In this section, we will derive $r$-weighted energy estimates for weighted
higher-order derivatives of $\psi$ of the form $(rL)^{k}\psi$, with
$k\in\mathbb{N}_{0}$. These will form the main tool for establishing time
decay of higher-order $T$-derivatives in the next section.

######  Lemma 7.1.

Let $N\in\mathbb{N}_{0}$ and let $\phi\in C^{\infty}(\mathcal{R})\cap
C^{1}(\widehat{\mathcal{R}})$ be a solution to (3.1). Then

$\begin{split}0=&\>-4\underline{L}L(rD^{-1}L)^{N}\psi+Dr^{-2}[\not{\Delta}_{\mathbb{S}^{2}}-\alpha+O_{\infty}(r^{-1})](rD^{-1}L)^{N}\psi-2N[r^{-1}+O_{\infty}(r^{-2})]L(rD^{-1}L)^{N}\psi\\\
&-\frac{N}{2}Dr^{-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi+N\sum_{n=0}^{N}O_{\infty}(r^{-2})(rD^{-1}L)^{n}\psi+N(N-1)\sum_{n=0}^{N-2}O_{\infty}(r^{-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi.\end{split}$
(7.1)

###### Proof.

We will prove the identity via induction. Note first of all that the $N=0$
case follows from (3.4). Now suppose (7.1) holds. Then we can multiply both
sides of the equation by $rD^{-1}$ and rearrange some terms to obtain:

$\begin{split}0=&\>-4\underline{L}(rD^{-1}L)^{N+1}\psi+r^{-1}[\not{\Delta}_{\mathbb{S}^{2}}-\alpha+O_{\infty}(r^{-1})](rD^{-1}L)^{N}\psi-2(N+1)[r^{-1}+O_{\infty}(r^{-2})](rD^{-1}L)^{N+1}\psi\\\
&-\frac{N}{2}r^{-1}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi+N\sum_{n=0}^{N}O_{\infty}(r^{-1})(rD^{-1}L)^{n}\psi+N(N-1)\sum_{n=0}^{N-2}O_{\infty}(r^{-1})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi.\end{split}$

Then, act with $L$ on both sides to obtain:

$\begin{split}0=&\>-4\underline{L}L(rD^{-1}L)^{N+1}\psi+Dr^{-2}[\not{\Delta}_{\mathbb{S}^{2}}-\alpha+O_{\infty}(r^{-1})](rD^{-1}L)^{N+1}\psi-[2(N+1)r^{-1}+O_{\infty}(r^{-2})](rD^{-1}L)^{N+1}\psi\\\
&-\frac{N+1}{2}Dr^{-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi+\sum_{n=0}^{N+1}O_{\infty}(r^{-2})(rD^{-1}L)^{n}\psi+N\sum_{n=0}^{N-1}O_{\infty}(r^{-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi.\end{split}$

Hence, (7.1) holds also with $N$ replaced by $N+1$ and the induction argument
is complete. ∎

######  Proposition 7.2 (Higher-order $r$-weighted energy estimates).

Let $N\in\mathbb{N}_{0}$ and let $\phi\in C^{\infty}(\mathcal{R})\cap
C^{1}(\widehat{\mathcal{R}})$ be a solution to (3.1). Let
$\max\\{1-\beta_{0},0\\}<p<\min\\{1+\beta_{0},2\\}$. Then, for $R>3M$ suitably
large, there exists a constant $C=C(M,h,N,R,\alpha,p)>0$ such that

$\begin{split}\sum_{0\leq n_{1}+n_{2}\leq
N}&\int_{\Sigma_{\tau_{2}}\cap\\{r\geq
R\\}}r^{p}(L(rL)^{n_{1}}T^{n_{2}}\psi)^{2}\,d\sigma dr\\\
&+\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{p-1}(L(rL)^{n_{1}}T^{n_{2}}\psi)^{2}+r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}(rL)^{n_{1}}T^{n_{2}}\psi|^{2}+r^{p-3}((rL)^{n_{1}}T^{n_{2}}\psi)^{2}\,d\sigma
drd\tau\\\ \leq&C\sum_{0\leq n_{1}+n_{2}\leq
N}\int_{\Sigma_{\tau_{1}}\cap\\{r\geq
R\\}}r^{p}(L(rL)^{n_{1}}T^{n_{2}}\psi)^{2}\,d\sigma dr+C\sum_{n\leq
N}\int_{\Sigma_{\tau_{1}}\cap\\{r\geq R\\}}r(XT^{n}\psi)^{2}\,d\sigma dr\\\
&+C\sum_{0\leq n\leq N}E_{T}[T^{n}\phi](\tau_{1}).\end{split}$ (7.2)

###### Proof.

Let $\chi:[M,\infty)\to\mathbb{R}_{\geq 0}$ be a smooth cut-off function, such
that $\chi(r)=1$ for $r\geq R$ and $\chi(r)=0$ for $r\leq 3M+\frac{R-3M}{2}$.

We multiply both sides of (7.1) with $-\chi r^{p}L(rD^{-1}L)^{N}\psi$ and
apply the Leibniz rule to obtain:

$\begin{split}0=&\>2\chi r^{p}\underline{L}((L(rD^{-1}L)^{N}\psi)^{2})-D\chi
r^{p-2}[\not{\Delta}_{\mathbb{S}^{2}}-\alpha+O_{\infty}(r^{-1})](rD^{-1}L)^{N}\psi
L(rD^{-1}L)^{N}\psi\\\
&+2N[r^{p-1}+O_{\infty}(r^{p-2})](L(rD^{-1}L)^{N}\psi)^{2}+\frac{N}{2}D\chi
r^{p-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi
L(rD^{-1}L)^{N}\psi\\\ &+N\sum_{n=0}^{N}\chi
O_{\infty}(r^{p-2})(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi+N(N-1)\sum_{n=0}^{N-2}\chi
O_{\infty}(r^{p-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi\\\ =&\>\underline{L}\left[2\chi
r^{p}(L(rD^{-1}L)^{N}\psi)^{2}\right]+L\left[\frac{1}{2}D\chi
r^{p-2}\left(|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}+(\alpha+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}\right)\right]\\\
&+\chi
r^{p-1}\left[p+2N+O_{\infty}(r^{-1})\right](L(rD^{-1}L)^{N}\psi)^{2}+\frac{1}{4}(2-p)\chi
r^{p-3}(1+O_{\infty}(r^{-1}))|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}\\\
&+\frac{1}{4}(2-p)\chi
r^{p-3}(\alpha+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}++\frac{N}{2}D\chi
r^{p-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi
L(rD^{-1}L)^{N}\psi\\\ &+N\sum_{n=0}^{N}\chi
O_{\infty}(r^{p-2})(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi+N(N-1)\sum_{n=0}^{N-2}\chi
O_{\infty}(r^{p-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi\\\
&+\textnormal{Err}_{\chi,N}+\textnormal{div}_{\mathbb{S}^{2}}(\ldots).\end{split}$
(7.3)

where $\textnormal{Err}_{\chi,N+1}$ denotes all terms multiplied by
derivatives of $\chi$ and squares of derivatives of $\phi$ up to order $N+1$.
We can further rewrite (7.3) as follows:

$\begin{split}0=&\>DX\left[\frac{1}{4}\chi
r^{p-2}\left(|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}+(\alpha+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}\right)-\chi
r^{p}(L(rD^{-1}L)^{N}\psi)^{2}\right]\\\
&+DT\left[\chi\tilde{h}r^{p}(L(rD^{-1}L)^{N}\psi)^{2}+\frac{h}{4D}\chi
r^{p-2}\left(|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}+(\alpha+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}\right)\right]\\\
&+\chi
Dr^{p-1}\left[p+2N+O_{\infty}(r^{-1})\right](L(rD^{-1}L)^{N}\psi)^{2}+\frac{1}{4}(2-p)\chi
r^{p-3}(1+O_{\infty}(r^{-1}))|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}\\\
&+\frac{1}{4}(2-p)D\chi
r^{p-3}(\alpha+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}+\frac{N}{2}D\chi
r^{p-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi
L(rD^{-1}L)^{N}\psi\\\ &+ND\sum_{n=0}^{N}\chi
O_{\infty}(r^{p-2})(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi+N(N-1)D\sum_{n=0}^{N-2}\chi
O_{\infty}(r^{p-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi\\\
&+\textnormal{Err}_{\chi,N+1}+\textnormal{div}_{\mathbb{S}^{2}}(\ldots).\end{split}$
(7.4)

We will prove (7.2) by induction. We first show that (7.2) holds for $N=0$. We
integrate (7.4) with $N=0$. We apply (2.10) to estimate

$\begin{split}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}&\frac{1}{4}(2-p)D\chi
r^{p-3}(\max\\{-\alpha,0\\}+O_{\infty}(r^{-1}))((rD^{-1}L)^{N}\psi)^{2}\,d\sigma
drd\tau\\\ \leq&\>\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\chi
r^{p-3}((2-p)^{-1}\max\\{-\alpha,0\\}+Cr^{-1})r^{p-1}(X(rD^{-1}L)^{N}\psi)^{2}\,d\sigma
drd\tau\\\
+&\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\textnormal{Err}_{\chi,N}\,d\sigma
drd\tau\\\
\leq&\>\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}4\chi((2-p)^{-1}\max\\{-\alpha,0\\}+Cr^{-1})r^{p-1}(L(rD^{-1}L)^{N}\psi)^{2}+Cr^{p-1}h^{2}(T(rD^{-1}L)^{N}\psi)^{2}\,d\sigma
drd\tau\\\
&+\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\textnormal{Err}_{\chi,N}\,d\sigma
drd\tau.\end{split}$

We can absorb the first term on the very right-hand side above into the
spacetime integral of $\chi Dpr^{p-1}(L(rD^{-1}L)^{N}\psi)^{2}$ provided that
$R>3M$ is chosen suitably large and moreover, when $\alpha<0$,
$1-\beta_{0}<p<1+\beta_{0}$. The second term on the very right-hand side,
which is only present in the $M>0$ case, can be estimated using (6.12) when
$N=0$. We moreover use (6.19) to estimate $\textnormal{Err}_{\chi,N+1}$, so
that we are left with (7.2) for $N=0$.

Now suppose we have already established (7.2) for $N$ replaced with $N-1$. We
will show that (7.2) holds also for $N$. We integrate (7.4) again and apply
(2.10) as above. In this case, however, we estimate the integral of
$r^{p-1}h^{2}(T(rD^{-1}L)^{N}\psi)^{2}$ by observing that

$r^{p-1}h^{2}(T(rD^{-1}L)^{N}\psi)^{2}\leq
Cr^{p-3}(L(rD^{-1})^{N-1}T\psi)^{2}$

and applying the induction hypothesis with $\psi$ replaced by $T\psi$, using
that $[\square_{g},T]=0$ by the Killing property of $T$. We moreover estimate

$\begin{split}ND&\sum_{n=0}^{N}\chi O_{\infty}(r^{p-2})(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi+N(N-1)D\sum_{n=0}^{N-2}\chi
O_{\infty}(r^{p-2})\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi
L(rD^{-1}L)^{N}\psi\\\ \leq&\>\epsilon\chi
r^{p-1}(L(rD^{-1}L)^{N}\psi)^{2}+\epsilon\chi
r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}+C\chi\epsilon^{-1}\sum_{n=0}^{N}r^{p-3}((rD^{-1}L)^{n}\psi)^{2}\\\
&+C\chi\sum_{n=0}^{N-2}\chi
O_{\infty}(r^{p-1})|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{n}\psi|^{2}+\textnormal{div}_{\mathbb{S}^{2}}(\ldots).\end{split}$

We can absorb the terms with a factor $\epsilon$ into $\chi
D(p+2N)r^{p-1}(L(rD^{-1}L)^{N}\psi)^{2}$ and we apply the induction hypothesis
together with (2.10) to absorb also the terms in the sums above.

We are left with estimating the spacetime integral of $\frac{N}{2}D\chi
r^{p-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi L(rD^{-1}L)^{N}\psi$.
We first apply the Leibniz rule to obtain:

$\begin{split}\frac{N}{2}&D\chi
r^{p-2}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi
L(rD^{-1}L)^{N}\psi=-L\left[\frac{N}{4}D\chi
r^{p-2}\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi\cdot\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\right]\\\
&+\frac{N}{4}D^{2}\chi
r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi|^{2}+O_{\infty}(r^{p-3})\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N-1}\psi\cdot\not{\nabla}_{\mathbb{S}^{2}}(rD^{-1}L)^{N}\psi.\end{split}$

Note that the second term on the right-hand side has a good sign, whereas the
third term can be estimated by applying Young’s inequality together with the
induction hypothesis.

The boundary integral terms along $\Sigma_{\tau_{2}}$ that are generated in
the above estimates either have a good sign or can be estimated by using the
induction hypothesis. ∎

##  8 Decay-in-time estimates for time derivatives

In this section, we extract from Proposition 7.2 a hierarchy of $r$-weighted
energy estimates, in order to obtain decay in time of ($r$-weighted) energies
for $T^{k}\phi$, with $k\in\mathbb{N}_{0}$. We will show that the larger $k$
is, the stronger the decay is that we can establish.

In the lemma below we show that we can express energies involving $T$
derivatives in terms of energies involving $L$ and
$\not{\nabla}_{\mathbb{S}^{2}}$ derivatives

######  Lemma 8.1.

Let $R>3M$. For all $p\in\mathbb{R}$, we have that there exists a constant
$C=C(R)>0$, such that in $\\{r\geq R\\}$:

$\begin{split}\sum_{n_{1}+n_{2}\leq N}&\int_{\Sigma\cap\\{r\geq
R\\}}r^{p+1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}LT(rL)^{n_{2}}\psi|^{2}\,d\sigma
dr\\\ \leq&\>C\sum_{\begin{subarray}{c}n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma\cap\\{r\geq
R\\}}r^{p-1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}\psi|^{2}+r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}+1}(rL)^{n_{2}}\psi|^{2}\,d\sigma
dr.\end{split}$ (8.1)

###### Proof.

We use that $T=L+\underline{L}$ to estimate in $\\{r\geq R\\}$:

$\begin{split}r^{p+1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}LT(rD^{-1}L)^{n_{2}}\psi|^{2}\leq&\>Cr^{p-1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rD^{-1}L)^{n_{2}+1}\psi|^{2}+Cr^{p-1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rD^{-1}L)^{n_{2}}\psi|^{2}\\\
&+r^{p+1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}\underline{L}L(rD^{-1}L)^{n_{2}}\psi|^{2}.\end{split}$

Then we apply (7.1) to estimate further:

$r^{p+1}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}\underline{L}L(rD^{-1}L)^{n_{2}}\psi|^{2}\leq
C\sum_{k=0}^{n_{2}}r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}\not{\Delta}_{\mathbb{S}^{2}}(rD^{-1}L)^{k}\psi|^{2}+C\sum_{k=0}^{n_{2}+1}r^{p-3}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rD^{-1}L)^{k}\psi|^{2}.$

The inequality (8.1) then follows by combining the above with (2.8). ∎

######  Proposition 8.2.

Let $N\in\mathbb{N}$ and let $\delta>0$ be arbitrarily small. Then for $R>3M$
suitably large, there exists a constant
$C=~{}C(M,h,\alpha,N,R,\delta)~{}>~{}0$ such that for all $\tau\geq 0$:

$\begin{split}E_{\mathbf{N}}[T^{N}\phi](\tau)\leq&\>C(1+\tau)^{-1-2N+\delta}\bBigg@{4}[\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N\\\ n_{1}+n_{2}\leq
N\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3N\\\ n_{1}\leq
<EMAIL_ADDRESS>(8.2)

and moreover,

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(LT^{N}\psi)\,d\sigma
dr\leq&\>C(1+\tau)^{1-2N+\delta}\bBigg@{4}[\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N-1\\\ n_{1}+n_{2}\leq
N\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3N-1\\\ n_{1}\leq
<EMAIL_ADDRESS>(8.3)

###### Proof.

We first consider the case $N=0$. Let $\\{\tau_{i}\\}$ be a dyadic sequence
with $\tau_{i}\to\infty$ as $i\to\infty$. Then we can apply the integrated
energy estimate (6.18) to conclude that

$\begin{split}\int_{\Sigma_{\tau_{i+1}}\cap\\{r\geq
R\\}}&r(L\psi)^{2}\,d\sigma
dr+\int_{\tau_{i}}^{\tau_{i+1}}E_{\mathbf{N}}[\phi](\tau)\,d\tau\\\
\leq&\>C\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r(L\psi)^{2}+r(LT\psi)^{2}\,d\sigma
dr+CE_{\mathbf{N}}[\phi](\tau_{i})+CE_{T}[T\phi](\tau_{i}).\end{split}$ (8.4)

Hence, after applying the mean-value theorem, we conclude that there exists a
sequence $\\{\tau_{i}^{\prime}\\}$ with
$\tau_{i}\leq~{}\tau_{i}\leq\tau_{i+1}$ along which

$E_{\mathbf{N}}[\phi](\tau_{i}^{\prime})\leq
C(1+\tau_{i}^{\prime})^{-1}\left[\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r(L\psi)^{2}+r(LT\psi)^{2}\,d\sigma
dr+E_{\mathbf{N}}[\phi](\tau_{i})+CE_{T}[T\phi](\tau_{i})\right].$

By applying (6.20), we then get that for all $\tau\geq 0$:

$E_{\mathbf{N}}[\phi](\tau)\leq
C(1+\tau)^{-1}\left[\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r(L\psi)^{2}+r(LT\psi)^{2}\,d\sigma
dr+E_{\mathbf{N}}[\phi](0)+CE_{T}[T\phi](0)\right].$ (8.5)

Now let $N=1$ and replace $\psi$ with $T\psi$. We then proceed to estimate
further: let $\\{\tau_{i}\\}$ be a dyadic sequence and $\gamma\geq 0$, then

$\begin{split}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r(LT\psi)^{2}\,d\sigma
drd\tau=&\>\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}\cap\\{r\leq R+{\tau_{i}}^{\gamma}\\}}r(LT\psi)^{2}\,d\sigma drd\tau\\\
&+\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R+{\tau_{i}}^{\gamma}\\}}r(LT\psi)^{2}\,d\sigma drd\tau\\\
\leq&\>C(1+\tau_{i})^{\gamma}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}(LT\psi)^{2}\,d\sigma drd\tau\\\
&+C(1+\tau_{i})^{-\gamma}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT\psi)^{2}\,d\sigma drd\tau.\end{split}$ (8.6)

Subsequently, we apply (8.1) with $N=1$ to estimate further

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(LT\psi)^{2}\,d\sigma
dr\leq&\>C\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}(L(rL)\psi)^{2}\,d\sigma dr\\\
&+C\sum_{k=0}^{1}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{k}X\phi|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k}\phi|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k+1}\phi|^{2}+r^{2}h\tilde{h}|\not{\nabla}_{\mathbb{S}^{2}}^{k}T\phi|^{2}\,d\sigma
dr.\end{split}$ (8.7)

We can therefore integrate (8.7) in $\tau$ and apply (7.2) with $N=1$ to
obtain:

$\begin{split}\int_{\tau_{i}}^{\tau_{i+1}}&\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT\psi)^{2}\,d\sigma drd\tau\leq C\sum_{n_{1}+n_{2}\leq
1}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r(L(rL)^{n_{1}}T^{n_{2}}\psi)^{2}\,d\sigma
dr+CE_{T}[\phi](\tau_{i})+CE_{T}[T\phi](\tau_{i})\\\
&+C\sum_{k=0}^{1}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{k}X\phi|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k}\phi|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k+1}\phi|^{2}+r^{2}h\tilde{h}|\not{\nabla}_{\mathbb{S}^{2}}^{k}T\phi|^{2}\,d\sigma
drd\tau.\end{split}$ (8.8)

When $k=0$, we can estimate the second integral on the right-hand side of
(8.8) by applying (6.17). When $k>0$, we use the property
$[\Omega_{i},\square_{g_{M}}-V_{\alpha}]=0$ to directly apply (6.17) to $\phi$
replaced with $\Omega_{j}\phi$, $j=1,2,3$. Then we can estimate additional
angular derivatives in terms of $\Omega_{j}$-derivatives via (2.5).

