On the Global Stability of the Wave-map Equation in Kerr Spaces with Small Angular Momentum

Abstract

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $\Phi$ defined from a fixed Kerr solution ${\mathcal K}(M,a)$, $0\le a \le M$, with values in the two dimensional hyperbolic space ${\mathbb H}^2$. A particular such wave map is given by the complex Ernst potential associated to the axial Killing vector-field $\mathbf{Z}$ of ${\mathcal K}(M,a)$. We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of ${\mathcal K}(M,a)$, for all $0\le a<M$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

Introduction

According to the general expectations the Kerr family ${\mathcal K}(a,M)$, in the sub-extremal regime $|a|<M$, is stable under general perturbations. More precisely, it is expected that:

Kerr Stability Conjecture

An initial data set $(\Sigma _0, g_0, k_0)$, sufficiently close to the initial data set of a fixed sub-extremal Kerr spacetime ${\mathcal K}(M_i, a_i)$, admits a maximal, vacuum, future, Cauchy development $(\mathbf{M}, \mathbf{g})$, with a complete future null infinity ${\mathcal I}^{+}$ and whose causal past $J^{-}({\mathcal I}^+)$ is bounded in the future by a smooth, complete, event horizon ${\mathcal H}^+$. Moreover $(\mathbf{M}, \mathbf{g})$ remains close to ${\mathcal K}(M_i, a_i)$ and approaches asymptotically another sub-extremal Kerr spacetime ${\mathcal K}(M_f, a_f)$.

Despite its extraordinary importance, in both mathematical and astrophysical¹ terms, and despite half a century of sustained efforts to settle it, the conjecture remains wide open. The main known mathematical arguments in favor of the conjecture are in fact few and, so far, not at all decisive.

• (1) We know that the Minkowski space, corresponding to $a=0$, $M=0$ is stable, see [8].

• (2) We know that, perturbatively, the Kerr family exhausts all stationary, smooth, solutions of the Einstein vacuum equations, see [15] and [1]. In other words, any stationary solution sufficiently close to a sub-extremal Kerr must belong to the Kerr family. A full review of rigidity results in the smooth setting is discussed in [16].

• (3) We possess a significantly large class of examples of dynamical black holes, settling down to a sub-extremal Kerr, constructed from infinity, see [13].

• (4) Most importantly, we have now a satisfactory understanding of the so called poor man linearization. More precisely, we have a general method for establishing boundedness and quantitative decay of solutions to the scalar wave equation $\square _{\mathbf{g}_{M, a}} \phi =0$, for all sub-extremal Kerr metrics $\mathbf{g}_{M, a}$. Such results were first established in Schwarzschild, see [3–6, 10, 19] and later extended for $|a| \ll M$ in [2, 12, 21]. The full sub-extremal regime was recently settled in [14].

• (5) We have results establishing the non-existence of exponentially growing modes for the more realistic linearized Teukolsky equations, see [20, 23]².

The goal of this paper is to provide additional evidence for the conjecture in the special case of axi-symetric perturbations.

A Non-linear Model Problem

As well known (see [22] ) the Ernst potential $\Phi =(\Phi ^1, \Phi ^2)$ of a Killing vector-field $\mathbf{Z}$ on a $3+1$ dimensional Einstein-vacuum manifold $(\mathbf{M}, \mathbf{g})$ can be interpreted as a wave map $\Phi :\mathbf{M}\longrightarrow {\mathbb H}^2$ where ${\mathbb H}^2$ denotes the upper-half Poincare space with constant negative curvature $K=-1$. More precisely,

$$\begin{aligned} \square _\mathbf{g}\Phi ^a+ \mathbf{g}^{\mu \nu }\Gamma ^a_{bc}(\Phi ){\partial }_\mu \Phi ^b {\partial }_\nu \Phi ^c=0, \end{aligned}$$

(1.1)

where $\Gamma$ denotes the Christoffel symbols of the metric h of ${\mathbb H}$. The full, axially symmetric, space-time metric $\mathbf{g}$ decomposes into its dynamic component $\Phi$ and a reduced $1+2$ metric $\hat{\mathbf{g}}$ defined on the orbit space $\hat{\mathbf{M}}=\mathbf{M}/\mathbf{Z}$ verifying,

$$\begin{aligned} \text{ Ric }(\hat{\mathbf{g}})_{{\alpha }{\beta }}=<{\partial }_{\alpha }\Phi , {\partial }_{\beta }\Phi >_h. \end{aligned}$$

(1.2)

Thus, in axial symmetry, the Einstein vacuum equations are equivalent³ to the coupled system (1.1)–(1.2), on the reduced space-time $\widehat{\mathbf{M}}$. A particular, stationary, solution of the system is provided by the pair $( \widehat{\mathbf{g}}_{M,a} ,\Phi _{M,a})$ denoting the decomposition of the Kerr metric $\mathbf{g}_{M,a}$ of a fixed Kerr spacetime $\mathbf{M}={\mathcal K}(M,a)$. The full problem of the stability of the Kerr solution, for axially symmetric perturbations, can be reformulated as a problem of stability of this special solution for the system (1.1)–(1.2). As this is still an extremely difficult problem we make one further important simplification by partially linearizing the system, that is we fix the reduced metric $\widehat{g}=\widehat{\mathbf{g}}_{M,a}$ but allow fully nonlinear perturbations of $\Phi _{M,a}$. It is easy to see that this amounts to the problem of stability of axially symmetric perturbations of the stationary solution $\Phi _{M,a}$ of the wave map system (1.1), where $\mathbf{g}$ is fixed to be the Kerr metric $\mathbf{g}_{M,a}$.

