forall y \geq B_0. \label{est:psinB0g}
\end{align}

Injecting (<ref>) and (<ref>) into (<ref>), we arrive at

\begin{align*}
A_{k,i,m} &\lesssim b_1^{-C\eta}\sum_{j = 0}^{m}\sum_{l = 0}^{i - m} \left(\int_{1 \leq y \leq B_0} \frac{|\py^j q|^2 |\py^l q|^2}{y^{4\Bbbk - 2j - 2l + 2 + 2\gamma}} +b_1^\alpha \int_{ y \geq B_0} \frac{|\py^j q|^2 |\py^l q|^2}{y^{4\Bbbk - 2j - 2l + 2 - 2\alpha}}\right),
\end{align*}

where $\alpha = k - i + \big(2\ell + 2(1 - \delta)\big)\sum_{i = 1}^{2\hbar +2\ell + 1}m_i$. Arguing as for the proof of (<ref>), we end up with

\begin{align*}
A_{k,i,m} &\lesssim b_1^{-C\eta} \left(b_1^{2L + 1 + \gamma + 2(1 - \delta)(1 -\eta) + \frac{(\gamma - 1)}{2}} + b_1^{2L + 1 + 2(1 - \delta)(1 -\eta) + \frac{(\gamma - 1)}{2}}\right)\\
&\qquad \lesssim b_1^{2L + 1 + 2(1 - \delta)(1 -\eta)},
\end{align*}

for $\eta$ small enough. This finishes the proof of (<ref>) as well as (<ref>). Since the proof of (<ref>) follows exactly the same lines as for the proof of (<ref>), we omit its proof here.

Inserting (<ref>), (<ref>) and (<ref>) into (<ref>) and recalling from Definition (<ref>) that $\Es_{2\Bbbk} \leq K b_1^{2L + 2(1 - \delta)(1 + \eta)}$, we arrive at

\begin{align}
\frac{1}{2}\frac{d}{dt}\left\{ \frac{\Es_{2\Bbbk}}{\lambda^{4\Bbbk - d}}\Big[1 + \Oc(b_1)\Big] \right\}
&\lesssim \frac{b_1}{\lambda^{4\Bbbk - d +2}}\left(\frac{\Es_{2\Bbbk}}{M^{2\delta}} + b_1^{L + (1 - \delta)(1 + \eta)}\sqrt{\Es_{2\Bbbk}} + b_1^{2L + 2(1 - \delta)(1 + \eta)}\right)\nonumber\\
&\qquad + \frac{1}{\lambda^{4\Bbbk - d +2}}\int \Ls^\Bbbk q \Ls^\Bbbk(\partial_s \xi_L). \label{eq:EnerID4}
\end{align}

Step 8: Time oscillations. In this step, we want to find the contribution of the last term in (<ref>) to the estimate (<ref>). Let us write

\begin{align}
\frac{1}{\lambda^{4\Bbbk - d +2}}\int \Ls^\Bbbk q \Ls^\Bbbk(\partial_s \xi_L) &= \frac{d}{ds} \left\{\frac{1}{\lambda^{4\Bbbk - d +2}}\left[\int \Ls^\Bbbk q \Ls^\Bbbk \xi_L - \frac{1}{2}\int |\Ls^\Bbbk \xi_L|^2 \right]\right \}\nonumber\\
&\quad +\frac{4\Bbbk - d + 2}{\lambda^{4\Bbbk - d +2}} \frac{\lambda_s}{\lambda} \left[\int \Ls^\Bbbk q \Ls^\Bbbk \xi_L + \frac{1}{2}\int |\Ls^\Bbbk \xi_L|^2\right]\nonumber\\
&\qquad  - \frac{1}{\lambda^{4\Bbbk - d +2}}\int \Ls^\Bbbk(\ps q - \ps \xi_L)  \Ls^\Bbbk \xi_L.\label{id:Aextract}
\end{align}

From (<ref>) and (<ref>), we have

\begin{equation}\label{est:LkxiL2}
\int|\Ls^\Bbbk \xi_L|^2 \lesssim b_1^{2\eta(1 - \delta)}\Es_{2\Bbbk}.
\end{equation}

This follows

\begin{align*}
\left|\int \Ls^\Bbbk q \Ls^\Bbbk \xi_L \right| &\lesssim \left(\int |\Ls^\Bbbk q|^2\right)^\frac{1}{2}\left(\int |\Ls^\Bbbk \xi_L|^2\right)^\frac{1}{2}\\
&\quad \lesssim \sqrt{\Es_{2\Bbbk}} \, b_1^{-(1-\delta)}\sqrt{\Es_{2\Bbbk}}\, b_1^{(1-\delta)(1+\eta)} = b_1^{\eta(1 - \delta)}\Es_{2\Bbbk}.
\end{align*}

Since $\frac{dt}{ds} = \lambda^2$, we then write

\begin{equation}\label{est:A1}
\frac{d}{ds} \left\{\frac{1}{\lambda^{4\Bbbk - d +2}}\Big[\int \Ls^\Bbbk q \Ls^\Bbbk \xi_L - \frac{1}{2}\int |\Ls^\Bbbk \xi_L|^2\Big]\right\} = \frac{d}{dt}\left(\frac{\Es_{2\Bbbk}}{\lambda^{4\Bbbk - d}}\mathcal{O}(b_1^{\eta(1 - \delta)}) \right).
\end{equation}

Noting from (<ref>) that $\left|\frac{\lambda_s}{\lambda}\right| \lesssim b_1$, this gives

\begin{equation}\label{est:A2}
\left|\frac{\lambda_s}{\lambda} \left[\int \Ls^\Bbbk q \Ls^\Bbbk \xi_L + \frac{1}{2}\int |\Ls^\Bbbk \xi_L|^2\right]\right| \lesssim b_1 b_1^{\eta(1 - \delta)}\Es_{2\Bbbk}.
\end{equation}

For the last term in (<ref>), we use equation (<ref>) and the decomposition (<ref>) to write

\begin{align}
\int \Ls^\Bbbk(\ps q - \ps \xi_L)  \Ls^\Bbbk \xi_L &= \left[-\int \Ls^\Bbbk q \Ls^{\Bbbk + 1}\xi_L + \frac{\lambda_s}{\lambda}\int \Lambda q \Ls^{2\Bbbk}\xi_L\right] \nonumber\\
&+ \int \Ls^{\Bbbk}\big[-\tilde \Psi_b - \widetilde{Mod} + \Hc(q) + \Nc(q)\big] \Ls^{\Bbbk}\xi_L.\label{eq:A3exp}
\end{align}

Using (<ref>), the admissibility of $T_L$ and the fact that $\Ls^k T_i = 0$ if $i < k$, we estimate

\begin{align*}
\int |\Ls^{\Bbbk +1}\xi_L|^2 & \lesssim \left|\frac{\left<\Ls^Lq, \chi_{B_0}\Lambda Q\right>}{\left<\Lambda Q, \chi_{B_0}\Lambda Q\right>} \right|^2\int|\Ls^{\Bbbk + 1}\big[(1 - \chi_{B_1})T_L\big]| ^2\\
&\lesssim b_1^{-2(1 - \delta)}\Es_{2\Bbbk}\int_{y \geq B_1}y^{2(2L - \gamma - 2(\Bbbk + 1))} y^{d-1}dy\\
&\quad \lesssim b_1^{-2(1 - \delta)}\Es_{2\Bbbk}b_1^{(4 - 2\delta)(1 + \eta)} \lesssim b_1^2b_1^{2\eta(1 - \delta)}\Es_{2\Bbbk}.
\end{align*}

from which we obtain

\begin{equation*}
\left|\int \Ls^\Bbbk q \Ls^{\Bbbk + 1}\xi_L\right| \lesssim b_1 b_1^{\eta(1 - \delta)}\Es_{2\Bbbk},
\end{equation*}

Similarly, we have the estimate

\begin{align*}
\int (1 + y^{4\Bbbk})|\Ls^{2\Bbbk}\xi_L|^2 &\lesssim b_1^{-2(1 - \delta)}\Es_{2\Bbbk}\int_{y \geq B_1}y^{4\Bbbk}y^{2(2L - \gamma - 4\Bbbk)}y^{d-1}dy\lesssim b_1^{2\eta(1-\delta)}\Es_{2\Bbbk},
\end{align*}

hence, using (<ref>) and (<ref>), we get

\begin{equation*}
\left|\frac{\lambda_s}{\lambda}\int \Lambda q \Ls^{2\Bbbk}\xi_L \right| \lesssim b_1 \left(\int \frac{|\py q|^2}{1 + y^{4\Bbbk - 2}} \right)^\frac{1}{2}\left(\int (1 + y^{4\Bbbk})|\Ls^{2\Bbbk}\xi_L|^2  \right)^\frac{1}{2} \lesssim b_1 b_1^{\eta(1 - \delta)}\Es_{2\Bbbk}.
\end{equation*}

From (<ref>), (<ref>) and (<ref>), we have

\begin{align*}
\left|\int \Ls^\Bbbk (\tilde{\Psi}_b + \widetilde{Mod}) \Ls^\Bbbk \xi_L\right| &\lesssim \left(\int |\Ls^\Bbbk \xi_L|^2 \right)^\frac{1}{2}\left(\int|\Ls^\Bbbk (\tilde \Psi_b + \widetilde{Mod})|^2 \right)^\frac{1}{2}\\
&\quad \lesssim b_1b_1^{\eta(1 - \delta)}\Es_{2\Bbbk} + b_1 b_1^{L + (1 - \delta)(1 + \eta)}\sqrt{\Es_{2\Bbbk}}.
\end{align*}

In the same manner, we have the estimate

\begin{align*}
\int (1 + y^{4})|\Ls^{\Bbbk + 1}\xi_L|^2 &\lesssim b_1^{-2(1 - \delta)}\Es_{2\Bbbk}\int_{y \geq B_1}y^{4}y^{2(2L - \gamma - 2(\Bbbk+1))}y^{d-1}dy\lesssim b_1^{2\eta(1-\delta)}\Es_{2\Bbbk},
\end{align*}

from which together (<ref>) and (<ref>), we get the bound

\begin{align*}
\left|\int \Ls^{\Bbbk - 1} (\Hc(q) + \Nc(q)) \Ls^{\Bbbk + 1}\xi_L \right|& \lesssim \left(\int \frac{|\Ls^{\Bbbk - 1}(\Hc(q) + \Nc(q))|^2}{1 + y^4} \right)^\frac{1}{2}\left(\int (1 + y^4)|\Ls^{\Bbbk + 1}\xi_L|^2\right)^\frac{1}{2}\\
&\quad \lesssim b_1 b_1^{\eta(1 - \delta)}\Es_{2\Bbbk} + b_1 b_1^{L + (1- \delta)(1 + \eta)}\sqrt{\Es_{2\Bbbk}}.
\end{align*}

Collecting these final bounds into (<ref>) yields

\begin{equation}\label{est:A3}
\left|\int \Ls^\Bbbk(\ps q - \ps \xi_L)  \Ls^\Bbbk \xi_L\right| \lesssim b_1 b_1^{\eta(1 - \delta)}\Es_{2\Bbbk} + b_1 b_1^{L + (1- \delta)(1 + \eta)}\sqrt{\Es_{2\Bbbk}}.
\end{equation}

Substituting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>) concludes the proof of (<ref>) as well as Proposition <ref>.

§.§ Conclusion of Proposition <ref>.

We give the proof of Proposition <ref> in this subsection in order to complete the proof of Theorem <ref>. Note that this section corresponds to Section 6.1 of [Merle et al., 2015]. Here we follow exactly the same lines as in [Merle et al., 2015] and no new ideas are needed. We divide the proof into 2 parts:

- Part 1: Reduction to a finite dimensional problem. Assume that for a given $K > 0$ large and an initial time $s_0 \geq 1$ large, we have $(b(s), q(s)) \in \Sc_K(s)$ for all $s \in [s_0, s_1]$ for some $s_1 \geq s_0$. By using (<ref>), (<ref>), (<ref>) and (<ref>), we derive new bounds on $\Vc_1(s)$, $b_k(s)$ for $\ell + 1 \leq k \leq L$ and $\Es_{2(\hbar + m)}$ for $1 \leq m \leq L+1$, which are better than those defining $\Sc_K(s)$ (see Definition <ref>). It then remains to control $(\Vc_2(s), \cdots, \Vc_\ell(s))$. This means that the problem is reduced to the control of a finite dimensional function $(\Vc_2(s), \cdots, \Vc_\ell(s))$ and then get the conclusion $(i)$ of Proposition <ref>.

- Part 2: Transverse crossing. We aim at proving that if $(\Vc_2(s), \cdots, \Vc_\ell(s))$ touches

$$\partial \hat \Sc _K(s):= \partial\left(-\frac{K}{s^{\frac{\eta}{2}}(1 - \delta)}, \frac{K}{s^{\frac{\eta}{2}}(1 - \delta)}\right)^{\ell - 1}$$

at $s = s_1$, it actually leaves $\partial \hat \Sc_K(s)$ at $s=s_1$ for $s_1 \geq s_0$, provided that $s_0$ is large enough. We then get the conclusion $(ii)$ of Proposition <ref>.