Subsequently, we apply the mean-value theorem and choose $\gamma=0$ to obtain
the existence of $\tau_{i}\leq\tau_{i}^{\prime}\leq\tau_{i+1}$ such that

$\begin{split}\int_{\Sigma_{\tau_{i}^{\prime}}\cap\\{r\geq
R\\}}r(LT\psi)^{2}\,d\sigma drd\tau\leq&\>C(1+\tau_{i})^{-1}\sum_{0\leq
n_{1}+n_{2}+n_{3}\leq 1}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\
&+C(1+\tau_{i})^{-1}\left[E_{T}[\phi](\tau_{i})+E_{T}[T\phi](\tau_{i})+E_{T}[\not{\nabla}_{\mathbb{S}^{2}}\phi](\tau_{i})\right].\end{split}$

Applying (7.2) with $p=1$ and $N=0$, allows us to infer that in fact for all
$\tau\geq 0$, we have that

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r(LT\psi)^{2}\,d\sigma
drd\tau\leq&\>C(1+\tau)^{-1}\sum_{0\leq n_{1}+n_{2}+n_{3}\leq
1}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\
&+C(1+\tau)^{-1}\left[E_{T}[\phi](0)+E_{T}[T\phi](0)+E_{T}[\not{\nabla}_{\mathbb{S}^{2}}\phi](0)\right].\end{split}$
(8.9)

We can use (8.9) to estimate the right-hand side of (8.4) further and $\phi$
is replaced by $T\phi$. We then obtain:

$E_{\mathbf{N}}[T\phi](\tau)\leq
C(1+\tau)^{-2}\left[\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq 2\\\
n_{1}+n_{2}\leq 1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3\\\ n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\right].$
(8.10)

Furthermore, we can split

$\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(LT\psi)^{2}\,d\sigma
dr=\int_{\Sigma_{\tau}\cap\\{R\leq r\leq R+\tau\\}}r^{2}(LT\psi)^{2}\,d\sigma
dr+\int_{\Sigma_{\tau}\cap\\{r\geq R+\tau\\}}r^{2}(LT\psi)^{2}\,d\sigma dr$

and estimate

$\int_{\Sigma_{\tau}\cap\\{R\leq r\leq R+\tau\\}}r^{2}(LT\psi)^{2}\,d\sigma
dr\leq C(1+\tau)\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r(LT\psi)^{2}\,d\sigma
dr$

and, applying (8.7) with $R$ replaced by $R+\tau$, we can moreover estimate

$\int_{\Sigma_{\tau}\cap\\{r\geq R+\tau\\}}r^{2}(LT\psi)^{2}\,d\sigma dr\leq
C(1+\tau)^{-1}\int_{\Sigma_{\tau}}r(L(rL)\psi)^{2}\,d\sigma
dr+CE_{T}[\phi](\tau)+CE_{T}[\not{\nabla}_{\mathbb{S}^{2}}\phi](\tau).$

Hence,

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(LT\psi)^{2}\,d\sigma
dr\leq&\>C(1+\tau)^{-1}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 1\\\ n_{1}+n_{2}\leq
1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 2\\\ n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\Bigg{]}+C(1+\tau)\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r(LT\psi)^{2}\,d\sigma dr.\end{split}$ (8.11)

Equipped with (8.10) and (8.9), we can now make a better choice of $\gamma$ to
improve the decay rate of $E_{\mathbf{N}}[T\phi](\tau)$: consider (8.6) with
$\gamma=\frac{1}{2}$ to obtain for all $\tau_{i}\leq\tau\leq\tau_{i+1}$:

$\begin{split}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}&r(LT\psi)^{2}\,d\sigma drd\tau\leq
C(1+\tau_{i})^{-1+\frac{1}{2}}\Bigg{[}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r(LT\psi)^{2}\,d\sigma dr+CE_{\mathbf{N}}[T\phi](\tau_{i})\Bigg{]}\\\
&+C(1+\tau_{i})^{-1-\frac{1}{2}}\sum_{n_{1}+n_{2}+n_{3}\leq
1}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\
&+C(1+\tau_{i})^{-1-\frac{1}{2}}\left[E_{T}[\phi](\tau_{i})+E_{T}[T\phi](\tau_{i})+E_{T}[\not{\nabla}_{\mathbb{S}^{2}}\phi](\tau_{i})\right]\\\
\leq&\>C(1+\tau_{i})^{-\frac{3}{2}}\left[\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 1\\\ n_{1}+n_{2}\leq
1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 2\\\ n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\right].\end{split}$

and hence

$\begin{split}E_{\mathbf{N}}[T\phi](\tau)\leq&\>C(1+\tau)^{-\frac{5}{2}}\left[\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2\\\ n_{1}+n_{2}\leq
1\end{subarray}}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr+C\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3\\\ n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\right].\end{split}$

We can now continue this process iteratively by choosing $\gamma=2^{-k}$ and
obtain:

$\displaystyle\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}$ $\displaystyle
r(LT\psi)^{2}\,d\sigma drd\tau\leq
C(1+\tau)^{-2+\frac{1}{2^{k}}}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 1\\\ n_{1}+n_{2}\leq
1\end{subarray}}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr$ $\displaystyle+C\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 2\\\
n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\Bigg{]},$
$\displaystyle E_{\mathbf{N}}[T\phi](\tau)\leq$
$\displaystyle\>C(1+\tau)^{-3+\frac{1}{2^{k}}}\left[\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2\\\ n_{1}+n_{2}\leq
1\end{subarray}}\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr+C\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3\\\ n_{1}\leq
1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\right].$

Given $\delta>0$, we can therefore take $k$ suitably charge to obtain (8.2)
for $N=1$ and (8.3) follows by additionally applying (8.11). Then, we apply
(8.11) to obtain moreover (8.3) for $N=1$.

The general $N>1$ case proceeds inductively. We further estimate the first
term on the right-hand side above after replacing $\phi$ with $T\phi$. In
general, we split

$\begin{split}\sum_{n_{1}+n_{2}+n_{3}\leq
N-1}&\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}+1}\psi|^{2}\,d\sigma
drd\tau\\\ \leq&\>C\sum_{1\leq n_{1}+n_{2}+n_{3}\leq
N-1}(1+\tau_{i})^{\gamma}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}+1}\psi|^{2}\,d\sigma
drd\tau\\\ &+C\sum_{1\leq n_{1}+n_{2}+n_{3}\leq
N}(1+\tau_{i})^{-\gamma}\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
drd\tau\end{split}$ (8.12)

after applying (8.1). As we pass from $N$ to $N+1$, we apply the mean-value
theorem an additional time and then optimize $\gamma$ as above in order to
increase the decay rates in (8.2) and (8.3) by 2. ∎

When $\alpha\geq 0$, we will moreover make use of the following stronger decay
estimates:

######  Proposition 8.3.

Let $N\in\mathbb{N}$ and let $\alpha\geq 0$ or let $\alpha<0$ and
$\phi=\phi_{\geq 1}$. Then for $R>3M$ suitably large, there exists a constant
$C=C(M,h,V_{\alpha},N,R)>0$ such that for all $\tau\geq 0$:

$\begin{split}E_{\mathbf{N}}[T^{N}\phi](\tau)\leq&\>C(1+\tau)^{-2-2N}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N+1\\\ n_{1}+n_{2}\leq
N\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3N+1\\\ n_{1}\leq
N\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\Bigg{]}.\end{split}$
(8.13)

and moreover,

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(LT^{N}\psi)\,d\sigma
dr\leq&\>C(1+\tau)^{-2N}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N\\\ n_{1}+n_{2}\leq
N\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\psi|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3N\\\ n_{1}\leq
N\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\phi](0)\Bigg{]}.\end{split}$
(8.14)

###### Proof.

As in the general $\alpha$ case, we have that (8.5) holds. When $\alpha>0$ or
$\phi=\phi_{\geq 1}$, we moreover have that (7.2) holds for $0<p\leq 2$.
Taking $p=2$ and $N=0$, we can therefore further estimate:

$\int_{\tau_{i}}^{\tau_{i+1}}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r(L\psi)^{2}\,d\sigma drd\tau\leq C\int_{\Sigma_{\tau_{i}}\cap\\{r\geq
R\\}}r^{2}(L\psi)^{2}\,d\sigma dr+CE_{\mathbf{N}}[\phi](\tau_{i}).$

Applying the mean-value theorem, we can therefore improve (8.5) to obtain:

$E_{\mathbf{N}}[\phi](\tau)\leq
C(1+\tau)^{-2}\left[\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2}(L\psi)^{2}+r(LT\psi)^{2}\,d\sigma
dr+E_{\mathbf{N}}[\phi](0)+E_{T}[T\phi](0)+E_{T}[T^{2}\phi](0)\right],$

which implies (8.13) with $N=0$. We moreover obtain from (7.2) with $p=2$ that

$\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{2}(L\psi)^{2}\,d\sigma dr\leq
C\left[\int_{\Sigma_{0}\cap\\{r\geq R\\}}r^{2}(L\psi)^{2}\,d\sigma
dr+E_{T}[\phi](0)\right],$

which implies (8.14) with $N=0$.

To obtain the estimates with $N\geq 1$, we then proceed as in the proof of
Proposition 8.2, by repeatedly applying (8.1). When $N=1$, for example, we use
(8.8) to apply the mean-value theorem once more and increase the decay rate of
$E_{\mathbf{N}}[T\phi](\tau)$ from $-2$ to $-3$. Applying (7.2) with $N=1$
together with the mean-value theorem again then results in a decay rate of
$-4$.

Similarly, (8.7), together with (7.2) with $N=1$ and $p=1$ and $p=2$ allows us
to obtain (8.14) with $N=1$. The general $N$ case then follows in a
straightforward manner. ∎

######  Remark 8.1.

In Propositions 8.2 and 8.3 there is a loss of total number of derivatives on
the right-hand side compared to the left-hand side. This is explained by the
loss of derivatives in the integrated energy estimate in (6.13) due the
presence of trapped null geodesics.

##  9 Time-inversion theory

In this section, we will show that, given a solution $\phi$ to (3.1) that
arises from appropriate initial data, we can construct initial data for (3.1)
that lead to a solution $\widetilde{\phi}$ satisfying the property that
$T\widetilde{\phi}=\phi$. We will refer to $\widetilde{\phi}$ as the _time
integral_ of $\phi$ (see the statement of Proposition 9.9 for a precise
definition). Formally,

$\widetilde{\phi}(\tau,r,\theta,\varphi)=-\int_{\tau}^{\infty}\phi(\tau^{\prime},r,\theta,\varphi)\,d\tau^{\prime},$

but since we have not yet established integrability of $\phi$ in $\tau$, we
will instead construct $\widetilde{\phi}$ from initial data by considering an
appropriate elliptic problem (and afterwards justify the above integral).

First, we develop the necessary elliptic theory in Section 9.1, which we then
apply in Section 9.2 to construct the time integral initial data.

###  9.1 Elliptic theory

We introduce the differential operator $\mathcal{L}$:

$\mathcal{L}u:=X(Dr^{2}Xu)+\not{\Delta}_{\mathbb{S}^{2}}u-r^{2}V_{\alpha}u.$

######  Proposition 9.1.

Let $u\in C^{\infty}_{c}(\Sigma)$. Let $-\beta_{0}<q<\beta_{0}$. Then there
exists a constant $C=C(M,V_{\alpha},q,h)>0$, such that

$\int_{\Sigma}r^{q}\left[r^{2}(Xu)^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u|^{2}+u^{2}\right]\,d\sigma
dr\leq C\int_{\Sigma}r^{q}(\mathcal{L}u)^{2}\,d\sigma dr.$ (9.1)

###### Proof.

We split $u=u_{\leq\ell_{0}-1}+u_{\geq\ell_{0}}$, where
$\ell_{0}\in\mathbb{N}$ will be chosen suitably large.

$u_{\leq\ell_{0}-1}$:

Consider first $u_{\ell}$, with $0\leq\ell\leq\ell_{0}-1$. The constants
appearing in the estimates for $u_{\ell}$ with $\ell\leq\ell_{0}-1$ below,
will depend on $\ell_{0}$.

We consider the weight functions $w_{\ell}$ from Lemma 3.3 and define:

$\check{u}_{\ell}:=w_{\ell}^{-1}u_{\ell}.$

Then

$\begin{split}w_{\ell}\mathcal{L}u_{\ell}=w_{\ell}\mathcal{L}(w_{\ell}\check{u}_{\ell})=&\>w_{\ell}X(Dr^{2}X(w_{\ell}\check{u}_{\ell}))-[\ell(\ell+1)+r^{2}V_{\alpha}]w_{\ell}^{2}\check{u}_{\ell}\\\
=&\>w_{\ell}^{2}X(Dr^{2}X\check{u}_{\ell})+2\frac{dw_{\ell}}{dr}X\check{u}_{\ell}+\check{u}_{\ell}w_{\ell}\left[\frac{d}{dr}\left(Dr^{2}\frac{dw_{\ell}}{dr}\right)-r^{2}V_{\alpha}w_{\ell}\right]\\\
=&\>X(Dr^{2}w_{\ell}^{2}X\check{u}_{\ell}),\end{split}$

where we applied (3.7) to arrive at the last equality.

Then it follows from multiple applications of the Leibniz rule that:

$\begin{split}r^{-q}X\check{u}_{\ell}\cdot
w_{\ell}\mathcal{L}u_{\ell}=&\>r^{-q}X\check{u}_{\ell}Dr^{2}w_{\ell}^{2}X^{2}\check{u}_{\ell}+(Dr^{2}w_{\ell}^{2})^{\prime}r^{-q}(X\check{u}_{\ell})^{2}\\\
=&\>X\left(\frac{1}{2}Dr^{2-q}w_{\ell}^{2}(X\check{u}_{\ell})^{2}\right)+\left[r-M+\frac{q}{2}(r-2M)+w_{\ell}^{-1}\frac{dw_{\ell}}{dr}r(r-2M)\right]r^{-q}w_{\ell}^{2}(X\check{u}_{\ell})^{2}.\end{split}$

We can moreover apply (3.8) to estimate

$\left|w_{\ell}^{-1}\frac{dw_{\ell}}{dr}-\frac{1}{2}(\beta_{\ell}-1)r^{-1}\right|\leq
C(r^{-\frac{1}{2}-\frac{1}{2}\beta_{\ell}}+r^{-\frac{3}{2}+\frac{1}{2}\beta_{\ell}}).$

We apply a weighted Young’s inequality to estimate

$\left|r^{-q}X\check{u}_{\ell}\cdot
w_{\ell}\mathcal{L}u_{\ell}\right|\leq\epsilon
r^{1-q}w_{\ell}^{2}(X\check{u}_{\ell})^{2}+\frac{1}{4\epsilon}r^{-1-q}(\mathcal{L}u_{\ell})^{2}.$

We first take $q>q_{\ell_{0}}$, with $q_{\ell_{0}}$ suitably large, so that in
$\\{r\leq R\\}$, with $3M<R<\infty$ arbitrarily large,

$r-M+\frac{q_{\ell_{0}}}{2}(r-2M)+w_{\ell}^{-1}\frac{dw_{\ell}}{dr}r(r-2M)>0.$

Then we integrate over $\Sigma\cap\\{r\leq R\\}$ to obtain

$\int_{\mathbb{S}^{2}}(X\check{u}_{\ell})^{2}\Big{|}_{r=R}\,d\sigma+\int_{\Sigma\cap\\{r\leq
R\\}}(X\check{u}_{\ell})^{2}\,d\sigma dr\leq C\int_{\Sigma\cap\\{r\leq
R\\}}(\mathcal{L}u_{\ell})^{2}\,d\sigma dr.$ (9.2)

Let $\epsilon>0$ be arbitrarily small. We then consider the region $\\{r\geq
R\\}$, with $R>3M$ suitably large so that

$r-M+\frac{q}{2}(r-2M)+w_{\ell}^{-1}\frac{dw_{\ell}}{dr}>\left(\frac{1}{2}+\frac{1}{2}q+\frac{1}{2}\beta_{\ell}-\epsilon\right)r.$

Given any $q>-1-\beta_{\ell}$, we can therefore take $\epsilon>0$ suitably
small, integrate over $\Sigma$ and apply (9.2) to obtain

$\int_{\Sigma}r^{\beta_{\ell}-q}(X\check{u}_{\ell})^{2}\,d\sigma dr\leq
C\int_{\Sigma}r^{-1-q}(\mathcal{L}u_{\ell})^{2}\,d\sigma dr.$ (9.3)

By restricting further to $-1-\beta_{\ell}<q<-1+\beta_{\ell}$ and applying
(2.10) together with compactness of the support of $u_{\ell}$, we obtain

$\int_{\Sigma}r^{\beta_{\ell}-q-2}\check{u}_{\ell}^{2}+r^{\beta_{\ell}-q}(X\check{u}_{\ell})^{2}\,d\sigma
dr\leq C\int_{\Sigma}r^{-1-q}(\mathcal{L}u_{\ell})^{2}\,d\sigma dr$

and hence,

$\int_{\Sigma}r^{-q-1}{u}_{\ell}^{2}+r^{-q+1}(X{u}_{\ell})^{2}\,d\sigma dr\leq
C\int_{\Sigma}r^{-q-1}(\mathcal{L}u_{\ell})^{2}\,d\sigma dr.$

$u_{\geq\ell_{0}}$:

Consider now $u_{\geq\ell_{0}}$. We have that

$(\mathcal{L}u_{\geq\ell_{0}})^{2}=(X(Dr^{2}Xu_{\geq\ell_{0}}))^{2}+((\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}}))^{2}+2X(Dr^{2}Xu_{\geq\ell_{0}})(\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}}).$

Let $-1-\beta_{0}<q<-1+\beta_{0}$. Then we integrate by parts to obtain:

$\begin{split}\int_{\Sigma}2r^{-q}X(Dr^{2}Xu_{\geq\ell_{0}})(\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}})\,d\sigma
dr=&\>\int_{\Sigma}X(2r^{-q}Dr^{2}Xu_{\geq\ell_{0}}(\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}}))\,d\sigma
dr\\\
&+\int_{\Sigma}2Dr^{2-q}(|\not{\nabla}_{\mathbb{S}^{2}}Xu_{\geq\ell_{0}}|^{2}+r^{2}V_{\alpha}(Xu_{\geq\ell_{0}})^{2})\,d\sigma
dr\\\
&+\int_{\Sigma}2qDr^{1-q}Xu_{\geq\ell_{0}}(\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}})\,d\sigma
dr\\\
&+\int_{\Sigma}2Dr^{2-q}Xu_{\geq\ell_{0}}\frac{d}{dr}(r^{2}V_{\alpha})u_{\geq\ell_{0}}\,d\sigma
dr.\end{split}$

We apply Young’s inequality to estimate:

$\begin{split}Xu_{\geq\ell_{0}}&\left[2qDr^{1-q}(\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})u_{\geq\ell_{0}}+2Dr^{2-q}\frac{d}{dr}(r^{2}V_{\alpha})u_{\geq\ell_{0}}\right]\\\
\leq&\>2(Cr^{-2}+q^{2})Dr^{2-q}(Xu_{\geq\ell_{0}})^{2}+\frac{1}{2}Dr^{-q}((\not{\Delta}_{\mathbb{S}^{2}}-r^{2}V_{\alpha})(u_{\geq\ell_{0}}))^{2}+\frac{1}{2}Dr^{-q}u_{\geq\ell_{0}}^{2},\end{split}$

for some suitably large $C>0$.

