$(\mathcal{A}-s\mathbf{1})^{-1}:=\mathcal{R}(s).$

######  Corollary 9.2.

Assume that for $s\in O$, with $O\subset\mathbb{C}$ a connected domain, either
$L_{s}^{-1}:H_{\sigma,R_{\infty}}^{N_{+}}\to
D^{N_{+}}_{\sigma,R_{\infty}}(L_{s_{0}})\subseteq
H_{\sigma,R_{\infty},2}^{N_{+}+1}$ is bounded with respect to the operator
norm $\|\cdot\|_{H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}}$, or $\ker L_{s}\neq\\{0\\}$. Assume also
that there exists a $s_{0}\in O$ such that $\ker L_{s_{0}}=\\{0\\}$.

Consider
$\mathcal{A}:\textnormal{ran}\,\mathcal{R}(s_{0})\to\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}$.
Then the spectrum ${\rm Spect}\,(\mathcal{A})$ satisfies:

${\rm Spect}\,(\mathcal{A})\cap O={\rm Spect}_{\rm point}\,(\mathcal{A})\cap
O=\\{s\in O\,|\,\ker L_{s}\neq\\{0\\}\\}.$

Furthermore, $(\Psi,\Psi^{\prime})\in\textnormal{ran}\,\mathcal{R}(s_{0})$ is
an eigenvector of $\mathcal{A}$ if and only if
$(\Psi,\Psi^{\prime})=(\widehat{\psi}_{s},s\widehat{\psi}_{s})$, with
$\widehat{\psi}_{s}\in\ker L_{s}$.

###### Proof.

In view of the form of $\mathcal{R}(s)$ from (ii) of Proposition 9.1,
boundedness of $\mathcal{R}(s)$ is equivalent to boundedness of $L_{s}^{-1}$.
Hence, $s\in{\rm Spect}\,(\mathcal{A})\cap O$ if and only if $L_{s}^{-1}$ is
not bounded. By assumption, the latter implies that $\ker L_{s}\neq\\{0\\}$.
Let $\widehat{\psi}_{s}\in\ker L_{s}$. Then

$\mathcal{A}(\widehat{\psi}_{s},s\widehat{\psi}_{s})=s\cdot(\widehat{\psi}_{s},s\widehat{\psi}_{s})\in\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}.$

and
$(f,g)=(\mathcal{A}-s_{0}\mathbf{1})(\widehat{\psi}_{s},s\widehat{\psi}_{s})=(s-s_{0})(\widehat{\psi}_{s},s\widehat{\psi}_{s})\in\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}$,
so $\mathcal{R}(s_{0})(f,g)=\begin{pmatrix}\widehat{\psi}_{s}\\\
s\widehat{\psi}_{s}\end{pmatrix}$ and therefore
$(\widehat{\psi},s\widehat{\psi}_{s})\in\textnormal{ran}\,\mathcal{R}(s_{0})$,
so $s\in{\rm Spect}_{\rm point}\,(\mathcal{A})\cap O$, with $\mathcal{A}$
defined on the domain $\textnormal{ran}\,\mathcal{R}(s_{0})$. ∎

The _Hermitian adjoint_ of
$L_{s}:D^{N_{+}}_{\sigma,R_{\infty}}(L_{s})\supset{\rm dom}\,(L_{s})\to
H_{\sigma,R_{\infty}}^{N_{+}}$, with respect to the
$H_{\sigma,R_{\infty}}^{N_{+}}$ inner product is defined as follows: let

$D_{\sigma,R_{\infty}}^{N_{+}}(L_{s}^{*}):=\left\\{f\in
H_{\sigma,R_{\infty}}^{N_{+}}\,,\,\langle
f,L_{s}(\cdot)\rangle_{H_{\sigma,R_{\infty}}^{N_{+}}}:D_{\sigma,R_{\infty}}^{N_{+}}(L_{s})\supset{\rm
dom}\,(L_{s})\to\mathbb{C}\>\textnormal{is bounded}\right\\}.$

By the Riesz representation theorem, we have that for all $f\in
D_{\sigma,R_{\infty}}^{N_{+}}(L_{s}^{*})$, there exists a unique $g\in
D^{N_{+}}_{\sigma,R_{\infty}}(L_{s})$, such that $\langle
g,\widehat{\psi}\rangle_{H_{\sigma,R_{\infty}}^{N_{+}}}=\langle
f,L_{s}\widehat{\psi}\rangle_{H_{\sigma,R_{\infty}}^{N_{+}}}$ for all
$\widehat{\psi}\in{\rm dom}\,(L_{s})$. Then we define the Hermitian adjoint of
$L_{s}$ by $L_{s}^{*}:D_{\sigma,R_{\infty}}^{N_{+}}(L_{s}^{*})\to
H_{\sigma,R_{\infty}}^{N_{+}}$, with $L_{s}^{*}f=g$, with $f,g$ as above.

Similarly, we define the Hermitian adjoint of $\mathcal{A}$, with respect to
the $\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}$ inner product, as follows: let
$s_{0}\in\mathbb{C}$ be such that $L_{s_{0}}^{-1}$ is well-defined, then

$\mathbf{D}_{\sigma,R_{\infty}}^{N_{+}}(\mathcal{A}^{*}):=\left\\{(f,g)\in\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}\,,\,\langle
f,\mathcal{A}(\cdot)\rangle_{\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}}:{\rm
ran}\,\mathcal{R}(s_{0})\to\mathbb{C}\>\textnormal{is bounded}\right\\}.$

By the Riesz representation theorem, we have that for all
$(f,g)\in\mathbf{D}_{\sigma,R_{\infty}}^{N_{+}}(\mathcal{A}^{*})$, there
exists a unique $(\Phi,\Phi^{\prime})\in{\rm ran}\,\mathcal{R}(s_{0})$, such
that
$\langle(\Phi,\Phi^{\prime}),(\Psi,\Psi^{\prime})\rangle_{\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}}=\langle(f,g),\mathcal{A}(\Psi,\Psi^{\prime})\rangle_{\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}}$
for all
$(\Psi,\Psi^{\prime})\in(C^{\infty}(\hat{\Sigma}))^{2}\cap\mathcal{R}(s_{0})$.
Then we define the Hermitian adjoint of $\mathcal{A}$ by
$\mathcal{A}^{*}:\mathbf{D}_{\sigma,R_{\infty}}^{N_{+}}(\mathcal{A}^{*})\to\mathbf{H}_{\sigma,R_{\infty}}^{N_{+}}$,
with $\mathcal{A}^{*}(f,g)=(\Phi,\Phi^{\prime})$, with $(f,g)$ and
$(\Phi,\Phi^{\prime})$ as above.

Taking the formal adjoint of the right-hand side of (9.1), we obtain

$\mathcal{A}^{{\dagger}}-\overline{s}\mathbf{1}:=\begin{pmatrix}\mathbf{1}&-\overline{s}\mathbf{1}\\\
0&\mathbf{1}\end{pmatrix}\begin{pmatrix}\frac{1}{2M^{2}h(0)-Mxb_{\tau\tau}}L_{s}^{*}&0\\\
0&\mathbf{1}\end{pmatrix}\begin{pmatrix}0&\mathbf{1}\\\
\mathbf{1}&P^{*}-\overline{s}\mathbf{1}\end{pmatrix}.$

From the above expression, it follows that $\mathcal{A}^{{\dagger}}(f,g)$ is
well-defined and agrees with $\mathcal{A}^{*}(f,g)$ if $g\in
D_{\sigma,R_{\infty}}^{N_{+}}(L_{s}^{*})\cap
D_{\sigma,R_{\infty}}^{N_{+}}(P^{*})$ and $f\in
H_{\sigma,R_{\infty}}^{N_{+}}$, with $D_{\sigma,R_{\infty}}^{N_{+}}(P^{*})$
the domain of the Hermitian adjoint $P^{*}$, with respect to the inner product
on $H_{\sigma,R_{\infty}}^{N_{+}}$.

A direct corollary of the standard orthogonality property between the kernel
of a linear operator and the range of its Hermitian adjoint results in the
following:

######  Corollary 9.3.

Assume additionally that there exists a $s_{0}\in\mathbb{C}$ such that $\ker
L_{s_{0}}=\\{0\\}$. We have that

$\displaystyle\textnormal{ran}(\mathcal{A}^{*}-\overline{s}\mathbf{1})\perp$
$\displaystyle\>\ker(\mathcal{A}-s\mathbf{1}),$
$\displaystyle\ker(\mathcal{A}^{*}-\overline{\tilde{s}}\mathbf{1})\perp$
$\displaystyle\>\ker(\mathcal{A}-s\mathbf{1})\quad\textnormal{if
$s\neq\tilde{s}$}.$

Let $\widehat{\psi}_{s}\in\ker L_{s}$ and denote
$(\Psi_{s},\Psi^{\prime}_{s})=(\widehat{\psi}_{s},s\widehat{\psi}_{s})\in\ker(\mathcal{A}-s\mathbf{1})$.
Let $g_{\tilde{s}}\in\ker L_{\tilde{s}}^{*}$ and denote
$(\tilde{\Psi}_{\tilde{s}},\tilde{\Psi}^{\prime}_{\tilde{s}})=(-(P^{*}-\overline{\tilde{s}})g_{\tilde{s}},g_{\tilde{s}})$.
Then
$(\tilde{\Psi}_{\tilde{s}},\tilde{\Psi}^{\prime}_{\tilde{s}})\in\ker(\mathcal{A}^{{\dagger}}-\overline{\tilde{s}}\mathbf{1})$
and

$\langle(\Psi_{s},\Psi^{\prime}_{s}),(\tilde{\Psi}_{\tilde{s}},\tilde{\Psi}^{\prime}_{\tilde{s}})\rangle_{\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}}=0$

if $s\neq\tilde{s}$.

##  10\. Precise statements of the main theorems

In this section, we give precise statements of the main theorems of the paper,
making use of the preliminaries introduced in §§6–9, and we give precise
definitions of the notions of _regularity quasinormal modes_ and _regularity
quasinormal frequencies_.

###  10.1. Regularity quasinormal frequencies

The following theorem provides a precise formulation of Theorem 1.1(i)–(ii)
and includes also an additional statement on orthogonality properties of
regularity quasinormal modes.

######  Theorem 10.1.

Let $\sigma\in\mathbb{R}$ and $N_{+}\in\mathbb{N}_{0}$. Let

$\displaystyle\Omega_{\sigma}:=$
$\displaystyle\>\left\\{x+iy\in\mathbb{C}\,|\,x\in\mathbb{R}_{-},\,y\in\mathbb{R},\,x^{2}+y^{2}<\sigma^{2},\,3y^{2}-5x^{2}>\sigma^{2}\right\\}$
$\displaystyle\cup\left\\{x+iy\in\mathbb{C}\,|\,x\in[0,\infty),\,\frac{\sigma^{2}}{4}<x^{2}+y^{2}<\sigma^{2}\right\\},$
$\displaystyle\Omega_{\sigma,N_{+}}:=$
$\displaystyle\>\Omega_{\sigma}\cap\left\\{\operatorname{\textnormal{Re}}z>-\left(\frac{1}{2}+N_{+}\right)\kappa_{+}\right\\}$

  1. _(i)_

Then there exists a suitably large $R_{\infty}>r_{+}$ such that the following
restrictions of the infinitesimal generator of time translations corresponding
to $\mathcal{S}(\tau)$ are closed linear operators:

$\mathcal{A}:\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}\supset
H^{N_{+}}_{\sigma,R_{\infty}}(L_{s})\oplus{H}_{\sigma,R_{\infty}}^{N_{+}}(P)\to\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}$

and

${\rm Spect}\,(\mathcal{A})\cap\Omega_{\sigma,N_{+}}={\rm Spect}_{\rm
point}\,(\mathcal{A})\cap\Omega_{\sigma,N_{+}}$

consists of isolated eigenvalues, which are independent of the precise choice
of $\tau$. We define regularity quasinormal frequencies to be the union of
eigenvalues:

$\mathscr{Q}_{\rm
reg}:=\bigcup_{N_{+}\in\mathbb{N}_{0},\sigma\in\mathbb{R}}{\rm Spect}_{\rm
point}\,(\mathcal{A})\cap\Omega_{\sigma,N_{+}}$

and regularity quasinormal modes to be the corresponding eigenvectors. The
sets ${\rm Spect}_{\rm point}\,(\mathcal{A})\cap\Omega_{\sigma,N_{+}}$ are
independent of the precise choice of foliation-defining function
$\mathbbm{h}$.

  2. _(ii)_

The union
$\Omega:=\bigcup_{N_{+}\in\mathbb{N}_{0},\sigma\in\mathbb{R}}\Omega_{\sigma,N_{+}}$
satisfies

$\Omega=\left\\{z\in\mathbb{C}\setminus\\{0\\}\,,\,|{\rm
arg\,z}|<\frac{2}{3}\pi\right\\}.$

The set of quasinormal frequencies $\mathscr{Q}_{\rm reg}$ has no accumulation
points in $\Omega$ and is independent of the choice of $R_{\infty}$.

  3. _(iii)_

The eigenspaces of $\mathcal{A}:\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}\supset
H^{N_{+}}_{\sigma,R_{\infty}}(L_{s})\oplus{H}_{\sigma,R_{\infty}}^{N_{+}}(P)\to\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}$
are finite dimensional.

  4. _(iv)_

We define regularity quasinormal co-modes to be eigenvectors of
$\mathcal{A}^{{\dagger}}$ in $((D_{\sigma,R_{\infty}}^{N_{+}})(L_{s}^{*})\cap
D_{\sigma,R_{\infty}}^{N_{+}}(P^{*}))\oplus H^{N_{+}}_{\sigma,R_{\infty}}$.
The following orthogonality relations hold with respect to the inner product
on $\mathbf{H}^{N_{+}}_{\sigma,R_{\infty}}$:

$\displaystyle\textnormal{ran}(\mathcal{A}^{{\dagger}}-\overline{s}\mathbf{1})\perp$
$\displaystyle\>\ker(\mathcal{A}-s\mathbf{1}),$
$\displaystyle\ker(\mathcal{A}^{{\dagger}}-\overline{\tilde{s}}\mathbf{1})\perp$
$\displaystyle\>\ker(\mathcal{A}-s\mathbf{1})\quad\textnormal{if
$s\neq\tilde{s}$}.$

###### Proof.

Theorem 10.1 follows directly from Theorem 10.2 below, in combination with
Corollaries 9.2 and 9.3. ∎

Figure 3. A (schematic) pictorial representation of the following subsets of
$\mathbb{C}$: $\Omega$ (the sector of the complex plane that is not dashed),
$\Omega_{\sigma}$ (the shaded region), and possible regularity quasinormal
frequencies represented by crosses.

The proof of Theorem 10.1 uses crucially the meromorphicity of the inverse
$L_{s}^{-1}$ of the Laplace transform of the wave operator in Kerr and
$L_{s,\Lambda}^{-1}$ in Kerr–de Sitter (see §18 for a definition of
$L_{s,\Lambda}$). Meromorphicity in this context is defined as follows:

######  Definition 10.1.

Let $O\subset\mathbb{C}$ be a connected and open subset, $X,Y$ Banach spaces
and $B(X,Y)$ the space of bounded linear operators from $X$ to $Y$. Let
$A:O\to B(X,Y)$ be a family of bounded operators. Then $A$ or $\\{A(s)\\}$ is
_meromorphic_ if for any $s_{0}\in O$ there exists a small neighbourhood $U$
of $s_{0}$, such that for all $s\in U$, we can write
$A(s)=A_{0}(s)+\sum_{k=1}^{m}(s-s_{0})^{-k}A_{k}$ in $U$, where $A_{0}(s)$ is
holomorphic at $s_{0}$, i.e. $\lim_{s\to
s_{0}}\frac{A_{0}(s)-A(s_{0})}{s-s_{0}}$ is well-defined with respect to the
operator topology on $B(X,Y)$. If $m>0$ for some $s_{0}\in O$, we refer to
$s_{0}$ as a _pole_ of $A$.

The orthogonality properties of regularity quasinormal modes are related to
the cokernel of $L_{s}$, which is defined as follows:

######  Definition 10.2.

Let $A:X\to Y$ be a linear operator between Banach spaces $X,Y$. Then the
_cokernel of $A$_ is defined as the quotient space
$\textnormal{coker}\,A=Y/\textnormal{ran}\,A$.

The following theorem is a precise version of Theorem 4.1(i)–(ii) and includes
an additional statement on the orthogonality relations between the kernel and
cokernel of $L_{s}$. _This theorem constitutes the core of the present paper._

######  Theorem 10.2.

Let $\sigma\in\mathbb{R}$ and $N_{+}\in\mathbb{N}_{0}$.

  1. _(i)_

For suitably large $R_{\infty}>r_{+}$ and suitably small $M^{2}\Lambda\geq 0$,
the family of operators
$\\{L_{s,\Lambda}^{-1}:H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}\\}$ with $s\in\Omega_{\sigma,N_{+}}$ is
meromorphic. We denote with $\mathscr{Q}_{\sigma,R_{\infty},N_{+}}$ the set of
poles of $\\{L_{s,\Lambda}^{-1}\\}$, which does not depend on the precise
choice of foliation-defining function $\mathbbm{h}$. Furthermore, there exists
a $C_{0}\geq 0$ such that the family of operators is holomorphic when
restricted to $s\in\\{\operatorname{\textnormal{Re}}z>C_{0}\\}$.

  2. _(ii)_

Let $\widetilde{R}_{\infty}\geq R_{\infty}$,
$\widetilde{N}_{+}\in\mathbb{N}_{0}$ and $\widetilde{\sigma}\in\mathbb{R}$.
Then

$\mathscr{Q}_{\sigma,\widetilde{R}_{\infty},\widetilde{N}_{+}}\cap\Omega_{\sigma,N_{+}}\cap\Omega_{\widetilde{\sigma},\widetilde{N}_{+}}=\mathscr{Q}_{\sigma,R_{\infty},N_{+}}\cap\Omega_{\sigma,N_{+}}\cap\Omega_{\widetilde{\sigma},\widetilde{N}_{+}}$

and
$\bigcup_{N_{+}\in\mathbb{N}_{0},\sigma\in\mathbb{R}}\mathscr{Q}_{\sigma,R_{\infty},N_{+}}$
has no accumulation points in $\Omega$.

  3. _(iii)_

Let $s\in\Omega_{\sigma,N_{+}}$. Then the space $\ker L_{s,\Lambda}$ is finite
dimensional with $\dim\ker
L_{s,\Lambda}=\dim\textnormal{coker}\,L_{s,\Lambda}$ and
$\textnormal{ran}\,L_{s,\Lambda}^{*}=(\ker L_{s,\Lambda})^{\perp}\subset
H_{\sigma,R_{\infty},2}^{N_{+}+1}$, with $L_{s,\Lambda}^{*}$ the Hermitian
adjoint and $\textnormal{coker}\,L_{s,\Lambda}^{*}$ isomorphic to $\ker
L_{s,\Lambda}$.

###### Proof.

By standard, local energy estimates for $\square_{g_{M,a}}\phi=0$ and a
Grönwall inequality, one can obtain uniform constants $C,C_{0}>0$ such that:

$\|(\phi,\partial_{\tau}\phi)\|_{H^{1}(\Sigma)\times L^{2}(\Sigma)}(\tau)\leq
Ce^{C_{0}\tau}\|(\phi,\partial_{\tau}\phi)\|_{H^{1}(\Sigma)\times
L^{2}(\Sigma)}(0).$ (10.1)

Suppose $\\{L_{s,\Lambda}^{-1}\\}$ had a pole in
$s_{0}\in\\{\operatorname{\textnormal{Re}}z>C_{0}\\}$. Then the solution
$\phi$ corresponding to initial data $(\phi_{s_{0}},s\phi_{s_{0}})$ with
$\psi_{s_{0}}=\varrho\phi_{s_{0}}\in\ker L_{s_{0},\Lambda}$ takes the form
$(\phi,\partial_{\tau}\phi)(\tau)=(e^{s_{0}\tau}\phi_{s_{0}},s\phi_{s_{0}})$
and hence

$\|(\phi,\partial_{\tau}\phi)\|_{H^{1}(\Sigma)\times
L^{2}(\Sigma)}(\tau)=e^{(\operatorname{\textnormal{Re}}s_{0})\tau}\|(\phi,\partial_{\tau}\phi)\|_{H^{1}(\Sigma)\times
L^{2}(\Sigma)}(0)$

for all $\tau\geq 0$, which is in contradiction with (10.1). Hence,
$\\{L_{s,\Lambda}^{-1}\\}$ is holomorphic when restricted to
$\\{\operatorname{\textnormal{Re}}z>C_{0}\\}$. In the $\Lambda=0$ case, the
remaining statement in (i) as well as (iii) follow from Proposition 16.2
below. Part (ii) follows from Proposition 16.4 below. This is then generalized
to $\Lambda\geq 0$ in Corollary 18.3. ∎

######  Remark 10.1.

Uniform boundedness (and decay) of solutions to the wave equation
$\square_{g_{M,a}}\phi=0$, established by Dafermos–Rodnianski–Shlapentokh-
Rothman in [DRSR16] implies that in the $\Lambda=0$ case, the set
$\mathscr{Q}_{\rm reg}$ is in fact entirely contained in the left-half plane
$\\{\operatorname{\textnormal{Re}}z<0\\}$ ($C_{0}$ in the proof of Theorem
10.2(i) can be set to $0$). The validity of Theorem 10.2, however, is
independent of a boundedness statement and applies to settings where such a
boundedness statement is false, like the class of wave equations with
potentials on Kerr studied in [Mos17].