Partial Stability Conjecture

The stationary solution $\Phi _{M,a}:{\mathcal K}(M,a)\longrightarrow {\mathbb H}$ of the wave map system (1.1) with $\mathbf{g}=\mathbf{g}_{M,a}$ the metric of ${\mathcal K}(M,a)$, $|a|<M$, is future asymptotically stable in the domain of outer communication of ${\mathcal K}(M,a)$, for all smooth, axially symmetric, admissible, perturbations⁴.

Remark 1.1

We note that the conjecture is consistent with the full nonlinear stability conjecture, for axially symmetric perturbations. More precisely the validity of the Kerr stability conjecture, for axially symmetric perturbations, implies (in principle) the validity of our partial stability conjecture, at least for initial data in the orthogonal complement of a finite dimensional space (corresponding to possible modulation). In this paper we produce convincing evidence that the conjecture is in fact true for all initial data.

We take the first step in proving the conjecture by deriving stability estimates for the linearized system. More precisely we introduce the linearized variables

$$\begin{aligned} \Phi =\Phi _{M,a}+ A\Psi , \qquad \Psi = (\phi ,\psi ). \end{aligned}$$

and show that the linearized equations in $\Psi$ possess a a coercive, conserved, energy quantity (for all $|a|\le M$) and verify, at least for a / M small, a Morawetz type estimate comparable to those derived in recent years, see [3–6, 10, 19], for the scalar wave equation $\square \phi =0$.

Remark 1.2

In the simplest case $a=0$ the system for $\Psi =(\phi ,\psi )$ is the decoupled system

$$\begin{aligned} \square \phi =0,\qquad \square \psi -\Big (\frac{4}{r^2(\sin \theta )^2}-\frac{8M}{r^3}\Big )\psi =0. \end{aligned}$$

(1.3)

Note the non-trivial nature of the potential for the $\psi$ equation, singular on the axis. The precise form of the potential is important in order to derive the needed stability estimates.

Kerr Metric

The domain of outer communications of the Kerr spacetime $\mathcal {K}(M,a)$, in standard Boyer–Lindquist coordinates, is given by

$$\begin{aligned} \mathbf{g}_{a,M}=-\frac{q^2\Delta }{\Sigma ^2}(dt)^2+\frac{\Sigma ^2(\sin \theta )^2}{q^2}\Big (d\phi -\frac{2aMr}{\Sigma ^2}dt\Big )^2 +\frac{q^2}{\Delta }(dr)^2+q^2(d\theta )^2, \end{aligned}$$

(1.4)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta =r^2+a^2-2Mr;\\ q^2=r^2+a^2(\cos \theta )^2;\\ \Sigma ^2=(r^2+a^2)q^2+2Mra^2(\sin \theta )^2=(r^2+a^2)^2-a^2(\sin \theta )^2\Delta . \end{array}\right. } \end{aligned}$$

(1.5)

Observe that

$$\begin{aligned} (2mr-q^2)\Sigma ^2=-q^4\Delta +4a^2m^2r^2(\sin \theta )^2. \end{aligned}$$

(1.6)

Note also the useful identities,

$$\begin{aligned} \frac{\Sigma ^2}{q^2}=q^2+(p+1)a^2(\sin \theta )^2,\qquad \Delta =q^2(1-p)+a^2(\sin \theta )^2,\qquad p:=\frac{2Mr}{q^2}. \end{aligned}$$

(1.7)

Thus the metric can also be written in the form,

$$\begin{aligned} \mathbf{g}_{ a, M}= & {} -\frac{\left( \Delta -a^2\sin ^2\theta \right) }{q^2}dt^2- \frac{ 4 aMr }{q^2} \sin ^2\theta dt d\phi \nonumber \\&+\frac{q^2}{\Delta }dr^2+ q^2 d\theta ^2+ \frac{ \Sigma ^2}{q^2}\sin ^2\theta d\phi ^2 \end{aligned}$$

(1.8)

and,

$$\begin{aligned} \mathbf{g}_{tt} \mathbf{g}_{\phi \phi }- \mathbf{g}_{t\phi }^2 =-\Delta \sin ^2 \theta . \end{aligned}$$

The volume element $d\mu$ of $\mathbf{g}$ is given by

$$\begin{aligned} d\mu =q^2 |\sin \theta | dt dr d\theta d\phi \end{aligned}$$

We also note that $\mathbf{T}={\partial }_t$, $\mathbf{Z}={\partial }_\phi$ are both Killing and $\mathbf{T}$ is only time-like in the complement of the ergoregion, i.e. $q^2> 2 Mr$.