We apply (2.6) to infer that:

$\int_{\Sigma}2(Cr^{-2}+q^{2})Dr^{2-q}(Xu_{\geq\ell_{0}})^{2}\,d\sigma
dr\leq\int_{\Sigma}2(\ell_{0}(\ell_{0}+1))^{-1}(Cr^{-2}+q^{2})Dr^{2-q}|\not{\nabla}_{\mathbb{S}^{2}}Xu_{\geq\ell_{0}}|^{2}\,d\sigma
dr$

and moreover

$\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+r^{2}V_{\alpha}u_{\geq\ell_{0}}^{2}\,d\sigma\geq\int_{\mathbb{S}^{2}}(\ell_{0}(\ell_{0}+1)+r^{2}V_{\alpha})u_{\geq\ell_{0}}^{2}\,d\sigma,$

so for $\ell_{0}>0$ suitably large depending on $q$ and $V_{\alpha}$, we can
absorb all unsigned terms in the integral of
$r^{-q}(\mathcal{L}u_{\geq\ell_{0}})^{2}$ over $\Sigma$ to obtain: for all
$-1-\beta_{0}<q<-1+\beta_{0}$,

$\begin{split}\int_{\Sigma}&r^{-q}(X(Dr^{2}Xu_{\geq\ell_{0}}))^{2}+r^{2-q}D|\not{\nabla}_{\mathbb{S}^{2}}Xu_{\geq\ell_{0}}|^{2}+r^{2-q}D(Xu_{\geq\ell_{0}})^{2}+r^{-q}(\not{\Delta}_{\mathbb{S}^{2}}u_{\geq\ell_{0}})^{2}+r^{-q}u_{\geq\ell_{0}}^{2}\,d\sigma
d\sigma\\\
\leq&\>C\int_{\Sigma}r^{-q}(\mathcal{L}u_{\geq\ell_{0}})^{2}\,d\sigma
dr.\end{split}$ (9.4)

In order to remover the degenerate factors $D$ that appear in front of the
$(Xu_{\geq L})^{2}$ term, we additionally consider:

$\begin{split}r\chi
Xu_{\geq\ell_{0}}\mathcal{L}u_{\geq\ell_{0}}=&\>X\left(\frac{1}{2}Dr^{3}\chi(Xu_{\geq\ell_{0}})^{2}\right)+\frac{1}{2}r\left((Dr^{2})^{\prime}-Dr\right)\chi(Xu_{\geq\ell_{0}})^{2}-\frac{1}{2}X(r\chi|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+\chi
r^{3}V_{\alpha}u_{\geq\ell_{0}}^{2})\\\
&+\frac{1}{2}\chi(|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+\frac{d}{dr}(r^{3}V_{\alpha})u_{\geq\ell_{0}}^{2})-\frac{1}{2}Dr^{3}\frac{d\chi}{dr}(Xu_{\geq\ell_{0}})^{2}+r\frac{d\chi}{dr}(|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+r^{2}V_{\alpha}u_{\geq\ell_{0}}^{2})\\\
&+\textnormal{div}_{\mathbb{S}^{2}}(\ldots),\end{split}$

with $\chi:[2M,\infty)\to\mathbb{R}_{>0}$ a smooth cut-off function, such that
$\chi=1$ for $r\leq r_{0}$ and $\chi=0$ for $r>r_{1}$, where
$2M<r_{0}<r_{1}<\infty$. Note first of all that $(Dr^{2})^{\prime}-Dr=r$ and

$\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+\frac{d}{dr}(r^{3}V_{\alpha})u_{\geq\ell_{0}}^{2}\,d\sigma\geq\frac{1}{2}\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+u_{\geq\ell_{0}}^{2}\,d\sigma$

by (2.6) and $\ell_{0}$ suitably large. Furthermore, we can apply Young’s
inequality to estimate:

$|r\chi Xu_{\geq\ell_{0}}\mathcal{L}u_{\geq\ell_{0}}|\leq
r^{2}(Xu_{\geq\ell_{0}})^{2}+(\mathcal{L}u_{\geq\ell_{0}})^{2}.$

We now integrate $r\chi Xu_{\geq\ell_{0}}\mathcal{L}u_{\geq\ell_{0}}$ over
$\Sigma$ and apply the above estimates to obtain

$\begin{split}\int_{\Sigma\cap\\{r\leq
r_{0}\\}}r^{2}(Xu_{\geq\ell_{0}})^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}+u_{\geq\ell_{0}}^{2}\,d\sigma
dr\leq&\>C\int_{\Sigma\cap\\{r\leq
r_{1}\\}}(\mathcal{L}u_{\geq\ell_{0}})^{2}\,d\sigma dr\\\
&+C\int_{\Sigma\cap\\{r_{0}\leq r\leq
r_{1}\\}}(Xu_{\geq\ell_{0}})^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u_{\geq\ell_{0}}|^{2}\,d\sigma
dr.\end{split}$

Finally, we combine the above equation with (9.4) to control (in particular):

$\int_{\Sigma}r^{2-q}(Xu_{\geq\ell_{0}})^{2}+r^{-q}u_{\geq\ell_{0}}^{2}\,d\sigma
d\sigma\leq C\int_{\Sigma}r^{-q}(\mathcal{L}u_{\geq\ell_{0}})^{2}\,d\sigma
dr.$

By combining all the above estimates and replacing $q$ with $-(q-1)$
everywhere above, we conclude that (9.1) must hold. ∎

######  Lemma 9.2.

The following identity holds:

$\left[\mathcal{L}+ND^{\prime}r^{2}X\right](rX)^{N}u=(rX)^{N}\mathcal{L}u+N\sum_{n=0}^{N}O_{\infty}(r^{-1})(rX)^{n}u.$
(9.5)

###### Proof.

We will prove (9.5) by induction. Note that (9.5) hold for $N=0$ by definition
of $\mathcal{L}$. Now we suppose as an induction hypothesis that (9.5) holds
for some $N\in\mathbb{N}_{0}$. We will show that (9.5) must then also hold
with $N$ replaced by $N+1$.

One can verify first of all that

$\begin{split}X(Dr^{2}X(rX)^{N+1}u)=&\>(rX+2)X(Dr^{2}X(rX)^{N}u)-(Dr^{2})^{\prime}X(rX)^{N+1}u-(Dr^{2})^{\prime\prime}(rX)^{N+1}u\\\
=&\>rX(Dr^{2}X(rX)^{N}u)-D^{\prime}r^{2}X(rX)^{N+1}u\end{split}$

Hence, for $M>0$:

$\begin{split}\mathcal{L}(rX)^{N+1}u=&\>X(Dr^{2}X(rX)^{N+1}u)+\not{\Delta}_{\mathbb{S}^{2}}(rX)^{N+1}u-r^{2}V_{\alpha}(rX)^{N+1}u\\\
=&\>rX\mathcal{L}(rX)^{N}u+rX(r^{2}V_{\alpha})(rX)^{N}u-D^{\prime}r^{2}X(rX)^{N+1}u\\\
=&\>(rX)^{N+1}\mathcal{L}u-(N+1)D^{\prime}r^{2}X(rX)^{N+1}u+\sum_{n=0}^{N+1}O_{\infty}(r^{-1})(rX)^{n}u,\end{split}$

where we applied the induction hypothesis to arrive at the last equality. We
then conclude (9.5) with $N$ replaced by $N+1$ by rearranging terms. ∎

We now obtain an analogue of Proposition 9.1 with higher-order angular
derivatives, as well as an analogue featuring the left-hand side of (9.5), the
latter of which will be important when we consider higher-order derivatives in
the $X$-direction.

We introduce first the space:

$C_{c,*}^{\infty}(\Sigma)=\left\\{u\in
C_{c}^{\infty}(\Sigma)\,|\,u|_{r=2M}\equiv 0\right\\}.$

######  Proposition 9.3.

Let $u\in C^{\infty}_{c}(\Sigma)$. Let $-\beta_{0}<q<\beta_{0}$. Then there
exists a constant $C=C(M,\alpha,q,h,N)>0$, such that

$\displaystyle\sum_{k=0}^{K}\int_{\Sigma}r^{q}\left[r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{k}Xu|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k+1}u|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{k}u|^{2}\right]\,d\sigma
dr\leq$
$\displaystyle\>C\sum_{k=0}^{K}\int_{\Sigma}r^{q}|\not{\nabla}_{\mathbb{S}^{2}}^{k}\mathcal{L}u|^{2}\,d\sigma
dr.$ (9.6)

Let $u\in C^{\infty}_{c}(\Sigma)$ if $N\geq 0$ and $u\in
C^{\infty}_{c,*}(\Sigma)$ if $N<0$. Let $-\beta_{0}<q<\beta_{0}$. Then there
exists a constant $C=C(M,V_{\alpha},q,h,N)>0$, such that

$\begin{split}\int_{\Sigma}&r^{q}\left[D^{2}r^{2}(X(rXu))^{2}+r^{2}(Xu)^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u|^{2}\right]\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma}r^{q}((\mathcal{L}-ND^{\prime}r^{2}X)u)^{2}+|N|r^{q}(u^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u|^{2}+D^{2}r^{2}(Xu)^{2})\,d\sigma
dr.\end{split}$ (9.7)

###### Proof.

The estimate (9.6) follows immediately applying (9.1) with $\phi$ replaced by
$\Omega^{I}\phi$, where $|I|\leq k$ and applying (2.5).

To obtain (9.7), we first consider for $q\in\mathbb{R}$ and $j\in\\{0,1\\}$:

$\begin{split}ND^{j}r^{q+1}Xu(\mathcal{L}u+ND^{\prime}r^{2}Xu)=&\>X\left(\frac{N}{2}D^{j+1}r^{3+q}(Xu)^{2}\right)+\frac{1}{2}D^{j}r^{q+1}\left[N(2N+1-j)D^{\prime}r-N(q-1)D\right](Xu)^{2}\\\
&-\frac{N}{2}X\left(r^{q+1}D^{j}|\not{\nabla}_{\mathbb{S}^{2}}u|^{2}+D^{j}r^{3+q}V_{\alpha}u^{2}\right)+\frac{N}{2}((q+1)D^{j}+jD^{\prime}r)r^{q}|\not{\nabla}_{\mathbb{S}^{2}}u|^{2}\\\
&+\frac{N}{2}\left(D^{j}\frac{d}{dr}(r^{q+3}V_{\alpha})+jD^{\prime}r\right)u^{2}+\textnormal{div}_{\mathbb{S}^{2}}(\ldots).\end{split}$
(9.8)

We have that $N(2N+1-j)>0$ for all $N\neq 0$. Furthermore, integrating (9.8)
along $\Sigma$, we only obtain a boundary term at $r=2M$ with a bad sign if
$N<0$ and $j=0$, in which case we appeal to the assumption that $u\in
C^{\infty}_{c,*}(\Sigma)$ to conclude that this boundary term actually
vanishes.

By taking $j=1$, we can therefore estimate:

$\int_{\Sigma}Dr^{q+2}(Xu)^{2}\,d\sigma dr\leq
C\int_{\Sigma}r^{q}((\mathcal{L}-ND^{\prime}r^{2}X)u)^{2}+|N|D^{2}r^{q+2}(Xu)^{2}+|N|r^{q}(u^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u|^{2})\,d\sigma
dr.$

Now, we take $j=0$ and we apply the above estimate to remove the $D$ factor on
the left-hand side:

$\int_{\Sigma}r^{q+2}(Xu)^{2}\,d\sigma dr\leq
C\int_{\Sigma}r^{q}((\mathcal{L}-ND^{\prime}r^{2}X)u)^{2}+|N|D^{2}r^{q+2}(Xu)^{2}+|N|r^{q}(u^{2}+|\not{\nabla}_{\mathbb{S}^{2}}u|^{2})\,d\sigma
dr.$

Finally, in order to control also the integral of $D^{2}r^{q+2}(X(rXu))^{2}$,
we repeat the elliptic estimate for $u_{\geq\ell_{0}}$ in the proof of
Proposition 9.1 with $\mathcal{L}u+ND^{\prime}r^{2}Xu$ replacing $\mathcal{L}$
and without the restriction to $\ell\geq\ell_{0}$, and we use that the
additional term $ND^{\prime}r^{2}Xu$ contributes as a lower-order term the
elliptic estimate. ∎

In order to define in a clean way the relevant Hilbert spaces on which
$\mathcal{L}$ is well-defined, we will modify $\mathcal{L}$ as follows: let
$q\in\mathbb{R}$ and $N\in\mathbb{Z}$ then

$\displaystyle\mathcal{L}_{q,N}v:=$
$\displaystyle\>r^{\frac{q}{2}}\mathcal{L}(r^{-\frac{q}{2}}v)+Nr^{\frac{q}{2}}D^{\prime}r^{2}X(r^{-\frac{q}{2}}v).$

Let $n\in\mathbb{N}_{0}$. We introduce the Hilbert spaces $\mathbf{H}_{n}$ as
completions of $C_{c}^{\infty}(\Sigma)$ under the norms $||\cdot||_{n}$
defined as follows:

$||u||^{2}_{n}:=\sum_{0\leq n_{1}+n_{2}\leq
n}\int_{\Sigma}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}u|^{2}\,d\sigma
dr.$

We denote with $D_{n}(\mathcal{L}_{q,N})$ the closure of
$C_{c}^{\infty}(\Sigma)$ with respect to the norm
$||\cdot||_{n}+||\mathcal{L}_{q,N}(\cdot)||_{n}$. Then

$\mathcal{L}_{q,N}:D_{n}(\mathcal{L}_{q,N})\to\mathbf{H}_{n}$

is a closed, densely-defined linear operator.

######  Corollary 9.4.

Let $v\in C^{\infty}_{c}(\Sigma)$ and let $n_{1},n_{2}\in\mathbb{N}_{0}$. Let
$-\beta_{0}<q<\beta_{0}$. Then there exists a constant
$C=C(M,V_{\alpha},q,n_{1},n_{2},h)>0$, such that:

$||D\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}v||_{2}^{2}+||\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}v||_{1}^{2}\leq
C||\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}\mathcal{L}_{q,\pm
n_{2}}(rX)^{n_{2}}v||^{2}_{0}+Cn_{2}||\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}v||_{0}^{2}+Cn_{2}||D\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}v||_{1}^{2}$
(9.9)

where to be able to take a minus sign in $\mathcal{L}_{q,\pm n_{2}}$, we need
to assume $v\in C^{\infty}_{c,*}(\Sigma)$.

###### Proof.

Let $u=r^{-\frac{q}{2}}v$ when $M>0$. Then (9.9) follows immediately from
Proposition 9.1 and Proposition 9.3. ∎

We will construct $\mathcal{L}_{q,N}^{-1}$ as an operator acting on
$\mathbf{H}_{n}$, with arbitrarily large $n\in N_{0}$, by first considering
the formal adjoint of $\mathcal{L}_{q,N}$ with respect to $\mathbf{H}_{0}$.

Let

$\mathcal{L}_{q,N}^{*}:\mathcal{D}_{0}(\mathcal{L}^{*}_{q,N})\to\mathbf{H}_{0}$

be the adjoint operator of $\mathcal{L}_{q,N}$, with
$w\in\mathcal{D}_{0}(\mathcal{L}^{*}_{q,N})$ if and only if there exists a
unique $f\in\mathbf{H}_{0}$ such that for all $u\in C_{c}^{\infty}(\Sigma)$

$\langle\mathcal{L}_{q,N}u,v\rangle_{L^{2}([2M,\infty)\times\mathbb{S}^{2})}=\langle
u,f\rangle_{L^{2}([2M,\infty)\times\mathbb{S}^{2})}.$

Then $\mathcal{L}_{q,N}^{*}v=f$.

We introduce moreover the operator
$\mathcal{L}^{{\dagger}}_{q,N}:=\mathcal{L}_{-q,-N}$ with domain
$\mathcal{D}_{n}(\mathcal{L}^{{\dagger}}_{q,N})$ defined as the closure of the
space $C_{*,c}^{\infty}(\Sigma)$ with respect to the norm
$||\cdot||_{n}+||\mathcal{L}^{{\dagger}}_{q,N}(\cdot)||_{n}$.

######  Lemma 9.5.

We have that $\mathcal{L}_{q,N}^{*}=\mathcal{L}_{q,N}^{{\dagger}}$ with
$\mathcal{D}_{0}(\mathcal{L}^{*}_{q,N})=\mathcal{D}_{0}(\mathcal{L}_{q,N}^{{\dagger}})$.

###### Proof.

We will show that $(\mathcal{L}_{q,N}^{{\dagger}})^{*}=\mathcal{L}_{q,N}$ and
$\mathcal{D}_{0}((\mathcal{L}_{q,N}^{{\dagger}})^{*})=\mathcal{D}_{0}(\mathcal{L}_{q,N})$.
It is easy to verify by definition of $\mathcal{L}^{{\dagger}}_{q,N}$ and
integration by parts that for all $u\in C_{c,*}^{\infty}(\Sigma)$ and $v\in
C_{c}^{\infty}(\Sigma)$:

$\langle\mathcal{L}_{q,N}^{{\dagger}}u,v\rangle_{L^{2}([2M,\infty)\times\mathbb{S}^{2})}=\langle
u,\mathcal{L}_{q,N}v\rangle_{L^{2}([2M,\infty)\times\mathbb{S}^{2})}.$ (9.10)

Since $\mathcal{L}_{q,N}$ and $\mathcal{L}_{q,N}^{{\dagger}}$ are both closed
operators, the identity (9.10) holds more generally for
$u\in\mathcal{D}_{0}(\mathcal{L}^{{\dagger}}_{q,N})$ and
$v\in\mathcal{D}_{0}(\mathcal{L}_{q,N})$. It follows that
$\mathcal{L}_{q,N}=(\mathcal{L}_{q,N}^{{\dagger}})^{*}$ as operators acting on
$\mathcal{D}_{0}(\mathcal{L}_{q,N})$, with
$\mathcal{D}_{0}(\mathcal{L}_{q,N})\subseteq\mathcal{D}_{0}((\mathcal{L}_{q,N}^{{\dagger}})^{*})$.