Closely related to uniform boundedness for $\square_{g_{M,a}}\phi=0$, is the
statement of “mode stability” [Whi89, SR15], which implies the absence of
regularity quasinormal modes of the form (10.3) below. Since regularity
quasinormal modes need not satisfy (10.3), mode stability alone is not enough
to ensure that $\mathscr{Q}_{\rm reg}$ is contained in the left-half plane
when $\Lambda=0$; see Remark 10.2 below.

For $\Lambda>0$ with $M^{2}\Lambda\ll 1$, the results of [Hin24] imply that
$\mathscr{Q}_{\rm reg}$ is contained in
$\\{\operatorname{\textnormal{Re}}z<0\\}$. Note that in the general
$\Lambda>0$ case however, not even mode stability is known; see [CT22] for
partial results in this direction.

######  Remark 10.2.

Consider the following angular ODE:

$-\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}S_{m\ell})+\left[\frac{1}{\sin^{2}\theta}m^{2}+a^{2}\omega^{2}\sin^{2}\theta\right]S_{m\ell}=\Lambda_{m\ell}(a\omega)S_{m\ell},$
(10.2)

with $S_{m\ell}(\theta;a\omega)$ in $L^{2}(\sin\theta d\theta)$ the _oblate
spheroidal harmonics_. When $a=0$, (10.2) is simply the Laplace equation on
$\mathbb{S}^{2}$ restricted to functions with azimuthal angular dependence
$e^{im\varphi}$ and the solutions are spherical harmonics.

When $\omega\in\mathbb{R}$, (10.2) admits solutions for a discrete set of
$\Lambda_{m\ell}(a\omega)\in\mathbb{R}$, labelled by
$\ell\in\mathbb{N}_{\geq|m|}$ such that
$\Lambda_{m\ell}(a\omega)\geq\Lambda_{m\ell^{\prime}}(a\omega)$ if
$\ell\geq\ell^{\prime}$. Furthermore, when $\omega\in\mathbb{R}$, the
functions
$\\{S_{m\ell}(\theta;a\omega)\\}_{\begin{subarray}{c}m\in\mathbb{Z},l\geq|m|\end{subarray}}$
form an orthonormal basis of $L^{2}(\sin\theta d\theta)$, the self-adjointness
of the differential operator on the left-hand side of (10.2) and the
compactness of its inverse.

The functions $\omega\mapsto\Lambda_{m\ell}(a\omega)$ can be analytically
continued to the rest of the complex plane as complex functions, with a finite
number of branch points, for which the corresponding (10.2) admits solutions
in $L^{2}(\sin\theta d\theta)$; see for example [MS54][Proposition 5 of
Chapter 1.5, Proposition 2 of Chapter 3.22] for a proof in the context of
general ODE and [TdC20][Proposition 2.1] for a discussion on this matter in
the context of (10.2). However, when $\omega\notin\mathbb{R}$,
$\\{S_{m\ell}(\theta;a\omega)\\}_{\begin{subarray}{c}m\in\mathbb{Z},l\geq|m|\end{subarray}}$
no longer need to form an orthonormal basis.

In much of the physics literature, see for example [Lea85], quasinormal mode
solutions $\phi$ to the wave equation (1.1) are required to take the separated
form

$\phi(t,r_{*},\theta,\varphi)=e^{-i\omega t}\frac{u_{\ell
m}(r_{*};\omega)}{\sqrt{r^{2}+a^{2}}}S_{m\ell}(\theta;a\omega)e^{im\varphi},$
(10.3)

with $S_{m\ell}(\theta;a\omega)$ a solution to (10.2) and $u_{\ell
m}(r;\omega)$ a solution to the ODE:

$u_{\ell m}^{\prime\prime}+(\omega^{2}-V_{m\ell}^{(a\omega)})u_{\ell m}=0,$
(10.4)

where $V_{m\ell}^{(a\omega)}(r_{*})$ is a potential function that depends on
$\Lambda_{m\ell}(a\omega)$.

In the present paper, we do not impose the restriction (10.3) and we instead
allow for more general solutions of the form:

$\phi(t,r,\theta,\varphi)=e^{-i\omega
t}\sum_{m\in\mathbb{Z}}\hat{\phi}_{m}(r,\theta)e^{im\varphi}.$

In view of the finite dimensionality of the space of regularity quasinormal
modes that follows from Theorem 10.1(iii), we can infer that quasinormal modes
corresponding to a single quasinormal frequency $s=-i\omega$ must be supported
on a finite number of $m$. If
$\\{S_{m\ell}(\theta;a\omega)\\}_{\begin{subarray}{c}m\in\mathbb{Z},l\geq|m|\end{subarray}}$
formed an orthonormal basis for $L^{2}(\mathbb{S}^{2})$, then we would also be
able to conclude that any regularity quasinormal mode can be written as a
finite linear combination of regularity quasinormal modes satisfying (10.3).
This is the case when $a=0$.

As mentioned above, we are not, however, guaranteed an orthonormal basis of
functions $S_{m\ell}$ when $a\neq 0$, so we cannot draw such a conclusion. The
set of regularity quasinormal frequencies could therefore in principle be
strictly larger than the set of frequencies corresponding to quasinormal modes
that can be separated as in (10.3).

###  10.2. Application 1: stability of the quasinormal spectrum and
convergence of Kerr–de Sitter quasinormal frequencies

We first give a definition of _stability_ of the spectrum of a family of
linear operators that is suitable for the setting of the present paper.

######  Definition 10.3 (Stability of the spectrum for a family of restricted
operators).

Let $O=\bigcup_{\alpha\in I}O_{\alpha}\subseteq\mathbb{C}$, with
$O_{\alpha}\subset\mathbb{C}$ open and $I$ an index set. Let $\mathbf{H}$ be a
Hilbert space and let $A:{\rm dom}(A)\subseteq\mathbf{H}\to\mathbf{H}$ be a
densely defined linear operator. Let
$\mathbf{H}^{\prime}_{\alpha}\subseteq\mathbf{H}_{\alpha}\subseteq\mathbf{H}$
be families of closed subspaces, such that the following restrictions of $A$
are densely defined operators: $A:\mathbf{H}^{\prime}_{\alpha}\supseteq{\rm
dom}(A_{\alpha})\to\mathbf{H}_{\alpha}$. Denote the corresponding restricted
operators $A_{\alpha}$.

Let $B$ be a linear operator, such that its restrictions to
$\mathbf{H}^{\prime}_{\alpha}$ results in operators
$B_{\alpha}:\mathbf{H}^{\prime}_{\alpha}\to\mathbf{H}_{\alpha}$ that are
bounded with respect to the operator norms
$\|\cdot\|_{\mathbf{H}_{\alpha}^{\prime}\to\mathbf{H}_{\alpha}}$ for all
$\alpha\in I$.

Then we say that $A$ has a _stable spectrum in $O$ in the direction $B$, with
respect to the the families of subspaces
$\\{\mathbf{H}_{\alpha},\mathbf{H}^{\prime}_{\alpha}\\}_{\alpha\in I}$_, if
the following holds: for any compact domain $K\subset O$ and for all
$\epsilon>0$, there exists a $\delta_{0}>0$ such that for all $\alpha\in I$
and for all $0<\delta\leq\delta_{0}$:

  1. (1)

${\rm Spect}(A_{\alpha}+\delta\cdot B_{\alpha})\cap O_{\alpha}\cap
K\subseteq\bigcup_{s\in{\rm Spect}(A_{\alpha})\cap
O_{\alpha}}D_{\epsilon}(s)$, with $D_{\epsilon}(s)$ the open disc of radius
$\epsilon$ around $s\in\mathbb{C}$.

  2. (2)

For all $s\in{\rm Spect}(A_{\alpha})\cap O_{\alpha}\cap K$,

${\rm Spect}(A_{\alpha}+\delta\cdot B_{\alpha})\cap K\cap
D_{\epsilon}(s)\neq\emptyset.$

Figure 4. A pictorial representation of stability of the spectrum in the case
of a spectrum of isolated eigenvalues of $A_{\alpha}$ represented by crosses.
The eigenvalues of $A_{\alpha}+\delta\cdot B_{\alpha}$ are represented by
squares and the shaded disks represent $D_{\epsilon}(s)$ with $s\in{\rm
Spect}(A_{\alpha})$.

Using the above definition, the following theorem gives a precise formulation
of Theorem 1.2(i):

######  Theorem 10.3.

Let $\mathcal{A}_{\delta\cdot Q}$ denote the infinitesimal generator of time
translations corresponding to the following perturbed wave operator:

$\square_{g_{M,a}}+\delta\cdot\varrho^{-3}Q(\varrho\cdot)$

with $Q$ the following second-order differential operator:

$Q=Q^{\tau\tau}(r,\vartheta,\varphi_{*})\partial_{\tau}^{2}+Q^{\tau
x}(r,\vartheta,\varphi_{*})\partial_{\tau}\partial_{x}+Q^{\tau\vartheta}(r,\vartheta,\varphi_{*})\partial_{\tau}\partial_{\vartheta}+Q^{\tau\varphi}(r,\vartheta,\varphi_{*})\partial_{\tau}\partial_{\varphi_{*}}+Q^{\tau}(r,\vartheta,\varphi_{*})\partial_{\tau}+\tilde{Q},$

where $Q^{\tau},Q^{\tau\tau},Q^{\tau x},Q^{\tau\vartheta},Q^{\tau\varphi_{*}}$
are bounded and $\tilde{Q}$ is a differential operator of order $\leq 2$
acting functions on $\Sigma$, such that
$\tilde{Q}_{s}=\tilde{Q}:H_{\sigma,R_{\infty},2}^{N_{+}}\to
H_{\sigma,R_{\infty}}^{N_{+}}$ is bounded for all $|\sigma|<\sigma_{0}$, with
$\sigma_{0}>0$ or $\sigma_{0}=\infty$. Let $B$ be the operator satisfying

$\mathcal{A}+\delta\cdot B=\mathcal{A}_{\delta\cdot Q}.$

Let $s_{0}\in\mathbb{C}$ be such that
$L_{s_{0}}^{-1}:H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}$ is well-defined. Then the spectrum of
$\mathcal{A}:{\rm dom}(\mathcal{A})\supset\mathbf{H}^{1}\to\mathbf{H}^{1}$ is
stable in
$\\{\operatorname{\textnormal{Re}}s>-(\frac{1}{2}+N_{+})\kappa_{+}\\}\cap
B_{\sigma_{0}^{2}}\cap\Omega=B_{\sigma_{0}^{2}}\cap\bigcup_{\sigma\in\mathbb{R}}\Omega_{\sigma,N_{+}}\subset\mathbb{C}$
with respect to the families of subspaces
$\\{\mathbf{H}_{\sigma}^{N_{+}},D_{\sigma,R_{\infty}}^{N_{+}}(L_{s_{0}})\oplus
H_{\sigma,R_{\infty},1}^{N_{+}+1})\\}_{\sigma\in(-\sigma_{0},\sigma_{0})}$ in
the direction $B$.

In particular, if $Q$ vanishes on $\Sigma\cap\\{r\geq R_{\infty}\\}$ and has
smooth coefficients, then the above stability result applies in the full
$\Omega$.

###### Proof.

By a slight modification of the proof of Proposition 9.1, we obtain the
following relation:

$\mathcal{A}_{\delta\cdot Q}-s\mathbf{1}=\begin{pmatrix}0&\mathbf{1}\\\
\mathbf{1}&P_{\delta\cdot
Q}-s\mathbf{1}\end{pmatrix}\begin{pmatrix}\frac{1}{2M^{2}h(0)-Mxb_{\tau\tau}-\delta\cdot
Q^{\tau\tau}}(L_{s}+\delta\cdot\tilde{Q})&0\\\
0&\mathbf{1}\end{pmatrix}\begin{pmatrix}\mathbf{1}&0\\\
-s\mathbf{1}&\mathbf{1}\end{pmatrix},$

with $P_{\delta\cdot
Q}=\frac{2M^{2}h(0)-Mxb_{\tau\tau}}{2M^{2}h(0)-Mxb_{\tau\tau}-\delta\cdot
Q^{\tau\tau}}P+\frac{\delta}{2M^{2}h(0)-Mxb_{\tau\tau}-\delta\cdot
Q^{\tau\tau}}\left(Q^{\tau
x}\partial_{x}+Q^{\tau\vartheta}\partial_{\vartheta}+Q^{\tau\varphi_{*}}\partial_{\varphi_{*}}+Q^{\tau}\mathbf{1}\right)$.

Note finally, that for all $\delta>0$ $P_{\delta\cdot Q}f\in
H_{\sigma,R_{\infty}}^{N_{+}}$ if $f\in H_{\sigma,R_{\infty},1}^{N_{+}+1}$, so
we can restrict the domain of $\mathcal{A}_{\delta\cdot Q}$ to
$D_{\sigma,R_{\infty}}^{N_{+}}(L_{s_{0}})\oplus
H_{\sigma,R_{\infty},1}^{N_{+}+1}$.

By the above expression for $\mathcal{A}_{\delta\cdot Q}-s\mathbf{1}$ and
Corollary 9.2, invertibility of $\mathcal{A}_{\delta\cdot Q}-s\mathbf{1}$ is
equivalent to invertibility of
$L_{s}+\delta\cdot\tilde{Q}:H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}$, so we can apply Proposition 17.1 below to
obtain the desired stability statement. ∎

The following theorem is a precise version of Theorem 4.2(ii) (which leads to
Theorem 1.2(ii)).

######  Theorem 10.4.

Let $\\{\Lambda_{n}\\}$ be a sequence of cosmological constants satisfying
$M^{2}\Lambda_{n}\ll 1$ and $\Lambda_{n}\downarrow 0$. Suppose that there
exists an $s_{*}\in\Omega$ with $\ker L_{s_{*}}^{-1}\neq\\{0\\}$ and consider
$\sigma\in\mathbb{R}$ and $N_{+}\in\mathbb{N}$ such that
$s_{*}\in\Omega_{\sigma,N_{+}}$.

Then there exists a sequence $s_{n}\in\Omega_{\sigma,N_{+}}$, such that $\ker
L_{s_{n},\Lambda_{n}}^{-1}\neq\\{0\\}$ and $s_{n}\to s_{*}$ as $n\to\infty$.
Furthermore, for all $\widehat{\psi}\in\ker L_{s_{*}}^{-1}$, there exists a
sequence $\\{\widehat{\psi}_{n}\\}$ in $H^{N_{+}}_{\sigma,R_{\infty}}$ with
$\widehat{\psi}_{n}\in\ker L_{s_{n},\Lambda_{n}}^{-1}$ and

$\|\widehat{\psi}-\widehat{\psi}_{n}\|_{H^{N_{+}}_{\sigma,R_{\infty}}}\to
0\quad(n\to\infty).$

Conversely, let $\\{s_{n}\\}$ be a convergent sequence of elements in $\ker
L_{s,\Lambda_{n}}$ and $\Lambda_{n}\to 0$ as $n\to\infty$, such that the limit
$s_{*}$ satisfies: $s_{*}\in\\{|{\rm arg}z|<\frac{2\pi}{3}\\}$. Then
$s_{*}\in\ker L_{s,0}$.

###### Proof.

In the context of the $\kappa$-regularized operator $L_{s,\kappa}$, this
proposition is proved in Lemma 16.3. This is then generalized to the true
Kerr–de Sitter operator $L_{s,\Lambda}$ in §18. ∎

###  10.3. Application 2: a characterization of scattering resonances

In order to justify the use of the term “regularity quasinormal frequencies”,
we need to make a connection with other definitions of “quasinormal
frequencies” that have previously appeared in the literature.

It is common to define “quasinormal frequencies” as scattering resonances
associated to the meromorphic continuation of the cut-off resolvent operator
corresponding to a “standard” time function whose level sets are
asymptotically flat hypersurfaces. In the case of Kerr, this is typically take
to be the Boyer–Lindquist time function $t$. One first considers the operator

$A(\omega)=e^{i\omega t}\square_{g_{M,a}}(e^{-i\omega t}\cdot(\cdot)),$

which can be inverted on suitable spaces of functions on
$\Sigma^{\prime}=\\{t=0\\}$ with so-called outgoing boundary conditions and
where $\omega\in\mathbb{C}$ with suitably large
$\operatorname{\textnormal{Im}}\omega>0$. The inverse, denoted $R(\omega)$, is
also called the _resolvent operator_.

While $R(\omega)$ fails to make sense in a region intersecting both
$\\{\operatorname{\textnormal{Im}}z>0\\}$ and
$\\{\operatorname{\textnormal{Im}}z<0\\}\subset\mathbb{C}$, one can conjugate
$R(\omega)$ with a multiplication operator $\chi\mathbf{1}$, where $\chi\in
C_{c}^{\infty}(\Sigma^{\prime})$ and try to meromorphically continue the _cut-
off resolvent_ :

$\chi R(\omega)\chi$

as an operator on suitable function spaces. The poles of the meromorphic
continuation of $\chi R(\omega)\chi$ are called _scattering resonances_.

The theorem below provides a more precise version of Theorem 1.1(iii). This
theorem demonstrates that scattering resonances form a _subset_ of the set of
regularity quasinormal frequences, thereby justifying the nomenclature.

######  Theorem 10.5.

Let $N_{+}\in\mathbb{N}_{0}$. The cut-off resolvent operator:

$\chi R(\omega)\chi:H^{N_{+}}_{\not{\nabla}}(\Sigma^{\prime})\to
H^{N_{+}+1}_{\not{\nabla}}(\Sigma^{\prime})$

is well-defined for any cut-off function $\chi\in
C_{c}^{\infty}(\Sigma^{\prime})$ and for all $\omega\in\mathbb{C}$ with
$\operatorname{\textnormal{Im}}\omega>\omega_{0}>0$ suitably large and
$\omega\mapsto\chi R(\omega)\chi$ can be meromorphically continued to
$i\Omega\cap\\{\operatorname{\textnormal{Im}}\omega>-\left(\frac{1}{2}+N_{+}\right)\kappa_{+}\\}$.
Let $i\mathscr{Q}_{\rm res}$ denote the union over the set of poles of
$\omega\mapsto\chi R(\omega)\chi$ over $N_{+}$. Then

$\mathscr{Q}_{\rm res}\subseteq\mathscr{Q}_{\rm reg}.$

Furthermore, for $\frac{|a|}{M}$ suitably small, or under the restriction to
functions with a bounded azimuthal number $m$,

$\chi R(\omega)\chi:L^{2}(\Sigma^{\prime})\to H^{2}(\Sigma^{\prime})$

is well-defined and can be meromorphically continued to $i\Omega$, with poles
in the set $\mathscr{Q}_{\rm res}$.