$\bullet\,$ Reduction to a finite dimensional problem. We give the proof of item $(i)$ of Proposition <ref> in this part. Given $K > 0$, $s_0 \geq 1$ and the initial data at $s = s_0$ as in Definition <ref>, we assume for all $s \in [s_0, s_1]$, $(b(s), q(s)) \in \Sc_K(s)$ for some $s_1 \geq s_0$. We claim that for all $s \in [s_0, s_1]$,

\begin{align}
&|\Vc_1(s)| \leq s^{-\frac \eta 2(1- \delta)},\label{est:impV1}\\
&|b_k(s)| \lesssim s^{-(k + \eta(1 - \delta))} \quad \text{for}\;\; \ell + 1 \leq k \leq L,\label{est:impbk}\\
&\Es_{2m} \leq \left\{ \begin{array}{ll}
\frac{K}{2} s^{-\frac{\ell(4m - d)}{2\ell - \gamma}} &\quad \text{for}\quad \hbar + 2 \leq m \leq \ell + \hbar,\\
\frac{1}{2}s^{-2(m - \hbar -1) - 2(1 - \delta) + K\eta} &\quad \text{for}\quad \ell + \hbar + 1 \leq m \leq \Bbbk - 1.
\end{array}\right.\label{est:impE2m}\\
&\Es_{2\Bbbk} \leq \frac{K}{2} s^{-(2L + 2(1 - \delta)(1 + \eta))},\label{est:impE2k}
\end{align}

Once these estimates are proved, it immediately follows from Definition <ref> of $\Sc_K$ that if $(b(s_1), q(s_1)) \in \partial\Sc_K(s_1)$, then $(\Vc_2, \cdots, \Vc_\ell))(s_1)$ must be in $\partial \hat \Sc_K(s_1)$, which concludes the proof of item $(i)$ of Proposition <ref>.

Before going to the proof of (<ref>)-(<ref>), let us compute explicitly the scaling parameter $\lambda$. To do so, let us note from (<ref>) and the a priori bound on $\Uc_1$ given in Definition <ref> that
$b_1(s) = \frac{c_1}{s} + \frac{\Uc_1}{s} = \frac{\ell}{(2\ell - \gamma)s} + \Oc\left(\frac{1}{s^{1 + c\eta}}\right).$
Using (<ref>) yields

\begin{equation}\label{eq:lam10}
-\frac{\lambda_s}{\lambda} = \frac{\ell}{(2\ell - \gamma)s} + \Oc\left(\frac{1}{s^{1 + c\eta}}\right),
\end{equation}

from which we write

\begin{equation*}
\left|\frac{d}{ds}\left\{\log \Big(s^{\frac{\ell}{2\ell - \gamma}}\lambda(s)\Big)\right\}\right| \lesssim \frac{1}{s^{1 + c\eta}}.
\end{equation*}

We now integrate by using the initial data value $\lambda(s_0) = 1$ to get

\begin{equation}\label{eq:Lamdas}
\lambda(s) = \left(\frac{s_0}{s}\right)^\frac{\ell}{2\ell - \gamma}\left[1 + \Oc\left(s^{-c\eta}\right)\right] \quad \text{for}\;\; s_0 \gg 1.
\end{equation}

This implies that

\begin{equation}\label{est:b1so_s}
s_0^{-\frac{\ell}{2\ell - \gamma}} \lesssim \frac{s^{-\frac{\ell}{2\ell - \gamma}} }{\lambda(s)}\lesssim s_0^{-\frac{\ell}{2\ell - \gamma}}.
\end{equation}

- Improved control of $\Es_{2\Bbbk}$: We aim at using (<ref>) to derive the improved bound (<ref>). To do so, we inject the bound of $\Es_{2\Bbbk}$ given in Definition <ref> into the monotonicity formula (<ref>) and integrate in time by using $\lambda(s_0) = 1$: for all $s \in [s_0, s_1)$,

\begin{align*}
\Es_{2\Bbbk}(s) \leq C \lambda(s)^{4\Bbbk - d}\left[\Es_{2\Bbbk}(s_0) + \left(\frac{K}{M^{2\delta}} + \sqrt K +  1\right)\int_{s_0}^s\frac{\tau^{-(2L + 1 + 2(1 - \delta)(1 + \eta))}}{\lambda(\tau)^{4\Bbbk - d}}d\tau\right].
\end{align*}

Using (<ref>), we estimate

\begin{align*}
&\lambda(s)^{4\Bbbk - d}\int_{s_0}^s\frac{\tau^{-(2L + 1 + 2(1 - \delta)(1 + \eta))}}{\lambda(\tau)^{4\Bbbk - d}}d\tau\\
& \quad \lesssim s^{- \frac{\ell(4\Bbbk - d)}{2\ell - \gamma}}\int_{s_0}^s \tau^{\frac{\ell(4\Bbbk - d)}{2\ell - \gamma} - (2L + 1 + 2(1 - \delta)(1 + \eta))}d\tau \lesssim s^{-(2L + 2(1 - \delta)(1 + \eta))}.
\end{align*}

Here we used the fact that the integral is divergent because

$$\frac{\ell(4\Bbbk - d)}{2\ell - \gamma} - [2L + 1 + 2(1 - \delta)(1 + \eta)] = \frac{2\gamma L}{2\ell - \gamma} + \Oc_{L \to +\infty}(1) \gg -1.$$

Using again (<ref>) and the initial bound (<ref>), we estimate

$$\lambda(s)^{4\Bbbk - d}\Es_{2\Bbbk}(s_0) \leq \left(\frac{s_0}{s}\right)^\frac{\ell(4\Bbbk - d)}{2\ell - \gamma} s_0^{-\frac{10L\ell}{2\ell - \gamma}} \lesssim s^{-(2L + 2(1 - \delta)(1 + \eta))},$$

for $L$ large enough. Therefore, we obtain

$$\Es_{2\Bbbk}(s) \leq C\left(\frac{K}{M^{2\delta}} + \sqrt K + 1\right)s^{-(2L + 2(1 - \delta)(1 + \eta))} \leq \frac{K}{2}s^{-(2L + 2(1 - \delta)(1 + \eta))}, $$

for $K = K(M)$ large enough. This concludes the proof of (<ref>).

- Improved control of $\Es_{2m}$. We can improve the control of $\Es_{2m}$ by using the monotonicity formula (<ref>). We distinguish two cases:

- Case 1: $\hbar +2 \leq m \leq \ell + \hbar$. From the bound of $\Es_{2m}$ given in Definition <ref> and $b_1(s) \sim \frac{1}{s}$, we integrate (<ref>) in time $s$ by using $\lambda(s_0) = 1$ to find that

\begin{align*}
\Es_{2m}(s) &\leq C \lambda(s)^{4m - d}\left[\Es_{2m}(s_0)  + \sqrt{K}\int_{s_0}^s \frac{ \tau^{-\frac{\ell}{2\ell - \gamma}\left(2m - \frac{d}{2}\right) -(m - \hbar + 1-\delta - C\eta)}}{\lambda(\tau)^{4m - d}}d\tau\right.\\
& \qquad \qquad \qquad \qquad \left. + \int_{s_0}^s \frac{ \tau^{-(2m - 2\hbar - 1 + 2(1 - \delta) - C\eta)}}{\lambda(\tau)^{4m - d}}d\tau\right]
\end{align*}

Using the initial bound (<ref>) and (<ref>), we estimate

$$C\lambda(s)^{4m - d}\Es_{2m}(s_0)\lesssim s^{-\frac{\ell}{2\ell - \gamma}(4m - d)},$$

for $s_0$ large.

Using (<ref>) and the identity

\begin{align*}
&\frac{\ell}{2\ell - \gamma}\left(2m - \frac{d}{2}\right) - (m - \hbar + 1 - \delta -C\eta) \\
& \quad = - \frac{\gamma}{2} - 1 + C\eta  + \frac{\gamma}{2\ell - \gamma}(m - \hbar - \delta - \frac{\gamma}{2}) \leq -1 - \frac{\gamma \delta}{2\ell - \gamma} + C\eta < -1,
\end{align*}

we estimate

\begin{align*}
&\lambda(s)^{4m - d}\int_{s_0}^s \frac{ \tau^{-\frac{\ell}{2\ell - \gamma}\left(2m - \frac{d}{2}\right) -(m - \hbar + 1-\delta - C\eta)}}{\lambda(\tau)^{4m - d}}d\tau \\
&\lesssim s^{-\frac{\ell}{2\ell - \gamma}(4m - d)}\int_{s_0}^s \tau^{\frac{\ell}{2\ell - \gamma}\left(2m -\frac d 2\right) - (m - \hbar + 1 - \delta - C\eta)} d\tau \\
& \lesssim s^{-\frac{\ell}{2\ell - \gamma}(4m - d)} \int_{s_0}^s \frac{d\tau}{\tau^{1 + \epsilon}} \lesssim  s^{-\frac{\ell}{2\ell - \gamma}(4m - d)}.
\end{align*}

Similarly, thanks to the identity

\begin{align*}
&\frac{\ell}{2\ell - \gamma}\left(4m - d\right) - (2m - 2\hbar - 1 +2(1- \delta) -C\eta) \\
& \quad = - \gamma - 1 + C\eta  + \frac{\gamma}{2\ell - \gamma}(2m - 2\hbar - 2\delta - \gamma) \leq -1 - \frac{2\gamma \delta}{2\ell - \gamma} + C\eta < -1,
\end{align*}

we obtain

$$\lambda(s)^{4m - d}\int_{s_0}^s \frac{ \tau^{-(2m - 2\hbar - 1 + 2(1 - \delta) - C\eta)}}{\lambda(\tau)^{4m - d}}d\tau \lesssim s^{-\frac{\ell}{2\ell - \gamma}(4m - d)}.$$

Therefore, we deduce that

$$\Es_{2m}(s) \leq C(1 + \sqrt{K})s^{-\frac{\ell}{2\ell - \gamma}(4m - d)} \leq \frac{K}{2}s^{-\frac{\ell}{2\ell - \gamma}(4m - d)},$$

for $K$ large, which yields the improved bound (<ref>) for $\hbar + 2 \leq m \leq \ell + \hbar$.

- Case 2: $\ell + \hbar +1 \leq m \leq \Bbbk - 1$. Proceeding as the previous case, we arrive at

\begin{align*}
\Es_{2m}(s) &\leq C \lambda(s)^{4m - d}\left[\Es_{2m}(s_0) + \int_{s_0}^s \frac{\tau^{-\left[2m - 2\hbar - 1 + 2(1 - \delta) - \left(C + \frac K2\right)\eta\right]}}{\lambda(\tau)^{4m - d}}d\tau\right].
\end{align*}

From the identity

\begin{align}
&\frac{\ell}{2\ell - \gamma}\left(4m - d\right) - (2m - 2\hbar - 1 +2(1- \delta) - \left(C + \frac K2 \right)\eta) \nonumber\\
& \quad = - \gamma - 1 + \left(C + \frac K2 \right)\eta)  + \frac{\gamma}{2\ell - \gamma}(2m - 2\hbar - 2\delta - \gamma)\nonumber\\
& \qquad \geq -1 + \frac{2\gamma (1 -\delta)}{2\ell - \gamma} + \left(C + \frac K2 \right)\eta) > -1,\label{ide2g}
\end{align}

together with (<ref>), we estimate

\begin{align*}
&\lambda(s)^{4m - d}\int_{s_0}^s \frac{\tau^{-\left[2m - 2\hbar - 1 + 2(1 - \delta) - \left(C + \frac K2\right)\eta\right]}}{\lambda(\tau)^{4m - d}}d\tau\\
& \quad \lesssim s^{-\frac{\ell(4m - d)}{2\ell - \gamma}}\int_{s_0}^s\tau^{\frac{\ell(4m - d)}{2\ell - \gamma} -\left[2m - 2\hbar - 1 + 2(1 - \delta) - \left(C + \frac K2\right)\eta\right]} d\tau\\
& \qquad \lesssim s^{-\left[2(m - \hbar - 1) + 2(1 - \delta) - \left(C + \frac K2\right)\eta\right]}\leq  \frac 14 s^{-\left[2(m - \hbar - 1) + 2(1 - \delta) - K\eta\right]}.
\end{align*}

Using (<ref>), (<ref>) and the initial bound (<ref>), we derive

\begin{align*}
C\lambda(s)^{4m - d}\Es_{2m}(s_0) \lesssim s^{-\frac{\ell(4m - d)}{2\ell - \gamma}} & \lesssim s^{-\left[2(m - \hbar - 1) + 2(1 - \delta) - \left(C + \frac K2\right)\eta\right]} \\
&\quad \leq  \frac 14 s^{-\left[2(m - \hbar - 1) + 2(1 - \delta) - K\eta\right]}.
\end{align*}

This concludes the proof of (<ref>).