It remains to show that
$\mathcal{D}_{0}((\mathcal{L}_{q,N}^{{\dagger}})^{*})\subseteq\mathcal{D}_{0}(\mathcal{L}_{q,N})$.
Observe that $\mathbf{H}_{0}=L^{2}([2M,\infty)_{r}\times\mathbb{S}^{2})$ and
the closure of $C_{c}^{\infty}(\Sigma)$ is
$L^{2}([2M,\infty)_{r}\times\mathbb{S}^{2})$ with respect to $||\cdot||_{0}$.
Let $v\in\mathcal{D}_{0}((\mathcal{L}_{q,N}^{{\dagger}})^{*})$, then by
$\eqref{eq:Ldag}$, $\mathcal{L}_{q,N}v\in\mathbf{H}_{0}$ exists in a weak
sense. Hence, by the above observations, both $v$ and $\mathcal{L}_{q,N}v$ lie
in the closure of $C_{c}^{\infty}(\Sigma)$ with respect to $||\cdot||_{0}$ and
therefore $v\in\mathcal{D}_{0}(\mathcal{L}_{q,N})$. ∎

######  Proposition 9.6.

Let $N\in\mathbb{N}_{0}$ and $f\in\mathbf{H}_{N}$. Let
$-\beta_{0}<q<\beta_{0}$. Then there exists a unique $v\in\mathbf{H}_{N+1}$
such that $\mathcal{L}_{q,0}v=f$.

###### Proof.

Suppose $N=0$. Then it follows from Corollary 9.4 with $n_{1}=n_{2}=0$ that
$\ker\mathcal{L}_{q,0}=\\{0\\}$. We will establish bijectivity by showing that
$\textnormal{Ran}\>\mathcal{L}_{q,0}=\mathbf{H}_{0}$.

Let $w\in(\textnormal{Ran}\>\mathcal{L}_{q,0})^{\perp}$ and $u\in
C^{\infty}_{c,*}(\Sigma)$. Then we have that

$0=\langle
w,\mathcal{L}_{q,0}u\rangle=\langle\mathcal{L}^{*}_{q,0}w,u\rangle,$

so it follows that $w\in\mathcal{D}_{0}(\mathcal{L}_{q,0}^{*})$ and
$w\in\ker\mathcal{L}_{q,0}^{*}$. Using now that by Lemma 9.5,
$\mathcal{L}^{*}_{q,0}=\mathcal{L}^{{\dagger}}_{q,0}=\mathcal{L}_{-q,0}$, we
can conclude from Corollary 9.4 with $n_{1}=n_{2}=0$ that
$\mathcal{L}^{*}_{q,0}$ is injective, so $\ker\mathcal{L}_{q}^{*}=\\{0\\}$ and
$w=0$. Hence, $(\textnormal{Ran}\>\mathcal{L}_{q,0})^{\perp}=\\{0\\}$, so
$\textnormal{Ran}\>\mathcal{L}_{q,0}=\mathbf{H}_{0}$.

So, for $f\in\mathbf{H}_{0}$ there exists a unique
$v\in\mathcal{D}_{0}(\mathcal{L}_{q,0})$ such that $\mathcal{L}_{q,0}v=f$. By
applying Corollary 9.4 once more, we in fact have that $v\in\mathbf{H}_{1}$
and $Dv\in\mathbf{H}_{2}$.

Now suppose $N=1$. Then we need to show that in fact we have the following
improved regularity statement: $v\in\mathbf{H}_{2}$. First of all, from the
above argument, together with Corollary 9.4 for $n_{1}=1$ and $n_{2}=0$, it
follows that $\not{\nabla}_{\mathbb{S}^{2}}v\in\mathbf{H}_{1}$.

By (9.5), we have that:

$\mathcal{L}_{q,1}(rXv)=rX\mathcal{L}_{q,0}v+r^{\frac{q}{2}}\sum_{n=0}^{1}O_{\infty}(r^{-1})(rX)^{n}(r^{-\frac{q}{2}}v),$

where the right-hand side is an element of $\mathbf{H}_{0}$ for
$f=\mathcal{L}v\in\mathbf{H}_{1}$, since we already established that
$v\in\mathbf{H}_{1}$.

We can therefore conclude that $rXv\in\mathbf{H}_{1}$, if we can establish
bijectivity of $\mathcal{L}_{q,1}$ and apply Corollary 9.4 with $n_{1}=0$ and
$n_{2}=1$. Bijectivity in turn follows as above, by replacing
$\mathcal{L}_{q,0}$ with $\mathcal{L}_{q,1}$ and $\mathcal{L}_{q,0}^{*}$ with
$\mathcal{L}_{q,1}^{*}$.

The general $N>0$ case follows in the same manner via an induction argument,
where we establish first that $v\in\mathbf{H}_{N}$ in order to conclude that
$v\in\mathbf{H}_{N+1}$. ∎

From the above, we can immediately conclude the following properties for
solutions to the equation $\mathcal{L}u=F$:

######  Corollary 9.7.

Let $-\beta_{0}<q<\beta_{0}$, $N\in\mathbb{N}_{0}$. Let
$F:\Sigma\to\mathbb{R}$ such that $r^{\frac{q}{2}}F\in\mathbf{H}_{N}$. Then
there exists a unique $u:\Sigma\to\mathbb{R}$ with
$r^{\frac{q}{2}}u\in\mathbf{H}_{N+1}$, such that

$\displaystyle\mathcal{L}u=$ $\displaystyle\>F,$
$\displaystyle||r^{\frac{q}{2}}u||_{\mathbf{H}_{N+1}}\leq$
$\displaystyle\>C||r^{\frac{q}{2}}F||_{\mathbf{H}_{N+1}},$

with $C=C(V_{\alpha},M,N,q)>0$ a constant.

We obtain stronger estimates when restricting $\mathcal{L}$ to functions
satisfiying $u=u_{\geq 1}$. In that case, it follows immediately that we can
take $\beta_{1}$ to replace the role of $\beta_{0}$ in the proof of
Proposition 9.1 and obtain:

######  Corollary 9.8.

Let $-\beta_{1}<q<\beta_{1}$, $N\in\mathbb{N}_{0}$. Let
$F:\Sigma\to\mathbb{R}$ such that $r^{\frac{q}{2}}F_{\geq
1}\in\mathbf{H}_{N}$. Then there exists a unique $u:\Sigma\to\mathbb{R}$, with
$u=u_{\geq 1}$ and with $r^{\frac{q}{2}}u_{\geq 1}\in\mathbf{H}_{N+1}$, such
that

$\displaystyle\mathcal{L}u=$ $\displaystyle\>F,$
$\displaystyle||r^{\frac{q}{2}}u_{\geq 1}||_{\mathbf{H}_{N+1}}\leq$
$\displaystyle\>C||r^{\frac{q}{2}}F_{\geq 1}||_{\mathbf{H}_{N+1}},$

with $C=C(V_{\alpha},M,N,q)>0$ a constant.

###  9.2 Time integrals along $\Sigma_{0}$

Let $\phi$ be a smooth solution to (3.1). Then by (3.2), we have that

$\displaystyle\mathcal{L}\phi=$
$\displaystyle\>F[T\phi],\quad\textnormal{with}$ $\displaystyle F[f]:=$
$\displaystyle\>2r(1-h)X(rf)-\frac{dh}{dr}r^{2}f+r^{2}h\tilde{h}Tf,$

for sufficiently regular functions $f$ on $\mathcal{R}$. We will denote the
projection of $F[f]$ to spherical harmonics of degree $\ell$ by $F_{\ell}[f]$
(see §2.2).

In the proposition below, we construct the time integral corresponding to
solutions $\phi$ to (3.1) arising from appropriate initial data.

######  Proposition 9.9.

Let $-\beta_{0}<q<\beta_{0}$, $N\in\mathbb{N}_{0}$. Consider initial data
$(\phi|_{\Sigma_{0}},T\phi|_{\Sigma_{0}})$ for (3.1), such that:

$\displaystyle r^{\frac{q}{2}+1}X(r\phi)|_{\Sigma_{0}}\in$
$\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{q}{2}}\phi|_{\Sigma_{0}}\in\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{q}{2}}T\phi|_{\Sigma_{0}}\in\mathbf{H}_{N}.$

Then there exists a unique solution $T^{-1}\phi$ to (3.1), such that
$T(T^{-1}\phi)=\phi$, with initial data
$(T^{-1}\phi|_{\Sigma_{0}},\phi|_{\Sigma_{0}})$, such that moreover:

$\displaystyle r^{\frac{q}{2}}X(rT^{-1}\phi)|_{\Sigma_{0}}\in$
$\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{q}{2}}T^{-1}\phi|_{\Sigma_{0}}\in\mathbf{H}_{N+1}\quad\textnormal{and}\quad
r^{\frac{q}{2}}T(T^{-1}\phi)|_{\Sigma_{0}}\in\mathbf{H}_{N+1},$

and there exists a constant $C=C(V_{\alpha},h,M,N,q)>0$, such that

$||r^{\frac{q}{2}}X(rT^{-1}\phi)|_{\Sigma_{0}}||_{\mathbf{H}_{N}}+||r^{\frac{q}{2}}T^{-1}\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}\leq
C||r^{\frac{q}{2}+1}X(r\phi)|_{\Sigma_{0}}||_{\mathbf{H}_{N}}+C||r^{\frac{q}{2}}\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}+C||r^{\frac{q}{2}}T\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}.$

We refer to $T^{-1}\phi$ as the _time integral_ of $\phi$.

###### Proof.

Given the choice of initial data in the proposition, we can apply Corollary
9.7 with the inhomogeneity $F=F[\phi]|_{\Sigma_{0}}$ to obtain a solution $u$
to $\mathcal{L}u=F$. We take as initial data for (3.1)
$(u,\phi|_{\Sigma_{0}})$, and we denote the resulting solution as
$\widetilde{\phi}$. By the choice of $F$ and by application of (3.1), it
follows then that

$(T\tilde{\phi}|_{\Sigma_{0}},T^{2}\tilde{\phi}|_{\Sigma_{0}})=(\phi|_{\Sigma_{0}},T\phi_{\Sigma_{0}}),$

so by uniqueness of solutions to (3.1), we must have that
$T\widetilde{\phi}=\phi$. ∎

######  Remark 9.1.

In Corollary 10.11 below, we provide an alternative characterization of
$T^{-1}\phi$ as an integral in time of $\phi$ along curves of constant
$r,\theta,\varphi$, which motivates the nomenclature.

In the corollary below, we show that for $\alpha\geq 0$ ($\beta_{0}\geq 1$),
we can construct multiple time integrals.

######  Corollary 9.10.

Let $\beta_{0}\geq 1$. Let $N\in\mathbb{N}_{0}$ and
$n_{\beta}=\lfloor\frac{\beta_{0}+1}{2}\rfloor\in\mathbb{N}_{0}$ and let
$0<\delta<\beta_{0}-n_{\beta}$ be arbitrarily small. Consider initial data
$(\phi|_{\Sigma},T\phi|_{\Sigma})$ for (3.1), such that:

$\displaystyle r^{\frac{\beta_{0}}{2}+1-\delta}X(r\phi)|_{\Sigma_{0}}\in$
$\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{0}}{2}-\delta}\phi|_{\Sigma_{0}}\in\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{0}}{2}-\delta}T\phi|_{\Sigma_{0}}\in\mathbf{H}_{N}.$

Then there exists a unique solution $T^{-1-n_{\beta}}\phi$ to (3.1), such that
$T^{1+n_{\beta}}(T^{-1-n_{\beta}}\phi)=\phi$, with initial data
$(T^{-1-n_{\beta}}\phi|_{\Sigma},T^{-n_{\beta}}\phi|_{\Sigma})$, such that:

$\displaystyle
r^{\frac{\beta_{0}}{2}-n_{\beta}-\delta}X(rT^{-1-n_{\beta}}\phi)|_{\Sigma_{0}}\in$
$\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{0}}{2}-n_{\beta}-\delta}T^{-1-n_{\beta}}\phi|_{\Sigma_{0}}\in\mathbf{H}_{N+1}$
and $\displaystyle\quad
r^{\frac{\beta_{0}}{2}-n_{\beta}-\delta}T(T^{-1-n_{\beta}}\phi)|_{\Sigma_{0}}\in\mathbf{H}_{N+1}.$

Furthermore, there exists a constant $C=C(V_{\alpha},h,M,N,\delta)>0$, such
that:

$\begin{split}||r^{\frac{\beta_{0}}{2}-n_{\beta}-\delta}X(rT^{-1-n_{\beta}}\phi)_{\Sigma_{0}}||_{\mathbf{H}_{N}}+||r^{\frac{\beta_{0}}{2}-n_{\beta}-\delta}T^{-1-n_{\beta}}\phi_{\Sigma_{0}}||_{\mathbf{H}_{N}}\leq&\>C||r^{\frac{\beta_{0}}{2}+1-\delta}X(r\phi)_{\Sigma_{0}}||_{\mathbf{H}_{N}}+C||r^{\frac{\beta_{0}}{2}-\delta}\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}\\\
&+C||r^{\frac{\beta_{0}}{2}-\delta}T\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}.\end{split}$

We refer to $T^{-k}\phi$, with $k\in\mathbb{N}$, as the _$k$ -th time
integral_ of $\phi$. If we assume additionally that

$\displaystyle r^{\frac{\beta_{1}}{2}+1-\delta}X(r\phi_{\geq 1})|_{\Sigma}\in$
$\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{1}}{2}-\delta}\phi_{\geq
1}|_{\Sigma}\in\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{1}}{2}-\delta}T\phi_{\geq 1}|_{\Sigma}\in\mathbf{H}_{N},$

then we obtain:

$\displaystyle
r^{\frac{\beta_{1}}{2}-n_{\beta}-\delta}X(rT^{-1-n_{\beta}}\phi_{\geq
1})|_{\Sigma}\in$ $\displaystyle\>\mathbf{H}_{N}\quad\textnormal{and}\quad
r^{\frac{\beta_{1}}{2}-n_{\beta}-\delta}T^{-1-n_{\beta}}\phi_{\geq
1}|_{\Sigma}\in\mathbf{H}_{N+1}$ and $\displaystyle\quad
r^{\frac{\beta_{1}}{2}-n_{\beta}-\delta}T(T^{-1-n_{\beta}}\phi_{\geq
1})|_{\Sigma}\in\mathbf{H}_{N+1}.$

Furthermore, there exists a constant $C=C(V_{\alpha},h,M,N,\delta)>0$, such
that:

$\begin{split}||r^{\frac{\beta_{1}}{2}-n_{\beta}-\delta}X(rT^{-1-n_{\beta}}\phi)_{\Sigma_{0}}||_{\mathbf{H}_{N}}+||r^{\frac{\beta_{1}}{2}-n_{\beta}-\delta}T^{-1-n_{\beta}}\phi_{\Sigma_{0}}||_{\mathbf{H}_{N}}\leq&\>C||r^{\frac{\beta_{1}}{2}+1-\delta}X(r\phi)_{\Sigma_{0}}||_{\mathbf{H}_{N}}+C||r^{\frac{\beta_{1}}{2}-\delta}\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}\\\
&+C||r^{\frac{\beta_{1}}{2}-\delta}T\phi|_{\Sigma_{0}}||_{\mathbf{H}_{N}}.\end{split}$

###### Proof.

We proceed as in the proof of Proposition 9.9 with $q=\beta_{0}-\delta$ and
$\delta>0$ sufficiently small to construct $T^{-1}\phi$. Then we repeat the
argument of Proposition 9.9 with the role of $\phi$ replaced with $T^{-1}\phi$
and $q=\beta_{0}-1-\delta$ to construct $T^{-2}\phi$. We can repeat this
procedure a total number $n_{\beta}$ times to obtain the desired estimates.

The improved estimates for $\phi_{\geq 1}$ follow as above, but we apply
Corollary 9.8 instead of Corollary 9.7. ∎

We will moreover need to determine the precise large-$r$ behaviour of
$T^{-1-n_{\beta}}\phi_{0}|_{\Sigma}$. In the proposition below we will assume
that $\beta_{0}\neq 2l+1$ for all $\ell\in\mathbb{N}_{0}$. This is equivalent
to the assumption that $\alpha\neq l(l+1)$ for all $l\in\mathbb{N}_{0}$. This
assumption is made for the sake of convenience. It is still possible to obtain
the precise large-$r$ behaviour when $\beta_{0}=2l+1$ for some
$l\in\mathbb{N}$, but the proof of Proposition 9.11 would have to be modified
slightly due to cancellations appearing in the asymptotics; see Remark 9.2 and
§4.2.

######  Proposition 9.11.

Let $\beta_{0}\neq 2l+1$ for all $\ell\in\mathbb{N}_{0}$. Consider smooth
initial data for (3.1) with corresponding solution $\phi$.