###### Proof.

Consider the extension of $\Sigma^{\prime}$ to the bifurcation sphere, denoted
$\overline{\Sigma^{\prime}}$. By standard local-in-time energy estimates (see
also the proof of Theorem 10.2), it follows that the time evolution map
$\mathcal{S}(\tau):H^{1}(\overline{\Sigma^{\prime}})\times
L^{2}(\overline{\Sigma^{\prime}})\to H^{1}(\overline{\Sigma^{\prime}})\times
L^{2}(\overline{\Sigma^{\prime}})$ corresponding to initial data
$(\phi,\partial_{t}\phi)$ on $\overline{\Sigma^{\prime}}$ is a bounded linear
operator for each $\tau$, with a norm that can be bounded by $e^{s_{0}\tau}$,
for suitably large $s_{0}>0$. By standard results of functional analysis, see
for example [Eva98][§7.4, Theorem 2] and [Hil48][Theorem 11.6.1], the
corresponding infinitesimal generator
$\mathcal{A}:H^{1}(\overline{\Sigma^{\prime}})\times
L^{2}(\overline{\Sigma^{\prime}})\supset{\rm dom}(\mathcal{A})\to
H^{1}(\overline{\Sigma^{\prime}})\times L^{2}(\overline{\Sigma^{\prime}})$ is
closed and densely defined with
$(\mathcal{A}-s\mathbf{1})^{-1}:H^{1}(\overline{\Sigma^{\prime}})\times
L^{2}(\overline{\Sigma^{\prime}})\to{\rm dom}(\mathcal{A})$ a well-defined
bounded operator for $\operatorname{\textnormal{Re}}s>s_{0}$.

In this case, we can apply (9.1), with $\frac{L_{s}}{(2Mh(0)-Mxb_{\tau\tau})}$
replaced with
$\frac{r^{2}A(\omega)}{\mathbbm{h}(2(r^{2}+a^{2})-\Delta\mathbbm{h})-a^{2}\sin^{2}\theta}$,
where $\mathbbm{h}=\frac{r^{2}+a^{2}}{\Delta}$ with $\omega=i\cdot s$ and $P$
replaced with an appropriate first order operator to conclude via (9.2) that:

$R(\omega):\frac{\Delta}{r^{2}}L^{2}(\overline{\Sigma^{\prime}})\to
H^{1}(\overline{\Sigma^{\prime}})$

is well-defined for suitably large $\operatorname{\textnormal{Im}}\omega>0$
and we can consider $\chi R(\omega)\chi:L^{2}(\Sigma^{\prime})\to
H^{1}(\Sigma^{\prime})$.

In Corollary 19.3 below, we prove the existence of the meromorphic
continuation of $\chi R(\omega)\chi$, defined on appropriate subspaces, and
its relation to $\mathscr{Q}_{\rm reg}$. ∎

######  Remark 10.3.

The results of Theorem 10.5 in fact apply also when considering cut-off
resolvents with respect to time functions whose level sets intersect
$\mathcal{H}^{+}$ to the future of the bifurcation sphere and are
asymptotically flat. We denote the level set corresponding to “time zero” in
this case by $\widetilde{\Sigma}$. Now, the cut-off functions $\chi$ are
allowed to be supported near $\mathcal{H}^{+}$ but the support remains
compact. In the domains and codomains of the maps in Theorem 10.5,
$\Sigma^{\prime}$ is then replaced with $\widetilde{\Sigma}$.

##  11\. Guide to the rest of the paper

We outline the structure of the rest of the paper and relate each section to
the steps sketched in §4, which concern the operator $L_{s}$. We first focus
on the $\Lambda=0$ setting.

#### Theorem 4.1(i):

  * •

In §12, we derive red-shift estimates and elliptic estimates (featuring a
finite number of derivatives) and couple them to obtain a single estimate in
the bounded region $\varrho\leq\eta^{-1}R_{\infty}$.

  * •

In §13, we derive Gevrey-type estimates (featuring an infinite number of
derivatives) in the region $\varrho\geq 2R_{\infty}$: estimates with higher-
order radial derivatives are derived in §13.1 and estimates with lower-order
radial derivatives are derived in §13.2.

  * •

The Gevrey estimates are coupled with the red-shift and elliptic estimates in
§14 to obtain a single, global estimate.

  * •

In §15, we establish the invertibility of $\mathfrak{L}_{s,\lambda,\kappa}$
with $\kappa>0$ and construct $\mathfrak{L}_{s,\lambda}^{-1}$ as a
$\kappa\downarrow 0$ limit.

  * •

In §16, we establish the meromorphicity and Fredholm properties of
$L_{s}^{-1}$.

Theorem 4.1(ii) and Theorem 4.2(ii):

  * •

In §16, we also establish the independence of the regularity quasinormal
spectrum with respect to a change in Hilbert spaces, corresponding to
different $N_{+}$, $R_{\infty}$ and $\sigma$. We also establish Theorem
4.2(ii) for the $\kappa$-regularized operator.

We subsequently prove Theorem 4.2(i):

  * •

In §17, we establish stability of the quasinormal spectrum under suitably
small perturbations at the level of $L_{s}$.

Since the analysis in the $\Lambda>0$ case actually follows, with minor
modifications, from the analysis that is already required for the $\Lambda=0$
case, we only take $\Lambda>0$ in §18, where we focus primarily on the
differences with the $\Lambda=0$ case. Theorem 4.1(i) for $\Lambda>0$ and
Theorem 4.2(ii) for the true Kerr–de Sitter operator follow from §18.

Finally, we establish the relation between regularity quasinormal frequencies
and scattering resonances:

  * •

Theorem 4.1(iii): In §19, we construct the meromorphic continuation of the
cut-off resolvent operator and relate scattering resonances to regularity
quasinormal frequencies.

##  12\. Red-shift and elliptic estimates

We will consider different types of estimates in the following two regions:

  1. (A)

$r_{+}\leq r\leq\frac{1}{2}R_{\infty}$,

  2. (B)

$\frac{1}{4}R_{\infty}\leq\varrho\leq\eta^{-1}R_{\infty}$,

with $\mathbb{R}\ni\eta<2$ and where $R_{\infty}M^{-1}$ will be chosen
suitably large. We will later also need to restrict to suitably small $\eta$.

The estimates in (A) are the spacelike analogues of the spacetime _red-shift
estimates_ of Dafermos–Rodnianski [DR09] and they establish, in particular,
control of the non-degenerate $H^{1}$ norm of $\hat{\psi}\in
C^{\infty}(\widehat{\Sigma})$ in terms of the $L^{2}$ norm of
$\mathfrak{L}_{s,\lambda}\hat{\psi}$. These estimates rely crucially on the
positivity of the surface gravity $\kappa_{+}$ associated to the event horizon
at $r=r_{+}$. In contrast with the spacetime red-shift estimates, which apply
only to regions where $r$ is sufficiently close to $r_{+}$, they hold in
regions of arbitrarily large $r$.

Note that for $M^{-1}R_{\infty}$ sufficiently large, the region (A) covers the
ergoregion of Kerr, where the vector field $T$ is not timelike, and therefore
“standard” (degenerate) elliptic estimates do not hold.

The estimates in (B) are more similar to standard elliptic estimates, in the
sense that they establish control of a (weighted) $H^{2}$ norm of
$\hat{\psi}\in C^{\infty}(\widehat{\Sigma})$ in terms of a (weighted) $L^{2}$
norm of $\mathfrak{L}_{s,\lambda}\hat{\psi}$ and rely on the fact that the
spacetime metric is sufficiently close to the Minkowski metric in appropriate
coordinates, by the asymptotic flatness property of the Kerr metric. The main
difference with standard elliptic estimates, is the addition of a degenerate
weight function in the norms, which will play an important role in the
coupling with the Gevrey estimates of §13.

###  12.1. Red-shift estimate in (A)

We first consider the region with:

$r_{+}\leq\varrho\leq\frac{1}{2}R_{\infty}.$

Recall from (7.10) that in this region $\varrho=r$, $\vartheta=\theta$ and we
can write:

$r^{-2}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}=r\Delta\partial_{r}^{2}(r^{-1}\cdot)+r\frac{d\Delta}{dr}\partial_{r}(r^{-1}\cdot)+2ar\partial_{r}\partial_{\varphi_{*}}(r^{-1}\cdot)+\not{\Delta}_{\mathbb{S}^{2}}(\cdot)+sr(4Mr-2(1-\mathbbm{h})\Delta)\partial_{r}(r^{-1}\cdot)-\lambda\pi_{\leq
l}.$

######  Proposition 12.1.

Let $\operatorname{\textnormal{Re}}s>-\frac{\kappa_{+}}{2}$ and
$\delta=1+2\kappa_{+}^{-1}\min\\{\operatorname{\textnormal{Re}}s,0\\}$. Then
there exists a constant $C_{s}=C_{s}(\kappa_{+},s)>0$ and a numerical constant
$C>0$ such that

$\int_{r_{+}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}\frac{1}{2}(r-M)(1+2\kappa_{+}^{-1}\min\\{\operatorname{\textnormal{Re}}s,0\\})r^{\delta}|\partial_{r}(r^{-1}\widehat{\psi})|^{2}\\\
+\frac{1}{2}(1+2\kappa_{+}^{-1}\min\\{\operatorname{\textnormal{Re}}s,0\\})r^{-3+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\psi}|^{2}+\lambda|\pi_{\leq
l}\widehat{\psi}|^{2}\right]\,d\upsigma dr\\\
+\int_{\mathbb{S}^{2}}r^{\delta-2}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\psi}|^{2}+\lambda|\pi_{\leq
l}\widehat{\psi}|^{2}\right]\Big{|}_{r=r_{+}}\,d\upsigma\\\
\leq\boxed{C\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-3+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\psi}|^{2}+\lambda|\pi_{\leq
l}\widehat{\psi}|^{2}\right]\,d\upsigma
dr}+C_{s}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-3+\delta}|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,d\upsigma
dr.$ (12.1)

###### Proof.

Let $\widehat{\phi}=r^{-1}\widehat{\psi}$. Without loss of generality, we will
assume $\widehat{\phi}=\pi_{\leq l}\widehat{\phi}$ in the derivations below.
The $\pi_{>l}\widehat{\phi}$ part of general $\widehat{\phi}$ can be estimated
in exactly the same way by simply taking $\lambda=0$.

We multiply $r^{-2}\rho^{2}\mathfrak{L}_{s,\lambda}\widehat{\psi}$ with
$\chi_{r\leq\frac{1}{3}R_{\infty}}r^{\delta-1}\overline{\partial_{r}\widehat{\phi}}$,
where $\delta>0$ will be fixed later, and where
$\chi_{r\leq\frac{1}{3}R_{\infty}}$ is a smooth cut-off function that is equal
to 1 for $r\leq\frac{1}{3}R_{\infty}$ and vanishes when
$r\geq\frac{1}{2}R_{\infty}$. We then apply the Leibniz rule to obtain:

$\operatorname{\textnormal{Re}}(\chi_{r\leq\frac{1}{3}R_{\infty}}r^{\delta-1}\overline{\partial_{r}\widehat{\phi}}\cdot
r^{-2}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})=\Delta
r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}(\overline{\partial_{r}\widehat{\phi}}\partial_{r}^{2}\widehat{\phi})+\frac{d\Delta}{dr}\chi_{r\leq\frac{1}{3}R_{\infty}}r^{\delta}|\partial_{r}\widehat{\phi}|^{2}\\\
+2ar^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}(\overline{\partial_{r}\widehat{\phi}}\partial_{r}\partial_{\varphi_{*}}\widehat{\phi})+r^{\delta}\operatorname{\textnormal{Re}}(\overline{\partial_{r}\widehat{\phi}}\not{\Delta}_{\mathbb{S}^{2}}\widehat{\phi})+(4Mr-2(\mathbbm{h}-1)\Delta))r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}{\operatorname{\textnormal{Re}}s}|\partial_{r}\widehat{\phi}|^{2}-\lambda
r^{\delta}\operatorname{\textnormal{Re}}(\overline{\partial_{r}\widehat{\phi}}\widehat{\phi})\\\
=\frac{1}{2}\partial_{r}\left(\Delta
r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\partial_{r}\widehat{\phi}|^{2}-r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}-\lambda\chi_{r\leq\frac{1}{3}R_{\infty}}|\widehat{\phi}|^{2}\right)+\partial_{\varphi_{*}}(\ldots)+\textnormal{div}_{\mathbb{S}^{2}}(\ldots)\\\
+\frac{1}{2}r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}\left[\frac{d\Delta}{dr}-\delta
r^{-1}\Delta+(8Mr-4(\mathbbm{h}-1)\Delta)){\operatorname{\textnormal{Re}}s}\right]|\partial_{r}\widehat{\phi}|^{2}+\frac{\delta}{2}r^{\delta-1}\chi_{r\leq\frac{1}{3}R_{\infty}}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}+\lambda|\widehat{\phi}|^{2}\right]\\\
-\frac{1}{2}\frac{d\chi_{r\leq\frac{1}{3}R_{\infty}}}{dr}\left(\Delta
r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\partial_{r}\widehat{\phi}|^{2}-r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}-\lambda\chi_{r\leq\frac{1}{3}R_{\infty}}|\widehat{\phi}|^{2}\right),$

where we employed the schematic notation
$\partial_{\varphi_{*}}(\ldots)+\textnormal{div}_{\mathbb{S}^{2}}(\ldots)$ for
total derivatives and divergences on $\mathbb{S}^{2}$, as these terms vanish
after integration over $\mathbb{S}^{2}$.

We first estimate via Young’s inequality:

$\left|\operatorname{\textnormal{Re}}\left[r^{\delta-1}\overline{\partial_{r}\widehat{\phi}}\cdot
r^{-2}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\right]\right|\leq\frac{\delta}{2}r^{\delta}(r-M)|\partial_{r}\widehat{\phi}|^{2}+\delta^{-1}r^{-2+\delta}(r-M)^{-1}|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}.$

Furthermore, using that $\mathbbm{h}\leq\frac{r^{2}+a^{2}}{\Delta}$, we obtain

$8Mr-4(\mathbbm{h}-1)\Delta\geq 8Mr-4(r^{2}+a^{2})+4\Delta=0$

so for $\operatorname{\textnormal{Re}}s\geq 0$, we have that
$(8Mr-4(\mathbbm{h}-1)\Delta)){\operatorname{\textnormal{Re}}s}\geq 0$.

For $\operatorname{\textnormal{Re}}s<0$, we use that $\mathbbm{h}\geq 1$ to
obtain

$(8Mr-4(\mathbbm{h}-1)\Delta)\operatorname{\textnormal{Re}}s\geq
8Mr\operatorname{\textnormal{Re}}s$

and we have that

$\frac{d\Delta}{dr}-\delta
r^{-1}\Delta+8Mr{\operatorname{\textnormal{Re}}s}-\frac{\delta}{2}(r-M)>2(r-M)\left(1-\frac{3}{4}\delta+2\frac{r_{+}-M}{r-M}\kappa_{+}^{-1}{\operatorname{\textnormal{Re}}s}\right),$
(12.2)

where we recall the following expression for the surface gravity $\kappa_{+}$:

$\kappa_{+}=\frac{r_{+}-M}{2Mr_{+}}.$

Hence, for ${\operatorname{\textnormal{Re}}s}>-\frac{\kappa_{+}}{2}$ and
$0<\frac{3}{4}\delta<1+\min\\{2\kappa_{+}^{-1}{\operatorname{\textnormal{Re}}s},0\\}$,
the right-hand side of (12.2) is non-negative definite.

Since $\frac{d\chi_{r\leq\frac{1}{3}R_{\infty}}}{dr}>0$ and there exists a
numerical constant $C>0$ such that
$-\frac{d\chi_{r\leq\frac{1}{3}R_{\infty}}}{dr}\leq CR_{\infty}^{-1}$, we can
estimate:

$-\frac{1}{2}\frac{d\chi_{r\leq\frac{1}{3}R_{\infty}}}{dr}\left(\Delta
r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\partial_{r}\widehat{\phi}|^{2}-r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}-\lambda\chi_{r\leq\frac{1}{3}R_{\infty}}|\widehat{\phi}|^{2}\right)\geq-
CR_{\infty}^{-1}r^{\delta}\left(|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}+\lambda|\widehat{\phi}|^{2}\right).$

After setting
$\delta=1+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\}$ and
integrating over $[r_{+},\frac{1}{2}R_{\infty}]\times\mathbb{S}^{2}$, we
obtain the following estimate: for
${\operatorname{\textnormal{Re}}s}>-\frac{\kappa_{+}}{2}$, there exists a
$C_{s}=C(\kappa_{+},{\operatorname{\textnormal{Re}}s},\delta)>0$, such that

$\int_{r_{+}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}\frac{1}{2}(r-M)(1+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})r^{\delta}|\partial_{r}\widehat{\phi}|^{2}\,d\upsigma
dr\\\
+\frac{1}{2}(1+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})r^{-1+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}+\lambda|\widehat{\phi}|^{2}\right]+\int_{\mathbb{S}^{2}}r^{\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}+\lambda|\widehat{\phi}|^{2}\right]\Big{|}_{r=r_{+}}\,d\upsigma\\\
\leq
C\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}+\lambda|\widehat{\phi}|^{2}\right]\,d\upsigma
dr+C_{s}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-3+\delta}|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,d\upsigma
dr.$

∎

In order to improve the range of allowed values of
${\operatorname{\textnormal{Re}}s}$, we need to consider higher-order
$r$-derivatives of $\widehat{\psi}$. In addition, we will consider higher
angular derivatives.

######  Lemma 12.2.

Let $k,m\in\mathbb{N}_{0}$ and $r_{+}\leq\varrho\leq\frac{1}{2}R_{\infty}$.
Then, for $\widehat{\phi}=r^{-1}\widehat{\psi}$:

$\partial_{r}^{k}\not{\Delta}_{\mathbb{S}^{2}}^{m}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})=\Delta\partial_{r}^{2+k}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}+(1+k)\frac{d\Delta}{dr}\partial_{r}^{k+1}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}+2a\partial_{\varphi_{*}}\partial_{r}^{k+1}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}+\not{\Delta}_{\mathbb{S}^{2}}^{1+m}\partial_{r}^{k}\widehat{\phi}\\\
+4Mrs\partial_{r}^{k+1}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}-(\lambda\pi_{\leq
l}-k(k+1)-4kMs)\partial_{r}^{k}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}-2\sum_{j=0}^{k}\frac{k!}{j!(k-j)!}\partial_{r}^{k-j}((\mathbbm{h}-1)\Delta)\partial_{r}^{j}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}.$
(12.3)

###### Proof.

We will first prove (12.3) with $m=0$ via induction. The $k=0$ case follows
from (7.10). Assume (12.3) holds for $k=K$. Then

$\partial_{r}^{K+1}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})=\partial_{r}\Big{(}\Delta\partial_{r}^{2+K}\widehat{\phi}+(1+K)\frac{d\Delta}{dr}\partial_{r}^{K+1}\widehat{\phi}+2a\partial_{\varphi_{*}}\partial_{r}^{K+1}\widehat{\phi}+\not{\Delta}_{\mathbb{S}^{2}}\partial_{r}^{K}\widehat{\phi}+4Mrs\partial_{r}^{K+1}\widehat{\phi}\\\
-(\lambda\pi_{\leq
l}-K(K+1)-4KMs)\partial_{r}^{K}\widehat{\phi}-2\sum_{j=0}^{K}\frac{K!}{j!(K-j)!}\partial_{r}^{K-j}((\mathbbm{h}-1)\Delta)\partial_{r}^{j}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}\Big{)}\\\
=\partial_{r}^{3+K}\widehat{\phi}+(K+2)\frac{d\Delta}{dr}\partial_{r}^{K+2}\widehat{\phi}+2a\partial_{\varphi_{*}}\partial_{r}^{K+2}\widehat{\phi}+\not{\Delta}_{\mathbb{S}^{2}}\partial_{r}^{K+1}\widehat{\phi}+4Mrs\partial_{r}^{K+2}\widehat{\phi}\\\
-\left(\lambda\pi_{\leq
l}-K(K+1)-\frac{d^{2}\Delta}{dr^{2}}(K+1)-4M(K+1)s\right)\partial_{r}^{K+1}\widehat{\phi}\\\
-2\sum_{j=0}^{K+1}\frac{(K+1)!}{j!(K+1-j)!}\partial_{r}^{K+1-j}((\mathbbm{h}-1)\Delta)\partial_{r}^{j}\not{\Delta}_{\mathbb{S}^{2}}^{m}\widehat{\phi}.$

Using that $K(K+1)+\frac{d^{2}\Delta}{dr^{2}}(K+1)=(K+2)(K+1)$, it therefore
follows that (12.3) with $m=0$ holds for $k=K+1$.