- Control of the stable modes $b_k$'s. We now end the control of the stable modes $(b_{\ell + 1}, \cdots, b_L)$, in particular, we prove (<ref>). We first treat the case when $k = L$. Let

$$\tilde{b}_L = b_L + \frac{\left<\Ls^Lq,\chi_{B_0}\Lambda Q \right>}{\left<\Lambda Q, \chi_{B_0}\Lambda Q \right>},$$

then from (<ref>) and (<ref>),

$$|\tilde b_L - b_L| \lesssim b_1^{-(1 - \delta)}\sqrt{\Es_{2\Bbbk}} \lesssim b_1^{L + \eta(1 - \delta)}.$$

It follows from the improved modulation equation (<ref>),

\begin{align*}
|(\tilde{b}_L)_s + (2L - \gamma)b_1\tilde{b}_L|&\lesssim b_1|\tilde b_L - b_L| + \frac{1}{B_0^{2\delta}}\left[C(M)\sqrt{\Es_{2\Bbbk}} + b_1^{L +(1 - \delta)}\right]\\
&\quad \lesssim b_1^{L + 1 + \eta(1 -\delta)}.
\end{align*}

This follows

$$\left|\frac{d}{ds} \left\{\frac{\tilde b_L}{\lambda^{2L - \gamma}} \right\}\right| \lesssim \frac{b_1^{L + 1 + \eta(1 - \delta)}}{\lambda^{2L - \gamma}}.$$

Integrating this identity in time from $s_0$ and recalling that $\lambda(s_0) = 1$ yields

\begin{align*}
\tilde{b}_L(s) \lesssim C\lambda(s)^{2L - \gamma}\left(\tilde{b}_L(s_0) + \int_{s_0}^s\frac{b_1(\tau)^{L + 1 + \eta(1 - \delta)}}{\lambda(\tau)^{2L - \gamma}} d\tau\right).
\end{align*}

Using (<ref>), $b_1(s) \sim \frac{1}{s}$, the initial bounds (<ref>) and (<ref>) together with (<ref>), we estimate

$$\lambda(s)^{2L - \gamma}\tilde{b}_L(s_0) \lesssim \left(\frac{s_0}{s}\right)^{\frac{\ell(2L - \gamma)}{2\ell - \gamma}}\left(s_0^{-\frac{5\ell(2L - \gamma)}{2\ell - \gamma}} + s_0^{\eta(1 - \delta)}s_0^{-\frac{5L\ell}{2\ell - \gamma}} \right) \lesssim s^{- L - \eta(1 - \delta)},$$

\begin{align*}
\lambda(s)^{2L - \gamma}\int_{s_0}^s\frac{b_1(\tau)^{L + 1 + \eta(1 - \delta)}}{\lambda(\tau)^{2L - \gamma}} d\tau &\lesssim s^{-\frac{\ell(2L - \gamma)}{2\ell - \gamma}}\int_{s_0}^s \tau ^{ \frac{\ell(2L - \gamma)}{2\ell - \gamma} - L - 1 - \eta(1 - \delta)}d\tau\\
&\quad \lesssim s^{-L - \eta(1 - \delta)}.
\end{align*}

$$b_L(s) \lesssim |\tilde{b}_L(s)| + |\tilde{b}_L(s) - b_L(s)| \lesssim s^{-L - \eta(1 - \delta)},$$

which concludes the proof of (<ref>) for $k = L$.
Now we will propagate this improvement that we found for the bound of $b_L$ to all $b_k$ for all $\ell + 1 \leq k \leq L - 1$. To do so we do a descending induction where the initialization is for $k=L$. Let assume  the bound

$$|b_k|\lesssim  b_1^{k + \eta(1 - \delta)},$$

for $k+1$ and let's prove it for $k$.
Indeed, from (<ref>) and the induction bound, we have

$$\left|(b_k)_s - (2k - \gamma)\frac{\lambda_s}{\lambda}b_k\right| \lesssim b_1^{L + 1} + |b_{k + 1}| \lesssim b_1^{k + 1 + \eta(1 - \delta)},$$

which follows

$$\left|\frac{d}{ds}\left\{\frac{b_k}{\lambda^{2k - \gamma}}\right\} \right| \lesssim \frac{b_1^{k + 1 + \eta(1 - \delta)}}{\lambda^{2k - \gamma}}.$$

Integrating this identity in time as for the case $k = L$, we end-up with

\begin{align*}
b_k(s) &\lesssim C\lambda(s)^{2k - \gamma}\left(b_k(s_0) + \int_{s_0}^s\frac{b_1(\tau)^{k + 1 + \eta(1 - \delta)}}{\lambda(\tau)^{2k - \gamma}} d\tau\right)\\
&\quad \lesssim s^{-k - \eta(1 - \delta)},
\end{align*}

where we used the initial bound (<ref>), (<ref>) and $k \geq \ell + 1$. This concludes the proof of (<ref>).

- Control of the stable mode $\Vc_1$. We recall from (<ref>) and (<ref>) that

$$b_k = b_k^e + \frac{\Uc_k}{s^k}, \quad 1 \leq k \leq \ell, \quad \Vc = P_\ell \Uc,$$

where $P_\ell$ diagonalize the matrix $A_\ell$ with spectrum (<ref>). From (<ref>), and (<ref>), we estimate for $1 \leq k \leq \ell - 1$,

$$|s(\Uc_k)_s - (A_\ell \Uc)_k| \lesssim s^{k + 1}|(b_k)_s + (2k - \gamma)b_1 b_k - b_{k + 1}| + |\Uc|^2 \lesssim s^{-L + k} + |\Uc|^2,$$

From (<ref>), (<ref>) and the improved bound (<ref>), we have

$$|s(\Uc_\ell)_s - (A_\ell \Uc)_\ell| \lesssim s^{\ell + 1}\left(|(b_k)_s + (2k - \gamma)b_1 b_\ell - b_{\ell + 1}| + |b_{\ell + 1}|\right) + |\Uc|^2 \lesssim s^{-\eta(1 - \delta)} + |\Uc|^2.$$

Using the diagonalization (<ref>), we obtain

\begin{equation}\label{eq:sVs}
s\Vc_s = D_\ell \Vc + \Oc(s^{-\eta(1 - \delta)}).
\end{equation}

Using (<ref>) again yields the control of the stable mode $\Vc_1$:

$$|(s\Vc_1)_s| \lesssim s^{-\eta(1 - \delta)}.$$

Thus from the initial bound (<ref>),

$$|s^{\eta(1 - \delta)}\Vc_1(s)| \leq \left(\frac{s_0}{s}\right)^{1 - \eta(1 - \delta)}s_0^{\eta(1 - \eta)}\Vc_1(s_0) + 1\lesssim s_0^{\eta(1 - \delta)},$$

which yields (<ref>) for $s_0 \geq s_0(\eta)$ large enough.

$\bullet \,$ Transverse crossing. We give the proof of item $(ii)$ of Proposition <ref> in this part. We compute from (<ref>) and (<ref>) at the exit time $s = s_1$:

\begin{align*}
&\frac{1}{2}\frac{d}{ds}\left(\sum_{k = 2}^\ell|s^{\frac{\eta}{2}(1 - \delta)}\Vc_k(s)|^2 \right)_{\big|_{s= s_1}} \\
&\quad = \left(s^{\eta(1 - \delta) - 1} \sum_{k = 2}^\ell \left[\frac{\eta}{2}(1 - \delta)\Vc_k^2(s) + s\Vc_k(\Vc_k)_s \right]\right)_{\big|_{s= s_1}}\\
&\qquad = \left(s^{\eta(1 - \delta) - 1} \Bigg[\sum_{k = 2}^\ell \left[\frac{k\gamma}{2k - \gamma} + \frac{\eta}{2}(1 - \delta)\right]\Vc_k^2(s) + \Oc\left(\frac{1}{s^{\frac{3}{2}\eta(1 - \delta)}}\right)\Bigg] \right)_{\big|_{s= s_1}}\\
&\quad \qquad \geq \frac{1}{s_1}\left[c(d,\ell) \sum_{k = 2}^\ell |s_1^{\frac{\eta}{2}(1 - \delta)}\Vc_k(s_1)|^2 +  \Oc\left(\frac{1}{s_1^{\frac{\eta}{2}(1 - \delta)}}\right)\right]\\
& \qquad \qquad \geq \frac{1}{s_1}\left[c(d,\ell) +  \Oc\left(\frac{1}{s_1^{\frac{\eta}{2}(1 - \delta)}}\right)\right] > 0,
\end{align*}

where we used item $(i)$ of Proposition <ref> in the last step. This completes the proof of Proposition <ref>.

§ COERCIVITY OF THE ADAPTED NORMS.

We give in this section the coercivity estimates for the operator $\Ls$ as well as the iterates of $\Ls$  under some suitable orthogonality condition. We first recall the standard Hardy type inequalities for the class of radially symmetric functions,

$$\Dc_{rad} = \{f \in \Cc_c^\infty(\Rd) \; \text{with radial symmetry}\}.$$

For simplicity, we write

$$\int f := \int_0^{+\infty} f(y)y^{d-1}dy.$$

$$D^k = \left\{\begin{array}{ll} \Delta^m &\quad \text{if}\;\; k = 2m,\\
\partial_y \Delta^m &\quad \text{if} \;\; k= 2m + 1.
\end{array}\right.$$

We have the following:
Let $d \geq 7$ and $f \in \Dc_{rad}$, then

$(i)$ (Hardy near the origin)

\begin{equation*}
\int_{0}^1 \frac{|\py f|^2}{y^{2i}} \geq \frac{(d - 2 - 2i)^2}{4}\int_{0}^1 \frac{f^2}{y^{2+2i}} - C(d)f^2(1), \quad i = 0, 1, 2.
\end{equation*}

$(ii)$ (Hardy away from zero for the non-critical exponent) Let $\alpha > 0, \alpha \ne \frac{d-2}{2}$, then

\begin{align*}
&\int_1^{+\infty} \frac{|\py f|^2}{y^{2\alpha}} \geq \left(\frac{d - (2\alpha + 2)}{2}\right)^2\int_1^{+\infty} \frac{f^2}{y^{2 + 2\alpha}} - C(\alpha,d)f^2(1).
\end{align*}

$(iii)$ (Hardy away from zero for the critical exponent) Let $\alpha = \frac{d-2}{2}$, then

\int_1^{+\infty} \frac{|\py f|^2}{y^{2\alpha}} \geq \frac 14 \int_1^{+\infty} \frac{f^2}{y^{2 + 2\alpha} (1 + \log y)^2} - C(d)f^2(1).$$

$(iv)$ (General weighted Hardy) For any $\mu > 0$, $k \geq 2$ be an integer and $1 \leq j \leq k-1$,

$$\int \frac{|D^jf|^2}{1 + y^{\mu + 2(k - j)}} \lesssim_{j,\mu} \int \frac{|D^k f|^2}{1 + y^\mu} + \int \frac{f^2}{1 + y^{\mu + 2k}}.$$

The proof can be found in [Merle et al., 2015], Lemma B.1.

From the Hardy type inequalities, we derive the following coercivity of $\As^*$:
Let $\alpha \geq 0$, there exists $c_\alpha > 0$ such that for all $f \in \Dc_{rad}$:

\begin{equation}\label{eq:coerAst}
\int \frac{|\As^* f|^2}{y^{2i}(1 + y^{2\alpha})} \geq c_\alpha\left(\int \frac{|\py f|^2}{y^{2i}(1 + y^{2\alpha})} + \int \frac{f^2}{y^{2i + 2}(1 + y^{2\alpha})}\right), \quad i = 0, 1,2.
\end{equation}

We proceed into two steps:

- Step 1: Subcoercive estimate for $\As^*$: We first prove the following subcoercive bound for $\As^*$: for $i = 0, 1, 2$ and $\alpha \geq 0$,

\begin{align}
\int\frac{|\As^* f|^2}{y^{2i}(1 + y^{2\alpha})} \gtrsim \int\frac{f^2}{y^{2i + 2}(1 + y^{2\alpha})} &+ \int \frac{|\py f|^2}{y^{2i}(1 + y^{2\alpha})}\nonumber\\
&\quad - f^2(1) - \int \frac{f^2}{1 + y^{2i + 2\alpha + 4}}.\label{eq:subcoerAst}
\end{align}

From the definition (<ref>) of $\As^*$ and the asymptotic of $V$ given in (<ref>), we use an integration by parts to estimate near the origin:

\begin{align*}
\int_{y \leq 1}\frac{|\As^* f|^2}{y^{2i}(1 + y^{2\alpha})} &\gtrsim \int_{y \leq 1}\frac{1}{y^{2i}}\left|\py f + \frac{d}{y}f + \Oc(|yf|)\right|^2\\
&\quad \gtrsim \int_{y \leq 1}\frac{|\py f|^2}{y^{2i}} + d\int_{y \leq 1}\frac{\py (f^2)}{y^{2i + 1}} + d^2\int_{y \leq 1}\frac{f^2}{y^{2i + 2}} + \Oc\left(\int_{y \leq 1}\frac{f^2}{y^{2i - 2}}\right)\\
&\qquad \gtrsim \int_{y\leq 1} \frac{|\py f|^2}{y^{2i}} + (2 + 2i)d\int_{y\leq 1}\frac{f^2}{y^{2i +2}} + df^2(1)+ \Oc\left(\int_{y\leq 1} \frac{f^2}{y^{2i - 2}}\right)\\
& \quad \qquad \gtrsim \int_{y \leq 1}\left(\frac{|\py f|^2}{y^{2i}} + \frac{f^2}{y^{2i + 2}}\right) - \int_{y \leq 1}y^2f^2.
\end{align*}

Away from the origin, we use (<ref>) to estimate:

\begin{align*}
\int_{y \geq 1}\frac{|\As^* f|^2}{y^{2i}(1 + y^{2\alpha})} &\gtrsim \int_{y \geq 1} \frac{1}{y^{2i + 2\alpha}}\left(\py f + \frac{d - 1 - \gamma}{y}f \right)^2 - \int_{y\geq 1}\frac{f^2}{y^{2i + 2\alpha + 4}}.
\end{align*}

We make the change of variable $g = y^{d - 1 - \gamma} f$ and use the Hardy inequality given in part $(ii)$ of Lemma <ref> to write

\begin{align*}
\int_{y \geq 1}\frac{|\py(y^{d - 1 - \gamma} f)|^2}{y^{2i + 2\alpha  +2(d - 1 - \gamma)}}dy &= \int_{y \geq 1}\frac{|\py g|^2}{y^{2i + 2\alpha + 2(d-1-\gamma)}}dy \gtrsim \int_{y \geq 1}\frac{g^2}{y^{2i + 2\alpha + 2(d-1-\gamma) + 2}}dy - g^2(1)\\
&\quad \gtrsim \int_{y \geq 1}\frac{f^2}{y^{2i + 2\alpha + 2}} - f^2(1).
\end{align*}

Gathering the above bounds together with the trivial bound from (<ref>),

$$\int_{y \geq 1} \frac{|\py f|^2}{y^{2i + 2\alpha}} \lesssim \int_{y \geq 1} \frac{|\As^* f|^2}{y^{2i + 2\alpha}} + \int_{y \geq 1}\frac{f^2}{y^{2i + 2\alpha + 2}},$$

yields the subcoercivity (<ref>).