  1. _(i)_

Let $\beta_{0}<1$. If $\int_{2M}^{\infty}w_{0}F_{0}[\phi]\,dr$ is finite and
$w_{0}F_{0}[\phi]$ is sufficiently regular at $r=\infty$ with respect to $rX$,
then

$\mathfrak{I}_{\beta_{0}}[\phi]:=\lim_{r\to\infty}r^{\frac{1}{2}+\frac{1}{2}\beta_{0}}X(rT^{-1}\phi_{0})=-\frac{1-\beta_{0}}{2\beta_{0}}\int_{2M}^{\infty}w_{0}F_{0}[\phi_{0}]\,dr^{\prime}.$

Furthermore, there exists a constant $C=C(M,\alpha,h)>0$ such that for all
$N\in\mathbb{N}_{0}$:

$\displaystyle\left|(rX)^{N}X(rT^{-1}\phi_{0})(0,r)-\mathfrak{I}_{\beta_{0}}[\phi](rX)^{N}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})\right|$
$\displaystyle\leq$
$\displaystyle\>Cr^{-\frac{1}{2}-\frac{3}{2}\beta_{0}}\sum_{n=0}^{N}\left(||(r^{2}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}+||(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}+||r(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}\right).$

  2. _(ii)_

Let $\beta_{0}>1$. Assume the initial data for (3.1) is compactly supported.
Then, for $n_{\beta}=\lfloor\frac{\beta_{0}+1}{2}\rfloor$:

$\mathfrak{I}_{\beta_{0}}[\phi]:=\lim_{r\to\infty}r^{\frac{1}{2}+(\frac{1}{2}\beta_{0}-n_{\beta})}X(rT^{-1-n_{\beta}}\phi_{0})=-\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2n_{\beta}+1-\beta_{0})}{2n_{\beta}!\beta_{0}(1-\beta_{0})\ldots(n_{\beta}-\beta_{0})}\int_{2M}^{\infty}w_{0}F_{0}[\phi_{0}]\,dr^{\prime}$

is finite and generically non-zero. Furthermore, there exists a constant
$C=C(M,\alpha,h)>0$ such that for all $N\in\mathbb{N}_{0}$:

$\displaystyle\left|(rX)^{N}X(rT^{-1-n_{\beta}}\phi_{0})(0,r)-\mathfrak{I}_{\beta_{0}}[\phi](rX)^{N}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}+n_{\beta}})\right|$
$\displaystyle\leq$
$\displaystyle\>Cr^{-\frac{3}{2}-\frac{1}{2}\beta_{0}+n_{\beta}}\Biggl{(}\sum_{n=0}^{n_{\beta}+N}||r^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}$
$\displaystyle+||r^{\frac{1}{2}+\frac{\beta_{0}}{2}}(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}+||r^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}\Biggr{)}.$

###### Proof.

We can express:

$X(Dr^{2}w_{0}^{2}XT^{-1}\check{\phi}_{0})=w_{0}F_{0}[\phi],$

with $F_{0}=\pi_{0}F$. By the regularity properties established in Proposition
9.9, it follows that

$\displaystyle Dr^{2}w_{0}^{2}XT^{-1}\check{\phi}_{0}(0,2M)=$
$\displaystyle\>0,$
$\displaystyle\lim_{r\to\infty}T^{-1}\check{\phi}_{0}(0,r)=$
$\displaystyle\>0.$

Hence, we can integrate in $r$ from $r=2M$ to obtain the explicit formula:

$Dr^{2}w_{0}^{2}XT^{-1}\check{\phi}_{0}(0,r)=\int_{2M}^{r}w_{0}F_{0}[\phi]\,dr,$

from which it follows that

$\displaystyle XT^{-1}\check{\phi}_{0}(0,r)=$
$\displaystyle\>(Dr^{2})^{-1}w_{0}^{-2}\int_{2M}^{r}w_{0}F_{0}[\phi]\,dr^{\prime},$
$\displaystyle T^{-1}\check{\phi}_{0}(0,r)=$
$\displaystyle-\int_{r}^{\infty}(Dr^{2})^{-1}w_{0}^{-2}\int_{2M}^{r}w_{0}F_{0}[\phi]\,dr.$

By the assumptions we made on our initial data, we moreover have that

$\displaystyle|d_{0}|<$ $\displaystyle\>\infty,$ $\displaystyle d_{0}:=$
$\displaystyle\>\int_{2M}^{\infty}w_{0}F_{0}[\phi]\,dr^{\prime}.$

Suppose first that $\beta_{0}<1$. Then for sufficiently regular
$w_{0}F_{0}[\phi]$ at $r=\infty$ with respect to $rX$:

$\displaystyle\int_{2M}^{r}w_{0}F_{0}[\phi]\,dr^{\prime}=$
$\displaystyle\>d_{0}D+DO_{\infty}(r^{\frac{1}{2}\beta_{0}-\frac{1}{2}}).$

We moreover have that

$\displaystyle w_{0}^{-2}=$
$\displaystyle\>r^{1-\beta_{0}}+O_{\infty}(r^{1-2\beta_{0}}).$

Hence,

$\displaystyle XT^{-1}\check{\phi}_{0}(0,r)=$
$\displaystyle[d_{0}r^{-2}+O_{\infty}(r^{\frac{1}{2}\beta_{0}-\frac{5}{2}})][r^{1-\beta_{0}}+O_{\infty}(r^{1-2\beta_{0}})],$
$\displaystyle T^{-1}\check{\phi}_{0}(0,r)=$ $\displaystyle-
d_{0}\beta_{0}^{-1}r^{-\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}(1+\beta_{0})})+O_{\infty}(r^{-2\beta_{0}}),$

so we obtain the following expansions:

$\displaystyle rT^{-1}\phi_{0}(0,r)=$ $\displaystyle-
d_{0}\beta_{0}^{-1}r^{\frac{1}{2}-\frac{1}{2}\beta_{0}}+O_{\infty}(1)+O_{\infty}(r^{\frac{1}{2}-\frac{3}{2}\beta_{0}}),$
$\displaystyle X(rT^{-1}\phi_{0})(0,r)=$
$\displaystyle-\frac{1}{2\beta_{0}}(1-\beta_{0})d_{0}r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{3}{2}\beta_{0}}),$

So

$\mathfrak{I}_{\beta_{0}}[\phi]=\lim_{r\to\infty}r^{\frac{1}{2}+\frac{1}{2}\beta_{0}}X(rT^{-1}\phi_{0})=-\frac{1}{2\beta_{0}}(1-\beta_{0})d_{0}$

is finite and generically non-zero. The inequalities in (i) follow by
expressing the terms appearing in $O_{\infty}(...)$ via derivatives of
$F_{0}[\phi]$.

Now suppose $\beta_{0}>1$. By the regularity properties established in
Corollary 9.10, it follows that

$\displaystyle Dr^{2}w_{0}^{2}XT^{-n}\check{\phi}_{0}(0,2M)=$
$\displaystyle\>0,$
$\displaystyle\lim_{r\to\infty}T^{-n}\check{\phi}_{0}(0,r)=$
$\displaystyle\>0,$

for all $1\leq n\leq n_{\beta}$. By the compactness assumption on the initial
data, we have in particular that

$\displaystyle\int_{2M}^{r}w_{0}F_{0}[\phi]\,dr^{\prime}=$
$\displaystyle\>d_{0}+DO_{\infty}(r^{-2}).$

We moreover have that

$\displaystyle w_{0}^{-2}=$
$\displaystyle\>r^{1-\beta_{0}}+O_{\infty}(r^{-\beta_{0}}).$

Hence, using that $\beta_{0}\neq 1$, since $\beta_{0}\neq 2l+1$ for all
$l\in\mathbb{N}_{0}$:

$\displaystyle T^{-1}\check{\phi}_{0}(0,r)=$ $\displaystyle-
d_{0}\beta_{0}^{-1}r^{-\beta_{0}}+O_{\infty}(r^{-\beta_{0}-1}),$

so we obtain:

$\displaystyle rT^{-1}\phi_{0}(0,r)=$ $\displaystyle-
d_{0}\beta_{0}^{-1}r^{\frac{1}{2}-\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}\beta_{0}-\frac{3}{2}}),$

and

$\displaystyle w_{0}F_{0}[T^{-1}{\phi}]=$
$\displaystyle-\beta_{0}^{-1}(1-\beta_{0})d_{0}+O_{\infty}(r^{-1}).$

We now consider $T^{-2}\check{\phi}$. From the above expressions, it follows
that

$\displaystyle\int_{2M}^{r}w_{0}F_{0}[T^{-1}\phi]\,dr^{\prime}=$
$\displaystyle\>-\beta_{0}^{-1}(1-\beta_{0})d_{0}Dr+D\log
rO_{\infty}(r^{-1}),$

and hence,

$\displaystyle T^{-2}\check{\phi}_{0}(0,r)=$ $\displaystyle-
d_{0}\beta_{0}^{-1}r^{-\beta_{0}+1}+O_{\infty}(r^{-\beta_{0}+\delta}),$

where we can take $\delta>0$ arbitrarily small.

We will establish analogous estimates for $T^{-k}\check{\phi}_{0}(0,r)$ with
$2\leq k\leq 1+n_{\beta}$ by induction. We take as the inductive assumption:

$T^{-k}\check{\phi}_{0}(0,r)=-d_{0}\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2k-3-\beta_{0})}{\beta_{0}(1-\beta_{0})\ldots(k-1-\beta_{0})}\frac{1}{(k-1)!}r^{k-1-\beta_{0}}+O_{\infty}(r^{k-2-\beta_{0}-\delta}),$

with $2\leq k\leq n_{\beta}$.

Note that the above estimate holds for $k=2$, by above. With the induction
assumption, we obtain:

$\displaystyle rT^{-k}{\phi}_{0}(0,r)=$ $\displaystyle-
d_{0}\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2k-3-\beta_{0})}{\beta_{0}(1-\beta_{0})\ldots(k-1-\beta_{0})}\frac{1}{(k-1)!}r^{k-\frac{1}{2}-\frac{1}{2}\beta_{0}}+O_{\infty}(r^{k-\frac{3}{2}-\frac{1}{2}\beta_{0}-\delta}),$
$\displaystyle w_{0}F[T^{-k}{\phi}_{0}]=$
$\displaystyle\>-d_{0}\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2k-3-\beta_{0})(2k-1-\beta_{0})}{\beta_{0}(1-\beta_{0})\ldots(k-1-\beta_{0})}\frac{1}{(k-1)!}r^{k-1}+O_{\infty}(r^{k-2+\delta}).$

Then it follows that

$\displaystyle X(T^{-k-1}\check{\phi}_{0}(0,r))=$ $\displaystyle-
d_{0}\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2k-3-\beta_{0})(2k-1-\beta_{0})}{\beta_{0}(1-\beta_{0})\ldots(k-1-\beta_{0})}\frac{1}{k(k-1)!}Dr^{k-1-\beta_{0}}+O_{\infty}(r^{k-2-\beta_{0}+\delta}),$
$\displaystyle T^{-k-1}\check{\phi}_{0}(0,r)=$ $\displaystyle-
d_{0}\frac{(1-\beta_{0})(3-\beta_{0})\ldots(2k-3-\beta_{0})(2k-1-\beta_{0})}{\beta_{0}(1-\beta_{0})\ldots(k-1-\beta_{0})(k-\beta_{0})}\frac{1}{k(k-1)!}Dr^{k-\beta_{0}}+O_{\infty}(r^{k-1-\beta_{0}+\delta})$

Hence, the induction assumption holds also with $k$ replaced by $k+1$.

We arrive at the inequality in (ii) by estimating the terms in
$O_{\infty}(\ldots)$ via $F_{0}[\phi]$ and its $X$-derivatives.∎

######  Remark 9.2.

If $\beta_{0}=2l+1$, for some $l\in\mathbb{N}_{0}$, it can be shown that,
given smooth and compactly supported initial data for $\phi$,
$T^{-l-2}\phi_{0}$ is well-defined and

$X(rT^{-l-2}\phi_{0})|_{\Sigma}=D_{l}r^{-1}+O_{\infty}(r^{-2}).$

Since we do not make use of this estimate, we will leave it to the reader to
prove this statement and determine the precise form of $D_{l}$.

##  10 Late-time asymptotics and time-decay estimates

We will now apply Proposition 8.2 with $N=n_{\beta}+1$ to the time integral
$T^{-n_{\beta}-1}\phi$, where $n_{\beta}=\lfloor\frac{\beta_{0}+1}{2}\rfloor$.
We will see that the estimates in Corollary 9.10 do not provide enough
regularity for $T^{-n_{\beta}-1}\phi$ to conclude that the weighted, higher-
order energy norms appearing on the right-hand side of (8.2) are finite.
Hence, the time-decay will be limited by the $r$-decay rates of
$T^{-n_{\beta}-1}\phi$ along $\Sigma_{0}$.

In order to go beyond (almost-)sharp time-decay estimates for $\phi$ to the
precise late-time asymptotics of $\phi$ as $\tau\to\infty$, we will consider
the difference

$\displaystyle\widehat{\phi}=$ $\displaystyle\>\phi-\Phi,$
$\displaystyle\Phi(\tau,r)=$
$\displaystyle\>\Phi_{0}w_{0}(r)(\tau+1+2r)^{-\frac{1}{2}-\frac{\beta_{0}}{2}}(\tau+1)^{-\frac{1}{2}-\frac{\beta_{0}}{2}},$
$\displaystyle\Phi_{0}\in$ $\displaystyle\>\mathbb{R}\setminus\\{0\\}.$

The goal of this section is to show that the global leading-order behaviour in
time of $\phi$ is actually determined by $\Phi$. We will show this as follows:

  1. 1.

We will establish sufficiently strong decay estimates for $\widehat{\phi}$,
which satisfies an inhomogeneous wave equation:

$(\square_{g}-V_{\alpha})\widehat{\phi}=-(\square_{g}-V_{\alpha})\Phi.$ (10.1)

In Lemma 10.1 below, we show that the choice of the function $\Phi$ ensures
that $(\square_{g}-V_{\alpha})\Phi$ has favourable decay properties in $\tau$
and $r$, which will allow us to apply the energy decay estimates of
Proposition 8.2 to $\widehat{\phi}$.

  2. 2.

We will choose the constant $\Phi_{0}$ such that the $(1+n_{\beta})$-th time
integral of the difference $\widehat{\phi}$ has _better_ decay properties in
$r$ along $\Sigma_{0}$ than the $(1+n_{\beta})$-th time integral of $\phi$.
This allows us to use Proposition 8.2 to show that $\widehat{\phi}$ decays
_faster_ than $\Phi$, so the late-time behaviour of $\phi$ is determined
(globally) by $\Phi$.

We define $\Phi^{(k)}$ as follows

$\Phi^{(k)}(\tau,r):=(-1)^{k}\int_{\tau}^{\infty}\int_{\tau_{1}}\ldots\int_{\tau_{k-1}}\Phi(\tau_{k},r)\,d\tau_{k}d\tau_{k-1}\ldots
d\tau_{1}.$

Then $T^{k}\Phi^{(k)}=\Phi$. We will also denote $T^{-k}\Phi=\Phi^{(k)}$.

Furthermore, we can integrate by parts $k-1$ times in $\tau$ to express:

$\Phi^{(k)}(\tau,r)=\frac{(-1)^{k}}{(k-1)!}\int_{\tau}^{\infty}(s-\tau)^{k-1}\Phi(s,r)\,ds$

for $k<\beta_{0}+1$.

We denote moreover

$T^{-k}\widehat{\phi}=T^{-k}\phi-\Phi^{(k)}.$

######  Lemma 10.1.

  1. _(i)_

For all $n_{1},n_{2}\in\mathbb{N}_{0}$ and $k\in\mathbb{N}_{0}$,
$k\leq\beta_{0}+1$, there exists a $C=C(M,V_{\alpha},n_{1},n_{2},k)>0$, such
that

$\displaystyle\left|(rX)^{n_{1}}T^{n_{2}}(\square_{g}-V_{\alpha})\Phi^{(k)}\right|\leq$
$\displaystyle\>C|\Phi_{0}|r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}-\min\\{1,\beta_{0}\\}}(\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}+k-n_{2}},$
(10.2)

  2. _(ii)_

Let $\beta_{0}\neq 2l+1$ for all $l\in\mathbb{N}_{0}$. For all
$k\in\mathbb{N}_{0}$, such that $k<\frac{\beta_{0}}{2}+\frac{1}{2}$, we can
expand

$X(r\Phi^{(k+1)})|_{\Sigma_{0}}=\frac{(-1)^{k}2^{k-\beta_{0}}\pi\Gamma(\beta_{0}-k)\Phi_{0}}{k!\Gamma\left(\frac{\beta_{0}+1}{2}\right)\Gamma\left(\frac{\beta_{0}-1-2k}{2}\right)\cos\left(\frac{\beta_{0}-2k}{2\pi}\right)}r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}+k}+O_{\infty}(r^{-\frac{3}{2}-\frac{1}{2}\beta_{0}+k})+O_{\infty}(r^{-2}).$
(10.3)

###### Proof.

We can write

$(\square_{g}-V_{\alpha})\Phi=r^{-2}X(Dr^{2}X\Phi)-2(1-h)XT\Phi+h\tilde{h}T^{2}\Phi+\left[-2r+\left(r^{2}h\right)^{\prime}\right]r^{-2}T\Phi-
V_{\alpha}\Phi.$

If we denote
$f(\tau,r)=(\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}$,
and we use (3.7), we can write:

$\begin{split}\Phi_{0}^{-1}(\square_{g}-V_{\alpha})\Phi=&\>w_{0}DX^{2}f-2(1-h)w_{0}XTf+\left[(Dr^{2})^{\prime}+2Dr^{2}w_{0}^{-1}w_{0}^{\prime}\right]r^{-2}w_{0}Xf\\\
&+\left[-2(1-h)r^{2}w_{0}^{-1}w_{0}^{\prime}-2r+\left(r^{2}h\right)^{\prime}\right]r^{-2}w_{0}Tf+h\tilde{h}w_{0}T^{2}f.\end{split}$

We will need the following properties of the function $f$:

$\displaystyle Xf=$
$\displaystyle\>-(\beta_{0}+1)(\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}},$
$\displaystyle Tf=$
$\displaystyle\>-\frac{1}{2}(\beta_{0}+1)((\tau+1+2r)^{-1}+(\tau+1)^{-1})(\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}$
$\displaystyle=$
$\displaystyle\>-(\beta_{0}+1)(\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}-(\beta_{0}+1)r(\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}},$
$\displaystyle X^{2}f=$
$\displaystyle\>(\beta_{0}+1)(\beta_{0}+3)(\tau+1+2r)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}},$
$\displaystyle XTf=$
$\displaystyle\>\frac{1}{2}(\beta_{0}+1)(\beta_{0}+3)(\tau+1+2r)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}+\frac{1}{2}(\beta_{0}+1)^{2}(\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}},$
$\displaystyle T^{2}f=$
$\displaystyle\>(\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}O_{\infty}(\tau+1)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}),$
$\displaystyle w_{0}=$
$\displaystyle\>r^{-\frac{1}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}}),$
$\displaystyle w_{0}^{\prime}w_{0}^{-1}=$
$\displaystyle\>\frac{1}{2}(\beta_{0}-1)r^{-1}+O_{\infty}(r^{-1-\beta_{0}})+O_{\infty}(r^{-2}),$

to obtain

$\begin{split}\Phi_{0}^{-1}\square_{g}\Phi=&\>(\beta_{0}+1)(\beta_{0}+3)[r^{-\frac{1}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}\\\
&-(\beta_{0}+1)(\beta_{0}+3)[r^{-\frac{1}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}\\\
&-(\beta_{0}+1)^{2}[r^{-\frac{1}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}\\\
&-(\beta_{0}+1)[(\beta_{0}+1)r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{3}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{5}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}\\\
&+(\beta_{0}+1)[(\beta_{0}+1)r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{3}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{5}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}\\\
&+(\beta_{0}+1)[(\beta_{0}+1)r^{-\frac{1}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{1}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}\\\
&+[r^{-\frac{5}{2}+\frac{1}{2}\beta_{0}}+O_{\infty}(r^{-\frac{5}{2}-\frac{1}{2}\beta_{0}})+O_{\infty}(r^{-\frac{7}{2}+\frac{1}{2}\beta_{0}})](\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}O_{\infty}((\tau+1)^{-\frac{5}{2}-\frac{1}{2}\beta_{0}}).\end{split}$

Note that most of the terms above cancel, so we are left with

$\Phi_{0}^{-1}(\square_{g}-V_{\alpha})\Phi=(\tau+1+2r)^{-\frac{1}{2}-\frac{1}{2}\beta_{0}}(\tau+1)^{-\frac{3}{2}-\frac{1}{2}\beta_{0}}O_{\infty}(r^{-\frac{3}{2}+\frac{1}{2}\beta_{0}-\min\\{1,\beta_{0}\\}}).$

We conclude that the estimates in (i) hold.