The case $m>0$ follows immediately from the fact that
$\not{\Delta}_{\mathbb{S}^{2}}$ and
$\partial_{r}^{K+1}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}(\cdot))$
commute. ∎

We state now a higher-order version of Proposition 12.1.

######  Corollary 12.3.

Let $N_{+},m\in\mathbb{N}_{0}$,
${\operatorname{\textnormal{Re}}s}>-\kappa_{+}\left(\frac{1}{2}+N_{+}\right)$
and
$\delta=1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\}$.
Assume that

$\lambda-
N_{+}(N_{+}+1)\gg\frac{N_{+}^{2}M^{2}|s|^{2}}{1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\}}.$

Then there exists a constant $C_{s}=C_{s}(\kappa_{+},s)>0$ and $C=C(N_{+})>0$
such that:

$\sum_{k=0}^{N_{+}}(1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})R_{\infty}^{2k}\int_{r_{+}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}(r-M)r^{\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k+1}(r^{-1}\widehat{\psi})|^{2}\\\
+r^{-3+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\psi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\psi}|^{2}\right]\,d\upsigma
dr\\\
\leq\boxed{C\sum_{k=0}^{N_{+}}R_{\infty}^{2k-2}\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\psi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\psi}|^{2}\right]\,d\upsigma
dr}\\\
+C_{s}R_{\infty}^{2N_{+}}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{N_{+}}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})|^{2}\,d\upsigma
dr.$ (12.4)

###### Proof.

We repeat the proof of Proposition 12.1, but with (7.10) replaced by the
higher-order equation (12.3) and, for $m>0$, we additionally integrate by
parts in the $\mathbb{S}^{2}$ direction. This gives for $k\in\mathbb{N}_{0}$:

$\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}(r-M)(1+2k+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k+1}\widehat{\phi}|^{2}\\\
+\frac{2}{3}(1+2k+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})r^{-1+\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\phi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\phi}|^{2}\right]\,d\upsigma
dr\\\
+\int_{\mathbb{S}^{2}}r^{\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\phi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\phi}|^{2}\right]\Big{|}_{r=r_{+}}\,d\upsigma\\\
\leq
C\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\phi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\phi}|^{2}\right]\,d\upsigma
dr\\\
+Ck^{2}M^{2}|s|^{2}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\phi}|^{2}\,d\upsigma
dr+C_{k}M^{2}|s|^{2}\sum_{j=0}^{k-1}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{j}\widehat{\phi}|^{2}\,d\upsigma
dr\\\
+C_{s}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})|^{2}\,d\upsigma
dr.$ (12.5)

Note that we can absorb the term with a factor $Ck^{2}M^{2}|s|^{2}$ into

$+\frac{2}{3}(1+2k+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})r^{-1+\delta}\chi_{r\leq\frac{1}{3}R_{\infty}}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\phi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\phi}|^{2}\right]\,d\upsigma
dr,$

if we take
$\lambda-k(k+1)\gg\frac{k^{2}|s|^{2}}{1+2k+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\}}$.
This gives the desired control for $k=N_{+}$ in the sum on the left-hand side
of (12.5).

In order to control the $k<N_{+}$ terms in the sum on the left-hand side of
(12.5) as well as bound the $k<N+1$ terms on the right-hand side of (12.5), we
will use that we can absorb these lower-order terms into the $k=N_{+}$ terms
by applying a standard Hardy inequality $N_{+}-k$ times, see for example
[AAG23a][Lemma 2.6, eq. (2.21)], and by taking $\lambda$ to be appropriately
large. For example,

$\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}\chi_{r\leq\frac{1}{3}R_{\infty}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k-1}\widehat{\phi}|^{2}\,d\upsigma
dr\leq
C\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}\chi_{r\leq\frac{1}{3}R_{\infty}}r^{1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k}\widehat{\phi}|^{2}\,d\upsigma
dr\\\
+C\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k-1}\widehat{\phi}|^{2}\,d\upsigma
dr\\\ \leq
CR_{\infty}^{2}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}\chi_{r\leq\frac{1}{3}R_{\infty}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k}\widehat{\phi}|^{2}\,d\upsigma
dr+C\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k-1}\widehat{\phi}|^{2}\,d\upsigma
dr.\qed$

In the sections below, we will consider a coupled system of estimates
consisting of (12.4) with $k+l\leq N_{+}$ and $l\geq 1$, and appropriate
elliptic estimates in the region $\\{r\geq\frac{1}{4}R_{\infty}\\}$. In this
way, we will be able to absorb the integral over the region
$\\{\frac{1}{3}R_{\infty}\leq r\leq\frac{1}{2}R_{\infty}\\}$ in the boxed term
on the right-hand side of (12.4).

We can in fact improve the estimate in (12.3) in the case of $|a|\ll M$ or
under the restriction to fixed azimuthal modes by applying a standard elliptic
estimate in $r\leq\frac{1}{2}R_{\infty}$ in order to include higher-order
derivative terms on the right-hand side, with weights that degenerate at
$r=r_{+}$. This will be relevant when considering the cut-off resolvent
operator.

######  Proposition 12.4.

Let $N_{+},m\in\mathbb{N}_{0}$,
${\operatorname{\textnormal{Re}}s}>-\kappa_{+}\left(\frac{1}{2}+N_{+}\right)$
and
$\delta=\frac{2}{3}(1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})$.
Assume that

$\lambda-
N_{+}(N_{+}+1)\gg\frac{N_{+}^{2}M^{2}|s|^{2}}{1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\}}.$

Then for $|a|\ll M$ or
$\widehat{\psi}=\sum_{m=-m_{0}}^{m_{0}}\widehat{\psi}_{m}$, with $m_{0}>0$ and
$\Phi\psi_{m}=im\psi_{m}$, there exists a constant
$C_{s}=C_{s}(a,m_{0},\kappa_{+},s)>0$ and $C=C(N_{+},m_{0})>0$ such that:

$\sum_{k=0}^{N_{+}}(1+2N_{+}+2\kappa_{+}^{-1}\min\\{{\operatorname{\textnormal{Re}}s},0\\})R_{\infty}^{2k}\int_{r_{+}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}(r-M)r^{\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k+1}(r^{-1}\widehat{\psi})|^{2}+\Delta^{2}r^{\delta-1}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k+2}(r^{-1}\widehat{\psi})|^{2}\\\
+r^{-3+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+2}\partial_{r}^{k}\widehat{\psi}|^{2}+\Delta|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k+1}\widehat{\psi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))^{2}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\psi}|^{2}\right]\,d\upsigma
dr\\\
\leq\boxed{C\sum_{k=0}^{N_{+}}R_{\infty}^{2k}\int_{\frac{1}{3}R_{\infty}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-3+\delta}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{m+1}\partial_{r}^{k}\widehat{\psi}|^{2}+(\lambda\pi_{\leq
l}-k(k+1))|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{k}\widehat{\psi}|^{2}\right]\,d\upsigma
dr}\\\
+C_{s}R_{\infty}^{2N_{+}}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-1+\delta}|\not{\nabla}_{\mathbb{S}^{2}}^{m}\partial_{r}^{N_{+}}(r^{-3}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})|^{2}\,d\upsigma
dr.$ (12.6)

###### Proof.

We complement the estimates in the proof of Proposition 12.3 with a standard
elliptic estimate. In the $N_{+}=0$ case, we integrate

$\chi|r^{-2}\rho^{2}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\phi}+\lambda\pi_{\leq
l}\widehat{\phi}-sr(4Mr-2(1-\mathbbm{h})\Delta)\partial_{r}\widehat{\phi}-r\frac{d\Delta}{dr}\partial_{r}\widehat{\phi}-2ar\partial_{r}\partial_{\varphi_{*}}\widehat{\phi}|^{2}=\chi|r\Delta\partial_{r}^{2}\widehat{\phi}+\not{\Delta}_{\mathbb{S}^{2}}\widehat{\phi}|^{2}$

and integrate by parts the mixed term arising from the square norm on the
right-hand side.

We moreover use that the integral of $\chi
r^{2}a^{2}|\partial_{r}\partial_{\varphi_{*}}\widehat{\phi}|^{2}$ can be
absorbed into the remaining terms of $|a|$ sufficiently small or for
$\widehat{\phi}$ supported on bounded set of azimuthal modes. ∎

###  12.2. Weighted elliptic estimates in (B)

Here we will use that

$R_{\infty}\gg M,$

so in the region $\varrho\geq\frac{1}{4}R_{\infty}$, the operator
$\mathfrak{L}_{s,\lambda,\kappa}$ is suitably close to the analogous operator
in Minkowski (i.e. the $M=0$ case).

To make the elliptic estimates compatible with the Gevrey estimates in §13.2,
we will need to add a sufficiently degenerate weight to the elliptic
estimates: $(\mathfrak{l}+1)^{-w_{p}}$, where
$w_{p}:[0,r_{+}^{-1}]\to\mathbb{R}$ was defined in (8.1). Recall,

$w_{p}(x)=\begin{cases}p&x\leq\frac{1}{4R_{\infty}},\\\
0&x\geq\frac{1}{3R_{\infty}}.\end{cases}$

We make the following observations regarding $w$:

  * •

We have that $w\geq 0$ and $w\equiv 0$ in the region $\varrho\leq 3R_{\infty}$
where $\mathfrak{L}_{s,\lambda,\kappa}$ does not agree with the corresponding
operator in Minkowski.

  * •

By employing standard cut-off functions, we can define $w$ such that

$\left|\left|\frac{dw_{p}}{dx}\right|\right|_{L^{\infty}[0,r_{+}^{-1}]}\leq
CpR_{\infty},$ (12.7)

for some numerical constant $C>0$.

From (7.12), it follows that we can split

$\displaystyle\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}=$
$\displaystyle\>\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi})+(\not{\Delta}_{\mathbb{S}^{2},\vartheta}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}+\mathfrak{R}_{s,\lambda},\ \textnormal{with}$
$\displaystyle\mathfrak{R}_{s,\lambda}\widehat{\psi}:=$
$\displaystyle\>2s\partial_{x}\widehat{\psi}+\chi_{\varrho\leq
R_{\infty}}(\mathcal{K}_{s}^{\infty}\widehat{\psi}+Mx\mathcal{B}_{s}^{\infty}\widehat{\psi})-\chi_{\varrho\leq\frac{1}{2}R_{\infty}}\mathcal{K}_{s}^{+}\widehat{\psi}.$

We will treat $\mathfrak{R}_{s,\lambda}\widehat{\psi}$ as an inhomogeneous
term that can be absorbed, because it will come with additional factors of $x$
compared to the terms on the left-hand side of the elliptic estimates and $x$
will be small in the regions under consideration. Note that
$\mathfrak{R}_{s,\lambda}\widehat{\psi}-2s\partial_{x}$ contains all
deviations from corresponding operator in Minkowski. This will follow from the
following lemma:

######  Lemma 12.5.

Let $k\in\mathbb{N}_{0}$. Then there exists a numerical constant $C=C(k)>0$,
such that

$\int_{\mathbb{S}^{2}}|\partial_{x}^{k}\mathfrak{R}_{s,\lambda}\widehat{\psi}|^{2}\,d\upsigma\leq
C\int_{\mathbb{S}^{2}}\Big{[}|s|^{2}|\partial_{x}^{k+1}\widehat{\psi}|^{2}+M^{2}\chi_{\varrho\leq
R_{\infty}}\Big{(}x^{6}|\partial_{x}^{k+2}\widehat{\psi}|^{2}+x^{4}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}^{k+1}\widehat{\psi}|^{2}+x^{2}|\not{\Delta}_{\mathbb{S}^{2}}\partial_{x}^{k}\widehat{\psi}|^{2}\\\
+x^{2}|\partial_{x}^{k+1}\widehat{\psi}|^{2}+|s|^{2}|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}^{k}\widehat{\psi}|^{2}+(|s|^{2}+x^{2})|\partial_{x}^{k}\widehat{\psi}|^{2}\Big{)}\Big{]}\,d\upsigma.$

###### Proof.

When $\varrho>R_{\infty}$, we simply have that

$\mathfrak{R}_{s,\lambda}\widehat{\psi}=2s\partial_{x}\widehat{\psi},$

so the estimate is immediate. When $\varrho\leq R_{\infty}$, we observe that
the top-order derivative terms in $x\mathcal{B}_{s}^{\infty}$ and
$\mathcal{K}_{s}^{+}$ have additional powers of $x$ compared to the top-order
derivative terms in the remaining terms of $\mathfrak{L}_{s,\lambda,\kappa}$.
∎

######  Proposition 12.6.

Let $0<\delta\leq 1$ and assume the following relations:

$\displaystyle 0<\eta<$ $\displaystyle\>\frac{1}{4},$ $\displaystyle\kappa\ll$
$\displaystyle\>\eta R_{\infty}^{-1},$ $\displaystyle M^{-1}R_{\infty}\gg$
$\displaystyle\>1,$ $\displaystyle\lambda\gg$
$\displaystyle\>1+|s|^{2}\eta^{-2}R_{\infty}^{2}+p.$

Then, there exists a numerical constant $C>0$ such that:

$\sum_{\ell\in\mathbb{N}_{0}}\int_{\eta
R_{\infty}^{-1}}^{3R_{\infty}^{-1}}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}\Bigg{(}|\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell})|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}_{\ell}|^{2}\\\ +2x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\left[|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}\widehat{\psi}_{\ell}|^{2}+\lambda\pi_{\leq
l}|\partial_{x}\widehat{\psi}_{\ell}|^{2}+\chi_{\varrho\geq
R_{\infty}}|\sqrt{S_{\lambda}}\pi_{\leq
l}\partial_{x}\widehat{\psi}_{\ell}|^{2}\right]\Bigg{)}\,d\upsigma dx\\\
\leq\boxed{C\int_{\frac{1}{4}R_{\infty}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-2+\delta}\left|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l})\widehat{\psi}\right|\cdot\left|\partial_{r}\widehat{\psi}\right|\,d\upsigma
dr}\\\
+\boxed{C(\mathfrak{l}+1)^{-p}\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}x^{2-\delta}(x+\kappa)\mathfrak{l}(\mathfrak{l}+1)|\widehat{\psi}_{\ell}|\cdot|\partial_{x}\widehat{\psi}_{\ell}|\,d\upsigma\Big{|}_{x=\eta
R_{\infty}^{-1}}}\\\ +C\sum_{\ell\in\mathbb{N}_{0}}\int_{\eta
R_{\infty}^{-1}}^{4R_{\infty}^{-1}}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx.$ (12.8)

###### Proof.

We split:

$\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi})+(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}=\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}-\mathfrak{R}_{s,\lambda}\widehat{\psi}$

Then we consider the $\ell$-th spherical harmonics $\widehat{\psi}_{\ell}$,
take the square norm of both sides, multiply both sides with the weight factor
$x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}$, with
$q\in\mathbb{R}$ and $\chi_{\varrho\geq\frac{1}{3}R_{\infty}}$ a smooth cut-
off function which equals 1 when $\varrho\geq\frac{1}{3}R_{\infty}$ and
vanishes for $\varrho\leq\frac{1}{4}R_{\infty}$, and we integrate over
$\mathbb{S}^{2}$ to obtain:

$\overbrace{\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\left(|\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell})|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}_{\ell}|^{2}\right)\,d\upsigma}^{=:J_{0}}\\\
+\int_{\mathbb{S}^{2}}2x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}\left((\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\overline{\widehat{\psi}}_{\ell}\cdot\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell})\right)\,d\upsigma\\\
=\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}-(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}|^{2}\,d\upsigma.$

For the sake of convenience, we will restrict to $\ell\leq l$. The $\ell>l$
case proceeds entirely analogously, but without the $\lambda$ and
$S_{\lambda}$ terms.

We integrate by parts on $\mathbb{S}^{2}$ to obtain:

$\int_{\mathbb{S}^{2}}2x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}\left((\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\overline{\widehat{\psi}}_{\ell}\cdot\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell}\right)\,d\upsigma\\\
=\overbrace{2\int_{\mathbb{S}^{2}}\partial_{x}\left[x^{q}(\mathfrak{l}+1)^{-w_{p}}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}\left((\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\overline{\widehat{\psi}}_{\ell}\cdot
x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell}\right)\right]\,d\upsigma}^{=:J_{1}}\\\
+\overbrace{2\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\left[|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}\widehat{\psi}_{\ell}|^{2}+\lambda|\partial_{x}\widehat{\psi}_{\ell}|^{2}+\chi_{\varrho\geq
R_{\infty}}|\sqrt{S_{\lambda}}\partial_{x}\widehat{\psi}_{\ell}|^{2}\right]\,d\upsigma}^{=:J_{2}}\\\
\overbrace{-2\int_{\mathbb{S}^{2}}\partial_{x}(x^{q}(\mathfrak{l}+1)^{-w_{p}})x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}\left((\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\overline{\widehat{\psi}}_{\ell}\cdot\partial_{x}\widehat{\psi}_{\ell}\right)\,d\upsigma}^{=:J_{3}}\\\
+\overbrace{2\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}(\chi_{\varrho\geq
R_{\infty}})\chi_{\varrho\geq\frac{1}{3}R_{\infty}}\operatorname{\textnormal{Re}}(S_{\lambda}\overline{\widehat{\psi}}_{\ell}\cdot\partial_{x}\widehat{\psi}_{\ell})\,d\upsigma}^{=:J_{4}}\\\
-\overbrace{2\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}(\chi_{\varrho\geq\frac{1}{3}R_{\infty}})\operatorname{\textnormal{Re}}\left((\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\overline{\widehat{\psi}}_{\ell}\cdot\partial_{x}\widehat{\psi}_{\ell}\right)\,d\upsigma}^{=:J_{5}}.$

Note first that $J_{2}$ is non-negative definite. We will show that we can
absorb $J_{3}$ and $J_{4}$ into $J_{0}$ and $J_{2}$ by taking $\lambda$
suitably large.

We apply Young’s inequality together with (12.7) to obtain for $x\in[\eta
R_{\infty}^{-1},4R^{-1}_{\infty}]$:

$|J_{3}|\leq\\\
\int_{\mathbb{S}^{2}}\left(|q|x^{-1}+\left|\frac{dw_{p}}{dx}\right|(\mathfrak{l}+1)^{-1}\right)x^{q+1}(x+\kappa\chi_{\varrho\geq
R_{\infty}})(\mathfrak{l}+1)^{-w_{p}}(\ell(\ell+1)+\lambda+2l\ell\chi_{\varrho\geq
R_{\infty}})|\widehat{\psi}_{\ell}\|\partial_{x}\widehat{\psi}_{\ell}|\,d\upsigma\\\
\leq\left(\frac{|q|}{\mathfrak{l}x}+CpR_{\infty}(\mathfrak{l}+1)^{-1}\mathfrak{l}^{-1}\right)\mathfrak{l}\int_{\mathbb{S}^{2}}x^{q+1}(x+\kappa\chi_{\varrho\geq
R_{\infty}})(\mathfrak{l}+1)^{-w_{p}}(\ell(\ell+1)+\lambda+2l\ell\chi_{\varrho\geq
R_{\infty}})|\widehat{\psi}_{\ell}\|\partial_{x}\widehat{\psi}_{\ell}|\,d\upsigma\\\
\leq\frac{1}{4}\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\widehat{\psi}_{\ell}|^{2}\,d\upsigma\\\
+C\left(\frac{|q|}{\mathfrak{l}x}+CpR_{\infty}(\mathfrak{l}+1)^{-1}\mathfrak{l}^{-1}\right)^{2}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\mathfrak{l}^{2}|\partial_{x}\widehat{\psi}_{\ell}|^{2}\,d\upsigma.$

The first term on the very right-hand side above can be absorbed into $J_{0}$.
The second term on the right-hand side can be absorbed into $J_{2}$ by taking
$\kappa\ll\eta R_{\infty}^{-1}$ and $\lambda\gg q^{2}+p$ to obtain:

$C\left(\frac{|q|}{\mathfrak{l}x}+CpR_{\infty}\mathfrak{l}^{-2}\right)^{2}x(x+\kappa\chi_{\varrho\geq
R_{\infty}})=C\left(|q|\mathfrak{l}^{-1}+CpxR_{\infty}\mathfrak{l}^{-2}\right)^{2}(1+\kappa
x^{-1}\chi_{\varrho\geq R_{\infty}})\leq\frac{1}{4}.$

and noting that

$\mathfrak{l}(\mathfrak{l}+1)=\ell(\ell+1)+l(l+1)+2l\ell\leq
2(\ell(\ell+1)+2l(l+1)).$

We conclude that $|J_{3}|\leq\frac{1}{4}J_{0}+\frac{1}{4}J_{2}$.