- Step 2: Coercivity of $\As^*$: We now argue by contradiction to show the coercivity of $\As^*$. Assume that (<ref>) does not hold. Up to a renormalization, we consider the sequence $f_n \in \Dc_{rad}$ with

\begin{equation}\label{eq:asumpfn}
\int \frac{f_n^2}{y^{2 i +2}(1 + y^{2\alpha})} + \int \frac{|\partial_y f_n|^2}{y^{2i}(1 + y^{2\alpha})} = 1 \quad \text{and} \quad \int \frac{|\As^* f_n|^2}{y^{2i}(1 + y^{2\alpha})} \leq \frac{1}{n}.
\end{equation}

This implies from (<ref>),

\begin{equation}\label{eq:boundfn}
f_n^2(1) + \int \frac{f_n^2}{1 + y^{2i + 2\alpha + 4}} \gtrsim 1.
\end{equation}

From (<ref>), the sequence $f_n$ is bounded in $H^1_{loc}$. Hence, from a standard diagonal extraction argument, there exists $f_\infty \in H^1_{loc}$ such that up to a subsequence,

$$f_n \rightharpoonup f_\infty \quad \text{in} \quad H^1_{loc},$$

and from the local compactness of one dimensional Sobolev embeddings:

$$f_n \rightarrow f_\infty \quad \text{in}\quad L^2_{loc}, \quad f_n(1) \rightarrow f_\infty(1).$$

This implies from (<ref>) and (<ref>),

\begin{equation}\label{eq:regufinA}
f_\infty^2(1) + \int \frac{f_\infty^2}{1 + y^{2i + 2\alpha + 4}} \gtrsim 1 \quad \text{and} \quad \int \frac{f_\infty^2}{y^{2i + 2}(1 + y^{2\alpha})} \lesssim 1,
\end{equation}

which means that $f_\infty \ne 0$.
On the other hand, from (<ref>) and the lower semi continuity of norms for the weak topology, we have

$$\As^* f_\infty = 0, $$

$$f_\infty = \frac{\beta}{y^{d-1} \Lambda Q} \quad \text{for some $\beta \ne 0$}.$$

Since $\Lambda Q \sim y$ near the origin, we have

$$\int_{y \leq 1} \frac{f_\infty^2}{y^{2i + 2}} \gtrsim \int_{y \leq 1}\frac{y^{d-1}}{y^{2d + 2i + 2}}dy = \int_{y \leq 1}\frac{dy}{y^{d+2i + 3}} = +\infty,$$

which contradicts the a priori regularity of $f_\infty$ given in (<ref>). This concludes the proof of Lemma <ref>.

We also need the following subcoercivity of $\As$.
Let $p \geq 0$ and $i = 0, 1, 2$ such that $|2p + 2i - (d - 2 - 2\gamma)| \ne 0$, where $\gamma \in (1,2]$ is defined by (<ref>), there holds:

\begin{align}
\int \frac{|\As f|^2}{y^{2i}(1 + y^{2p})} \gtrsim \int \frac{|\py f|^2}{y^{2i}(1 + y^{2p})} & + \int \frac{f^2}{y^{2i + 2}(1 + y^{2p})} \nonumber\\
&\quad - \left[f^2(1) + \int \frac{f^2}{1 +y^{2i + 2p + 4}} \right].\label{eq:subcoerA}
\end{align}

Assume in addition that

$$\left<f, \Phi_M\right> = 0 \quad \text{if}\quad 2i + 2p > d - 2\gamma - 2,$$

where $\Phi_M$ is defined in (<ref>), we have

\begin{equation}\label{eq:coerA}
\int \frac{|\As f|^2}{y^{2i}(1 + y^{2p})} \gtrsim \int \frac{|\py f|^2}{y^{2i}(1 + y^{2p})} + \int \frac{f^2}{y^{2i + 2}(1 + y^{2p})}.
\end{equation}

The proof is very similar to the proof of Lemma <ref>. We proceed into two steps. The first step is to derive the subcoercive estimate (<ref>). In the second step, we use a compactness argument to show the coercivity of $\As$ under a suitable condition.

- Step 1: Subcoercive estimate for $\As$. From the definition (<ref>) of $\As$ and the asymptotic of $V$ given in (<ref>), we estimate near the origin

\begin{align*}
\int_{y \leq 1} \frac{|\As f|^2}{y^{2i}(1 + y^{2p})} &\gtrsim \int_{y \geq 1} \frac 1 {y^{2i}}\left|-\py f + \frac{f}{y} + \Oc(|yf|)\right|^2\\
& \quad \gtrsim \int_{y \leq 1} \frac{|\py f|^2}{y^{2i}} + \int_{y \leq 1} \frac{f^2}{y^{2i+2}} - \int_{y \leq 1}\frac{\py(f^2)}{y^{2i+1}} - \int_{y\leq 1}\frac{f^2}{y^{2i-2}}\\
& \qquad \gtrsim \int_{y \leq 1} \frac{|\py f|^2}{y^{2i}} + (d-2i-1)\int_{y \leq 1} \frac{f^2}{y^{2i+2}} - f^2(1) - \int_{y\leq 1}\frac{f^2}{y^{2i-2}}\\
& \qquad \quad \gtrsim \int_{y \leq 1} \frac{|\py f|^2}{y^{2i}} + \int_{y \leq 1} \frac{f^2}{y^{2i+2}} - f^2(1) - \int_{y\leq 1}y^2f^2.
\end{align*}

Away from the origin, we estimate from (<ref>):

$$\int_{y \geq 1} \frac{|\As f|^2}{y^{2i}(1 + y^{2p})} \gtrsim \int_{y \geq 1}\frac{1}{y^{2i + 2p }}\left(\py f + \frac{\gamma}{y}f\right)^2 - \int_{y \geq 1}\frac{f^2}{y^{2i + 2p + 4}}.$$

We make the change of variable $g = y^\gamma f$. From the assumption $|2i + 2p - (d - 2 - 2\gamma)| \ne 0$, we use the Hardy inequality given in part $(ii)$ of Lemma <ref> to write

\begin{align*}
\int_{y \geq 1}\frac{|\partial_y(y^\gamma f)|^2}{y^{2i + 2p  + 2\gamma}} = \int_{y \geq 1}\frac{|\py g|^2}{y^{2i + 2p+ 2\gamma}} &\gtrsim \int_{y \geq 1}\frac{g^2}{y^{2i + 2p + 2 + 2\gamma}} - g^2(1)\\
&\quad \gtrsim \int_{y \geq 1}\frac{f^2}{y^{2i + 2p + 2}} - f^2(1).
\end{align*}

Note also that we have the trivial bound from (<ref>),

$$\int_{y \geq 1} \frac{|\As f|^2}{y^{2i + 2p}} + \int_{y \geq 1}\frac{f^2}{y^{2i + 2p + 2}} \gtrsim \int_{y \geq 1} \frac{|\py f|^2}{y^{2i + 2p}}.$$

The collection of the above bounds yields the subcoercivity (<ref>).

- Step 2: Coercivity of $\As$: Arguing as the proof of (<ref>), we end up with the existence of $f_\infty \ne 0$ such that

$$ \int \frac{f_\infty^2}{y^{2i + 2}(1 + y^{2p})} \lesssim 1 \quad \text{and} \quad \As f_\infty = 0.$$

Hence, from the definition (<ref>) of $\As$, we have

$$f_\infty = \beta \Lambda Q \quad \text{for some $\beta \ne 0$.}$$

If $2i + 2p > d - 2\gamma - 2$, we use the orthogonality condition to deduce that

$$0 = \left<f_\infty, \Phi_M\right> = \beta\left<\Lambda Q, \chi_M \Lambda Q\right>,$$

thus, $\beta = 0$. If $2i + 2p \leq d - 2\gamma - 2$, we use the fact that $\Lambda Q \sim \frac{1}{y^\gamma}$ as $y \to +\infty$ to estimate

$$\int_{y \geq 1}\frac{|\Lambda Q|^2 y^{d-1}dy}{y^{2i + 2}(1 + y^{2p})} \gtrsim \int_{y \geq 1} y^{d - 1 - 2\gamma - 2i - 2p - 2} dy  \gtrsim \int_{y \geq 1} y^{-1}dy= +\infty,$$

which contradicts with the regularity of $f_\infty$. This concludes the proof of Lemma <ref>.

From the coercivities of $\As$ and $\As^*$, we claim the following coercivity for $\Ls$:
Let $k \in \mathbb{N}$, $i = 0, 1, 2$ and $M = M(k)$ large enough, then there exists $c_{M,k} > 0$ such that for all $f \in \Dc_{rad}$ satisfying the orthogonality

$$\left<f, \Phi_M\right> = 0 \quad \text{if}\;\; 2i + 2k > d - 2\gamma - 4,$$

where $\Phi_M$ is defined by (<ref>), $\hbar$ is given in (<ref>), there holds:

\begin{equation}\label{eq:coerL}
\int \frac{|{\Ls} f|^2}{y^{2i}(1 + y^{2k})} \geq c_{M,k} \int\left( \frac{|\partial_{yy}f|^2}{y^{2i}(1 + y^{2k})} +  \frac{|\py f|^2}{y^{2i}(1 + y^{2k + 2})} +  \frac{|f|^2}{y^{2i + 2}(1 + y^{2k + 2})}\right).
\end{equation}

\begin{equation}\label{eq:coerL1}
\int \frac{|{\Ls} f|^2}{y^{2i}(1 + y^{2k})} \geq c_{M,k} \int \left(\frac{|\As f|^2}{y^{2i + 2}(1 + y^{2k})} + \int \frac{|f|^2}{y^{2i}(1 + y^{2k + 4})} \right).
\end{equation}

We proceed into two steps:

- Step 1: Subcoercivity of $\Ls$.  We apply Lemma <ref> to $\As f$ with $\alpha = k$ and note that

$$\py (\As f) = \As(\py f) + \py \left(\frac Vy\right) f,$$

to write

\begin{align}
\int \frac{|{\Ls} f|^2}{y^{2i}(1 + y^{2k})} &\gtrsim \int \frac{|\As f|^2}{y^{2i + 2}(1 + y^{2k})} + \int \frac{|\partial_y (\As f)|^2}{y^{2i}(1 + y^{2k})}\label{eq:tmpcoerLA}\\
& \quad \gtrsim \int \frac{|\As f|^2}{y^{2i}(1 + y^{2k + 2})} + \int \frac{|\partial_y (\As f)|^2}{y^{2i}(1 + y^{2k})}\nonumber\\
&\qquad \gtrsim \int \frac{|\As f|^2}{y^{2i}(1 + y^{2k + 2})} + \int \frac{|\As (\py f)|^2}{y^{2i}(1 + y^{2k})} - \int \frac{|f|^2}{y^{2i + 2}(1 + y^{2k})}.\label{eq:subcoerLax1}
\end{align}

Applying Lemma <ref> to $f$ with $p = k + 1$ and noting that the condition $|2(k + 1) + 2i - (d - 2 - 2\gamma)| \neq 0$ is always satisfied (if not, we have $d = 4 + 2\sqrt{(k + 1 + i)^2 + 2} \not \in \mathbb{N}$), we have

\begin{align*}
\int \frac{|\As f|^2}{y^{2i}(1 + y^{2k  +2})} \gtrsim \int \frac{|\py f|^2}{y^{2i}(1 + y^{2k  +2})} & + \int \frac{f^2}{y^{2i+2}(1 + y^{2k + 2})}\\
&\quad - \left[f^2(1) + \int \frac{f^2}{1 +y^{2k + 2i + 6}} \right].
\end{align*}