We can verify by differentiating in time $\tau$ and appealing to standard
properties of hypergeometric functions $\,{}_{2}F_{1}\left(a,b,c;z\right)$
(see for example [Olv74]) that:

$\begin{split}&X(rr^{\frac{-1+\beta_{0}}{2}}w_{0}^{-1}\Phi^{(k+1)})(\tau,r)=\frac{(-1)^{k+1}}{(k+1)!}\frac{\pi(1+\beta_{0})}{2\cos\left(\frac{\pi}{2}(\beta_{0}-2k)\right)}r^{\frac{1}{2}(\beta_{0}-1)}(1+2r+\tau)^{-\beta_{0}}\\\
\cdot&\Bigg{[}-\frac{\Gamma(\beta_{0}-k)}{\Gamma\left(\frac{3+\beta_{0}}{2}\right)\Gamma\left(\frac{\beta_{0}-1-2k}{2}\right)}(2+2r+\tau)^{k}\,_{2}F_{1}\left(\frac{1}{2}(\beta_{0}-1),\beta_{0}-k,\frac{1}{2}(\beta_{0}-1-2k);\frac{\tau+1}{\tau+2r+1}\right)\\\
&+\frac{\Gamma(1+k)}{\Gamma\left(\frac{\beta_{0}-1}{2}\right)\Gamma\left(\frac{5-\beta_{0}-1+k}{2}\right)}(2+2r+\tau)^{\frac{3-\beta_{0}+k}{2}}(1+\tau)^{\frac{3-\beta_{0}+2k}{2}}\,_{2}F_{1}\left(\frac{1}{2}(3+\beta_{0}),1+k,\frac{5-\beta_{0}+k}{2};\frac{\tau+1}{\tau+2r+1}\right)\Bigg{]}.\end{split}$
(10.4)

To obtain (ii), we use that

$r^{\frac{-1+\beta_{0}}{2}}w_{0}^{-1}=1+O_{\infty}(r^{-\min\\{\beta_{0},1\\}})$

and we evaluate the expression on the right-hand side of (10.4) at $\tau=0$
and expand the hypergeometric functions in $\frac{1}{r}$. ∎

######  Remark 10.1.

If we take $M=0$ in the Schwarzschild metric $g_{M}$ and we fix
$V_{\alpha}=\alpha r^{-2}$, we in fact have that:
$(\square_{g}-V_{\alpha})\Phi=0$.

######  Remark 10.2.

If $\beta_{0}=2l+1$ for some $l\in\mathbb{N}_{0}$, it can be shown that the
following asymptotics hold:

$X(r\Phi^{(l+1)})(0,r)=(-1)^{l+1}\frac{l!}{2^{l+1}(2l+1)!}\Phi_{0}r^{-1}+O_{\infty}(r^{-2}).$

We now fix the constant $\Phi_{0}$:

$\Phi_{0}=(-1)^{n_{\beta}}n_{\beta}!\frac{\Gamma\left(\frac{\beta_{0}+1}{2}\right)\Gamma\left(\frac{\beta_{0}-1-2n_{\beta}}{2}\right)\cos\left(\frac{\beta_{0}-2n_{\beta}}{2\pi}\right)}{2^{n_{\beta}-\beta_{0}}\pi\Gamma(\beta-
n_{\beta})}\mathfrak{I}_{\beta}[\phi].$ (10.5)

With the choice (10.5), it follows immediately that we can obtain improved
decay properties in $r$ for $T^{-1-n_{\beta}}\widehat{\phi}_{0}$ compared to
Proposition 9.11, since this choice of $\Phi_{0}$ ensures a cancellation
between the leading-order terms in $r^{-1}$ of
$X(rT^{-1-n_{\beta}}\widehat{\phi}_{0})$ and $X(r\Phi^{(1+n_{\beta})})$:

######  Corollary 10.2.

Assume $\beta_{0}\neq 2l+1$ for all $l\in\mathbb{N}_{0}$. Let $\Phi_{0}$
satisfy (10.5) and assume smooth initial data for (3.1).

  1. _(i)_

Let $\beta_{0}<1$. If $\lim_{r\to\infty}r^{2}X(r\phi)|_{\Sigma}<\infty$. Then
for all $N\in\mathbb{N}_{0}$, there exists a constant
$C=~{}C(M,\alpha,h,N)>~{}0$ such that:

$\displaystyle\left|(rX)^{N}X(rT^{-1}\widehat{\phi}_{0})(0,r)\right|$
$\displaystyle\leq$
$\displaystyle\>Cr^{-\frac{1}{2}-\frac{3}{2}\beta_{0}}\sum_{n=0}^{N}\left(||(r^{2}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}+||(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}+||r(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}\right).$

  2. _(ii)_

Let $\beta_{0}>1$. Assume the initial data are compactly supported. Then, for
$n_{\beta}=\lfloor\frac{\beta_{0}+1}{2}\rfloor$ and for all
$N\in\mathbb{N}_{0}$, there exists a constant $C=C(M,\alpha,h,N)>0$, such that
:

$\begin{split}&\left|(rX)^{N}X(rT^{-1-n_{\beta}}\widehat{\phi}_{0})(0,r)\right|\\\
\leq&\>Cr^{-\frac{3}{2}-\frac{1}{2}\beta_{0}+n_{\beta}}\Biggl{(}\sum_{n=0}^{n_{\beta}+N}||r^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}\\\
&+||r^{\frac{1}{2}+\frac{\beta_{0}}{2}}(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}+||r^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}\Biggr{)}.\end{split}$

###  10.1 Energy decay estimates for $\widehat{\phi}=\phi-\Phi$

In this section and in Section 10.2 below, we establish decay-in-time
estimates for ($r$-weighted) energies of $\widehat{\phi}$ along
$\Sigma_{\tau}$, which are necessary for obtaining the $L^{\infty}$ decay-in-
time estimates in Section 10.3.

We will first treat separately the cases $\alpha<0$ and $\alpha>0$.

####  10.1.1 $\alpha<0$

In order to obtain time-decay for $T^{-1}\widehat{\phi}$, which for $\alpha$
such that $3\beta_{0}\leq 1$ decays too slowly in $r$ to apply directly the
(inhomogeneous analogues of) the energy decay estimates of Proposition 8.2, we
first introduce $\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$, which
we define to be the solution to

$(\square_{g}-V_{\alpha})\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}=-(\square_{g}-V_{\alpha})\Phi^{(1)},$

arising from the cut-off initial data:

$(\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|_{\Sigma_{0}},T\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|_{\Sigma_{0}})=(\chi_{r_{\infty}}T^{-1}\widehat{\phi}|_{\Sigma_{0}},\chi_{r_{\infty}}\widehat{\phi}|_{\Sigma_{0}})$

with $r_{\infty}>3M$ and $\chi_{r_{\infty}}$ a smooth cut-off function such
that $\chi_{r_{\infty}}=1$ for $r\leq r_{\infty}$ and $\chi_{r_{\infty}}=0$
for $r\geq 2r_{\infty}$.

By the domain of dependence property of (3.1), we have that

$T^{k-1}\widehat{\phi}=T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}\quad\textnormal{in}\quad\left\\{v\leq
v|_{\tau=0}(r_{\infty}):=v_{r_{\infty}}\right\\}$

for all $k\in\mathbb{N}_{0}$.

We first prove analogues of the non-degenerate energy boundedness estimate
(6.20) and the integrated energy estimates in Corollary 6.8 for
$\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$, which take into account
the inhomogeneity in the equation for $\widehat{\phi}$.

######  Proposition 10.3.

Let $\alpha<0$ and $3\beta_{0}\leq 1$, $N\in\mathbb{N}$ and let $\delta>0$ be
arbitrarily small. Then there exists a constant
$C=C(M,h,V_{\alpha},N,\delta)>0$ such that for all $0\leq\tau_{1}<\tau_{2}$

  1. _(i)_

$\begin{split}E_{\mathbf{N}}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{2})\leq&\>CE_{\mathbf{N}}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+C(1+\tau_{1})^{-1}\int_{\Sigma_{\tau_{1}}\cap\\{r\geq
R\\}}r(LT^{k}\widehat{\psi}_{\textrm{\char 34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr\\\ &+C\Phi_{0}^{2}(1+\tau_{1})^{-3\beta_{0}-2k+2\delta},\end{split}$ (10.6)

  2. _(ii)_

$\begin{split}E_{\mathbf{N}}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{2})+&\int_{\Sigma_{\tau_{2}}\cap\\{r\geq
R\\}}r(LT^{k}\widehat{\psi}_{\textrm{\char 34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr+\int_{\tau_{1}}^{\tau_{2}}E_{\mathbf{N}}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}]\,d\sigma drd\tau\\\
\leq&\>CE_{\mathbf{N}}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+CE_{\mathbf{N}}[T^{k+1}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+C\int_{\Sigma_{\tau_{1}}\cap\\{r\geq
R\\}}r(LT^{k}\widehat{\psi}_{\textrm{\char 34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr\\\
&+C\Phi_{0}^{2}\max\\{(1+\tau_{1})^{1-3\beta_{0}-2k+2\delta},(1+\tau_{2})^{1-3\beta_{0}-2k+2\delta}\\},\end{split}$
(10.7)

  3. _(iii)_

$\begin{split}\int_{\tau_{1}}^{\tau_{2}}&\int_{\Sigma_{\tau}\cap\\{R_{0}\leq
r\leq R_{1}\\}}\Bigg{[}\sum_{n_{1}+n_{2}+n_{3}\leq
N}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X^{n_{2}}T^{n_{3}+k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}+1}X^{n_{2}}T^{n_{3}+k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|^{2}\\\
&+|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X^{n_{2}+2}T^{n_{3}+k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|^{2}+h_{0}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X^{n_{2}}T^{n_{3}+1+k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|^{2}\Bigg{]}\,d\sigma drd\tau\\\
\leq&\>C\sum_{n=1}^{N+k}E_{T}[T^{n_{2}}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+C\int_{\Sigma_{\tau_{1}}\cap\\{r\geq
R\\}}r(LT^{n}\widehat{\psi}_{\textrm{\char 34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr\\\
&+C\Phi_{0}^{2}\max\\{(1+\tau_{1})^{1-3\beta_{0}-2k+2\delta},(1+\tau_{2})^{1-3\beta_{0}-2k+2\delta}\\}.\end{split}$
(10.8)

###### Proof.

Let $0\leq\tau_{1}<\tau_{2}$. We will first prove __(i)__ with $E_{T}$ instead
of $E_{\mathbf{N}}$ on the right-hand side. Note that (5.1) no longer holds
due to the presence of an inhomogeneity in the equation for
$\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$. In the estimate for
(5.1) we instead encounter the additional term:

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{2}T\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}\cdot(\square_{g}-V_{\alpha})\Phi^{(1)}\,d\sigma dr.$

After integrating by parts in $\tau$ and applying Young’s inequality with
appropriate weights depending on $\tau$ and $r$ and $\epsilon>0$ and
$\delta>0$ arbitrarily small, we can estimate the norm of the above integral
by:

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\epsilon(1+\tau)^{-2}r^{1-\delta}(\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}+\frac{1}{4}\epsilon^{-1}r^{3+\delta}(1+\tau)^{2}((\square_{g}-V_{\alpha})T\Phi^{(1)})^{2}\,d\sigma
drd\tau\\\
+\sum_{i=1}^{2}\int_{\Sigma_{\tau_{i}}}\epsilon(\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}+\frac{1}{4}\epsilon^{-1}r^{2-2\delta}((\square_{g}-V_{\alpha})\Phi^{(1)})^{2}\,d\sigma
dr.$

We have that

$\sum_{i=1}^{2}\int_{\Sigma_{\tau_{i}}}\epsilon(\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr\leq\epsilon\left[E_{T}[\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+E_{T}[\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{2})\right]$

and by (2.10), it follows that

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\epsilon(1+\tau)^{-2}r^{1-\delta}(\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma drd\tau\leq
C\epsilon(1+\tau_{1})^{-1}\sup_{\tau_{1}\leq\tau\leq\tau_{2}}\int_{\Sigma_{\tau}}r^{1-\delta}(X\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr.$

By applying moreover the estimates in __(i)__ of Lemma 10.1, we can also
further estimate for $\delta>0$ suitably small:

$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\frac{1}{4}\epsilon^{-1}r^{3+\delta}(1+\tau)^{2}((\square_{g}-V_{\alpha})T\Phi^{(1)})^{2}\,d\sigma
drd\tau\leq$
$\displaystyle\>C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{-3\beta_{0}+\delta},$
$\displaystyle\sum_{i=1}^{2}\int_{\Sigma_{\tau_{i}}}\frac{1}{4}\epsilon^{-1}r^{2-2\delta}((\square_{g}-V_{\alpha})\Phi^{(1)})^{2}\,d\sigma
dr\leq$
$\displaystyle\>C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{-2-2\beta_{0}}.$

We can similarly, replace $\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}$ by $T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}$ above and use that

$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}\frac{1}{2}\epsilon^{-1}r^{3+\delta}(1+\tau)^{2}((\square_{g}-V_{\alpha})T^{k+1}\Phi^{(1)})^{2}\,d\sigma
drd\tau\leq$
$\displaystyle\>C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{-3\beta_{0}-2k+\delta},$
$\displaystyle\sum_{i=1}^{2}\int_{\Sigma_{\tau_{i}}}\frac{1}{4}\epsilon^{-1}r^{2-2\delta}((\square_{g}-V_{\alpha})T^{k}\Phi^{(1)})^{2}\,d\sigma
dr\leq$
$\displaystyle\>C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{-2-2\beta_{0}-2k}.$

Hence, for $\epsilon>0$ arbitrarily small, we can estimate

$\begin{split}E_{T}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{2})\leq&\>(1+\epsilon)E_{T}[T^{k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau_{1})+\epsilon(1+\tau_{1})^{-1}\sup_{\tau_{1}\leq\tau\leq\tau_{2}}\int_{\Sigma_{\tau}}r^{1-\delta}(LT^{k}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr\\\
&+C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{-3\beta_{0}-2k+\delta}.\end{split}$
(10.9)

In order to replace $E_{T}$ on the left-hand side with $E_{\mathbf{N}}$ we
proceed as in the derivation of (6.16), but due to the presence of an the
inhomogeneity, we need to include the following additional term on the right-
hand side:

$C\epsilon^{-1}\int_{\Sigma_{\tau_{2}}\cap\\{r\leq(2+\delta)M\\}}((\square_{g}-V_{\alpha})\Phi^{(1)})^{2}\,d\sigma
drd\tau\leq C\epsilon^{-1}\Phi_{0}^{2}(1+\tau)^{-1-2\beta_{0}}.$

Then, we apply the proof of Corollary 6.9 to conclude that (10.9) holds with
$E_{T}$ replaced with $E_{\mathbf{N}}$ on both sides of the equation.

In order to arrive at the integrated energy estimates in Corollary 6.8 with
$\phi$ replaced by $\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$, we
similarly have to estimate the following main additional term involving the
extra inhomogeneity in the arguments leading to a proof of Corollary 6.8:

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r|L\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)}|\cdot r|(\square_{g}-V_{\alpha})\Phi^{(1)}|\,d\sigma
dr\leq\epsilon\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}(1+\tau)^{-1-\delta}r(L\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma drd\tau\\\
+\frac{1}{4}\epsilon^{-1}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{3}(1+\tau)^{1+\delta}((\square_{g}-V_{\alpha})\Phi^{(1)})^{2}\,d\sigma
dr.$

We estimate the right-hand side as above via __(i)__ of Lemma 10.1, noting
that

$\frac{1}{4}\epsilon^{-1}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{3}(1+\tau)^{1+\delta}((\square_{g}-V_{\alpha})\Phi^{(1)})^{2}\,d\sigma
dr\leq
C\epsilon^{-1}\Phi_{0}^{2}\max\\{(1+\tau_{2})^{1-3\beta_{0}},(1+\tau_{1})^{1-3\beta_{0}+2\delta}\\}$

and similarly, for $k\geq 1$:

$\frac{1}{4}\epsilon^{-1}\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}}r^{3}(1+\tau)^{1+\delta}((\square_{g}-V_{\alpha})T^{k}\Phi^{(1)})^{2}\,d\sigma
dr\leq C\epsilon^{-1}\Phi_{0}^{2}(1+\tau_{1})^{1-2k-3\beta_{0}+2\delta}.$

By combining (10.9) with the above integrated energy estimates, we can remove
the supremum over $\tau$ appearing on the right hand side of (10.9) to
conclude that __(i)__ holds, and we can also conclude that __(ii)__ holds.

Finally, __(iii)__ follows from __(ii)__ , without loss of $T$-derivatives on
the right-hand side by applying elliptic estimates as in the proof of
Corollary 6.8, using that the additional terms involving the inhomogeneity can
be bounded straightforwardly by applying __(i)__ of Lemma 10.1 again. ∎

In the proposition below we obtain energy decay estimates for
$\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$ in terms of initial
energies for $\widehat{\phi}$.

######  Proposition 10.4.

Let $\alpha<0$, $N\in\mathbb{N}$ and let $\delta>0$ be arbitrarily small. Then
there exists a constant $C=C(M,h,V_{\alpha},N,\delta)>0$ such that for all
$0\leq k\leq 1$:

$\begin{split}&E_{\mathbf{N}}[T^{N+1-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau)\\\
\leq&\>C(1+\tau)^{-3-2N+2k+\delta}r_{\infty}^{\max\\{1-3\beta_{0},0\\}+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]\Bigg{]}+C|\Phi_{0}|^{2}(1+\tau)^{-2-2N-3\beta_{0}+2k+2\delta}\end{split}$
(10.10)

and moreover

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT^{N+1-k}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{-1-2N+2k+\delta}r_{\infty}^{\max\\{1-3\beta_{0},0\\}+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]\Bigg{]}+C|\Phi_{0}|^{2}(1+\tau)^{-2N-3\beta_{0}+2k+2\delta}.\end{split}$
(10.11)

When $\phi=\phi_{\geq 1}$, we can take the $r$-weight inside the integrals on
the right-hand sides above to be $r$ rather than
$r^{\min\\{3\beta_{0},1\\}-\delta}$ and we can omit the factor
$r_{\infty}^{\max\\{1-3\beta_{0},0\\}+\delta}$ and the term involving
$|\Phi_{0}|^{2}$.

###### Proof.

We proceed as in the proof of Proposition 8.2 and we estimate the additional
terms present as a result of the inhomogeneity present in the equation for
$\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$ as in the proof of
Proposition 10.3, via the application of the estimates in __(i)__ of Lemma
10.1, using moreover the energy boundedness and integrated energy estimates of
Proposition 10.3, in the case $3\beta_{0}\leq 1$.