To estimate $|J_{4}|$, we first note that there exists a constant $C>0$ such
that $\left|\frac{d\chi_{\varrho\geq R_{\infty}}}{d\varrho}\right|\leq
CR_{\infty}^{-1}$, so that

$\left|\frac{d\chi_{\varrho\geq R_{\infty}}}{dx}\right|\leq CR_{\infty}.$

By repeating the estimates for $|J_{3}|$ and taking $\lambda\gg
M^{-1}R_{\infty}$, we conclude that
$|J_{4}|\leq\frac{1}{4}J_{0}+\frac{1}{4}J_{2}$. We conclude that

$\sum_{j=0}^{4}J_{j}\geq\frac{1}{2}(J_{0}+J_{2}).$

We can estimate $J_{5}$ by observing that
$\partial_{x}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}$ is supported in
$\frac{1}{4}R_{\infty}\leq\varrho\leq\frac{1}{3}R_{\infty}$, where
$\varrho=r$, $w(x)=0$ and there exists a numerical constant $C>0$ such that
$|\partial_{x}\chi_{\varrho\geq\frac{1}{3}R_{\infty}}|\leq CR_{\infty}$. We
obtain:

$|J_{5}|\leq\int_{\mathbb{S}^{2}}r^{-q+1}|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda)\widehat{\psi}_{\ell}|\cdot|\partial_{r}\widehat{\psi}_{\ell}|\,d\upsigma$

and $\operatorname{\textnormal{supp}}J_{5}\subset\\{\frac{1}{4}R_{\infty}\leq
r\leq\frac{1}{3}R_{\infty}\\}$.

We apply Young’s inequality to estimate:

$\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}-(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}|^{2}\,d\upsigma\\\
\leq
2\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma+2\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}|^{2}\,d\upsigma.$

Note that $(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}$ does not need to
agree with $(\mathfrak{R}_{s,\lambda}(\widehat{\psi}_{\ell}))$. However, since
$w\equiv 0$ in $\\{\varrho\leq 2R_{\infty}\\}$ and
$\mathfrak{R}_{s,\lambda}\widehat{\psi}=2s\partial_{x}\widehat{\psi}$ in
$r\geq R_{\infty}$, we can conclude that:

$\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}|^{2}\,d\upsigma=\begin{cases}\int_{\mathbb{S}^{2}}|\mathfrak{R}_{s,\lambda}\widehat{\psi}|^{2}\,d\upsigma\quad&\textnormal{if}\quad
x\geq\frac{1}{2R_{\infty}},\\\
\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}(\mathfrak{l}+1)^{-w_{p}}|\mathfrak{R}_{s,\lambda}(\widehat{\psi}_{\ell})|^{2}\,d\upsigma\quad&\textnormal{if}\quad
x<\frac{1}{2R_{\infty}}.\end{cases}$

By applying Lemma 12.5, we can take $MR_{\infty}^{-1}\ll 1$ and
$\lambda\gg|s|^{2}\eta^{-2}R_{\infty}^{2}$ to obtain:

$2\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{R}_{s,\lambda}\widehat{\psi})_{\ell}|^{2}\,d\upsigma\leq\frac{1}{4}\sum_{\ell\in\mathbb{N}_{0}}(J_{0}+J_{2}).$

We now integrate over $x$ to obtain:

$\frac{1}{4}\sum_{\ell\in\mathbb{N}_{0}}\int_{x(\eta^{-1})}^{x(\frac{1}{4}R_{\infty})}x^{q}(\mathfrak{l}+1)^{-w_{p}}\Bigg{(}|\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}_{\ell})|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda-\chi_{\varrho\geq
R_{\infty}}S_{\lambda})\widehat{\psi}_{\ell}|^{2}\\\
+2x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\left[|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}\widehat{\psi}_{\ell}|^{2}+\lambda|\partial_{x}\widehat{\psi}_{\ell}|^{2}+\chi_{\varrho\geq
R_{\infty}}|\sqrt{S_{\lambda}}\partial_{x}\widehat{\psi}_{\ell}|^{2}\right]\Bigg{)}\,d\upsigma
dx\\\ \leq
C\int_{\frac{1}{4}R_{\infty}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-q-1}\left|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda)\widehat{\psi}\right|\cdot\left|\partial_{r}\widehat{\psi}\right|\,d\upsigma
dr\\\
+2(\mathfrak{l}+1)^{-p}\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}x^{q+1}(x+\kappa)\mathfrak{l}(\mathfrak{l}+1)|\widehat{\psi}_{\ell}|\cdot|\partial_{x}\widehat{\psi}_{\ell}|\,d\upsigma\Big{|}_{x=\eta
R_{\infty}^{-1}}\\\
+2\sum_{\ell\in\mathbb{N}_{0}}\int_{x(\eta^{-1})}^{x(\frac{1}{4}R_{\infty})}x^{q}(\mathfrak{l}+1)^{-w_{p}}|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx,$

and we conclude that (12.8) holds after taking $q=1-\delta$. ∎

We will also consider higher-order analogues of (12.8). For this it is useful
to define:

$\widehat{\psi}^{(n)}:=\partial_{x}^{n}\widehat{\psi}.$

Then it can be easily verified that:

$\partial_{x}^{n}(\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi})+(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi})=\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}^{(n)}\\\
+(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq l}-\chi_{\varrho\geq
R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}^{(n)}+2(k+1)\left(x+\frac{1}{2}\kappa\chi_{\varrho\geq
R_{\infty}}\right)\partial_{x}\widehat{\psi}^{(n)}\\\
+\kappa\sum_{k=0}^{n+1}{n+1\choose k}\partial_{x}^{n+1-k}(x\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}^{(k)}-\sum_{k=0}^{n}{n\choose
k}\partial_{x}^{n-k}(\chi_{\varrho\geq
R_{\infty}})S_{\lambda}\widehat{\psi}^{(k)}.$

Define

$\mathfrak{R}^{(n)}_{s,\lambda}:=\partial_{x}^{n}\mathfrak{R}_{s,\lambda}-2(k+1)\left(x+\frac{1}{2}\kappa\chi_{\varrho\geq
R_{\infty}}\right)\partial_{x}\widehat{\psi}^{(n)}-\kappa\sum_{k=0}^{n+1}{n+1\choose
k}\partial_{x}^{n+1-k}(x\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}^{(k)}\\\ +\sum_{k=0}^{n}{n\choose
k}\partial_{x}^{n-k}(\chi_{\varrho\geq R_{\infty}})\pi_{\leq
l}S_{\lambda}\widehat{\psi}^{(k)}.$

Then

$\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})=\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}\widehat{\psi}^{(n)})+(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})\widehat{\psi}^{(n)}+\mathfrak{R}^{(n)}_{s,\lambda}.$

We obtain the following analogue of Proposition 12.8, including additional
commutation with vector fields $(x^{2}\partial_{x})^{k}$.

######  Corollary 12.7.

Let $N\in\mathbb{N}_{0}$, $0<\delta<1$ and $p\geq 0$ and assume the following
smallness relations:

$\displaystyle 0<\eta<$ $\displaystyle\>\frac{1}{4},$ $\displaystyle\kappa\ll$
$\displaystyle\>\eta R_{\infty}^{-1},$ $\displaystyle M^{-1}R_{\infty}\gg$
$\displaystyle\>1,$ $\displaystyle\lambda-N(N+1)\gg$
$\displaystyle\>1+|s|^{2}\eta^{-2}R_{\infty}^{2}+p.$

Then, there exists a numerical constant $C=C(N)>0$ such that:

$\sum_{k=0}^{N}\sum_{\ell\in\mathbb{N}_{0}}\int_{x(\eta^{-1})}^{3R_{\infty}^{-1}}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}\Bigg{(}|\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell})|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}\\\
+2x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\left[|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}+\lambda\pi_{\leq
l}|\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}+\chi_{\varrho\geq
R_{\infty}}|\sqrt{S_{\lambda}\pi_{\leq
l}}\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}\right]\Bigg{)}\,d\upsigma
dx\\\
\leq\sum_{k=0}^{N}\boxed{C\int_{\frac{1}{4}R_{\infty}}^{\frac{1}{3}R_{\infty}}\int_{\mathbb{S}^{2}}r^{-2+\delta}\left|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l})\partial_{r}^{k}\widehat{\psi}\right|\cdot\left|\partial_{r}^{k+1}\widehat{\psi}\right|\,d\upsigma
dr}\\\
+\boxed{C(\mathfrak{l}+1)^{-p}\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}x^{2+4k-\delta}(x+\kappa)\mathfrak{l}(\mathfrak{l}+1)|\widehat{\psi}_{\ell}^{(k)}|\cdot|\partial_{x}\widehat{\psi}_{\ell}^{(k)}|\,d\upsigma\Big{|}_{x=\eta
R_{\infty}^{-1}}}\\\
+C\sum_{\ell\in\mathbb{N}_{0}}\int_{x(\eta^{-1})}^{4R_{\infty}^{-1}}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}|(x^{2}\partial_{x})^{k}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx.$ (12.9)

###### Proof.

We repeat the proof of Proposition 12.6 with $\widehat{\psi}^{(k)}$ replacing
$\widehat{\psi}$ and with $q=1-\delta+4k$, sum over $k$, and use that all the
terms coming from $\mathfrak{R}^{(n)}_{s,\lambda}$ can be absorbed in the
terms $J_{0}$ and $J_{2}$. ∎

###  12.3. Combining the estimates in (A) and (B)

In this section we combine the estimates from Corollary 12.3 and Corollary
12.7 by taking $\lambda$ suitably large.

######  Corollary 12.8.

Let $N_{+}\in\mathbb{N}_{0}$, $0<\delta<1$, $p\geq 0$, and
${\operatorname{\textnormal{Re}}s}>-\kappa_{+}\left(\frac{1}{2}+N_{+}\right)$,
and assume the following smallness relations:

$\displaystyle 0<\eta<$ $\displaystyle\>\frac{1}{4},$ $\displaystyle\kappa\ll$
$\displaystyle\>\eta R_{\infty}^{-1},$ $\displaystyle M^{-1}R_{\infty}\gg$
$\displaystyle\>1,$ $\displaystyle\lambda-N_{+}(N_{+}+1)\gg$
$\displaystyle\>p+N_{+}|s|^{2}(1+2N_{+}+2\kappa_{+}^{-1}{\min\\{\operatorname{\textnormal{Re}}s,0\\}})^{-1}+(1+2N_{+}+2\kappa_{+}^{-1}{\min\\{\operatorname{\textnormal{Re}}s,0\\}})^{-2}$
$\displaystyle+|s|^{2}\eta^{-2}R_{\infty}^{2}+(M^{-1}R_{\infty})^{2(1+N_{+})}.$

Then, there exists a constant $C=C(N_{+})>0$ such that:

$\sum_{k=0}^{N_{+}}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}|\partial_{r}^{1+k}\not{\nabla}_{\mathbb{S}^{2}}\widehat{\psi}|^{2}+\lambda|\partial_{r}^{1+k}\widehat{\psi}|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l})\widehat{\psi}|^{2}+\chi_{r\geq\frac{1}{3}R_{\infty}}|\partial_{r}^{2+k}\psi|^{2}\,d\upsigma
dr\\\ +\sum_{k=0}^{N_{+}}\sum_{\ell\in\mathbb{N}_{0}}\int_{\eta
R_{\infty}^{-1}}^{x(\frac{1}{2}R_{\infty})}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}\Bigg{(}|\partial_{x}(x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell})|^{2}+|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l}-\chi_{\varrho\geq R_{\infty}}S_{\lambda}\pi_{\leq
l})(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}\\\
+2x(x+\kappa\chi_{\varrho\geq
R_{\infty}})\left[|\not{\nabla}_{\mathbb{S}^{2}}\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}+\lambda\pi_{\leq
l}|\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}+\chi_{\varrho\geq
R_{\infty}}|\sqrt{S_{\lambda}\pi_{\leq
l}}\partial_{x}(x^{2}\partial_{x})^{k}\widehat{\psi}_{\ell}|^{2}\right]\Bigg{)}\,d\upsigma
dx\\\
\leq\boxed{C(\mathfrak{l}+1)^{-p}\sum_{k=0}^{N_{+}}\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}x^{2+4k-\delta}(x+\kappa)\mathfrak{l}(\mathfrak{l}+1)|\widehat{\psi}_{\ell}^{(k)}|\cdot|\partial_{x}\widehat{\psi}_{\ell}^{(k)}|\,d\upsigma\Big{|}_{x=\eta
R_{\infty}^{-1}}}\\\
+C_{s}\sum_{k=0}^{N_{+}}\int_{r_{+}}^{\frac{1}{2}R_{\infty}}|\partial_{r}^{k}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}+\chi_{r\leq\frac{1}{3}R_{\infty}}(|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}+\lambda\pi_{\leq
l}|\partial_{r}^{k}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2})\,d\upsigma
dr\\\ +C_{s}\sum_{k=0}^{N_{+}}\sum_{\ell\in\mathbb{N}_{0}}\int_{\eta
R_{\infty}^{-1}}^{x(\frac{1}{2}R_{\infty})}x^{1-\delta}(\mathfrak{l}+1)^{-w_{p}}|(x^{2}\partial_{x})^{k}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx.$ (12.10)

Furthermore, for suitably small $\frac{|a|}{M}$ or for
$\widehat{\phi}=\sum_{m=-m_{0}}^{m_{0}}\widehat{\phi}_{m}$, with $m_{0}>0$, we
can omit the
$|\not{\nabla}_{\mathbb{S}^{2}}\partial_{r}^{k}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}$
term on the right-hand side.

###### Proof.

We apply Corollary 12.3 with arbitrary $N_{+}\in\mathbb{N}_{0}$ and
$m\in\\{0,1\\}$ and we also apply Corollary 12.7 with the same value of
$N_{+}$. We estimate the first term on the right-hand side of (12.9) as
follows: let $\epsilon>0$ be arbitrarily small, then we apply Young’s
inequality and use that $\widehat{\psi}=r\phi$ in the region under
consideration to obtain:

$C\sum_{k=0}^{N_{+}}\int_{\mathbb{S}^{2}}r^{-2+\delta}\left|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l})\partial_{r}^{k}\widehat{\psi}\right|\cdot\left|\partial_{r}^{k+1}\widehat{\psi}\right|\,d\upsigma\leq\sum_{k=0}^{N_{+}}\int_{\mathbb{S}^{2}}\epsilon
r^{-1+\delta}\left|(\not{\Delta}_{\mathbb{S}^{2}}-\lambda\pi_{\leq
l})\partial_{r}^{k}\widehat{\phi}\right|^{2}+\frac{C^{2}}{4\epsilon\lambda}\lambda
r^{1+\delta}\left|\partial_{r}^{k+1}\widehat{\phi}\right|^{2}\,d\upsigma.$

We take $\lambda$ suitably large so that
$\frac{C^{2}}{4\epsilon\lambda}<\epsilon$ and
$\epsilon\ll(1+2N_{+}+2\kappa_{+}^{-1}{\operatorname{\textnormal{Re}}s})$.
Then we can absorb the right-hand side above into the left-hand side of (12.4)
with $m=0$, and $m=1$, after multiplying the $m=0$ estimates by a factor
$\lambda$.

By taking $\epsilon$ suitably small compared to the constant
$CR_{\infty}^{2N_{+}}$ appearing on the right-hand side of (12.4), we can also
absorb the first integral on the right-hand side of (12.4) into the left-hand
side of (12.9). ∎

######  Remark 12.1.

Note that the estimate (12.10) still has $\widehat{\psi}$ and its derivatives
appearing in the boxed boundary term at $x=\eta R_{\infty}^{-1}$. In the next
section we will show that we can absorb this term into the left-hand side of
appropriate Gevrey-type estimates.

##  13\. Gevrey estimates

In this section, we consider estimates in the region $x\leq x_{0}$, where we
define:

$x_{0}:=\frac{1}{2R_{\infty}}.$

By (7.11), we can express in the region $x\leq x_{0}$:

$(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}=\partial_{x}(x(\kappa+x)\partial_{x}\widehat{\psi}_{\ell})+2s\partial_{x}\widehat{\psi}_{\ell}-\mathfrak{l}(\mathfrak{l}+1)\widehat{\psi}_{\ell}.$

Hence, all the estimates in this section are estimates for the rescaled
Laplace-transformed wave operator in Minkowski, restricted to fixed spherical
harmonic modes with angular frequencies $\geq\mathfrak{l}$. We will moreover
require these estimates to be uniform in $\mathfrak{l}$.

Denote $\widehat{\psi}^{(n)}=\partial_{x}^{n}\widehat{\psi}$. Then it can be
easily verified that:

$(\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}=x(\kappa+x)\partial_{x}\widehat{\psi}^{(n+1)}_{\ell}+2(n+1)\left(x+\frac{\kappa}{2}\right)\partial_{x}\widehat{\psi}^{(n)}+2s\partial_{x}\widehat{\psi}^{(n)}_{\ell}+(n(n+1)-\mathfrak{l}(\mathfrak{l}+1))\widehat{\psi}_{\ell}^{(n)}.$
(13.1)

We will estimate higher-order energy norms of $\widehat{\psi}$ in terms of
similar energy norms of $\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}$ in two
steps.

  * Step 1:

We consider
$\widehat{\psi}^{(n)}_{\ell}=\partial_{x}^{n}\widehat{\psi}_{\ell}$ and derive
$\dot{H}^{1}$ estimates for $\widehat{\psi}^{(n)}_{\ell}$ with
$n\geq\mathfrak{l}$. We then sum the estimates for different values of $n$ to
obtain an $L^{2}$-based Gevrey-type estimate. _Here it is important that we
will not see any terms $\widehat{\psi}^{(n)}_{\ell}$ with $n<\mathfrak{l}$._

  * Step 2:

We estimate weighted $\dot{H}^{1}$ norms of $\widehat{\psi}^{(n)}_{\ell}$ with
$n<\mathfrak{l}$ in terms of a weighted $\dot{H}^{1}$ norm of
$\widehat{\psi}^{(n)}_{\ell}$. To be able to sum these lower-order estimates,
we apply elliptic-type estimates in the region $x\leq x_{0}$.

###  13.1. Step 1: higher-order derivatives

We will first derive weighted $L^{2}$ estimates for $\widehat{\psi}^{(n)}$
with $n\geq\mathfrak{l}$. We obtain these estimates by simply rearranging the
terms in (13.1) to obtain:

$x(\kappa+x)\partial_{x}\widehat{\psi}^{(n+1)}_{\ell}+2(n+1)\left(x+\frac{\kappa}{2}\right)\partial_{x}\widehat{\psi}_{\ell}^{(n)}+2s\partial_{x}\widehat{\psi}^{(n)}_{\ell}=(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\widehat{\psi}_{\ell}^{(n)}+(\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}.$
(13.2)

and by taking the square norm on both sides, treating the right-hand side as
an inhomogeneous term. By considering appropriate weighted sums in $n$ of
these $L^{2}$ estimates, we will then be able to absorb the terms arising from
$(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\widehat{\psi}_{\ell}^{(n)}$ on the
right-hand side of (13.2).