We apply again Lemma <ref> to $\py f$ with $p = k$ to estimate

\begin{align*}
\int \frac{|\As (\py f)|^2}{y^{2i}(1 + y^{2k})} \gtrsim \int \frac{|\partial_{yy} f|^2}{y^{2i}(1 + y^{2k })} & + \int \frac{|\py f|^2}{y^{2i + 2}(1 + y^{2k})}\\
&\quad - \left[|\py f(1)|^2 + \int \frac{|\py f|^2}{1 +y^{2k + 2i + 4}} \right].
\end{align*}

Injecting these bounds into (<ref>) yields the subcoercive estimate for $\Ls$,

\begin{align}
\int \frac{|{\Ls} f|^2}{y^{2i}(1 + y^{2k})} &\gtrsim \int \frac{|\partial_{yy}f|^2}{y^{2i}(1 + y^{2k})} + \int \frac{|\py f|^2}{y^{2i}(1 + y^{2k + 2})} + \int \frac{f^2}{y^{2i + 2}(1 + y^{2k + 2})}\nonumber\\
& \quad - \left[ f^2(1) + |f_y(1)|^2 + \int \frac{|f_y|^2}{1 + y^{2k + 2i + 4}} + \int \frac{f^2}{1 + y^{2k + 2i + 6}} \right]. \label{eq:subcoerL}
\end{align}

- Step 2: Coercivity of $\Ls$. We argue by contradiction. Assume that (<ref>) does not hold. Up to a renormalization, there exists a a sequence of functions $f_n \in \Dc_{rad}$ such that

\begin{equation}\label{eq:asumpfnL}
\int \frac{|\Ls f_n|^2}{y^{2i}(1 + y^{2k})} \leq \frac{1}{n}, \quad \int \frac{|\partial_{yy}f_n|^2}{y^{2i}(1 + y^{2k})} + \int \frac{|\partial_y f_n|^2}{y^{2i}(1 + y^{2k + 2})} + \int \frac{|f_n|^2}{y^{2i + 2}(1 + y^{2k + 2})} = 1.
\end{equation}

This implies from (<ref>),

\begin{equation}\label{eq:boundfnL}
f_n^2(1) + |\partial_y f_n(1)|^2 + \int \frac{|\partial_y f_n|^2}{1 + y^{2k + 2i + 4}} + \int \frac{f_n^2}{y^2(1 + y^{2k + 2i + 6})} \gtrsim 1.
\end{equation}

From (<ref>), the sequence $f_n$ is bounded in $H^2_{loc}$. Hence, from a standard diagonal extraction argument, there exists $f_\infty \in H^2_{loc}$ such that up to a subsequence,

$$f_n \rightharpoonup f_\infty \quad \text{in} \quad H^2_{loc},$$

and from the local compactness of one dimensional Sobolev embeddings:

$$f_n \rightarrow f_\infty \quad \text{in}\quad H^1_{loc},$$

$$f_n(1) \rightarrow f_\infty(1),\quad  \partial_y f_n(1) \rightarrow \partial_y f_\infty(1).$$

This implies from (<ref>) and (<ref>),

$$f_\infty^2(1) + |\partial_y f_\infty(1)|^2 + \int \frac{|\partial_y f_\infty|^2}{1 + y^{2k + 2i + 4}} + \int \frac{f_\infty^2}{y^2(1 + y^{2k + 2i + 6})} \gtrsim 1,$$

which means that $f_\infty \ne 0$.
On the other hand, from (<ref>) and the lower semi continuity of norms for the weak topology, we deduce that $f_\infty$ is a non trivial function in the kernel of $\Ls$, namely that

$$ \Ls f_\infty = 0, $$

which follows

$$f_\infty =  \mu\Gamma + \beta \Lambda Q,$$

where $\mu$ and $\beta$ two real numbers.

From (<ref>) and the lower semicontinuity, we have

$$\int \frac{f_\infty^2}{y^{2 i + 2}(1 + y^{2k+2})} < +\infty.$$

Recall from (<ref>) that $\Gamma \sim \frac{1}{y^{d-1}}$ as $y \to 0$. This yields the estimate

$$\int_{y \leq 1} \frac{\Gamma^2}{y^{2i + 2}(1 + y^{2k + 2})} \gtrsim \int_{y \leq 1} \frac{dy}{y^{2i + 2 + d - 1}} = +\infty,$$

hence, $\mu = 0$.

From (<ref>), we have $\Lambda Q \sim \frac{1}{y^{\gamma}}$ as $y \to +\infty$. If $2i + 2k \leq d - 2\gamma - 4$, we have

$$\int_{y \geq 1} \frac{|\Lambda Q|^2 y^{d-1}dy }{y^{2i + 2}(1 + y^{2k + 2})} \gtrsim \int_{y \geq 1} y^{d - 1 - 2i - 2k - 4 - 2\gamma} dy \gtrsim \int_{y \geq 1} y^{-1}dy = + \infty,$$

hence, $\beta = 0$. If $2i + 2k > d - 2\gamma - 4$, we use the orthogonality condition to deduce

$$0= \left<f_\infty, \Phi_M\right> = \beta\left<\Lambda Q, \chi_M \Lambda Q\right>,$$

which yields $\beta = 0$, hence $f_\infty = 0$. The contradiction then follows and the coercivity (<ref>) is proved. The estimate (<ref>) simply follows from (<ref>) and (<ref>). This finished the proof of Lemma <ref>.

We are now in a position to  prove the coercivity of ${\Ls}^k$ under a suitable orthogonality condition.  We claim the following:
Let $k \in \mathbb{N}$ and $M = M(k)$ large enough, then there exists $c_{M,k} > 0$ such that for all $f \in \Dc_{rad}$ satisfying the orthogonal condition

$$\left<f, \Ls^m \Phi_M\right> = 0, \quad 0 \leq m \leq k - \hbar,$$

where $\hbar$ is defined as in (<ref>), there holds:

\begin{align} \label{eq:coerLk}
\Es_{2k + 2}(f) = \int |\Ls^{k+1} f|^2 &\geq c_{M,k}\left\{\int \frac{|\As (\Ls^k f)|^2}{y^2} \right.\nonumber \\
& \quad +\left. \sum_{m = 0}^k \int \frac{|\Ls^m f|^2}{y^4(1 + y^{4(k - m)})} + \sum_{m = 0}^{k-1}\frac{|\As(\Ls^{m}f)|^2}{y^6(1 + y^{4(k - m - 1)})}\right\}.
\end{align}

We argue by induction on $k$. For $k = 0$, we apply Lemma <ref> to $\As f$ with $i = 0$ and $\alpha = 0$, then  Lemma <ref> to $f$ with $i = 1$ and $p = 0$ to write

$$\Es_2(f) = \int |\Ls f|^2 \gtrsim \int \frac{|\As f|^2}{y^2} \gtrsim \int \frac{|\As f|^2}{y^2} + \int \frac{f^2}{y^4}.$$

Note that we had to use the orthogonality condition $\left< f, \Phi_M\right>$ when $\hbar = 0$. In fact, the case $\hbar = 0$ only happens when $d = 7$. In this case, the condition $2 > d - 2\gamma - 2$ is fulfilled when applying Lemma <ref> with $i = 1$ and $p = 0$.

We now assume the claim for $k \geq 0$ and prove it for $k + 1$. We have the orthogonality condition

$$\left<f, \Ls^{m}\Phi_M\right> = 0, \quad 0 \leq m \leq k + 1-\hbar.$$

Let $g = \Ls f$, then we have

$$\left<g, \Ls^m \Phi_M\right> = 0, \quad 0 \leq m \leq k - \hbar.

By induction hypothesis, we write

\begin{align*}
\int |\Ls ^{k+2} f|^2 &= \int |\Ls^{k+1} g|^2\\
&\gtrsim \int \frac{|\As (\Ls^k g)|^2}{y^2} + \sum_{m = 0}^k \int \frac{|\Ls^m g|^2}{y^4(1 + y^{4(k-m)})} + \sum_{m = 0}^{k-1}\frac{|\As(\Ls^{m}g)|^2}{y^6(1 + y^{4(k - m - 1)})}\\
&= \int \frac{|\As (\Ls^{k+1}f)|^2}{y^2} + \sum_{m = 1}^{k+1} \int \frac{|\Ls^m f|^2}{y^4(1 + y^{4(k + 1 -m)})} + \sum_{m = 1}^{k}\frac{|\As(\Ls^{m}f)|^2}{y^6(1 + y^{4(k - m)})}.
\end{align*}

Note that we have the orthogonality condition $\left<f, \Phi_M\right> = 0$ when $k \geq \hbar - 1$. The case $k \leq \hbar - 2$ implies that

$$4 + 4k \leq 4 + 4\left(\frac{d}{4} - \frac{\gamma}{2} - \delta\right) - 8 \leq d - 2\gamma - 4.$$

Hence, we use the coercivity bound (<ref>) to derive

$$\int \frac{|\Ls f|^2}{y^4(1 + y^{4k})} \gtrsim \int \frac{|\As f|^2}{y^6(1  +y^{4k})} + \int \frac{f^2}{y^4(1 + y^{4k + 4})},$$

which concludes the proof of Lemma <ref>.

§ INTERPOLATION BOUNDS.

We derive in this section interpolation bounds on $q$ which are the consequence of the coercivity property given in Lemma <ref>. We have the following:

$(i)$ Weighted bounds for $q_i$: for $1 \leq m \leq \Bbbk$,

\begin{equation}\label{eq:qmbyE2k}
\int |q_{2m}|^2 + \sum_{i = 0}^{2k - 1}\int \frac{|q_i|^2}{y^2(1 + y^{4m - 2i - 2})} \leq C(M)\Es_{2m}.
\end{equation}

$(ii)$ Development near the origin:

\begin{equation}\label{eq:expandqat0}
q = \sum_{i = 1}^{\Bbbk}c_i T_{\Bbbk - i} + r_q,
\end{equation}

with bounds

$$|c_i| \lesssim \sqrt{\Es_{2\Bbbk}},$$

$$|\py^j r_q| \lesssim y^{2\Bbbk - \frac{d}{2} - j}|\ln(y)|^{\Bbbk}\sqrt{\Es_{2\Bbbk}}, \quad 0 \leq j \leq 2\Bbbk - 1, \;\; y < 1.$$

$(iii)$ Bounds near the origin for $q_i$ and $\py^i q$: for $y \leq \frac 12$,

\begin{align*}
|q_{2i}| + |\py^{2i}q| &\lesssim y^{-\frac{d}{2} + 2}|\ln y|^\Bbbk\sqrt{\Es_{2\Bbbk}}, \quad \text{for}\quad 0 \leq i \leq \Bbbk - 1,\\
|q_{2i - 1}| + |\py^{2i-1}q|& \lesssim y^{-\frac{d}{2} + 1}|\ln y|^\Bbbk\sqrt{\Es_{2\Bbbk}}, \quad \text{for} \quad 1 \leq i \leq \Bbbk.
\end{align*}

$(iv)$ Weighted bounds for $\py^iq$: for $1 \leq m \leq \Bbbk$,

\begin{equation}\label{eq:weipykq}
\sum_{i = 0}^{2m}\int \frac{|\py^i q|^2}{1 + y^{4m - 2i}} \lesssim \Es_{2m}.
\end{equation}

Moreover, let $(i,j) \in \mathbb{N}\times \mathbb{N}^*$ with $2 \leq i + j \leq 2\Bbbk$, then

\begin{equation}\label{eq:weipyijq}
\int \frac{|\py^i q|^2}{1 + y^{2j}} \lesssim \left\{\begin{array}{ll}
\Es_{2m} &\quad \text{for}\quad i+j = 2m, \;\; 1 \leq m \leq \Bbbk,\\
\sqrt{\Es_{2m}}\sqrt{\Es_{2(m+1)}}&\quad \text{for}\quad  i+j = 2m + 1,\;\; 1 \leq m \leq \Bbbk - 1.
\end{array} \right.
\end{equation}

$(v)$ Pointwise bound far away. Let $(i,j) \in \mathbb{N}\times \mathbb{N}$ with $1 \leq i + j \leq 2\Bbbk - 1$, we have for $y \geq 1$,

\begin{equation}\label{eq:pointwise_yg1}
\left|\frac{\py^i q}{y^{j}}\right|^2 \lesssim \frac 1{y^{d-2}}\left\{\begin{array}{ll}
\Es_{2m} &\quad \text{for}\quad i+j + 1 = 2m, \;\; 1 \leq m \leq \Bbbk,\\
\sqrt{\Es_{2m}}\sqrt{\Es_{2(m+1)}}&\quad \text{for}\quad  i+j = 2m,\;\; 1 \leq m \leq \Bbbk - 1.
\end{array} \right.
\end{equation}

$(i)$ The estimate (<ref>) directly follows from Lemma <ref>.