Then we estimate the energies along $\Sigma_{0}$ for
$T^{k}\widehat{\phi}_{\textrm{\char 34},r_{\infty}}^{(1)}$ in terms of initial
energies for $T^{k}(T^{-1}\widehat{\phi})$. The only non-trivial step here is
the following, when $3\beta_{0}<1$:

$\begin{split}\int_{\Sigma_{0}}r(L(rL)^{k}T^{l}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma
dr=&\>\int_{\Sigma_{0}}r(L(rL)^{k}\chi_{r_{\infty}}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr\\\ \leq&\>C\int_{\Sigma_{0}\cap\\{r\leq
2r_{\infty}\\}}\chi_{r_{\infty}}^{2}r(L(rL)^{k}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr\\\ &+C\sum_{k_{1}+k_{2}=k}\int_{\Sigma_{0}\cap\\{r\leq
2r_{\infty}\\}}r|(rX)^{k_{1}}X(\chi_{r_{\infty}})|^{2}((rL)^{k_{2}}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr\\\ \leq&\>Cr_{\infty}^{1-3\beta_{0}+\delta}\int_{\Sigma_{0}\cap\\{r\leq
2r_{\infty}\\}}\chi_{r_{\infty}}^{2}r^{3\beta_{0}-\delta}(L(rL)^{k}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr\\\ &+Cr_{\infty}^{1-3\beta_{0}+\delta}\sum_{k_{2}\leq
k}\int_{\Sigma_{0}\cap\\{R\leq r\leq
2r_{\infty}\\}}r^{-2+3\beta_{0}-\delta}((rL)^{k_{2}}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr\\\ \leq&\>Cr_{\infty}^{1-3\beta_{0}+\delta}\left[\sum_{k_{2}\leq
k}\int_{\Sigma_{0}}r^{3\beta_{0}-\delta}(L(rL)^{k_{2}}T^{l-1}\widehat{\psi})^{2}\,d\sigma
dr+\sum_{k_{1}=0}^{k}E_{T}[T^{k_{1}+l-1}\widehat{\phi}]\right],\end{split}$

where we applied (2.10) to estimate the $k_{2}=0$ term in the sum on the
right-hand side of the second inequality and arrive at the last inequality,
estimating the boundary terms at $r=R$ in terms of the $T$-energy by averaging
appropriately around $r=R$. ∎

####  10.1.2 $\alpha>0$ and $\beta_{0}\neq 2l+1$

If we assume $\alpha>0$, we define $({\phi}_{\textrm{\char
34},r_{\infty}}^{(1+n_{\beta})})_{\geq 1}$ to be the solution to (3.1) arising
from initial data:

$((\phi_{\textrm{\char 34},r_{\infty}}^{(1+n_{\beta})})_{\geq
1}|_{\Sigma_{0}},T(\phi_{\textrm{\char 34},r_{\infty}}^{(1+n_{\beta})})_{\geq
1}|_{\Sigma_{0}})=(\chi_{r_{\infty}}T^{-1-n_{\beta}}{\phi}_{\geq
1}|_{\Sigma_{0}},\chi_{r_{\infty}}T^{-n_{\beta}}{\phi}_{\geq
1}|_{\Sigma_{0}}).$

We emphasize that in contrast with ${\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}$ above, $({\phi}_{\textrm{\char
34},r_{\infty}}^{(1+n_{\beta})})_{\geq 1}$ satisfies the _homogeneous_
equation (3.1), so the energy boundedness and integrated energy decay
estimates of Propositions 5.1 and 6.8 apply directly, without the need for an
analogue of Proposition 10.3. Note moreover that, in contrast with the
$\alpha<0$ case, we have the additional, stronger, energy decay estimates in
Proposition 8.3 at our disposal.

######  Proposition 10.5.

Let $\alpha>0$ and $\beta_{0}\neq 2l+1$ for all $l\in\mathbb{N}_{0}$. Let
$N\in\mathbb{N}$, and let $\delta>0$ be arbitrarily small. Then there exists a
constant $C=C(M,h,V_{\alpha},N,\delta)>0$ such that for all $0\leq k\leq
1+n_{\beta}$:

  1. _(i)_

$\begin{split}&E_{\mathbf{N}}[T^{N-k}\widehat{\phi}_{0}](\tau)\\\
\leq&\>C(1+\tau)^{-3-2n_{\beta}-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+1+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{0}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{0}]\Bigg{]}+C|\Phi_{0}|^{2}(1+\tau)^{-4-\beta_{0}-2N+2k+\delta}\end{split}$
(10.12)

and moreover

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT^{N-k}\widehat{\psi_{0}})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{-1-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{0}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{0}]\Bigg{]}+C|\Phi_{0}|^{2}(1+\tau)^{-4-\beta_{0}-2N+2k+2\delta}.\end{split}$
(10.13)

  2. _(ii)_

if $\beta_{1}>2n_{\beta}$

$\begin{split}&E_{\mathbf{N}}[T^{N+n_{\beta}+1-k}({\phi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta}+1)})_{\geq 1}](\tau)\\\
\leq&\>C(1+\tau)^{-3-2n_{\beta}-2N+2k+\delta}r_{\infty}^{\max\\{2n_{\beta}+1-\beta_{1},0\\}+\delta}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+1+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{1,\beta_{1}-2n_{\beta}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.14)

and moreover

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT^{N+n_{\beta}+1-k}({\psi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta}+1)})_{\geq 1})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{-1-2n_{\beta}-2N+2k+\delta}r_{\infty}^{\max\\{2n_{\beta}+1-\beta_{1},0\\}+\delta}\\\
\cdot&\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+n_{\beta}+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{1,\beta_{1}-2n_{\beta}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.15)

  3. _(iii)_

if $\beta_{1}\leq 2n_{\beta}$

$\begin{split}&E_{\mathbf{N}}[T^{N+n_{\beta}-k}({\phi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta})})_{\geq 1}](\tau)\\\
\leq&\>C(1+\tau)^{-2-2n_{\beta}-2N+2k}r_{\infty}^{2n_{\beta}-\beta_{1}+\delta}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+n_{\beta})\\\ n_{1}\leq
N+n_{\beta}\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.16)

and moreover

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}r^{2}(LT^{N+n_{\beta}-k}({\psi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta})})_{\geq 1})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{-2n_{\beta}-2N+2k}r_{\infty}^{2n_{\beta}-\beta_{1}+\delta}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+n_{\beta})\\\ n_{1}\leq
N+n_{\beta}\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]}.\end{split}$ (10.17)

###### Proof.

Consider first the spherical harmonic mode $\phi_{0}$. We can repeat the steps
in the proof of Proposition 10.4, using (10.2) to estimate the terms coming
from the inhomogeneity in the equation for $\widehat{\phi}$. When considering
$\phi_{0}$ in the case $\beta_{0}>1$, we carry out estimates directly for
$\widehat{\phi}_{0}$, in contrast with Proposition 10.4.

Consider now the remaining spherical harmonic modes $\phi_{\geq 1}$. We will
derive estimates for $({\phi}_{\textrm{\char
34},r_{\infty}}^{(1+n_{\beta})})_{\geq 1}$. Since
$(\square_{g}-V_{\alpha})({\phi}_{\textrm{\char
34},r_{\infty}}^{(1+n_{\beta})})_{\geq 1}=0$, the energy boundedness estimate
(6.20) and the integrated energy estimates in Corollary 6.8 apply.

Note that since $\beta_{1}>\beta_{0}$, we also have that

$\beta_{1}>2n_{\beta}-1.$

Suppose that $\beta_{1}>2n_{\beta}$. Then we proceed again as in the proof of
Proposition 10.4, but with $T^{1+n_{\beta}-k}({\phi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta}+1)})_{\geq 1}$ replacing $\phi_{0}$, but there is
no inhomogeneity present in the relevant wave equation. The only difference is
the following estimate along $\Sigma_{0}$:

$\begin{split}&\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}({\phi}_{\textrm{\char
34},r_{\infty}}^{(1+n_{\beta})})_{\geq 1}|^{2}\,d\sigma dr\\\
\leq&\>Cr_{\infty}^{2n_{\beta}-\beta_{1}+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+n_{\beta}+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{1,\beta_{1}-2n_{\beta}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\
&+E_{T}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]}.\end{split}$

Now suppose that $\beta_{1}\leq 2n_{\beta}$. Then we apply the energy decay
estimates in Proposition 8.3 to $({\phi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta})})_{\geq 1}$ instead of the energy decay estimates
in Proposition 8.2. In this case, we estimate the corresponding weighted
energy along $\Sigma_{0}$ as follows:

$\begin{split}&\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}({\phi}_{\textrm{\char
34},r_{\infty}}^{(n_{\beta})})_{\geq 1}|^{2}\,d\sigma dr\\\
\leq&\>Cr_{\infty}^{\max\\{2n_{\beta}+1-\beta_{1},0\\}}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\
&+E_{T}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]}.\qed\end{split}$

###  10.2 Additional $r$-weighted energy decay estimates

In Section 10.1 we established energy decay estimates for (inhomogeneous)
solutions to (3.1) arising from cut-off initial data with growing weights in
the quantity $r_{\infty}$, which appears in the support of the cut-off. In
this section we promote these estimates to genuine energy decay estimates for
$\widehat{\phi}$ by interpolating between the decay estimates for the energies
$E_{\mathbf{N}}$ and the energies with additional growing $r$-weights in the
integrands. We moreover derive decay estimates for appropriate $r$-weighted
energies for $\widehat{\phi}$, both for positive and for negative $r$-weights.

We first establish energy decay estimates in the $\alpha<0$ case:

######  Proposition 10.6.

Let $\alpha<0$, $N\in\mathbb{N}$, $p\geq 0$ and $p+\max\\{1-3\beta_{0},0\\}<2$
and let $\delta>0$ be arbitrarily small. Then there exists a constant
$C=C(M,h,V_{\alpha},N,\delta,p)>0$ such that for all $0\leq k\leq 1$

$\begin{split}E_{\mathbf{N}}[T^{N-k}\widehat{\phi}]\\\
\leq&\>C(1+\tau)^{\max\\{1-3\beta_{0},0\\}-3-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Bigg{]}\end{split}$
(10.18)

and

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}&r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{p+\max\\{1-3\beta_{0},0\\}-3-2N+2k+\delta}\Biggl{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Biggr{]}.\end{split}$
(10.19)

When restricting to $\phi=\phi_{\geq 1}$, the above estimates hold for $0\leq
p<2$ if we replace $\min\\{3\beta_{0},1\\}$ and $\max\\{1-3\beta_{0},0\\}$
with 1 and 0, respectively and we omit the term $|\Phi_{0}|^{2}$.

###### Proof.

First, recall that $T^{-1}\widehat{\phi}=\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}$ in $v\leq v_{r_{\infty}}$.

We split

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}&r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr=\int_{\Sigma_{\tau}\cap\\{R\leq
r\leq\tau+R\\}}r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr+\int_{\Sigma_{\tau}\cap\\{r\geq\tau+R\\}}r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr.\end{split}$

First, choose $r_{\infty}=R+B\tau$, for some suitably large constant $B>0$.
Then we can apply Proposition 10.4 to obtain

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{R\leq
r\leq\tau+R\\}}r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr=\int_{\Sigma_{\tau}\cap\\{v\leq v_{\infty}\\}\cap\\{R\leq
r\leq\tau+R\\}}r^{p}(LT^{N+1-k}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{p}E_{\mathbf{N}}[T^{N+1-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau)\\\
\leq&\>C(1+\tau)^{p-3+\max\\{1-3\beta_{0},0\\}-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Bigg{]}.\end{split}$

Note that we can moreover estimate analogously:

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\leq\tau+R\\}}&r^{2}(XT^{N-k}\widehat{\phi})^{2}+r^{2}\tilde{h}h(T^{N+1-k}\widehat{\phi})^{2}+|\not{\nabla}_{\mathbb{S}^{2}}T^{N-k}\widehat{\phi}|^{2}\,d\sigma
dr\\\ =&\>\int_{\Sigma_{\tau}\cap\\{v\leq
v_{\infty}\\}\cap\\{r\leq\tau+R\\}}r^{2}(XT^{N+1-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}+r^{2}\tilde{h}h(T^{N+2-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}+|\not{\nabla}_{\mathbb{S}^{2}}T^{N+1-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}|^{2}\,d\sigma dr\\\
\leq&\>E_{\mathbf{N}}[T^{N+1-k}\widehat{\phi}_{\textrm{\char
34},r_{\infty}}^{(1)}](\tau)\\\
\leq&\>C(1+\tau)^{-3+\max\\{1-3\beta_{0},0\\}-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Bigg{]}.\end{split}$

We now partition the interval $[R+\tau,\infty)$ into dyadic intervals
$[r_{i},r_{i+1}]$, with $r_{0}=R+\tau$ and, taking $B>0$ suitably large,
independently of $i$, we set $r_{\infty}=Br_{i+1}$. Then

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r_{i}\leq r\leq
r_{i+1}\\}}r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr=\int_{\Sigma_{\tau}\cap\\{v\leq v_{\infty}\\}\cap\\{r_{i}\leq r\leq
r_{i+1}\\}}r^{p}(LT^{N+1-k}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr\\\
\leq&\>Cr_{i}^{p-2}\int_{\Sigma_{\tau}\cap\\{v\leq
v_{\infty}\\}\cap\\{r_{i}\leq r\leq
r_{i+1}\\}}r^{2}(LT^{N+1-k}\widehat{\psi}_{\textrm{\char
34},r_{\infty}}^{(1)})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{-1-2N+2k+\delta}r_{i}^{\max\\{1-3\beta_{0},0\\}+p-2+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Bigg{]}.\end{split}$

Using that $r_{i}\geq R+\tau$, for $p<2-\max\\{1-3\beta_{0},0\\}$ we can find
$\delta>0$ suitably small, such that

$\displaystyle r_{i}^{\max\\{1-3\beta_{0},0\\}+p-2+\delta}\leq
C(1+\tau)^{-2+\max\\{1-3\beta,0\\}+p+2\delta}r_{i}^{-\delta}.$

After summing over $i$ and using the convergence of geometric series, we are
then left with

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq\tau+R\\}}r^{p}(LT^{N-k}\widehat{\psi})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{p-3+\max\\{1-3\beta_{0},0\\}-2N+2k+2\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+1)\\\ n_{1}+n_{2}\leq
N+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+|\Phi_{0}|^{2}\Bigg{]}.\qed\end{split}$

We now establish energy decay estimates in the $\alpha>0$ case:

######  Proposition 10.7.

Let $\alpha>0$, $\beta_{0}\neq 2l+1$ for all $l\in\mathbb{N}_{0}$. Let
$N\in\mathbb{N}$, $0\leq p<2-\max\\{2n_{\beta}+1-\beta_{1},0\\}$ and let
$\delta>0$ be arbitrarily small. Then there exists a constant
$C=C(M,h,V_{\alpha},N,\delta)>0$ such that for all $0\leq k\leq n_{\beta}+1$:

  1. _(i)_

for all $0\leq p\leq 2$,

$\begin{split}\int_{\Sigma_{\tau}\cap\\{r\geq
R\\}}&r^{p}(LT^{N-k}\widehat{\psi}_{0})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{p-3-2n_{\beta}-2N+2k+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+n_{\beta}+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{0}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{0}]+\Phi_{0}^{2}\Bigg{]}.\end{split}$
(10.20)

  2. _(ii)_

if $\beta_{1}>2n_{\beta}$

$\begin{split}&E_{\mathbf{N}}[T^{N-k}\phi_{\geq 1}](\tau)\\\
\leq&\>C(1+\tau)^{-3-2n_{\beta}+2k+\max\\{2n_{\beta}+1-\beta_{1},0\\}+\delta-2N}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+1+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{1,\beta_{1}-2n_{\beta}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.21)

and moreover, for $0\leq p<2-\max\\{2n_{\beta}+1-\beta_{1},0\\}$,

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{p}(LT^{N-k}\psi_{\geq
1})^{2}\,d\sigma dr\\\
\leq&\>C(1+\tau)^{p-3-2n_{\beta}+2k+\max\\{2n_{\beta}+1-\beta_{1},0\\}+\delta-2N}\\\
\cdot&\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+n_{\beta}+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{1,\beta_{1}-2n_{\beta}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+1)\\\ n_{1}\leq
N+n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.22)

  3. _(iii)_

if $\beta_{1}\leq 2n_{\beta}$

$\begin{split}&E_{\mathbf{N}}[T^{N-k}\phi_{\geq 1}](\tau)\\\
\leq&\>C(1+\tau)^{-2-\beta_{1}+2k+\delta-2N}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+n_{\beta})\\\ n_{1}\leq
N+n_{\beta}\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]},\end{split}$ (10.23)

and moreover, for $0\leq p<2-2n_{\beta}+\beta_{1}$,

$\begin{split}&\int_{\Sigma_{\tau}\cap\\{r\geq R\\}}r^{p}(LT^{N-k}\psi_{\geq
1})^{2}\,d\sigma dr\\\ \leq&\>C(1+\tau)^{p-2-\beta_{1}+2k+\delta-2N}\\\
&\cdot\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+n_{\beta})+1\\\ n_{1}+n_{2}\leq
N+n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+n_{\beta})\\\ n_{1}\leq
N+n_{\beta}\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}{\phi}_{\geq
1}]\Bigg{]}.\end{split}$ (10.24)

###### Proof.

We partition the interval $[R,\infty)$ and apply the same interpolation
argument as in the proof of Proposition 10.6, appealing to the decay estimates
in Proposition 10.5. We omit the details. ∎

Now we can established improved decay rates for $r$-weighted energies with
_decaying_ weights in $r$.

We first introduce the following weighted initial energies along $\Sigma_{0}$
involving time integrals of $\phi$:

$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}^{(1)}[\phi]:=$
$\displaystyle\>\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq 2(N+2)\\\
n_{1}+n_{2}\leq N+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}_{0}|^{2}+r^{1-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}{\psi}_{\geq
1}|^{2}\,d\sigma dr$ $\displaystyle+\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}\leq 3(N+2)\\\ n_{1}\leq
N+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+\mathfrak{I}_{\beta}^{2}[\phi]\quad\textnormal{when
$\beta_{0}<1$},$
$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}^{(n_{\beta}+1)}[\phi]:=$
$\displaystyle\>\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+2n_{\beta}+2)\\\ n_{1}+n_{2}\leq
N+2n_{\beta}+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{0}|^{2}$
$\displaystyle+r^{\min\\{\beta_{1}-2n_{\beta},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}{\psi}_{\geq
1}|^{2}+r^{2+\min\\{\beta_{1}-2n_{\beta},0\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}{\psi}_{\geq
1}|^{2}\,d\sigma dr$ $\displaystyle+\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}\leq 3(N+2n_{\beta}+2)\\\ n_{1}\leq
N+2n_{\beta}+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}]+\mathfrak{I}_{\beta}^{2}[\phi]\quad\textnormal{when
$\beta_{0}>1$}.$

The above energies will play a role in the energy decay estimates in
Proposition 10.8.