######  Proposition 13.1.

Let $\kappa\geq 0$, $M=1$, $\ell,l\in\mathbb{N}_{0}$,
$N_{\infty}\in\mathbb{N}_{0}$, with $N_{\infty}>\mathfrak{l}$, and let
$\alpha,\beta,\nu,\tilde{\sigma},\epsilon>0$, with $\mu\in(0,1)$. Then there
exists a constant $C>0$ such that:

$\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}(1-\mu\nu^{-1})x^{2}(x+\kappa)^{2}|\hat{\psi}_{\ell}^{(n+2)}|^{2}\,dx\\\
+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\left[4-\beta-(1-\mu)(\alpha+\tilde{\sigma}^{-2}\alpha^{-1})-6(n+1)^{-1}\right]\\\
\times(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}_{\ell}^{(n+1)}|^{2}\,dx\\\
+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\left(1-\mu\nu-4\beta^{-1}\left(\frac{\max\\{-\operatorname{\textnormal{Re}}s,0\\}}{|s|}\right)^{2}-(1+\epsilon)\tilde{\sigma}^{2}\left(1+2(n-1)^{-1}\right)^{2}\right)\\\
\times|2s|^{2}|\hat{\psi}_{\ell}^{(n+1)}|^{2}\,dx\\\
\boxed{-\frac{|2s|^{2(N_{\infty}+1)}\tilde{\sigma}^{2(N_{\infty}+1)}}{(N_{\infty}+1)!^{2}(N_{\infty}+2)!^{2}}\int_{0}^{x_{0}}(1-\mu)\alpha^{-1}\tilde{\sigma}^{-2}(N_{\infty}+2)^{2}\left(x+\frac{1}{x^{-1}+2\kappa^{-1}}\right)^{2}|\hat{\psi}_{\ell}^{(N_{\infty}+2)}|^{2}\,dx}\\\
+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}x_{0}^{3}(n+1)|\widehat{\psi}^{(n+1)}|^{2}\big{|}_{x=x_{0}}\\\
\leq
C(1+\epsilon^{-1})\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,dx.$
(13.3)

###### Proof.

We will omit the subscript $\ell$ in $\widehat{\psi}_{\ell}$ for the sake of
convenience. Rearranging (13.1) and taking the square norm on both sides, we
obtain:

$x^{2}(x+\kappa)^{2}|\hat{\psi}^{(n+2)}|^{2}+4(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}+4|s|^{2}|\hat{\psi}^{(n+1)}|^{2}\\\
+T_{1}+T_{2}+T_{3}=\left|[\mathfrak{l}(\mathfrak{l}+1)-n(n+1)]\hat{\psi}^{(n)}+\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}\right|^{2},$

with

$\displaystyle T_{1}:=$
$\displaystyle\>2(n+1)x\left(x+\frac{1}{2}\kappa\right)(x+\kappa)\partial_{x}(|\hat{\psi}^{(n+1)}|^{2}),$
$\displaystyle T_{2}:=$
$\displaystyle\>4\operatorname{\textnormal{Re}}(2s)(n+1)\left(x+\frac{1}{2}\kappa\right)|\hat{\psi}^{(n+1)}|^{2},$
$\displaystyle T_{3}:=$
$\displaystyle\>2x(x+\kappa)\operatorname{\textnormal{Re}}\left(\hat{\psi}^{(n+2)}2\overline{s}\overline{\hat{\psi}^{(n+1)}}\right).$

We integrate $T_{1}$ by parts to obtain:

$\int_{0}^{x_{0}}T_{1}\,dx\geq
2(n+1)x_{0}\left(x_{0}+\frac{1}{2}\kappa\right)(x_{0}+\kappa)|\hat{\psi}^{(n+1)}|^{2}|_{x=x_{0}}\\\
-\int_{0}^{x_{0}}(n+1)(6x^{2}+6\kappa
x+\kappa^{2})|\hat{\psi}^{(n+1)}|^{2}\,dx\\\ \geq
2(n+1)x_{0}^{3}|\hat{\psi}^{(n+1)}|^{2}|_{x=x_{0}}-\int_{0}^{x_{0}}6(n+1)\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}\,dx.$

If ${\operatorname{\textnormal{Re}}s}\geq 0$, then $T_{2}\geq 0$. If
${\operatorname{\textnormal{Re}}s}<0$, then we introduce a constant $\beta>0$
and apply Young’s inequality to split:

$|T_{2}|\leq\beta(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}+4\beta^{-1}\left(\frac{|\operatorname{\textnormal{Re}}s|}{|s|}\right)^{2}|2s|^{2}|\hat{\psi}^{(n+1)}|^{2}.$

We introduce additionally $0\leq\mu\leq 1$ and split

$\begin{split}|T_{3}|\leq
2x(x+\kappa)|2s||\hat{\psi}^{(n+1)}\|\hat{\psi}^{(n+2)}|=&\>\mu\cdot
2x(x+\kappa)|2s||\hat{\psi}^{(n+1)}\|\hat{\psi}^{(n+2)}|\\\ &+(1-\mu)\cdot
2x(x+\kappa)|2s||\hat{\psi}^{(n+1)}\|\hat{\psi}^{(n+2)}|.\end{split}$

We further estimate via Young’s inequality with an additional weight $\nu>0$:

$\mu\cdot
2x(x+\kappa)|2s||\hat{\psi}^{(n+1)}\|\hat{\psi}^{(n+2)}|\leq\mu\nu|2s|^{2}|\hat{\psi}^{(n+1)}|^{2}+\mu\nu^{-1}x^{2}(x+\kappa)^{2}|\hat{\psi}^{(n+2)}|^{2},$

and we estimate the remaining term with a slightly different Young’s
inequality, involving the weight $\alpha$ and an additional factor
$\frac{|2s|^{2}\tilde{\sigma}^{2}}{(n+1)^{2}(n+2)^{2}}$ that appears in the
summation below:

$\begin{split}(1-\mu)\cdot&2x(x+\kappa)|2s\|\hat{\psi}^{(n+1)}\|\hat{\psi}^{(n+2)}|\leq\alpha(1-\mu)(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}\\\
&+(1-\mu)\alpha^{-1}\tilde{\sigma}^{-2}\cdot\frac{|2s|^{2}\tilde{\sigma}^{2}}{(n+1)^{2}(n+2)^{2}}(n+1+1)^{2}\frac{x^{2}(x+\kappa)^{2}}{\left(x+\frac{1}{2}\kappa\right)^{2}}|\hat{\psi}^{(n+1+1)}|^{2}.\end{split}$

We moreover have that
$\frac{x^{2}(x+\kappa)^{2}}{\left(x+\frac{1}{2}\kappa\right)^{2}}=(x+\frac{1}{x^{-1}+2\kappa^{-1}})^{2}\leq(x+\frac{1}{2}\kappa)^{2}$
Finally, we estimate

$|[\mathfrak{l}(\mathfrak{l}+1)-n(n+1)]\hat{\psi}^{(n)}+\partial_{x}^{n}((\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell})|^{2}\leq(1+\epsilon)(n(n+1)-\mathfrak{l}(\mathfrak{l}+1))^{2}|\hat{\psi}^{(n+1-1)}|^{2}\\\
+(1+\epsilon^{-1})|(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}.$

Now we integrate the above estimates over the interval $[0,x_{0}]$ to obtain:

$\int_{0}^{x_{0}}(1-\mu\nu^{-1})x^{2}(x+\kappa)^{2}|\hat{\psi}^{(n+2)}|^{2}\,dx+2(n+1)x_{0}^{3}|\hat{\psi}^{(n+1)}|^{2}|_{x=x_{0}}\\\
+\int_{0}^{x_{0}}\left[4-\beta-\alpha(1-\mu)-6(n+1)^{-1}\right](n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}\,dx\\\
\boxed{-\int_{0}^{x_{0}}(1-\mu)\alpha^{-1}\tilde{\sigma}^{-2}\cdot\frac{|2s|^{2}\tilde{\sigma}^{2}}{(n+1)^{2}(n+2)^{2}}(n+1+1)^{2}\left(x+\frac{1}{x^{-1}+2\kappa^{-1}}\right)^{2}|\hat{\psi}^{(n+1+1)}|^{2}\,dx}\\\
+\int_{0}^{x_{0}}\left(1-\mu\nu-4\beta^{-1}\left(\frac{\max\\{-\operatorname{\textnormal{Re}}s,0\\}}{|s|}\right)^{2}\right)|2s|^{2}|\hat{\psi}^{(n+1)}|^{2}\,dx\\\
\boxed{-\int_{0}^{x_{0}}(1+\epsilon)|2s|^{2}\tilde{\sigma}^{2}\left(\frac{n(n+1)-\mathfrak{l}(\mathfrak{l}+1)}{n(n-1)}\right)^{2}\frac{n^{2}(n-1)^{2}}{|2s|^{2}\tilde{\sigma}^{2}}|\hat{\psi}^{(n+1-1)}|^{2}\,dx}\\\
\leq(1+\epsilon^{-1})\int_{0}^{x_{0}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,dx.$

We can sum the above estimate over $n$ with the following weights:

$\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{|2s|^{2n}\tilde{\sigma}^{2n}}{n!^{2}(n+1)!^{2}}\left[\cdot\right],$

and absorbing all but the top order term with a bad sign into the terms with
good signs, we obtain (13.3). ∎

In order for the unboxed terms on the left-hand side of (13.3) to be non-
negative definite, the coefficients $\tilde{\sigma},\alpha,\beta,\mu,\nu$ need
to satisfy the following compatibility conditions: when
${\operatorname{\textnormal{Re}}s}<0$

$\displaystyle 0<\mu<$ $\displaystyle\>\nu,$ (13.4) $\displaystyle
4-\beta-(1-\mu)(\alpha+\tilde{\sigma}^{-2}\alpha^{-1})>$ $\displaystyle\>0,$
(13.5) $\displaystyle
1-\mu\nu-4\beta^{-1}\left(\frac{|{\operatorname{\textnormal{Re}}s}|}{|s|}\right)^{2}-\tilde{\sigma}^{2}>$
$\displaystyle\>0.$ (13.6)

For ${\operatorname{\textnormal{Re}}s}\geq 0$, we can take $\mu=0$, $\beta=1$
and $\alpha=2$, so we instead need

$\displaystyle\frac{1}{4}<\tilde{\sigma}^{2}<1.$ (13.7)

Indeed, if (13.4)–(13.7) are satisfied, then we can take $\lambda$
sufficiently large and $\epsilon$ sufficiently small, to guarantee non-
negative definiteness of the unboxed terms on the left-hand side of (13.3).

######  Lemma 13.2.

There exist $\tilde{\sigma}$, $\alpha,\beta,\mu,\nu$ satisfying (13.4)–(13.7)
if and only if:

  1. _(i)_

$-{\operatorname{\textnormal{Re}}s}\leq\frac{1}{2}|s|$,

  2. _(ii)_

For fixed $2|s|\tilde{\sigma}=\sigma$, with $\sigma\in\mathbb{R}$, we have
$s\in\Omega_{\sigma}$, where

$\Omega_{\sigma}:=\left\\{x+iy\in\mathbb{C}\,|\,x\in\mathbb{R}_{-},\,y\in\mathbb{R},\,x^{2}+y^{2}<\sigma^{2},\,3y^{2}-5x^{2}>\sigma^{2}\right\\}\\\
\cup\left\\{x+iy\in\mathbb{C}\,|\,x\in[0,\infty),\,\frac{\sigma^{2}}{4}<x^{2}+y^{2}<\sigma^{2}\right\\}.$

  3. _(iii)_

For $s\in\\{-\frac{1}{2}|s|\leq\operatorname{\textnormal{Re}}s<0\\}$.

$\frac{1}{4}<\tilde{\sigma}^{2}<\frac{3}{4}-2\left(\frac{\operatorname{\textnormal{Re}}s}{|s|}\right)^{2}.$

Furthermore,

$\Omega:=\bigcup_{\sigma\in\mathbb{R}}\Omega_{\sigma}=\left\\{z\in\mathbb{C}\,,\,|\arg(z)|<\frac{2\pi}{3}\right\\}.$

###### Proof.

Appendix A. ∎

By combining Proposition 13.1 with Lemma 13.2, we immediately obtain:

######  Corollary 13.3.

Let $\kappa\geq 0$, $M=1$, $\ell,l\in\mathbb{N}_{0}$,
$N_{\infty}\in\mathbb{N}_{0}$, with $N_{\infty}>\mathfrak{l}$. Let
$\sigma\in\mathbb{R}$ and let $s\in\Omega_{\sigma}$. Then, there exists a
constant $C_{s,\sigma}=C(s,\sigma)>0$, such that:

$\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}x^{2}(x+\kappa)^{2}|\hat{\psi}_{\ell}^{(n+2)}|^{2}\,dx+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}(n+1)x_{0}^{3}|\widehat{\psi}^{(n+1)}|^{2}\big{|}_{x=x_{0}}\\\
+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\left[|2s|^{2}+(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}\right]|\hat{\psi}_{\ell}^{(n+1)}|^{2}\,dx\\\
\boxed{-C_{s,\sigma}\frac{|2s|^{2(N_{\infty}+1)}\tilde{\sigma}^{2(N_{\infty}+1)}}{(N_{\infty}+1)!^{2}(N_{\infty}+2)!^{2}}\int_{0}^{x_{0}}(N_{\infty}+2)^{2}\left(x+\frac{1}{x^{-1}+2\kappa^{-1}}\right)^{2}|\hat{\psi}_{\ell}^{(N_{\infty}+2)}|^{2}\,dx}\\\
\leq
C_{s,\sigma}\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,dx.$
(13.8)

We now restrict to the case $\kappa>0$ to absorb the boxed boundary term in
(13.9) and arrive at a closed, uniform estimate in $\kappa$:

######  Corollary 13.4.

Let $\kappa>0$, $M=1$, $\ell,l\in\mathbb{N}_{0}$,
$N_{\infty}\in\mathbb{N}_{0}$, with $N_{\infty}>\mathfrak{l}$. Let
$\sigma\in\mathbb{R}$ and let $s\in\Omega_{\sigma}$. Then there exists a
constant $C_{s,\sigma}=C(s,\sigma)>0$ and a $N_{\kappa}>\mathfrak{l}$, such
that for all $N_{\infty}\geq N_{\kappa}$:

$\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}x^{2}(x+\kappa)^{2}|\hat{\psi}_{\ell}^{(n+2)}|^{2}\,dx+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}(n+1)x_{0}^{3}|\widehat{\psi}^{(n+1)}|^{2}\big{|}_{x=x_{0}}\\\
+\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\left[|2s|^{2}+(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}\right]|\hat{\psi}_{\ell}^{(n+1)}|^{2}\,dx\\\
\leq
C_{s,\sigma}\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,dx.$
(13.9)

###### Proof.

Note that

$C_{s,\sigma}\frac{|2s|^{2}\tilde{\sigma}^{2}}{(N_{\infty}+1)^{2}(N_{\infty}+2)^{2}}(N_{\infty}+2)^{2}\left(x+\frac{1}{x^{-1}+2\kappa^{-1}}\right)^{2}<4\frac{C_{s,\sigma}|2s|^{2}\tilde{\sigma}^{2}\kappa^{-2}}{(N_{\infty}+1)^{2}}\kappa^{2}x^{2},$

so, for
$N_{\infty}+1>4\sqrt{C_{s,\sigma}}|2s\|\tilde{\sigma}|\kappa^{-1}=:N_{\kappa}+1$,
we can absorb the boxed term in (13.9) into the first term on the right-hand
side of (13.9). ∎

######  Remark 13.1.

Note that the estimate (13.9) is _closed_ , i.e. there are only terms
involving $(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}$ on the
right-hand side. This means that we can control weighted $L^{2}$-norms of
$\widehat{\psi}^{(n)}_{\ell}$ with $n\geq\mathfrak{l}+1$ in $x\leq x_{0}$
independently from $\widehat{\psi}^{(n)}_{\ell}$ with $n\leq\mathfrak{l}$.

###  13.2. Step 2: lower-order derivatives

In this section, we will establish estimates for appropriately weighted
$\dot{H}^{1}$ estimates in the region $\\{0<x\leq x_{0}\\}$ for
$\widehat{\psi}^{(n)}$ with $n\leq\mathfrak{l}$ in terms of a $\dot{H^{1}}$
norm for $\widehat{\psi}^{(\mathfrak{l}+1)}$ and lower-order boundary terms.

The estimates in this section rely on a different rearranging of (13.1),
compared to Step 1:

$x(\kappa+x)\partial_{x}\widehat{\psi}^{(n+1)}_{\ell}+2(n+1)\left(x+\frac{\kappa}{2}\right)\partial_{x}\widehat{\psi}_{\ell}^{(n)}-(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\widehat{\psi}_{\ell}^{(n)}=-2s\partial_{x}\widehat{\psi}^{(n)}_{\ell}+(\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}.$

The main estimates are obtained by considering the square norm of both sides
of the equation and summing over $n$ with a appropriately weighted sum.

######  Proposition 13.5.

Let $\kappa\leq x_{0}$ and $\gamma>1$. Then there exists a constant
$C=C(\gamma)>0$ such that

$\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}(x^{4}+\kappa^{2}x^{2})|\hat{\psi}^{(n+2)}|^{2}+(x^{2}(\mathfrak{l}+1)^{2}+\kappa^{2}n(n+1)+|2s|^{2})|\hat{\psi}^{(n+1)}|^{2}\\\
+(\mathfrak{l}-n)^{2}(\mathfrak{l}+n+1)^{2}|\hat{\psi}^{(n)}|^{2}\,d\upsigma
dx\\\ \leq
C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\hat{\psi}^{(\mathfrak{l}+1)}|^{2}\,dx+C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{\mathbb{S}^{2}}|\hat{\psi}^{(\mathfrak{l})}|^{2}|_{x=x_{0}}\,d\upsigma\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}(\mathfrak{l}-n)(\mathfrak{l}+n+1)(\mathfrak{l}+1)^{2}x_{0}^{-1}\int_{\mathbb{S}^{2}}|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}d\upsigma\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,d\upsigma
dx.$ (13.10)

Furthermore, there exists a constant $C=C(\gamma,|s|)>0$ such that

$(\mathfrak{l}+1)^{-(4N_{+}+1)}\sum_{n=0}^{N_{+}+1}\int_{\mathbb{S}^{2}}x|\hat{\psi}^{(n)}|^{2}|_{x=\eta
x_{0}}d\upsigma\leq
C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\hat{\psi}^{(\mathfrak{l}+1)}|^{2}\,dx\\\
+C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{\mathbb{S}^{2}}|\hat{\psi}^{(\mathfrak{l})}|^{2}|_{x=x_{0}}\,d\upsigma\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}(\mathfrak{l}-n)(\mathfrak{l}+n+1)(\mathfrak{l}+1)^{2}x_{0}^{-1}\int_{\mathbb{S}^{2}}|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}d\upsigma\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,d\upsigma
dx.$ (13.11)

###### Proof.

Consider (13.1), suppress the $\ell$ subscript and rearrange the terms to
obtain:

$|(\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})-2s\widehat{\psi}^{(n+1)}|^{2}=|x(\kappa+x)\partial_{x}^{2}\widehat{\psi}^{(n)}+2(n+1)\left(x+\frac{\kappa}{2}\right)\partial_{x}\widehat{\psi}^{(n)}-(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\widehat{\psi}^{(n)}|^{2}\\\
=x^{2}(x+\kappa)^{2}|\hat{\psi}^{(n+2)}|^{2}+4(n+1)^{2}\left(x+\frac{1}{2}\kappa\right)^{2}|\hat{\psi}^{(n+1)}|^{2}+(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))^{2}|\hat{\psi}^{(n)}|^{2}\\\
+T_{1}+T_{2}+T_{3},$ (13.12)

with

$\displaystyle T_{1}:=$
$\displaystyle\>2(n+1)x\left(x+\frac{1}{2}\kappa\right)(x+\kappa)\partial_{x}(|\hat{\psi}^{(n+1)}|^{2}),$
$\displaystyle T_{2}:=$
$\displaystyle\>-2(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))x(\kappa+x)\operatorname{\textnormal{Re}}(\partial_{x}^{2}\widehat{\psi}^{(n)}\cdot\overline{\widehat{\psi}^{(n)}}),$
$\displaystyle T_{3}:=$
$\displaystyle\>-2(n+1)(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\left(x+\frac{\kappa}{2}\right)\partial_{x}(|\widehat{\psi}^{(n)}|^{2}).$

We now integrate both sides of (13.12) over $[0,x_{0}]\times\mathbb{S}^{2}$,
but we suppress the integration over $\mathbb{S}^{2}$ in the notation.