$(ii)$ We claim that for $1 \leq m \leq \Bbbk$, $q_{2\Bbbk -2m}$ admits the Taylor expansion at the origin

\begin{equation}\label{eq:expandq2k}
q_{2\Bbbk - 2m} = \sum_{i = 1}^m c_{i,m}T_{m - i} + r_{2m},
\end{equation}

with the bounds

$$|c_{i,m}| \lesssim \sqrt{\Es_{2\Bbbk}},$$

$$|\py^j r_{2m}| \lesssim y^{2m - \frac{d}{2} - j}|\ln(y)|^m\sqrt{\Es_{2\Bbbk}}, \quad 0 \leq j \leq 2m - 1, \;\; y < 1,$$

The expansion (<ref>) then follows from (<ref>) with $m = \Bbbk$.

We proceed by induction in $m$ for the proof of (<ref>). For $m = 1$, we write from the definition (<ref>) of $\As^*$,

$$r_1(y) = q_{2\Bbbk - 1}(y) = \frac{1}{y^{d-1}\Lambda Q}\int_0^y q_{2\Bbbk} \Lambda Q x^{d-1}dx + \frac{d_1}{y^{d-1}\Lambda Q}.$$

Note from (<ref>) that $\int \frac{|q_{2\Bbbk - 1}|^2}{y^2} \lesssim \Es_{2\Bbbk}$ and from (<ref>) that $\Lambda Q \sim y$ as $y \to 0$, we deduce that $d_1 = 0$. Using the Cauchy-Schwartz inequality, we derive the pointwise estimate

$$|r_1(y)| \leq \frac{1}{y^d} \left(\int_0^y |q_{2\Bbbk}|^2 x^{d-1}dx\right)^\frac{1}{2}\left(\int_0^y x^2x^{d-1} dx\right)^\frac{1}{2} \lesssim y^{-\frac{d}{2} + 1}\sqrt{\Es_{2\Bbbk}}, \quad y < 1.$$

We remark that there exists $a \in (1/2,1)$ such that

$$|q_{2\Bbbk - 1}(a)|^2 \lesssim \int_{y \leq 1} |q_{2\Bbbk - 1}|^2 \lesssim \Es_{2\Bbbk}.$$

We then define

$$r_2(y) = -\Lambda Q \int_a^y \frac{r_1}{\Lambda Q}dx,$$

and obtain from the pointwise estimate of $r_1$,

$$|r_2(y)| \lesssim y y^{-\frac{d}{2} + 1}\sqrt{\Es_{2\Bbbk}}\int_a^y \frac{dx}{x} \lesssim y^{-\frac{d}{2} + 2}|\ln(x)|\sqrt{\Es_{2\Bbbk}}, \quad y < 1.$$

By construction and the definition (<ref>) of $\As$, we have

$$\As r_2 = r_1 = q_{2\Bbbk - 1}, \quad \Ls r_2 = \As^*q_{2\Bbbk-1} = q_{2\Bbbk} = \Ls q_{2\Bbbk - 2}.$$

Recall that $\text{span}(\Ls) = \{\Lambda Q, \Gamma\}$ where $\Gamma$ admits the singular behavior (<ref>). From (<ref>), we have $\int\frac{|q_{2\Bbbk - 2}|^2}{y^4} \lesssim \Es_{2\Bbbk} < +\infty$. This implies that there exists $c_2 \in \Rb$ such that

$$q_{2\Bbbk - 2} = c_2 \Lambda Q + r_2.$$

Moreover, there exists $a \in (1/2,1)$ such that

$$|q_{2\Bbbk - 2}(a)|^2 \lesssim \int_{|y| \leq 1} |q_{2\Bbbk - 2}|^2\lesssim \Es_{2\Bbbk},$$

which follows

$$|c_2| \lesssim \sqrt{\Es_{2\Bbbk}}, \quad |q_{2\Bbbk - 2}| \lesssim y^{-\frac{d}{2} + 2}|\ln(y)|\sqrt{\Es_{2\Bbbk}}, \quad y < 1.$$

Since $\As r_2 = r_1$, we then write from the definition (<ref>) of $\As$,

$$|\py r_2| \lesssim |r_1| + \left|\frac{r_2}{y}\right| \lesssim y^{-\frac{d}{2} +2}|\ln(y)|\sqrt{\Es_{2\Bbbk}}, \quad y < 1.$$

This concludes the proof of (<ref>) for $m = 1$.

We now assume that (<ref>) holds for $m \geq 1$ and prove it for $m + 1$. The term $r_{2m}$ is built as follows:

$$r_{2m - 1} = \frac{1}{y^{d-1}\Lambda Q}\int_0^y r_{2m - 2} \Lambda Q x^{d-1}dx, \quad r_{2m} = - \Lambda Q\int_a^y \frac{r_{2m -1}}{\Lambda Q} dx, \quad a\in (1/2,1).$$

We now use the induction hypothesis to estimate

\begin{align*}
|r_{2m + 1}| &= \left|\frac{1}{y^{d-1} \Lambda Q}\int_0^y r_{2m} \Lambda Q x^{d-1}dx \right| \\
&\quad \lesssim \frac{1}{y^d}\sqrt{\Es_{2\Bbbk}}\int_0^yx^{2m + \frac{d}{2}}|\ln(x)|^m dx\\
&\qquad \lesssim y^{2m - \frac{d}{2}} \sqrt{\Es_{2\Bbbk}}\int_0^y |\ln(x)|^m dx \\
& \qquad \quad  \lesssim y^{2m - \frac{d}{2} + 1}|\ln(y)|^m \sqrt{\Es_{2\Bbbk}}.
\end{align*}

Here we used the following identity

$$I_m = \int_0^y [\ln(x)]^mdx \lesssim y|\ln(y)|^m, \quad m \geq 1, \;\; y < 1.$$

Indeed, we have $I_1 = \int_0^y \ln (x) dx = y\ln(y) - y \lesssim y|\ln(y)|$ for $y < 1$. Assume the claim for $m \geq 1$, we use an integration by parts to estimate for $m +1$

\begin{align*}
I_{m+1} &= \int_0^y[\ln(x)]^m (x\ln(x) - x)'dx\\
&\quad  = y[\ln(y)]^{m+1} - y[\ln(y)]^m - m(I_m - I_{m-1})\lesssim y|\ln(y)|^{m+1}.
\end{align*}

Using an integration by parts yields $\int_{a}^y \frac{[\ln(x)]^m}{x}dx = \frac{[\ln(y)]^{m+1} - [\ln(a)]^{m+1}}{m+1}$. Hence, we have the estimate

\begin{align*}
|r_{2m + 2}| = \left|\Lambda Q\int_a^y \frac{r_{2m +1}}{\Lambda Q} dx\right| &\lesssim y^{2m - \frac{d}{2} + 2}\sqrt{\Es_{2\Bbbk}}\int_a^y \frac{|\ln(x)|^m}{x}dx\\
&\quad \lesssim y^{2m - \frac{d}{2} + 2} |\ln(y)|^{m+1} \sqrt{\Es_{2\Bbbk}}.
\end{align*}

By construction, we have

$$\As r_{2m+2} = r_{2m + 1}, \quad \Ls r_{2m+2} = r_{2m}.$$

From the induction hypothesis and the definition (<ref>) of $T_k$, we write

$$\Ls q_{2\Bbbk - 2(m + 1)} = q_{2\Bbbk - 2m} = \sum_{i = 1}^m c_{i,m}T_{m - i} + r_{2m} = \sum_{i = 1}^mc_{i,m}\Ls T_{m + 1 - i} + \Ls r_{2m + 2}.$$

The singularity (<ref>) of $\Gamma$ at the origin and the bound $\int_{y \leq 1} \frac{|q_{2 \Bbbk - 2(m+1)}|^2}{y^4} \lesssim \Es_{2\Bbbk}$ allows us to deduce that

$$q_{2\Bbbk - 2(m + 1)} = \sum_{i=1}^m c_{i,m}T_{m +1 - i} + c_{2m + 2}\Lambda Q + r_{2m + 2}.$$

From (<ref>), we see that there exists $a \in (1/2,1)$ such that

$$|q_{2\Bbbk - 2(m+1)}(a)|^2 \lesssim \int_{y \leq 1}|q_{2\Bbbk - 2(m+1)}|^2 \lesssim \Es_{2\Bbbk}.$$

Together with the induction hypothesis $|c_{i, m}| \lesssim \sqrt{\Es_{2\Bbbk}}$ and the pointwise estimate on $r_{2m + 2}$, we get the bound $|c_{2m + 2}| \leq \sqrt{\Es_{2\Bbbk}}$.

A brute force computation using the definitions of $\As$ and $\As^*$ and the asymptotic behavior (<ref>)  ensure that for any function $f$,

\begin{equation}\label{eq:pyjf}
\py^j f = \sum_{i = 0}^j P_{i,j}f_i, \quad |P_{i,j}| \lesssim \frac{1}{y^{j - i}},
\end{equation}

and we estimate

\begin{align*}
|\py^j r_{2m + 2}|&\lesssim \sum_{i = 0}^j \frac{|r_{2m + 2 - i}|}{y^{j-i}}\\
&\quad \lesssim \sqrt{\Es_{2\Bbbk}}\sum_{i = 0}^j \frac{y^{2m + 2 - i - \frac{d}{2}}|\ln(y)|^{m+1}}{y^{j - i}} \lesssim y^{2m + 2 -\frac{d}{2} - j}|\ln(y)|^{m+1} \sqrt{\Es_{2\Bbbk}}.
\end{align*}

This concludes the proof of (<ref>) as well as (<ref>).

$(iii)$ The proof of $(iii)$ directly follows from (<ref>).

$(iv)$ We have from (<ref>),

$$|\py^kq| \lesssim \sum_{j = 0}^k \frac{|q_j|}{y^{k - j}},$$

and thus, using (<ref>) and the pointwise bounds given in part $(iii)$ yields

\begin{align*}
\sum_{i = 0}^{2m}\int \frac{|\py^i q|^2}{1 + y^{4m - 2i}} & \lesssim \Es_{2m} + \sum_{i = 0}^{2m - 1} \int_{y < 1} |\py^i q|^2 + \sum_{i = 0}^{2m - 1}\int_{y > 1}\frac{|\py^i q|^2}{y^{4m - 2i}}\\
&\quad \lesssim \Es_{2m} + \Es_{2\Bbbk}\int_{y< 1}y|\ln y|^\Bbbk dy  + \sum_{i = 0}^{2m - 1} \sum_{j = 0}^i \int_{y > 1}\frac{|q_j|^2}{y^{4m - 2j}} \lesssim \Es_{2m},
\end{align*}

which concludes the proof of (<ref>).

The estimate (<ref>) simply follows from (<ref>). Indeed, if $i + j = 2m$ with $1 \leq m \leq \Bbbk$, we have

$$\int \frac{|\py^i q|^2}{1 + y^{2j}} = \int \frac{|\py^iq|^2}{1 + y^{4m - 2i}} \lesssim \Es_{2m}.$$

If $i + j = 2m + 1$ with $1 \leq m \leq \Bbbk - 1$, we write

\begin{align*}\int \frac{|\py^i q|^2}{1 + y^{2j}} = \int \frac{|\py^iq|^2}{1 + y^{4m - 2i + 2}} &\lesssim \left(\int \frac{|\py^iq|^2}{1 + y^{4m - 2i}} \right)^\frac 12\left(\int \frac{|\py^iq|^2}{1 + y^{4m - 2i + 4}} \right)^\frac 12\\
&\quad \lesssim \sqrt{\Es_{2m}} \sqrt{\Es_{2(m+1)}}.
\end{align*}

$(v)$ Let $i,j \geq 0$ with $1 \leq i + j \leq 2\Bbbk -1$, then $2 \leq i + j + 1 \leq 2\Bbbk$ and we conclude from (<ref>) that for $y \geq 1$,

\begin{align*}
\left|\frac{\py^i q}{y^j} \right|^2 &\lesssim \left| \int_y^{+\infty} \partial_x \left(\frac{(\partial_x^i q)^2}{x^{2j}} \right)dx \right| \lesssim \frac{1}{y^{d-2}} \left\{\int_y^{+\infty} \frac{|\partial_x^i q|^2}{x^{2j + 2}} + \int_y^{+\infty} \frac{|\partial_x^{i+1} q|^2}{x^{2j}} \right\}\\
&\quad \lesssim \frac 1{y^{d-2}}\left\{\begin{array}{ll}
\Es_{2m} &\quad \text{for}\quad i+j + 1 = 2m, \;\; 1 \leq m \leq \Bbbk,\\
\sqrt{\Es_{2m}}\sqrt{\Es_{2(m+1)}}&\quad \text{for}\quad  i+j+1 = 2m + 1,\;\; 1 \leq m \leq \Bbbk - 1.
\end{array} \right.
\end{align*}

This ends the proof of Lemma <ref>.

§ PROOF OF (<REF>).