######  Proposition 10.8.

Let $\beta_{0}\notin 2\mathbb{N}_{0}+1$ and $\delta>0$ arbitrarily small. Then
there exists a constant $C=C(M,h,V_{\alpha},N,\delta)>0$ such that for all
$0\leq k\leq n_{\beta}+1$:

  1. _(i)_

when $\alpha<0$ $(\beta_{0}<1)$:

$\int_{\Sigma_{\tau}}r^{-1+2\beta_{0}}(XT^{N-k}w_{0}^{-1}\widehat{\phi}_{0})^{2}\,d\sigma
dr\leq
C(1+\tau)^{\max\\{1-3\beta_{0},0\\}+\beta_{0}-5+2k-2N+\delta}\mathbf{D}_{\beta_{0},N,\delta}^{(1)}[\phi],$
(10.25)

and

$\begin{split}\int_{\Sigma_{\tau}}r^{-\beta_{0}+\delta}(T^{N-k}\widehat{\phi}_{\geq
1})^{2}+r^{2-\beta_{0}+\delta}(XT^{N-k}\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\leq&\>C(1+\tau)^{-3-\beta_{0}+2k-2N+2\delta}\mathbf{D}_{\beta_{0},N,\delta}^{(1)}[\phi],\end{split}$

  2. _(ii)_

and when $\alpha>0$ $(\beta_{0}>1)$:

$\begin{split}\int_{\Sigma_{\tau}}r^{-\beta_{0}+\delta}(T^{N-k}\widehat{\phi})^{2}+r^{2-\beta_{0}+\delta}(XT^{N-k}\widehat{\phi})^{2}\,d\sigma
dr\leq&\>C(1+\tau)^{-3-\beta_{0}-2n_{\beta}+2k+\max\\{2n_{\beta}+1-\beta_{1},0\\}-2N+2\delta}\mathbf{D}_{\beta_{0},N,\delta}^{(n_{\beta}+1)}[\phi].\end{split}$
(10.26)

###### Proof.

We prove the $k=0$ case here. The $1\leq k\leq 1+n_{\beta}$ case proceeds
entirely analogously. Note that $\widehat{\phi}$ satisfies the equation:

$\displaystyle\mathcal{L}\widehat{\phi}|_{\Sigma_{\tau}}=$
$\displaystyle\>\widehat{F}[T{\phi}|_{\Sigma_{\tau}}],$ $\displaystyle
F[T\widehat{\phi}|_{\Sigma_{\tau}}]:=$
$\displaystyle\>2r(1-h)XT\widehat{\psi}|_{\Sigma_{\tau}}+\frac{dh}{dr}r^{2}T\widehat{\phi}|_{\Sigma_{\tau}}+r^{2}h\tilde{h}T^{2}\widehat{\phi}|_{\Sigma_{\tau}}+r^{2}\square_{g}\Phi|_{\Sigma_{\tau}}.$

Consider first $\beta_{0}>1$. By Corollary 9.7 with $q=-\beta_{0}+\delta$, we
have that

$\begin{split}\int_{\Sigma_{\tau}}r^{-\beta_{0}+\delta}\widehat{\phi}^{2}+r^{2-\beta_{0}+\delta}(X\widehat{\phi})^{2}\,d\sigma
dr\leq&\>C\int_{\Sigma_{\tau}}r^{-\beta_{0}+\delta}(\mathcal{L}\widehat{\phi})^{2}\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{2-\beta_{0}+\delta}(XT\widehat{\psi})^{2}+r^{-2-\beta_{0}+\delta}(T\widehat{\phi})^{2}+r^{-\beta_{0}+\delta}(r^{2}\tilde{h}h)^{2}(T^{2}\widehat{\phi})^{2}\\\
&+r^{4-\beta_{0}+\delta}(\square_{g}\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr.\end{split}$

Note that

$\int_{\Sigma_{\tau}}r^{4-\beta_{0}+\delta}(\square_{g}\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr\leq C\Phi_{0}^{2}(1+\tau)^{-4-2\beta_{0}+2\delta}.$

We apply Corollary 9.7 once more, with $q=-\beta_{0}+\delta+2$ to estimate
further

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(L\hat{\psi})^{2}\,d\sigma
dr\leq
C\int_{\Sigma_{\tau}}r^{2-\beta_{0}+\delta}(XT\widehat{\psi})^{2}+r^{-2-\beta_{0}+\delta}(T\widehat{\phi})^{2}+r^{-\beta_{0}+\delta}(r^{2}\tilde{h}h)^{2}(T^{2}\widehat{\phi})^{2}\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{4-\beta_{0}+\delta}(XT^{2}\widehat{\psi})^{2}+r^{-\beta_{0}+\delta}(T^{2}\widehat{\phi})^{2}+r^{2-\beta_{0}+\delta}(r^{2}\tilde{h}h)^{2}(T^{3}\widehat{\phi})^{2}\,d\sigma
dr\\\
&+C\int_{\Sigma_{\tau}}r^{6-\beta_{0}+\delta}(\square_{g}T\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr,\end{split}$

and we have that

$\int_{\Sigma_{\tau}}r^{6-\beta_{0}+\delta}(\square_{g}T\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr\leq C\Phi_{0}^{2}(1+\tau)^{-4-2\beta_{0}+2\delta}.$

By definition of $n_{\beta}$, we have that
$\beta_{0}-1<2n_{\beta}<\beta_{0}+1$. If $2n_{\beta}-\beta_{0}>0$, we apply
Corollary 9.7 a total of $n_{\beta}$ times to obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}\widehat{\phi}^{2}+r^{2-\beta_{0}+\delta}(X\widehat{\phi})^{2}\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{2n_{\beta}-\beta_{0}+\delta}(XT^{n_{\beta}}\widehat{\psi})^{2}+r^{2n_{\beta}-4-\beta_{0}+\delta}(T^{n_{\beta}}\widehat{\phi})^{2}+r^{2n_{\beta}-2-\beta_{0}+\delta}(r^{2}\tilde{h}h)^{2}(T^{1+n_{\beta}}\widehat{\phi})^{2}\,d\sigma
dr\\\ &+C\Phi_{0}^{2}(1+\tau)^{-4-2\beta+2\delta}\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{2n_{\beta}-\beta_{0}+\delta}(XT^{n_{\beta}}\widehat{\psi})^{2}\,d\sigma
dr+CE_{\mathbf{N}}[T^{n_{\beta}}\widehat{\phi}](\tau)+C\Phi_{0}^{2}(1+\tau)^{-4-2\beta_{0}+2\delta}.\end{split}$

If $2n_{\beta}-\beta_{0}<0$, we apply Corollary 9.7 a total of $n_{\beta}+1$
times to obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}\widehat{\phi}^{2}+r^{2-\beta_{0}+\delta}(X\widehat{\phi})^{2}\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{2n_{\beta}+2-\beta_{0}+\delta}(XT^{n_{\beta}+1}\widehat{\psi})^{2}+r^{2n_{\beta}-2-\beta_{0}+\delta}(T^{n_{\beta}+1}\widehat{\phi})^{2}+r^{2n_{\beta}-\beta_{0}+\delta}(r^{2}\tilde{h}h)^{2}(T^{2+n_{\beta}}\widehat{\phi})^{2}\,d\sigma
dr\\\ &+C\Phi_{0}^{2}(1+\tau)^{-4-2\beta+2\delta}\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{2n_{\beta}-\beta_{0}+2+\delta}(XT^{n_{\beta}+1}\widehat{\psi})^{2}\,d\sigma
dr+CE_{\mathbf{N}}[T^{n_{\beta}+1}\widehat{\phi}](\tau)+C\Phi_{0}^{2}(1+\tau)^{-4-2\beta_{0}+2\delta}.\end{split}$

We can therefore apply the $r$-weighted energy decay estimates in Proposition
10.7 and __(i)__ of Proposition 10.5, with $p=2n_{\beta}-\beta_{0}+\delta$ and
$N=n_{\beta}$ (if $2n_{\beta}-\beta_{0}>0$) or
$p=2n_{\beta}-\beta_{0}+2+\delta$ and $N=n_{\beta}+1$ (if
$2n_{\beta}-\beta_{0}<0$).

We then obtain the following estimate for $\phi_{0}$ with $\beta_{0}>1$:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(T^{N}\widehat{\phi}_{0})^{2}+r^{2-\beta_{0}+\delta}(XT^{N}\widehat{\phi}_{0})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{-3-\beta_{0}-2n_{\beta}+2\delta-2N}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+2n_{\beta}+2)\\\ n_{1}+n_{2}\leq
N+2n_{\beta}+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{0}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+2n_{\beta}+2)\\\
n_{1}\leq
N+2n_{\beta}+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{0}]+\Phi_{0}^{2}\Bigg{]}+C\Phi_{0}^{2}(1+\tau)^{-4-2\beta_{0}-2N+2\delta}.\end{split}$

To estimate the $\phi_{\geq 1}$ part of $\phi$, we consider separately the
following three cases: 1) $\beta_{0}<2n_{\beta}$ and $\beta_{1}>2n_{\beta}$,
2) $\beta_{0}<2n_{\beta}$ and $\beta_{1}\leq 2n_{\beta}$, 3) $\beta_{0}\geq
2n_{\beta}$ and $\beta_{1}>2n_{\beta}$. Since $\widehat{\phi}_{\geq
1}=\phi_{\geq 1}$, we can omit the terms above involving $\Phi_{0}^{2}$ when
restricting to $\phi_{\geq 1}$.

_Case 1):_

We apply the estimates in __(ii)__ of Proposition 10.7 with
$p=2n_{\beta}-\beta_{0}+\delta$, $N$ replaced by $N+n_{\beta}$ and $\delta>0$
appropriately small. Note that this choice of $p$ lies in the allowed range of
$p$ in __(ii)__ of Proposition 10.7, since $\beta_{0}>2n_{\beta}-1$ and
$\beta_{1}>2n_{\beta}-1$. We then obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(T^{N}\widehat{\phi}_{\geq
1})^{2}+r^{2-\beta_{0}+\delta}(XT^{N}\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{-3-\beta_{0}-2n_{\beta}+\max\\{2n_{\beta}+1-\beta_{1},0\\}+2\delta-2N}\\\
\cdot&\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+2n_{\beta}+1)\\\ n_{1}+n_{2}\leq
N+2n_{\beta}+1\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{\beta_{1}-2n_{\beta},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+2n_{\beta}+1)\\\ n_{1}\leq
2n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{\geq
1}]\Bigg{]}.\end{split}$

_Case 2):_

We apply the estimates in __(iii)__ of Proposition 10.7 with
$p=2n_{\beta}-\beta_{0}+\delta$, $N$ replaced by $N+n_{\beta}$ and $\delta>0$
appropriately small. Note that this choice of $p$ lies in the allowed range of
$p$ in __(ii)__ of Proposition 10.7, since $\beta_{0}>2n_{\beta}-1$ and
$\beta_{1}>2n_{\beta}-1$ by assumption. We obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(T^{N}\widehat{\phi}_{\geq
1})^{2}+r^{2-\beta_{0}+\delta}(XT^{N}\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\\\ \leq&\>C(1+\tau)^{-2-\beta_{0}-\beta_{1}+2\delta}\\\
\cdot&\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+2n_{\beta})\\\ n_{1}+n_{2}\leq
N+2n_{\beta}\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\beta_{1}-2n_{\beta}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-n_{\beta}}\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+2n_{\beta})\\\ n_{1}\leq
N+2n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-n_{\beta}}\widehat{\phi}_{\geq
1}]\Bigg{]}.\end{split}$

_Case 3):_

We apply the estimates in __(ii)__ of Proposition 10.7 with
$p=2+2n_{\beta}-\beta_{0}+\delta$, $N$ replaced by $N+n_{\beta}+1$ and
$\delta>0$ appropriately small. Note that this choice of $p$ lies in the
allowed range of $p$ in __(ii)__ of Proposition 10.7, since
$\beta_{0}>2n_{\beta}$ and $\beta_{1}>2n_{\beta}$ by assumption. We obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(T^{N}\widehat{\phi}_{\geq
1})^{2}+r^{2-\beta_{0}+\delta}(XT^{N}\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{-3-\beta_{0}-2n_{\beta}+\max\\{2n_{\beta}+1-\beta_{1},0\\}+2\delta-2N}\\\
\cdot&\Bigg{[}\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}+n_{3}\leq
2(N+2n_{\beta}+2)\\\ n_{1}+n_{2}\leq
N+2n_{\beta}+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{\beta_{1}-2n_{\beta},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+2n_{\beta}+2)\\\ n_{1}\leq
2n_{\beta}+1\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{\geq
1}]\Bigg{]}.\end{split}$

We conclude that __(ii)__ must hold.

Consider now $\beta_{0}<1$ and restrict first to $\phi_{0}$. By (9.3) with
$\ell=0$ and a suitably choice of $q$, it follows that for $\delta>0$
arbitrarily small

$\begin{split}\int_{\Sigma_{\tau}}r^{-1+2\beta_{0}}(X(w_{0}^{-1}\widehat{\phi}_{0}))^{2}\,d\sigma
dr\leq&\>C\int_{\Sigma_{\tau}}r^{-2+\beta_{0}}(\mathcal{L}\widehat{\phi}_{0})^{2}\,d\sigma
dr\\\
\leq&\>C\int_{\Sigma_{\tau}}r^{\beta_{0}}(XT\widehat{\psi}_{0})^{2}+r^{-4+\beta_{0}}(T\widehat{\phi}_{0})^{2}+r^{-2+\beta_{0}}(r^{2}\tilde{h}h)^{2}(T^{2}\widehat{\phi}_{0})^{2}+r^{2+\beta_{0}}(\square_{g}\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr\\\
\leq&\>CE_{\mathbf{N}}[T\widehat{\phi}_{0}](\tau)+C\int_{\Sigma_{\tau}}r^{\beta_{0}}(XT\widehat{\psi}_{0})^{2}+r^{2}(\square_{g}\Phi|_{\Sigma_{\tau}})^{2}\,d\sigma
dr.\end{split}$

Hence, we can apply Proposition 10.6 with $p=\beta_{0}$ and $N$ replaced with
$N+1$, together with the above estimate applied to $T^{N}\phi_{0}$ to obtain

$\begin{split}\int_{\Sigma_{\tau}}&r^{-1+2\beta_{0}}(XT^{N}w_{0}^{-1}\widehat{\phi}_{0})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{\max\\{1-3\beta_{0},0\\}+\beta_{0}-5-2N+\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+2)\\\ n_{1}+n_{2}\leq
N+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}\widehat{\psi}_{0}|^{2}\,d\sigma
dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq 3(N+2)\\\ n_{1}\leq
N+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1}\widehat{\phi}]+\Phi_{0}^{2}\Bigg{]}+C|\Phi_{0}|^{2}(1+\tau)^{-4-2\beta_{0}-2N+\delta}.\end{split}$

In order to obtain an estimate for $\phi_{\geq 1}$, we instead start as in the
$\beta_{0}>1$ case, using that $\phi_{\geq 1}=\widehat{\phi}_{\geq 1}$, to
obtain:

$\begin{split}\int_{\Sigma_{\tau}}r^{-\beta_{0}+\delta}\widehat{\phi}_{\geq
1}^{2}+r^{2-\beta_{0}+\delta}(X\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\leq&\>C\int_{\Sigma_{\tau}}r^{2-\beta_{0}+\delta}(XT\widehat{\psi}_{\geq
1})^{2}+CE_{\mathbf{N}}[T\widehat{\phi}_{\geq 1}](\tau).\end{split}$

Then we apply Proposition 10.6 with $p=2-\beta_{0}+\delta$ and $\phi_{\geq 1}$
replaced by $T^{N}\phi_{\geq 1}$ to obtain:

$\begin{split}\int_{\Sigma_{\tau}}&r^{-\beta_{0}+\delta}(T^{N}\widehat{\phi}_{\geq
1})^{2}+r^{2-\beta_{0}+\delta}(XT^{N}\widehat{\phi}_{\geq 1})^{2}\,d\sigma
dr\\\
\leq&\>C(1+\tau)^{-3-\beta_{0}-2N+2\delta}\Bigg{[}\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2(N+2)\\\ n_{1}+n_{2}\leq
N+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1-n_{\beta}}\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr\\\ &+\sum_{\begin{subarray}{c}0\leq n_{1}+n_{2}\leq
3(N+2)\\\ n_{1}\leq
N+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}-1-n_{\beta}}\widehat{\phi}_{\geq
1}]\Bigg{]}.\end{split}$

By combining the above estimates, we conclude __(i)__. ∎

Since $\mathbf{D}_{\beta_{0},N,\delta}^{(1)}[\phi]$ and
$\mathbf{D}_{\beta_{0},N}^{(n_{\beta}+1)}[\phi]$ are expressed in terms of
time integrals of $\phi$ along $\Sigma_{0}$, it remains to show that they can
be bounded in terms of initial data for $\phi$ itself.

######  Proposition 10.9.

There exists a constant $C=C(M,V_{\alpha},h,N,\delta)>0$, such that

$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}^{(1)}[\phi]\leq$
$\displaystyle\>C\mathbf{D}_{\beta_{0},N,\delta}[\phi]\quad(\beta_{0}<1),$
$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}^{(n_{\beta}+1)}[\phi]\leq$
$\displaystyle\>C\mathbf{D}_{\beta_{0},N,\delta}[\phi]\quad(\beta_{0}>1),$

with

$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}[\phi]:=$
$\displaystyle\>\sum_{n\leq
N+2}\left(||((r+1)^{2}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}^{2}+||(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}^{2}+||r(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}^{2}\right)$
$\displaystyle+\sum_{n_{1}+n_{2}\leq N+2}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{3-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X(rX)^{n_{2}}\widehat{\psi}_{\geq
1}|^{2}+r^{-1-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}(rX)^{n_{2}}T\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr$ $\displaystyle+\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N+3\\\ n_{1}+n_{2}\leq
N+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{\min\\{3\beta_{0},1\\}-\delta}|L(rL)^{n_{2}}T^{n_{3}}\widehat{\psi}_{0}|^{2}+r^{1-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}-1}{\psi}_{\geq
1}|^{2}\,d\sigma dr$ $\displaystyle+\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}\leq 3N+5\\\ n_{1}\leq
N+2\end{subarray}}E_{\mathbf{N}}[\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}T^{n_{2}}\widehat{\phi}]+\mathfrak{I}_{\beta}^{2}[\phi]\qquad(\beta_{0}<1),$
$\displaystyle\mathbf{D}_{\beta_{0},N,\delta}[\phi]:=$
$\displaystyle\>\sum_{n\leq
N+3n_{\beta}+2}\Biggl{(}||(r+1)^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}X(r\phi_{0})||_{L^{\infty}(\Sigma_{0})}^{2}$
$\displaystyle+M^{4}||r^{\frac{1}{2}+\frac{\beta_{0}}{2}}(rX)^{n}\phi_{0}||_{L^{\infty}(\Sigma_{0})}^{2}+M^{4}||r^{\frac{3}{2}+\frac{\beta_{0}}{2}}(rX)^{n}T\phi_{0}||_{L^{\infty}(\Sigma_{0})}^{2}\Biggr{)}$
$\displaystyle+\sum_{n_{1}+n_{2}\leq
N+2n_{\beta}+2}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r^{2+\min\\{2n_{\beta}+1,\beta_{1}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X(rX)^{n_{2}}\widehat{\psi}_{\geq
1}|^{2}$
$\displaystyle+M^{4}r^{-2+\min\\{2n_{\beta}+1,\beta_{1}\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X(rX)^{n_{2}}T\widehat{\psi}_{\geq
1}|^{2}\,d\sigma dr$ $\displaystyle+\sum_{\begin{subarray}{c}0\leq
n_{1}+n_{2}+n_{3}\leq 2N+3n_{\beta}+5\\\ n_{1}+n_{2}\leq
N+2n_{\beta}+2\end{subarray}}\int_{\Sigma_{0}\cap\\{r\geq
R\\}}r|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}\widehat{\psi}_{0}|^{2}$
$\displaystyle+r^{\min\\{\beta_{1}-2n_{\beta},1\\}-\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}L(rL)^{n_{2}}T^{n_{3}}{\psi}_{\geq