We first estimate the integral of $T_{1}$ by integrating by parts:

$\int_{0}^{x_{0}}T_{1}\,dx=-\int_{0}^{x_{0}}(n+1)(6x^{2}+6\kappa
x+\kappa^{2})|\widehat{\psi}^{(n+1)}|^{2}\,dx\\\
+2(n+1)x_{0}\left(x_{0}+\frac{1}{2}\kappa\right)(x_{0}+\kappa)|\hat{\psi}^{(n+1)}|^{2}|_{x=x_{0}}\\\
\geq-\int_{0}^{x_{0}}6(n+1)(x(x+\kappa)+\frac{1}{6}\kappa^{2})|\hat{\psi}^{(n+1)}|^{2}\,dx+2(n+1)x_{0}\left(x_{0}+\frac{1}{2}\kappa\right)(x_{0}+\kappa)|\hat{\psi}^{(n+1)}|^{2}|_{x=x_{0}}$

To estimate $T_{2}$, we integrate by parts twice:

$\int_{0}^{x_{0}}T_{2}\,dx=\int_{0}^{x_{0}}2(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))x(\kappa+x)|\widehat{\psi}^{(n+1)}|^{2}\,dx-\int_{0}^{x_{0}}(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))(\kappa+2x)\partial_{x}(|\widehat{\psi}^{(n)}|^{2})\,dx\\\
-2(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))x_{0}(\kappa+x_{0})\operatorname{\textnormal{Re}}(\widehat{\psi}^{(n+1)}\cdot\overline{\widehat{\psi}^{(n)}})|_{x=x_{0}}\\\
\geq\int_{0}^{x_{0}}2(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))x(\kappa+x)|\widehat{\psi}^{(n+1)}|^{2}\,dx+2\int_{0}^{x_{0}}(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))|\widehat{\psi}^{(n)}|^{2}\,dx\\\
-2(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))x_{0}(\kappa+x_{0})\operatorname{\textnormal{Re}}(\widehat{\psi}^{(n+1)}\cdot\overline{\widehat{\psi}^{(n)}})|_{x=x_{0}}-(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))(\kappa+2x_{0})|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}$

We also integrate $T_{3}$ by parts to obtain:

$\int_{0}^{x_{0}}T_{3}\,dx\geq\int_{0}^{x_{0}}2(n+1)(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))|\widehat{\psi}^{(n)}|^{2}\,dx-2(n+1)(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))\left(x_{0}+\frac{\kappa}{2}\right)|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}.$

Finally, we apply Young’s inequality to estimate for $\epsilon>0$:

$|(\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})-2s\widehat{\psi}^{(n+1)}|^{2}\leq(1+\epsilon)|2s|^{2}|\widehat{\psi}^{(n+1)}|^{2}+(1+\epsilon^{-1})|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}.$

We combine the above estimates to obtain for $\gamma>1$:

$\int_{0}^{x_{0}}x^{2}(x+\kappa)^{2}|\hat{\psi}^{(n+2)}|^{2}\,dx\\\
+\int_{0}^{x_{0}}\left[(2(n+1)(n-1)+2\mathfrak{l}(\mathfrak{l}+1))x(\kappa+x)+(n+1)^{2}\kappa^{2}\right]|\hat{\psi}^{(n+1)}|^{2}\,dx\\\
+\int_{0}^{x_{0}}(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))(\mathfrak{l}(\mathfrak{l}+1)-n(n+1)+2(n+2))|\hat{\psi}^{(n)}|^{2}\,dx\\\
\boxed{-\int_{0}^{x_{0}}\frac{|2s|^{2}\gamma^{2}}{(\mathfrak{l}(\mathfrak{l}+1)-(n+1)(n+2))^{2}}\gamma^{-2}(1+\epsilon)(\mathfrak{l}(\mathfrak{l}+1)-(n+1)(n+2))^{2}|\hat{\psi}^{(n+1)}|^{2}\,dx}\\\
\leq
C(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))(\mathfrak{l}+1)^{2}x_{0}^{-1}|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}+(1+\epsilon^{-1})\int_{0}^{x_{0}}|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,dx.$

In order to absorb the boxed term above, we will sum over $n$ with the
following weights:

$\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}}{(\mathfrak{l}(\mathfrak{l}+1))^{2}(\mathfrak{l}(\mathfrak{l}+1)-1(1+1))^{2}\ldots(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))^{2}}.$

Using that
$\mathfrak{l}(\mathfrak{l}+1)-n(n+1)=(\mathfrak{l}-n)(\mathfrak{l}+n+1)$. We
can simplify for $n\leq\mathfrak{l}-1$:

$(\mathfrak{l}(\mathfrak{l}+1))^{2}(\mathfrak{l}(\mathfrak{l}+1)-1(1+1))^{2}\ldots(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))^{2}=\frac{(\mathfrak{l}+n+1)!^{2}}{(\mathfrak{l}-(n+1))!^{2}},$

so that the summation becomes

$\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}.$

We obtain:

$\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}x^{4}|\hat{\psi}^{(n+2)}|^{2}\,dx\\\
+\int_{0}^{x_{0}}\left[((n+1)^{2}+\mathfrak{l}(\mathfrak{l}+1))x^{2}+|2s|^{2}\right]|\hat{\psi}^{(n+1)}|^{2}\,dx\\\
+\int_{0}^{x_{0}}(1-(1+\epsilon)\gamma^{-2})(\mathfrak{l}(\mathfrak{l}+1)-n(n+1))^{2}|\hat{\psi}^{(n)}|^{2}\,dx\\\
\leq
C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}|\hat{\psi}^{(\mathfrak{l})}|^{2}\,dx\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}(\mathfrak{l}-n)(\mathfrak{l}+n+1)(\mathfrak{l}+1)^{2}x_{0}^{-1}|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}\\\
+C(1+\epsilon^{-1})\sum_{n=0}^{\mathfrak{l}-1}\int_{0}^{x_{0}}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,dx.$

By applying a Hardy inequality we can further estimate:

$\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}|\hat{\psi}^{(\mathfrak{l})}|^{2}\,dx\leq
4x_{0}^{2}\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}|\hat{\psi}^{(\mathfrak{l}+1)}|^{2}\,dx+2\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}x_{0}|\widehat{\psi}^{(\mathfrak{l})}|^{2}|_{x=x_{0}}.$

We conclude (13.10).

By integrating by parts $T_{3}$ in the region $[\eta
x_{0},x_{0}]\times\mathbb{S}^{2}$ in the argument above and using the integral
estimates above, we can moreover estimate boundary terms at $x=\eta x_{0}$. We
obtain in particular also:

$(\mathfrak{l}+1)^{-4(N_{+}+1)}\sum_{n=0}^{N_{+}+1}x|\hat{\psi}^{(n)}|^{2}|_{x=\eta
x_{0}}\leq
C\frac{|2s|^{2(\mathfrak{l}-1)}\gamma^{2(\mathfrak{l}-1)}}{(2\mathfrak{l}!)^{2}}\int_{0}^{x_{0}}|\hat{\psi}^{(\mathfrak{l})}|^{2}\,dx\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}(\mathfrak{l}-n)(\mathfrak{l}+n+1)(\mathfrak{l}+1)^{2}x_{0}^{-1}|\widehat{\psi}^{(n)}|^{2}|_{x=x_{0}}\\\
+C(1+\epsilon^{-1})\sum_{n=0}^{\mathfrak{l}-1}\int_{0}^{x_{0}}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}|\partial_{x}^{n}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}|^{2}\,dx.$

∎

We now estimate the boundary terms at $x=x_{0}$ that appear on the right-hand
sides of the estimates in Proposition 13.5.

######  Lemma 13.6.

Let $\kappa\leq x_{0}$ and $\mathfrak{l}^{2}\gg|s|^{2}x_{0}^{-2}$. Then there
exists constants $C>0$ suitably large and $d_{0}>0$ suitably small, with
$d_{0}$ independent of $|s|$, $\mathfrak{l}$ and $x_{0}$, such that for all
$n\in\mathbb{N}_{0}$:

$\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\hat{\psi}^{(n+2)}_{\ell}|^{2}\big{|}_{x=x_{0}}\leq
Cx_{0}^{-4}\left[(\mathfrak{l}+1)^{4}|\hat{\psi}_{\ell}|^{2}+(x_{0}^{2}+|2s|^{2})|\hat{\psi}^{(1)}_{\ell}|^{2}\right]\big{|}_{x=x_{0}}\\\
+Cx_{0}^{-4}\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\big{|}_{x=x_{0}}.$
(13.13)

Furthermore, for any $\gamma>0$, with $\mathfrak{l}\gg
d_{0}^{-1}x_{0}^{-1}\gamma|s|$, we can estimate

$\sum_{n=0}^{\mathfrak{l}-1}\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}|\hat{\psi}^{(n+2)}_{\ell}|^{2}\big{|}_{x=x_{0}}\leq
Cx_{0}^{-4}\left[|\hat{\psi}_{\ell}|^{2}+(x_{0}^{2}+|2s|^{2})(\mathfrak{l}+1)^{-4}|\hat{\psi}^{(1)}_{\ell}|^{2}\right]\big{|}_{x=x_{0}}\\\
+C(\mathfrak{l}+1)^{-4}x_{0}^{-4}\sum_{n=0}^{\mathfrak{l}-1}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\big{|}_{x=x_{0}}.$
(13.14)

###### Proof.

We will first establish the following estimate inductively: for
$n\in\mathbb{N}_{0}$

$\frac{(d_{0}^{\prime}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\hat{\psi}^{(n+2)}_{\ell}|^{2}\big{|}_{x=x_{0}}\leq
Cx_{0}^{-4}\left[(\mathfrak{l}+1)^{4}|\hat{\psi}_{\ell}|^{2}+(x_{0}^{2}+|2s|^{2})|\hat{\psi}^{(1)}_{\ell}|^{2}\right]\\\
+Cx_{0}^{-4}\sum_{k=0}^{n}\frac{(d_{0}^{\prime}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,k+1\\})^{2k}}|\partial_{x}^{k}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\big{|}_{x=x_{0}}.$
(13.15)

with $d_{0}<d_{0}^{\prime}$. Note that the $n=0$ case follows immediately from
(13.1) with $n=0$. Now suppose (13.15) holds for $n\leq N$. By applying (13.1)
with $n=N+1$, we obtain

$\frac{(d_{0}^{\prime}x_{0})^{2(N+1)}}{(\max\\{\mathfrak{l}+1,N+2\\})^{2(N+1)}}|\hat{\psi}^{(N+3)}|^{2}\big{|}_{x=x_{0}}\leq
C\frac{(d_{0}^{\prime}x_{0})^{2(N+1)}}{(\max\\{\mathfrak{l}+1,N+1\\})^{2N}}\\\
\times\Bigg{[}((N+2)^{2}x_{0}^{-2}+|2s|^{2})|\hat{\psi}^{(N+2)}|^{2}\big{|}_{x=x_{0}}+x_{0}^{-4}(\mathfrak{l}(\mathfrak{l}+1)-(N+1)(N+2))^{2}|\hat{\psi}^{(N+1)}|^{2}\big{|}_{x=x_{0}}\\\
+x_{0}^{-4}|\partial_{x}^{N+1}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\big{|}_{x=x_{0}}\Bigg{]}\\\
\leq
C\frac{(d_{0}^{\prime}x_{0})^{2N}}{(\max\\{\mathfrak{l}+1,N+1\\})^{2N}}\frac{d_{0}^{\prime
2}((N+2)^{2}+|2s|^{2}x_{0}^{-2})}{(\max\\{\mathfrak{l}+1,N+1\\})^{2}}|\hat{\psi}^{(N+2)}|^{2}\big{|}_{x=x_{0}}\\\
+C\frac{(d_{0}^{\prime}x_{0})^{2(N-1)}}{(\max\\{\mathfrak{l}+1,N\\})^{2(N-1)}}\frac{d_{0}^{\prime
4}(\mathfrak{l}(\mathfrak{l}+1)-(N+1)(N+2))^{2}}{(\max\\{\mathfrak{l}+1,N+1\\})^{2}(\max\\{\mathfrak{l}+1,N+2\\})^{2}}|\hat{\psi}^{(N+1)}|^{2}\big{|}_{x=x_{0}}\\\
+C\frac{(d_{0}^{\prime}x_{0})^{2(N+1)}}{(\max\\{\mathfrak{l}+1,N+2\\})^{2(N+1)}}x_{0}^{-4}|\partial_{x}^{N+1}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\\\
\leq\frac{C}{2}\frac{(d_{0}^{\prime}x_{0})^{2N}}{(\max\\{\mathfrak{l}+1,N+1\\})^{2N}}|\hat{\psi}^{(N+2)}|^{2}\big{|}_{x=x_{0}}+\frac{C}{2}\frac{(d_{0}^{\prime}x_{0})^{2(N-1)}}{(\max\\{\mathfrak{l}+1,N\\})^{2(N-1)}}|\hat{\psi}^{(N+1)}|^{2}\big{|}_{x=x_{0}}\\\
+C\frac{(d_{0}^{\prime}x_{0})^{2(N+1)}}{(\max\\{\mathfrak{l}+1,N+2\\})^{2(N+1)}}x_{0}^{-4}|\partial_{x}^{N+1}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2},$

where we arrived at the final inequality by taking $d_{0}^{\prime}$ suitably
small, $|2s|^{2}x_{0}^{-2}\leq\mathfrak{l}^{2}$. We now conclude (13.15) with
$n=N+1$ by applying (13.15) with $n=N$ and $n=N-1$.

To obtain (13.13), we let $0<{d_{0}}<{d_{0}}^{\prime}$ and we sum both sides
of (13.15) to obtain:

$\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\hat{\psi}^{(n+2)}_{\ell}|^{2}\big{|}_{x=x_{0}}\leq
C\sum_{n=0}^{\infty}\left(\frac{d_{0}}{d_{0}^{\prime}}\right)^{n}x_{0}^{-4}\left[|\hat{\psi}_{\ell}|^{2}+(x_{0}^{2}+|2s|^{2})(\mathfrak{l}(\mathfrak{l}+1))^{-2}|\hat{\psi}^{(1)}_{\ell}|^{2}\right]\\\
+Cx_{0}^{-4}\sum_{n=0}^{\infty}\left(\frac{d_{0}}{d_{0}^{\prime}}\right)^{2(n-k)}\sum_{k=0}^{\infty}\frac{(d_{0}x_{0})^{2k}}{(\max\\{\mathfrak{l}+1,k+1\\})^{2k}}|\partial_{x}^{k}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\big{|}_{x=x_{0}}\\\
\leq
Cx_{0}^{-4}\left[|\hat{\psi}_{\ell}|^{2}+(x_{0}^{2}+|2s|^{2})(\mathfrak{l}(\mathfrak{l}+1))^{-2}|\hat{\psi}^{(1)}_{\ell}|^{2}\right]\\\
+Cx_{0}^{-4}\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2},$

where we used that
$\sum_{j=0}^{\infty}\left(\frac{d_{0}}{d_{0}^{\prime}}\right)^{2j}<\infty$. We
now conclude (13.13).

By applying Stirling’s approximation in the following form: for any
$k\in\mathbb{N}$

$k!\sim k^{k+\frac{1}{2}}e^{-k},$

we can estimate for $n\leq\mathfrak{l}-1$:

$\frac{\gamma^{2n}|2s|^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+(n+1))!^{2}}\leq
C\gamma^{2n}|2s|^{2n}\frac{(\mathfrak{l}-(n+1))^{2\mathfrak{l}-2(n+1)+1}}{(\mathfrak{l}+(n+1))^{2\mathfrak{l}+2(n+1)+1}}e^{4(n+1)}\\\
=C\gamma^{2n}|2s|^{2n}e^{4(n+1)}\left(1-2\frac{n+1}{\mathfrak{l}+(n+1)}\right)^{4(n+1)}(\mathfrak{l}+(n+1))^{-4(n+1)}\\\
\leq
C\left(e^{-2}d_{0}x_{0}|2s|^{-1}\gamma^{-1}(\mathfrak{l}+1)\right)^{-2(n+1)}(d_{0}x_{0})^{2n}(\mathfrak{l}+1)^{-2n-4}\leq(\mathfrak{l}+1)^{-4}(d_{0}x_{0})^{2n}(\mathfrak{l}+1)^{-2n}$

for $\mathfrak{l}\gg d_{0}^{-1}x_{0}^{-1}\gamma|2s|$. Hence, (13.14) follows.
∎

###  13.3. Combining all derivatives

We combine Corollary 13.3 with Proposition 13.5 and Lemma 13.6 to obtain an
estimate in the region $\\{x\leq x_{0}\\}$, with on the right-hand side only
the terms
$\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}$ and
boundary terms at $x=x_{0}$ involving only the lowest two derivatives
$\widehat{\psi}$ and $\widehat{\psi}^{(1)}$:

######  Corollary 13.7.

Let $M=1$, $N_{+}\in\mathbb{N}_{0}$, $\mathfrak{l}>|s|x_{0}^{-1}$ and
$0<\kappa<x_{0}$ and let $d_{0}$ be the constant from Lemma 13.6. Let
$s\in\Omega_{\sigma}$. Then, for $\lambda$ suitably large, there exists a
constant $C_{s,\sigma}=C(s,\sigma)>0$, an numerical constant $C>0$, such that
for all $\mathbb{N}_{0}\ni N_{\infty}\geq N_{\kappa}$:

$\sum_{n=0}^{\mathfrak{l}-1}\frac{(4\sigma)^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}(x^{4}+\kappa^{2}x^{2})|\hat{\psi}^{(n+2)}|^{2}+(x^{2}(\mathfrak{l}+1)^{2}+(n+1)^{2}\kappa^{2}+|2s|^{2})|\hat{\psi}^{(n+1)}|^{2}\\\
+(\mathfrak{l}-n)^{2}(\mathfrak{l}+n+1)^{2}|\hat{\psi}^{(n)}|^{2}\,d\upsigma
dx\\\
+(\mathfrak{l}+1)^{3}\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}x^{4}|s|^{-2}|\widehat{\psi}^{(n+2)}|^{2}+\left[1+(n+1)^{2}x^{2}|s|^{-2}\right]|\hat{\psi}_{\ell}^{(n+1)}|^{2}\,d\upsigma
dx\\\
+\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}\int_{\mathbb{S}^{2}}|\widehat{\psi}^{(n+2)}|^{2}|_{x=x_{0}}d\upsigma+(\mathfrak{l}+1)^{-4(N_{+}+1)}\sum_{n=0}^{N_{+}+1}\int_{\mathbb{S}^{2}}x|\hat{\psi}^{(n)}|^{2}|_{x=\eta
x_{0}}d\upsigma\\\
\leq\boxed{C\int_{\mathbb{S}^{2}}|\hat{\psi}_{\ell}|^{2}+(\mathfrak{l}+1)^{-4}|\hat{\psi}^{(1)}_{\ell}|^{2}\big{|}_{x=x_{0}}\,d\upsigma}\\\
+C\sum_{n=0}^{\mathfrak{l}-1}\frac{(4\sigma)^{2n}(\mathfrak{l}-(n+1))!^{2}}{(\mathfrak{l}+n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx\\\
+C(\mathfrak{l}+1)^{3}\sum_{n=\mathfrak{l}}^{N_{\infty}}\frac{\sigma^{2n}}{n!^{2}(n+1)!^{2}}\int_{0}^{x_{0}}\int_{\mathbb{S}^{2}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}\,d\upsigma
dx\\\
+C\sum_{n=0}^{\infty}\frac{(d_{0}x_{0})^{2n}}{(\max\\{\mathfrak{l}+1,n+1\\})^{2n}}\int_{\mathbb{S}^{2}}|\partial_{x}^{n}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})_{\ell}|^{2}|_{x=x_{0}}d\upsigma.$
(13.16)

###### Proof.

Note first that the boundary terms at $x=x_{0}$ on the right-hand sides of
(13.10) and (13.11) can be estimated directly via (13.14).