We give here the proof of (<ref>). Before going to the proof, we need the following Leibniz rule for $\Ls^k$.
Let $\phi$ be a smooth function and $k \in \mathbb{N}$, we have

\begin{equation}\label{eq:LeibnizLk}
\Ls^{k + 1}(\phi f) = \sum_{m = 0}^{k+1}f_{2m}\phi_{2k+2, 2m} + \sum_{m = 0}^k f_{2m + 1}\phi_{2k + 2, 2m + 1},
\end{equation}

\begin{equation}\label{eq:LeibnizALk}
\As\Ls^{k}(\phi f) = \sum_{m = 0}^{k}f_{2m + 1}\phi_{2k+1, 2m + 1} + \sum_{m = 0}^k f_{2m}\phi_{2k+1, 2m},
\end{equation}

- for $k = 0$,

\begin{align*}
&\phi_{1,0} = -\py \phi, \quad \phi_{1,1} = \phi,\\
&\phi_{2,0} = -\py^2 \phi - \frac{d-1 +2V}{y}\py \phi, \quad \phi_{2,1}= 2\py \phi, \quad \phi_{2,2} = \phi,
\end{align*}

- for $k \geq 1$

\begin{align*}
&\phi_{2k + 1, 0} = -\py \phi_{2k,0},\\
&\phi_{2k + 1, 2i} = - \py \phi_{2k, 2i} - \phi_{2k, 2i - 1}, \quad 1 \leq i \leq k,\\
&\phi_{2k+1, 2i + 1}= \phi_{2k, 2i} + \frac{d-1 + 2V}{y}\phi_{2k, 2i + 1} - \py\phi_{2k, 2i+1}, \quad 0 \leq i \leq k-1,\\
&\phi_{2k + 1, 2k + 1} = \phi_{2k, 2k} = \phi,\\
& \quad\\
&\phi_{2k + 2, 0} = \py \phi_{2k+1,0} + \frac{d-1 + 2V}{y}\phi_{2k+1, 0},\\
&\phi_{2k + 2, 2i} = \phi_{2k + 1, 2i - 1} + \py \phi_{2k + 1, 2i} + \frac{d-1 + 2V}{y}\phi_{2k + 1, 2i},\quad 1 \leq i \leq k,\\
&\phi_{2k+2, 2i + 1}= -\phi_{2k + 1,2i} + \py\phi_{2k+1, 2i + 1}, \quad 0 \leq i \leq k,\\
&\phi_{2k + 2, 2k + 2} = \phi_{2k + 1, 2k + 1} = \phi.
\end{align*}

We use the following relation,

\begin{align*}
\As(\phi f) = \phi \As f - \py \phi f, \quad  \As^*(\phi f) = \phi \As^* f + \py \phi f, \quad \As f + \As^*f = \frac{d-1 + 2V}{y}f,
\end{align*}

to compute

\begin{align*}
\As(\phi f) &= f_1 \phi + f(-\py \phi),\\
\Ls(\phi f) &= \As^*\As(\phi f) = f_2 \phi + f_1(2\py \phi) + f\left(-\py^2 \phi - \frac{d-1 + 2V}{y}\py \phi\right),
\end{align*}

which is the conclusion of (<ref>) and (<ref>) for $k = 0$.

Assume that (<ref>) and (<ref>) hold for $k \in \mathbb{N}$,  let us compute for $k \to k+1$. Using (<ref>), we write

\begin{align*}
\As\Ls^{k+1}(\phi f) &= \sum_{m = 0}^{k+1}\As \big[f_{2m}\phi_{2k+2, 2m}\big] + \sum_{m = 0}^k \left[-\As^* + \frac{d-1 +2V}{y}\right]f_{2m + 1}\phi_{2k + 2, 2m + 1}\\
&= \sum_{m = 0}^{k+1}\big\{f_{2m + 1} \phi_{2k + 2, 2m} + f_{2m}(-\py \phi_{2k+2, 2m})\big\}\\
& \quad + \sum_{m = 0}^k \Big\{f_{2m+2} (-\phi_{2k + 2, 2m + 1}) + f_{2m + 1} (-\py \phi_{2k + 2, 2m + 1})\\
&\qquad \qquad \qquad \qquad  \left.+ f_{2m + 1}\left(\frac{d-1 + 2V}{y}\phi_{2k + 2, 2m + 1}\right)\right\}\\
& = \sum_{m = 0}^k f_{2m + 1} \left(\phi_{2k + 2, 2m} - \py\phi_{2k + 2, 2m+1} + \frac{d-1 + 2V}{y} \phi_{2k+2, 2m + 1}\right)\\
& \quad + \sum_{m = 1}^k f_{2m}\left(-\py \phi_{2k+2, 2m} - \phi_{2k + 2, 2m + 1}\right) + f_{2k + 3} \phi_{2k+2, 2k + 2} + f (-\py \phi_{2k + 2, 0}),
\end{align*}

which yields the recurrence relation for $\phi_{2k + 3, j}$ with $0 \leq j \leq 2k + 3$.

Similarly, we write $\Ls^{k + 2}(\phi f) = \As^*\big[\As \Ls^{k+1}(\phi f)\big]$ and use the formula (<ref>) with $k+1$ to obtain the recurrence relation for $\phi_{2k + 4, j}$ with $0 \leq j \leq 2k + 4$. This concludes the proof of Lemma <ref>.

Let us now give the proof of (<ref>). By induction and the definition (<ref>), we have

$$[\pt, \Ls_\lambda^{\Bbbk - 1}]v = \sum_{m = 0}^{\Bbbk - 2} \Ls_\lambda^m \left([\pt, \Ls_\lambda] \Ls_\lambda^{\Bbbk - 2 - m}v\right) = \sum_{m = 0}^{\Bbbk - 2} \Ls_\lambda^m \left(\frac{\pt Z_\lambda}{r^2} \Ls_\lambda^{\Bbbk - 2 - m}v\right).$$

Noting that $\frac{\pt Z_\lambda}{r^2} = \frac{b_1 \Lambda Z}{\lambda^4 y^2}$, we make a change of variables to obtain

\begin{align*}
\int \frac{1}{\lambda^2(1 + y^2)}\left|[\pt , \Ls_{\lambda}^{\Bbbk - 1}]v\right|^2 &= \frac{b_1^2}{\lambda^{4\Bbbk - d + 2}}\int \frac{1}{1 + y^2} \left|\sum_{m = 0}^{\Bbbk - 2}\Ls^m\left(\frac{\Lambda Z}{y^2} \Ls^{\Bbbk - 2 - m}q \right) \right|^2\\
& \lesssim \frac{b_1^2}{\lambda^{4\Bbbk - d + 2}} \sum_{m = 0}^{\Bbbk - 2}\int \frac{1}{1 + y^2}\left|\Ls^m\left(\frac{\Lambda Z}{y^2} \Ls^{\Bbbk - 2 - m}q \right) \right|^2.
\end{align*}

For $m = 0$, we use (<ref>) and (<ref>) to estimate

$$\int \frac{1}{1 + y^2}\left|\left(\frac{\Lambda Z}{y^2} \Ls^{\Bbbk - 2}q \right) \right|^2 \lesssim \int \frac{|q^2_{2\Bbbk - 4}|}{1 + y^{10}} \lesssim \Es_{2\Bbbk}.$$

For $m = 1, \cdots, \Bbbk - 2$, we apply (<ref>) with $\phi = \frac{\Lambda Z}{y^2} = \frac{(d-1)\Lambda \cos(2Q)}{y^2}$ and note from (<ref>) that

$$|\phi_{k, i}| \lesssim \frac{1}{1 + y^{2\gamma + 2 + (2k - i)}} \lesssim \frac{1}{1 + y^{4 + (2k - i)}}, \quad k \in \mathbb{N}^*, \;\; 0 \leq i \leq 2k,$$

which yields

\begin{align*}
\int \frac{1}{1 + y^2}\left|\Ls^m\left(\frac{\Lambda Z}{y^2} \Ls^{\Bbbk - 2 - m}q \right) \right|^2 \lesssim \sum_{i = 0}^{2m}\int\frac{q^2_{2\Bbbk - 4 - 2m - i}}{(1 + y^{10 + (4m - 2i)})} \lesssim \Es_{2\Bbbk}.
\end{align*}

$$\int \frac{1}{\lambda^2(1 + y^2)}\left|[\pt , \Ls_{\lambda}^{\Bbbk - 1}]v\right|^2 \lesssim \frac{b_1^2}{\lambda^{4\Bbbk - d + 2}} \Es_{2\Bbbk}.$$

Similarly, we use (<ref>) to get the estimate

$$\int \left|\As[\pt , \Ls_{\lambda}^{\Bbbk - 1}]v\right|^2 \lesssim  \frac{b_1^2}{\lambda^{4\Bbbk - d + 2}}\Es_{2\Bbbk}.$$

This concludes the proof of (<ref>).

[Biernat, 2015]
P. Biernat.
Non-self-similar blow-up in the heat flow for harmonic maps in higher
Nonlinearity, 280 (1):0 167–185, 2015.
ISSN 0951-7715.
URL <http://dx.doi.org/10.1088/0951-7715/28/1/167>.

[Biernat and Bizoń, 2011]
P. Biernat and P. Bizoń.
Shrinkers, expanders, and the unique continuation beyond generic
blowup in the heat flow for harmonic maps between spheres.
Nonlinearity, 240 (8):0 2211–2228, 2011.
ISSN 0951-7715.
URL <http://dx.doi.org/10.1088/0951-7715/24/8/005>.

[Biernat and Seki, 2016]
P. Biernat and Y. Seki.
Type II blow-up mechanism for supercritical harmonic map heat flow.
arXiv:1601.01831, 2016.
URL <http://arxiv.org/abs/1601.01831>.

[Biernat et al., 2016]
P. Biernat, R. Donninger, and B. Schörkhuber.
Stable self-similar blowup in the supercritical heat flow of harmonic
arXiv:1610.09497, 2016.

[Bizoń and Wasserman, 2015]
P. Bizoń and A. Wasserman.
Nonexistence of shrinkers for the harmonic map flow in higher
Int. Math. Res. Not. IMRN, 0 (17):0
7757–7762, 2015.
ISSN 1073-7928.
URL <http://dx.doi.org/10.1093/imrn/rnu176>.

[Bressan, 1990]
A. Bressan.
On the asymptotic shape of blow-up.
Indiana Univ. Math. J., 390 (4):0 947–960,
ISSN 0022-2518.
URL <http://dx.doi.org/10.1512/iumj.1990.39.39045>.

[Bressan, 1992]
A. Bressan.
Stable blow-up patterns.
J. Differential Equations, 980 (1):0 57–75,
ISSN 0022-0396.
URL <http://dx.doi.org/10.1016/0022-0396(92)90104-U>.

[Bricmont and Kupiainen, 1994]
J. Bricmont and A. Kupiainen.
Universality in blow-up for nonlinear heat equations.
Nonlinearity, 70 (2):0 539–575, 1994.
ISSN 0951-7715.
URL <http://stacks.iop.org/0951-7715/7/539>.

[C., 2016]
Collot C.
Non radial type II blow up for the energy supercritical semilinear
heat equation.
arXiv:1604.02856, 2016a.
URL <http://arxiv.org/abs/1604.02856>.

[C., 2016]
Collot C.
Type II blow up manifolds for the energy supercritical wave
arXiv:1407.4525, 2016b.
URL <http://arxiv.org/abs/1407.4525>.

[C. et al., 2016]
Collot C., Merle F., and Raphaël P.
Dynamics near the ground state for the energy critical nonlinear heat
equation in large dimensions.
arXiv:1604.08323, 2016.
URL <http://arxiv.org/abs/1604.08323>.

[Chang et al., 1992]
K.-C. Chang, W. Y. Ding, and R. Ye.
Finite-time blow-up of the heat flow of harmonic maps from surfaces.
J. Differential Geom., 360 (2):0 507–515,
ISSN 0022-040X.
URL <http://projecteuclid.org/euclid.jdg/1214448751>.

[Chen and Ding, 1990]
Y. M. Chen and W. Y. Ding.
Blow-up and global existence for heat flows of harmonic maps.
Invent. Math., 990 (3):0 567–578, 1990.
ISSN 0020-9910.
URL <http://dx.doi.org/10.1007/BF01234431>.

[Coron and Ghidaglia, 1989]
J.-M. Coron and J.-M. Ghidaglia.
Explosion en temps fini pour le flot des applications harmoniques.
C. R. Acad. Sci. Paris Sér. I Math., 3080
(12):0 339–344, 1989.
ISSN 0249-6291.

[Côte and Zaag, 2013]
R. Côte and H. Zaag.
Construction of a multisoliton blowup solution to the semilinear wave
equation in one space dimension.
Comm. Pure Appl. Math., 660 (10):0 1541–1581,
ISSN 0010-3640.
URL <http://dx.doi.org/10.1002/cpa.21452>.

[Duyckaerts et al., 2013]
T. Duyckaerts, C. Kenig, and F. Merle.
Classification of radial solutions of the focusing, energy-critical
wave equation.
Camb. J. Math., 10 (1):0 75–144, 2013.
ISSN 2168-0930.
URL <http://dx.doi.org/10.4310/CJM.2013.v1.n1.a3>.

[Eells and Sampson, 1964]
J. Eells and J. H. Sampson.
Harmonic mappings of Riemannian manifolds.
Amer. J. Math., 86:0 109–160, 1964.
ISSN 0002-9327.

[Fan, 1999]
H. Fan.
Existence of the self-similar solutions in the heat flow of harmonic
Sci. China Ser. A, 420 (2):0 113–132, 1999.
ISSN 1006-9283.
URL <http://dx.doi.org/10.1007/BF02876563>.