Furthermore, the $x$-integral of $\widehat{\psi}^{(\mathfrak{l}+1)}$ on the
right-hand sides of (13.10) and (13.11) can be controlled by the $x$-integral
of $\widehat{\psi}^{(\mathfrak{l}+1)}$ on the left-hand side of (13.9),
provided we multiply both sides of (13.9) with the factor

$|s|^{-2}\frac{\gamma^{2\mathfrak{l}}(\mathfrak{l}+1)!^{2}\mathfrak{l}!^{2}}{\tilde{\sigma}^{2\mathfrak{l}}(2\mathfrak{l})!^{2}}=|s|^{-2}(\mathfrak{l}+1)^{2}\frac{\gamma^{2\mathfrak{l}}}{\tilde{\sigma}^{2\mathfrak{l}}}\frac{\mathfrak{l}!^{4}}{(2\mathfrak{l})!^{2}}$

By applying Stirling’s approximation, we have that

$\frac{\mathfrak{l}!^{4}}{(2\mathfrak{l})!^{2}}\sim\frac{\mathfrak{l}^{4\mathfrak{l}+2}e^{-4\mathfrak{l}}}{(2\mathfrak{l})^{4\mathfrak{l}+1}e^{-4\mathfrak{l}}}=2^{-4\mathfrak{l}}\mathfrak{l}.$

By taking $\gamma=4\tilde{\sigma}$, we therefore obtain:

$|s|^{2}\frac{\gamma^{2\mathfrak{l}}(\mathfrak{l}+1)!^{2}\mathfrak{l}!^{2}}{\tilde{\sigma}^{2\mathfrak{l}}(2\mathfrak{l})!^{2}}\sim|s|^{-2}(\mathfrak{l}+1)^{3}.\qed$

##  14\. A global estimate for $\mathfrak{L}_{s,\lambda,\kappa}$

We combine Corollaries 12.8 and 13.7, to obtain a global estimate for an
$\widehat{\psi}$ in terms of $\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}$.
Recall the Hilbert spaces $H^{N_{+}}_{\sigma,\delta}$ and
$H^{N_{+}+1}_{\sigma,R_{\infty},2}$ introduced in §8.

######  Proposition 14.1.

Let $M=1$, $N_{+}\in\mathbb{N}_{0}$, $0<\kappa\ll\eta x_{0}$. Take
$s\in\Omega_{\sigma}\cap\\{{\operatorname{\textnormal{Re}}s}>(\frac{1}{2}+N_{+})\kappa_{+}\\}$
and $p=4(N_{+}+1)+2$. Then, for $M^{-1}R_{\infty}$ and $\lambda$ suitably
large, there exists a constant $C_{s,\lambda,\delta,R_{\infty}}>0$ and a
constant $N_{\kappa}>\mathfrak{l}$, such that for
$\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\in H^{N_{+}}_{\sigma,\delta}$:

$\|\widehat{\psi}\|_{H^{N_{+}+1}_{\sigma,R_{\infty},2}}^{2}+\kappa^{2}(\|x\partial_{x}^{2}\widehat{\psi}\|_{G_{\sigma,R_{\infty}}}^{2}+\|\not{\Delta}_{\mathbb{S}^{2}}\widehat{\psi}\|_{G_{\sigma,R_{\infty}}}^{2})\leq
C_{s,\sigma}\|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\|_{H^{N_{+}}_{\sigma,R_{\infty}}}^{2}.$
(14.1)

###### Proof.

We need to show that:

$\sum_{k=0}^{1}\|\not{\nabla}_{\mathbb{S}^{2}}^{k}\widehat{\psi}\|_{H^{N_{+}+1}(\Sigma\cap\\{\varrho\leq\frac{1}{3}R_{\infty}\\})}^{2}+\|\widehat{\psi}\|_{H^{N_{+}+2}(\Sigma\cap\\{\frac{1}{3}R_{\infty}\leq\varrho\leq
3R_{\infty}\\})}^{2}\\\
+\left|\left\|(-\not{\Delta}_{\mathbb{S}^{2}}+\mathbf{1})^{-\frac{w_{p}}{4}}\widehat{\psi}\right|\right|_{H^{N_{+}+2}(\Sigma\cap\\{3R_{\infty}\leq\varrho\leq
4R_{\infty}\\})}^{2}+\|\widehat{\psi}\|_{B_{R_{\infty},2}}^{2}+\|\widehat{\psi}\|_{G_{\sigma,R_{\infty},2}}^{2}+\kappa^{2}\|x\partial_{x}^{2}\widehat{\psi}\|_{G_{\sigma,R_{\infty}}}^{2}\\\
\leq
C_{s,\lambda,R_{\infty}}\Biggl{[}\|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\|_{H^{N_{+}}(\Sigma\cap\\{\varrho\leq
3R_{\infty}\\})}^{2}+\|\not{\nabla}_{\mathbb{S}^{2}}\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\|_{H^{N_{+}}(\Sigma\cap\\{\varrho\leq\frac{1}{3}R_{\infty}\\})}^{2}\\\
+\|(-\not{\Delta}_{\mathbb{S}^{2}}+\mathbf{1})^{-\frac{w_{p}}{4}}(\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi})\|_{H^{N_{+}}(\Sigma\cap\\{3R_{\infty}\leq\varrho\leq
4R_{\infty}\\})}^{2}+\|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\|_{B_{R_{\infty}}}^{2}+\|\mathfrak{L}_{s,\lambda,\kappa}\widehat{\psi}\|_{G_{\sigma,R_{\infty}}}^{2}\Biggr{]}.$
(14.2)

We consider (12.10) and we commute additionally with
$\not{\Delta}_{\mathbb{S}^{2}}$ to control additional angular derivatives on
the left-hand side and obtain control of the desired $H^{N_{+}}$,
$H^{N_{+}+1}$ and $H^{N_{+}+2}$ norms.

By using that $M^{-1}R_{\infty}\ll\sqrt{\lambda}$, we can absorb the boxed
term on the right-hand side of (13.16) into the left-hand side of (12.10). In
order to subsequently absorb the boxed terms that appear on the right-hand
side of (12.10) into the left-hand side of (13.16), we take $p=4(N_{+}+1)+2$.
Finally, we note that

$\sum_{\ell=\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}(\mathfrak{l}+1)^{-w_{p}}|f_{\ell}|^{2}\,d\upsigma\leq\sum_{\ell\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}2^{\frac{w_{p}}{2}}(\ell(\ell+1)+\lambda)^{-\frac{p}{2}}|f_{\ell}|^{2}\,d\upsigma\leq
2^{\frac{w_{p}}{2}}(1+\lambda^{-\frac{w_{p}}{2}})\int_{\mathbb{S}^{2}}|(-\not{\Delta}_{\mathbb{S}^{2}}+\mathbf{1})^{-\frac{w_{p}}{4}}f|^{2}\,d\upsigma.$

We also have that

$\sum_{\ell=\in\mathbb{N}_{0}}\int_{\mathbb{S}^{2}}(\mathfrak{l}+1)^{-w_{p}}|f_{\ell}|^{2}\,d\upsigma=\sum_{\ell=0}^{l}\int_{\mathbb{S}^{2}}(\ell+l+1)^{-w_{p}}|f_{\ell}|^{2}\,d\upsigma+\sum_{\ell=l+1}^{\infty}(\ell+1)^{-w_{p}}|f_{\ell}|^{2}\,d\upsigma\\\
\geq((2l+1)^{w_{p}}+1)\sum_{\ell=0}^{\infty}(1+\ell(\ell+1))^{\frac{-w_{p}}{2}}|f_{\ell}|^{2}\,d\upsigma\geq((2l+1)^{w_{p}}+1)\int_{\mathbb{S}^{2}}|(-\not{\Delta}_{\mathbb{S}^{2}}+\mathbf{1})^{-\frac{w_{p}}{4}}f|^{2}\,d\upsigma.$

Finally, we observe that in the case $\ell>l$, we have that
$\mathfrak{l}=\ell$, whereas in the case $\ell\leq l$, we use boundedness of
$\ell$ to ensure that the $\mathfrak{l}$-dependent norms in (12.10) are
comparable to the norms appearing in (14.2). ∎

##  15\. Inverting $\mathfrak{L}_{s,\lambda,\kappa}$

In this section, we will construct $\mathfrak{L}_{s,\lambda,\kappa}^{-1}$ for
$\kappa\geq 0$ with an appropriate Hilbert space domain and with a suitably
large $\lambda$.

We first make use of the regularizing effect of $\kappa>0$ to invert
$\mathfrak{L}_{s,\lambda,\kappa}$ as an operator on suitable Sobolev spaces.
We then appeal to the uniform-in-$\kappa$ estimate from Proposition 14.1 to
construct $\mathfrak{L}_{s,\lambda,\kappa=0}^{-1}$ as a limit of a sequence of
operators $\mathfrak{L}_{s,\lambda,\kappa_{n}}^{-1}$, with $\kappa_{n}>0$,
$\kappa_{n}\to 0$ as $n\to\infty$.

Define $H^{k}(\mathfrak{L}_{s,\lambda,\kappa})$ as the closure of
$C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})$ with respect to the graph
norm:

$\|\cdot\|_{H^{k}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})}+\|\mathfrak{L}_{s,\lambda,\kappa}(\cdot)\|_{H^{k}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})}.$

Then the map:

$\mathfrak{L}_{s,\lambda,\kappa}:H^{k}(\mathfrak{L}_{s,\lambda,\kappa})\to
H^{k}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})$

is a well-defined bounded linear operator for all $k\in\mathbb{N}_{0}$.

######  Theorem 15.1.

Let $N\geq N_{+}\in\mathbb{N}_{0}$ and
${\operatorname{\textnormal{Re}}s}>-(N+\frac{1}{2})\min\\{\kappa_{+},\kappa\\}$.
Then, there exists a $\lambda>0$ suitably large, such that the inverse map:

$\mathfrak{L}_{s,\lambda,\kappa}^{-1}:H^{N}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})\cap\left\\{f\in
L^{2}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})\,|\,\exists
m\in\mathbb{Z}\,:\,\partial_{\varphi}f=imf\right\\}\\\ \to
H^{N}(\mathfrak{L}_{s,\lambda,\kappa})\subseteq
H^{N+1}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})$

is a bounded linear operator.

###### Proof.

Since we are considering a domain restricted to finite azimuthal modes (with
$\varphi$-dependence $e^{im\varphi}$), we can apply a straightforward
modification of [War15][Theorem 5.7] to obtain the existence of the desired
inverse. ∎

######  Remark 15.1.

The ability to restrict to fixed azimuthal modes on Kerr is not crucial. One
can in principle derive an analogue of Theorem 15.1 without making use of this
assumption and instead make use of red-shift estimates depending on $\kappa$
in the large-$r$-region, instead of Gevrey-type estimates. See also the
related results in [Vas13].

By applying Proposition 14.1, which is uniform in $\kappa$ and $m$, we can
obtain the existence of $\mathfrak{L}_{s,\lambda,\kappa}^{-1}$ with domain
$H_{\sigma,R_{\infty}}^{N_{+}}$.

######  Corollary 15.2.

Let $\sigma\in\mathbb{R}$ and take
$s\in\Omega_{\sigma}\cap\\{{\operatorname{\textnormal{Re}}s}>(\frac{1}{2}+N_{+})\kappa_{+}\\}$.
Then, there exists a suitably small $\kappa>0$ and a suitably large
$R_{\infty}$ and $\lambda\in\mathbb{N}_{0}$, such that:

$\mathfrak{L}_{s,\lambda,\kappa}^{-1}:H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}$

is a bounded linear operator and there exists a constant
$C_{s,\sigma,R_{\infty}}>0$ such that:

$\|\mathfrak{L}_{s,\lambda,\kappa}^{-1}\|_{H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}}\leq C_{s,\sigma,R_{\infty}}.$ (15.1)

Furthermore, for all $f\in H_{\sigma,R_{\infty}}^{N_{+}}$,

$\kappa\|x\partial_{x}^{2}\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f)\|_{G_{\sigma,R_{\infty}}}+\kappa^{2}\|\not{\Delta}_{\mathbb{S}^{2}}\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f)\|_{G_{\sigma,R_{\infty}}}\leq
C_{s,\sigma}\|f\|_{H_{\sigma,R_{\infty}}^{N_{+}}}.$ (15.2)

###### Proof.

Let $\kappa_{+}>\kappa>0$ and
$s\in\Omega_{\sigma}\cap\\{{\operatorname{\textnormal{Re}}z}>(\frac{1}{2}+N_{+})\kappa_{+}\\}$.
Then there exists a $N_{\kappa}>N_{+}$ such that
${\operatorname{\textnormal{Re}}s}>-(N_{\kappa}+\frac{1}{2})\kappa$. Let
$f_{m}\in C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}))\cap
H_{\sigma,R_{\infty}}^{N_{+}}$, such that $\partial_{\varphi}f_{m}=imf_{m}$
for some $m\in\mathbb{Z}$. Then we can apply Theorem 15.1 to conclude that
$\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f_{m})\in
C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})$ is well-defined for
suitably large $\lambda$.

By Proposition 14.1, it then follows that for $\lambda$ suitably large and
$\kappa$ suitably small, we can estimate

$\|\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f_{m})\|_{H^{N_{+}+1}_{\sigma,R_{\infty},2}}^{2}\leq
C_{s,\sigma,\lambda}\|f_{m}\|_{H^{N_{+}}_{\sigma,R_{\infty}}}^{2}.$

Let $f\in C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}))\cap
H_{\sigma,R_{\infty}}^{N_{+}}$. Then we can decompose
$f=\sum_{m\in\mathbb{Z}}f_{m}$, with $\partial_{\varphi}f_{m}=imf_{m}$. Since
the constant in the above estimate does not depend on $m$, we can sum over $m$
to conclude that for general $f\in
C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}))\cap
H_{\sigma,R_{\infty}}^{N_{+}}$,
$\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f)=\sum_{m\in\mathbb{Z}}\mathfrak{L}_{s,\lambda,\kappa}^{-1}(f_{m})$
is well defined.

Hence,
$\mathfrak{L}_{s,\lambda,\kappa}^{-1}(C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})\cap
H_{\sigma,R_{\infty}}^{N_{+}})\subseteq H_{\sigma,R_{\infty},2}^{N_{+}+1}$.
Since $C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})\cap
H_{\sigma,R_{\infty}}^{N_{+}}$ is dense in $H_{\sigma,R_{\infty}}^{N_{+}}$, we
can extend $\mathfrak{L}_{s,\lambda,\kappa}^{-1}$ to act on
$H_{\sigma,R_{\infty}}^{N_{+}}$ and we must have that
$\mathfrak{L}_{s,\lambda,\kappa}^{-1}(H_{\sigma,R_{\infty}}^{N_{+}})\subseteq
H_{\sigma,R_{\infty},2}^{N_{+}+1}$. ∎

######  Proposition 15.3.

Let $\sigma\in\mathbb{R}$ and take
$s\in\Omega_{\sigma}\cap\\{{\operatorname{\textnormal{Re}}z}>-(\frac{1}{2}+N_{+})\kappa_{+}\\}$.
Then, there exists a suitably large $\lambda\in\mathbb{N}_{0}$, such that:

$\mathfrak{L}_{s,\lambda}^{-1}=\mathfrak{L}_{s,\lambda,0}^{-1}:H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}$ (15.3)

is a bounded linear operator and there exists a constant $C_{s,\sigma}>0$ such
that:

$\|\mathfrak{L}_{s,\lambda}^{-1}\|_{H_{\sigma,R_{\infty}}^{N_{+}}\to
H_{\sigma,R_{\infty},2}^{N_{+}+1}}\leq C_{s,\sigma}.$

Furthermore, for any sequence $\\{\kappa_{n}\\}$, with $\kappa_{n}\to 0$ as
$n\to\infty$

$\mathfrak{L}_{s,\lambda,\kappa_{n}}^{-1}\to\mathfrak{L}_{s,\lambda}^{-1}$

as $n\to\infty$, with respect to the operator norm on
$H_{\sigma,R_{\infty}}^{N_{+}}$.

###### Proof.

Note that there exist $\sigma_{2}^{2}>\sigma_{1}^{2}>\sigma^{2}$, such that
$s\in\Omega_{\sigma_{1}}\cap\Omega_{\sigma_{2}}$. Let $f\in
C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2})\cap
H_{\sigma_{2},R_{\infty}}^{N_{+}+1}$.

Let $\kappa_{n}$ be a monotonically non-increasing sequence with
$\kappa_{n}>0$ and $\kappa_{n}\to 0$ as $n\to\infty$. By Corollary 15.2,
$\widehat{\psi}_{n}:=\mathfrak{L}_{s,\lambda,\kappa_{n}}^{-1}(f)$ is a well-
defined, uniformly bounded sequence in
$H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}$. By (B.2),
$H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}\subset\joinrel\subset
H_{\sigma_{1},R_{\infty},2}^{N_{+}+1}$, so $\widehat{\psi}_{n}$ admits a
convergent subsequence $\\{\widehat{\psi}_{n_{j}}\\}$ with respect to
$H_{\sigma_{1},R_{\infty},2}^{N_{+}+1}$, with limit $\widehat{\psi}\in
H_{\sigma_{1},R_{\infty},2}^{N_{+}+1}$. In fact, since
$\\{\widehat{\psi}_{n_{j}}\\}$ is a bounded sequence in
$H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}$, it admits a further subsequence with
a weak limit in $H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}$. By the uniqueness of
limits, this limit must also be $\widehat{\psi}$, so in fact
$\widehat{\psi}\in H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}$.

We will now show that $\mathfrak{L}_{s,\lambda,0}(\widehat{\psi})=f$. We have
that

$\mathfrak{L}_{s,\lambda,0}(\widehat{\psi})=f-(\mathfrak{L}_{s,\lambda,\kappa_{n_{j}}}-\mathfrak{L}_{s,\lambda,0})(\widehat{\psi})+\mathfrak{L}_{s,\lambda,\kappa_{n}}(\widehat{\psi}-\widehat{\psi}_{n_{j}}).$
(15.4)

Note that:

$(\mathfrak{L}_{s,\lambda,\kappa_{n_{j}}}-\mathfrak{L}_{s,\lambda,0})(\widehat{\psi})=\kappa_{n_{j}}\chi_{\varrho\geq
R_{\infty}}(x\partial_{x}^{2}\widehat{\psi}+\partial_{x}\widehat{\psi}).$

Hence, by Proposition C.1

$\|(\mathfrak{L}_{s,\lambda,\kappa_{n_{j}}}-\mathfrak{L}_{s,\lambda,0})(\widehat{\psi})\|_{H^{N_{+}}_{\sigma,R_{\infty}}}\leq
2\kappa_{n_{j}}(\|\chi_{\varrho\geq
R_{\infty}}\widehat{\psi}^{(1)}\|_{H^{N_{+}}_{\sigma,R_{\infty}}}+\|x\chi_{\varrho\geq
R_{\infty}}\widehat{\psi}^{(2)}\|_{H^{N_{+}}_{\sigma,R_{\infty}}})\\\ \leq
C\kappa_{n_{j}}(\|\widehat{\psi}\|_{H^{N_{+}+2}_{\sigma,R_{\infty},2}}+\|x\widehat{\psi}^{(2)}\|_{G_{\sigma,R_{\infty}}}).$

To bound $\|x\widehat{\psi}^{(2)}\|_{G_{\sigma,R_{\infty}}}$, we first apply
Lemma C.3, from which it follows that

$\|x\widehat{\psi}^{(2)}\|_{G_{\sigma,R_{\infty}}}\leq
C\|\widehat{\psi}^{(2)}\|_{G_{\sigma,R_{\infty}}}\leq
C\|\widehat{\psi}^{(1)}\|_{G_{\sigma,R_{\infty},2}}.$

Then we observe that we can use that $\sigma_{1}>\sigma$ to further estimate:

$\|\widehat{\psi}^{(1)}\|_{G_{\sigma,R_{\infty},2}}\leq
C\|\widehat{\psi}\|_{G_{\sigma_{1},R_{\infty},2}}.$

We can similarly bound:

$\|\mathfrak{L}_{s,\lambda,\kappa_{n_{j}}}(\widehat{\psi}-\widehat{\psi}_{n_{j}})\|_{H^{N_{+}}_{\sigma,R_{\infty}}}\leq
C\|\widehat{\psi}-\widehat{\psi}_{n_{j}}\|_{H^{N_{+}+2}_{\sigma_{1},R_{\infty},2}}\to
0\quad\textnormal{as $j\to\infty$}.$

Taking the limit $j\to\infty$ in (15.4) therefore gives
$\mathfrak{L}_{s,\lambda,0}(\widehat{\psi})=f$. Hence,
$\mathfrak{L}_{s,\lambda,0}^{-1}:C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}))\cap
H_{\sigma_{2},R_{\infty}}^{N_{+}}\to H_{\sigma_{2},R_{\infty},2}^{N_{+}+1}$ is
well-defined. Using density of
$C^{\infty}([0,r_{+}^{-1}]_{x}\times\mathbb{S}^{2}))\cap
H_{\sigma_{2},\delta_{2}}^{N_{+}}$ in $H_{\sigma_{2},\delta_{2}}^{N_{+}}$, we
conclude (15.3).