[Filippas et al., 2000]
S. Filippas, M. A. Herrero, and J. J. L. Velázquez.
Fast blow-up mechanisms for sign-changing solutions of a semilinear
parabolic equation with critical nonlinearity.
R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 4560
(2004):0 2957–2982, 2000.
ISSN 1364-5021.
URL <http://dx.doi.org/10.1098/rspa.2000.0648>.

[Germain and Rupflin, 2011]
P. Germain and M. Rupflin.
Selfsimilar expanders of the harmonic map flow.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 280
(5):0 743–773, 2011.
ISSN 0294-1449.
URL <http://dx.doi.org/10.1016/j.anihpc.2011.06.004>.

[Germain et al., 2016]
P. Germain, T. Ghoul, and H. Miura.
On uniqueness for the harmonic map heat flow in supercritical
arXiv:1601.06601, 2016.

[Ghoul and Masmoudi, 2016]
T. Ghoul and N. Masmoudi.
Stability of infinite time blow up for the Patlak Keller Segel
arXiv:1610.00456, 2016.

[Ghoul et al., 2016]
T. Ghoul, V. T. Nguyen, and H. Zaag.
Construction and stability of blowup solutions for a non-variational
parabolic system.
arXiv:1610.09883, 2016.
URL <https://arxiv.org/abs/1610.09883>.

[Ghoul et al., 2017]
T. Ghoul, V. T. Nguyen, and H. Zaag.
Blowup solutions for the exponential reaction-diffusion equation
involving a critical power nonlinear gradient term.
J. Differential Equation, to appear, 2017.
URL <https://arxiv.org/abs/1611.02474>.

[Giga and Kohn, 1987]
Y. Giga and R. V. Kohn.
Characterizing blowup using similarity variables.
Indiana Univ. Math. J., 360 (1):0 1–40, 1987.
ISSN 0022-2518.
URL <http://dx.doi.org/10.1512/iumj.1987.36.36001>.

[Giga et al., 2004]
Y. Giga, S. Matsui, and S. Sasayama.
Blow up rate for semilinear heat equations with subcritical
Indiana Univ. Math. J., 530 (2):0 483–514,
ISSN 0022-2518.
URL <http://dx.doi.org/10.1512/iumj.2004.53.2401>.

[Herrero and Velázquez, 1994]
Miguel A. Herrero and Juan J. L. Velázquez.
Explosion de solutions d'équations paraboliques semilinéaires
C. R. Acad. Sci. Paris Sér. I Math., 3190
(2):0 141–145, 1994.
ISSN 0764-4442.

[Hillairet and Raphaël, 2012]
M. Hillairet and P. Raphaël.
Smooth type II blow-up solutions to the four-dimensional
energy-critical wave equation.
Anal. PDE, 50 (4):0 777–829, 2012.
ISSN 2157-5045.
URL <http://dx.doi.org/10.2140/apde.2012.5.777>.

[Jost, 1981]
J. Jost.
Ein Existenzbeweis für harmonische Abbildungen, die ein
Dirichletproblem lösen, mittels der Methode des Wärmeflusses.
Manuscripta Math., 340 (1):0 17–25, 1981.
ISSN 0025-2611.
URL <http://dx.doi.org/10.1007/BF01168706>.

[Koch and Lamm, 2012]
H. Koch and T. Lamm.
Geometric flows with rough initial data.
Asian J. Math., 160 (2):0 209–235, 2012.
ISSN 1093-6106.
URL <http://dx.doi.org/10.4310/AJM.2012.v16.n2.a3>.

[Lin and Wang, 2008]
F. Lin and C. Wang.
The analysis of harmonic maps and their heat flows.
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
ISBN 978-981-277-952-6; 981-277-952-3.
URL <http://dx.doi.org/10.1142/9789812779533>.

[Martel et al., 2014]
Y. Martel, F. Merle, and P. Raphaël.
Blow up for the critical generalized Korteweg–de Vries equation.
I: Dynamics near the soliton.
Acta Math., 2120 (1):0 59–140, 2014.
ISSN 0001-5962.
URL <http://dx.doi.org/10.1007/s11511-014-0109-2>.

[Martel et al., 2015]
Y. Martel, F. Merle, and P. Raphaël.
Blow up for the critical gKdV equation III: exotic regimes.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 140
(2):0 575–631, 2015a.
ISSN 0391-173X.

[Martel et al., 2015]
Y. Martel, F. Merle, and P. Raphaël.
Blow up for the critical gKdV equation. II: Minimal mass
J. Eur. Math. Soc. (JEMS), 170 (8):0
1855–1925, 2015b.
ISSN 1435-9855.
URL <http://dx.doi.org/10.4171/JEMS/547>.

[Masmoudi and Zaag, 2008]
N. Masmoudi and H. Zaag.
Blow-up profile for the complex Ginzburg-Landau equation.
J. Funct. Anal., 2550 (7):0 1613–1666, 2008.
ISSN 0022-1236.
URL <http://dx.doi.org/10.1016/j.jfa.2008.03.008>.

[Matano and Merle, 2004]
H. Matano and F. Merle.
On nonexistence of type II blowup for a supercritical nonlinear
heat equation.
Comm. Pure Appl. Math., 570 (11):0 1494–1541,
ISSN 0010-3640.
URL <http://dx.doi.org/10.1002/cpa.20044>.

[Matano and Merle, 2009]
Hiroshi Matano and Frank Merle.
Classification of type I and type II behaviors for a
supercritical nonlinear heat equation.
J. Funct. Anal., 2560 (4):0 992–1064, 2009.
ISSN 0022-1236.
URL <http://dx.doi.org/10.1016/j.jfa.2008.05.021>.

[Merle and Raphael, 2003]
F. Merle and P. Raphael.
Sharp upper bound on the blow-up rate for the critical nonlinear
Schrödinger equation.
Geom. Funct. Anal., 130 (3):0 591–642, 2003.
ISSN 1016-443X.
URL <http://dx.doi.org/10.1007/s00039-003-0424-9>.

[Merle and Raphael, 2004]
F. Merle and P. Raphael.
On universality of blow-up profile for $L^2$ critical nonlinear
Schrödinger equation.
Invent. Math., 1560 (3):0 565–672, 2004.
ISSN 0020-9910.
URL <http://dx.doi.org/10.1007/s00222-003-0346-z>.

[Merle and Raphael, 2005]
F. Merle and P. Raphael.
The blow-up dynamic and upper bound on the blow-up rate for critical
nonlinear Schrödinger equation.
Ann. of Math. (2), 1610 (1):0 157–222,
ISSN 0003-486X.
URL <http://dx.doi.org/10.4007/annals.2005.161.157>.

[Merle and Raphael, 2005]
F. Merle and P. Raphael.
Profiles and quantization of the blow up mass for critical nonlinear
Schrödinger equation.
Comm. Math. Phys., 2530 (3):0 675–704,
ISSN 0010-3616.
URL <http://dx.doi.org/10.1007/s00220-004-1198-0>.

[Merle and Zaag, 1997]
F. Merle and H. Zaag.
Stability of the blow-up profile for equations of the type
$u_t=\Delta u+\vert u\vert ^{p-1}u$.
Duke Math. J., 860 (1):0 143–195, 1997.
ISSN 0012-7094.
URL <http://dx.doi.org/10.1215/S0012-7094-97-08605-1>.

[Merle et al., 2011]
F. Merle, P. Raphaël, and I. Rodnianski.
Blow up dynamics for smooth equivariant solutions to the energy
critical Schrödinger map.
C. R. Math. Acad. Sci. Paris, 3490 (5-6):0
279–283, 2011.
ISSN 1631-073X.
URL <http://dx.doi.org/10.1016/j.crma.2011.01.026>.

[Merle et al., 2013]
F. Merle, P. Raphaël, and I. Rodnianski.
Blowup dynamics for smooth data equivariant solutions to the critical
Schrödinger map problem.
Invent. Math., 1930 (2):0 249–365, 2013.
ISSN 0020-9910.
URL <http://dx.doi.org/10.1007/s00222-012-0427-y>.

[Merle et al., 2015]
F. Merle, P. Raphaël, and I. Rodnianski.
Type II blow up for the energy supercritical NLS.
Camb. Jour. Math., 30 (4):0 439 – 617, 2015.
URL <http://dx.doi.org/10.4310/CJM.2015.v3.n4.a1>.

[Mizoguchi, 2004]
N. Mizoguchi.
Type-II blowup for a semilinear heat equation.
Adv. Differential Equations, 90 (11-12):0
1279–1316, 2004.
ISSN 1079-9389.

[Mizoguchi, 2007]
N. Mizoguchi.
Rate of type II blowup for a semilinear heat equation.
Math. Ann., 3390 (4):0 839–877, 2007.
ISSN 0025-5831.
URL <http://dx.doi.org/10.1007/s00208-007-0133-z>.

[Nguyen and Zaag, 2016]
V. T. Nguyen and H. Zaag.
Finite degrees of freedom for the refined blow-up profile for a
semilinear heat equation.
Ann. Scient. Éc. Norm. Sup. to appear, 2016a.
URL <https://arxiv.org/abs/1509.03520>.

[Nguyen and Zaag, 2016]
V. T. Nguyen and H. Zaag.
Construction of a stable blow-up solution for a class of strongly
perturbed semilinear heat equations.
Ann. Scuola Norm. Sup. Pisa Cl. Sci., 160
(4):0 1275–1314, 2016b.
URL <http://dx.doi.org/10.2422/2036-2145.201412_001>.

[Nouaili and Zaag, 2015]
N. Nouaili and H. Zaag.
Profile for a simultaneously blowing up solution to a complex valued
semilinear heat equation.
Comm. Partial Differential Equations, 400
(7):0 1197–1217, 2015.
ISSN 0360-5302.
URL <http://dx.doi.org/10.1080/03605302.2015.1018997>.

[Raphaël and Rodnianski, 2012]
P. Raphaël and I. Rodnianski.
Stable blow up dynamics for the critical co-rotational wave maps and
equivariant Yang-Mills problems.
Publ. Math. Inst. Hautes Études Sci., pages 1–122, 2012.
ISSN 0073-8301.
URL <http://dx.doi.org/10.1007/s10240-011-0037-z>.

[Raphaël and Schweyer, 2013]
P. Raphaël and R. Schweyer.
Stable blowup dynamics for the 1-corotational energy critical
harmonic heat flow.
Comm. Pure Appl. Math., 660 (3):0 414–480,
ISSN 0010-3640.
URL <http://dx.doi.org/10.1002/cpa.21435>.

[Raphaël and Schweyer, 2014]
P. Raphaël and R. Schweyer.
Quantized slow blow-up dynamics for the corotational energy-critical
harmonic heat flow.
Anal. PDE, 70 (8):0 1713–1805,
ISSN 2157-5045.
URL <http://dx.doi.org/10.2140/apde.2014.7.1713>.

[Raphaël and Schweyer, 2014]
P. Raphaël and R. Schweyer.
On the stability of critical chemotactic aggregation.
Math. Ann., 3590 (1-2):0 267–377,
ISSN 0025-5831.
URL <http://dx.doi.org/10.1007/s00208-013-1002-6>.

[Schweyer, 2012]
R. Schweyer.
Type II blow-up for the four dimensional energy critical semi
linear heat equation.
J. Funct. Anal., 2630 (12):0 3922–3983,
ISSN 0022-1236.
URL <http://dx.doi.org/10.1016/j.jfa.2012.09.015>.

[Schweyer, 2012]
R. Schweyer.
Type II blow-up for the four dimensional energy critical semi
linear heat equation.
J. Funct. Anal., 2630 (12):0 3922–3983,
ISSN 0022-1236.
URL <http://dx.doi.org/10.1016/j.jfa.2012.09.015>.

[Struwe, 1988]
M. Struwe.
On the evolution of harmonic maps in higher dimensions.
J. Differential Geom., 280 (3):0 485–502,
ISSN 0022-040X.
URL <http://projecteuclid.org/euclid.jdg/1214442475>.

[Struwe, 1996]
M. Struwe.
Geometric evolution problems.
In Nonlinear partial differential equations in differential
geometry (Park City, UT, 1992), volume 2 of IAS/Park City Math.
Ser., pages 257–339. Amer. Math. Soc., Providence, RI, 1996.

[van den Berg et al., 2003]
J. B. van den Berg, J. Hulshof, and J. R. King.
Formal asymptotics of bubbling in the harmonic map heat flow.
SIAM J. Appl. Math., 630 (5):0 1682–1717
(electronic), 2003.
ISSN 0036-1399.
URL <http://dx.doi.org/10.1137/S0036139902408874>.

[Wang, 2011]
C. Wang.
Well-posedness for the heat flow of harmonic maps and the liquid
crystal flow with rough initial data.
Arch. Ration. Mech. Anal., 2000 (1):0 1–19,
ISSN 0003-9527.
URL <http://dx.doi.org/10.1007/s00205-010-0343-5>.

[Zaag, 1998]
H. Zaag.
Blow-up results for vector-valued nonlinear heat equations with no
gradient structure.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 150
(5):0 581–622, 1998.
ISSN 0294-1449.
URL <http://dx.doi.org/10.1016/S0294-1449(98)80002-4>.