The domain of outer communication of ${\mathcal K}(M,a)$ is given by,

$$\begin{aligned} {\mathcal R}=\{(\theta ,r,t,\varphi )\in (-\pi ,\pi )\times (\,r_{{\mathcal H}},\infty )\times \mathbb {R}\times \mathbb {S}^1\}, \end{aligned}$$

where $\,r_{{\mathcal H}}:=M+\sqrt{M^2-a^2}$, the larger root of $\Delta$, corresponds to the event horizon. The metric posesses the Killing v-fields $\mathbf{T}={\partial }_t$ and $\mathbf{Z}={\partial }_\phi$.

The Ernst potential $\mathring{\Phi }=(A, B)$ associated to the Killing vector-field $\mathbf{Z}={\partial }_\varphi$, is given explicitly by the formula,

$$\begin{aligned} A+iB:= & {} \frac{\Sigma ^2(\sin \theta )^2}{q^2}-i \Big [2aM(3\cos \theta -(\cos \theta )^3)+\frac{2a^3M(\sin \theta )^4\cos \theta }{q^2}\Big ],\nonumber \\&A=\mathbf{g}(\mathbf{Z},\mathbf{Z}). \end{aligned}$$

(1.9)

One can easily check⁵ that (A, B) verify the system,

$$\begin{aligned} \begin{aligned}&A\square A=\mathbf{D}^\mu A\mathbf{D}_\mu A-\mathbf{D}^\mu B\mathbf{D}_\mu B,\\&A\square B=2\mathbf{D}^\mu A\mathbf{D}_\mu B. \end{aligned} \end{aligned}$$

(1.10)

where $\square =\square _{g_{M,a}}$ denotes the usual wave operator with respect to the metric. We can interpret $\mathring{\Phi }:=(A, B)$ as a stationary, axisymmetric, wave map from ${\mathcal K}(M,a)$ to the hyperbolic space ${\mathbb H}^2 =({\mathbb R}_+^2, h)$ with the metric h given by,

$$\begin{aligned} ds^2=\frac{1}{A^2} \big (\textit{dA}^2+ \textit{dB}^2\big ) \end{aligned}$$

Reinterpreting the Conjecture

As mentioned above the goal of this paper is to investigate the future global asymptotic stability, in the exterior region of ${\mathcal K}(M,a)$, of the special stationary map $\mathring{\Phi }=(A,B)$, under general axially symmetric perturbations. In other words we consider solutions $\Phi =(X,Y)$ of the wave map system,

$$\begin{aligned} \begin{aligned}&X\square X=\mathbf{D}^\mu X\mathbf{D}_\mu X-\mathbf{D}^\mu Y\mathbf{D}_\mu Y,\\&X\square Y=2\mathbf{D}^\mu X\mathbf{D}_\mu Y. \end{aligned} \end{aligned}$$

(1.11)

which are $\mathbf{Z}$-invariant, i.e. $\mathbf{Z}(\Phi ^1)=\mathbf{Z}(\Phi ^2)=0$, and whose initial conditions on a given space-like hypersurface in ${\mathcal R}$ are a small perturbation of the initial data of $\mathring{\Phi }$. We have to be careful however that the perturbed map $\Phi =(X, Y)$ has the same axis of rotation as $\mathring{\Phi }=(A,B)$, i.e. $\Phi =\mathring{\Phi }$ on the axis of symmetry of ${\mathcal K}(M, a )$, i.e. $\sin ^2\theta =0$. To make sure that this latter condition is satisfied we search for solutions $\Phi =(X, Y)$ of the form,

$$\begin{aligned} \Phi =\mathring{\Phi }+ A\Psi , \qquad \Psi = (\phi ,\psi ). \end{aligned}$$

(1.12)

with $\psi$ vanishing on the axis of symmetry ${\mathcal A}$. With these notation we can interpret the system (1.11) as a nonlinear system of equations for $\Psi$, depending also on the fixed $\mathring{\Phi }$, of the form,

$$\begin{aligned} {\mathcal F}(\mathring{\Phi }; \Psi )=0. \end{aligned}$$

(1.13)

Our Conjecture can thus be interpreted as a statement on the stability of the trivial solution $\Psi \equiv 0$ for the nonlinear system (1.13) .

Conjecture

The trivial solution $\Psi =0$ of the nonlinear system (1.13) is future asymptotically stable in the exterior region $r\ge r_{\mathcal H}$ for arbitrary, smooth, axially symmetric, admissible (i.e. such that $\psi =0$ on the axis ${\mathcal A}$) initial conditions on a $\mathbf{Z}$-invariant spacelike hypersurface.

Main Difficulties

A simple comparison with the far simpler case of nonlinear systems of wave equations in Minkowski space shows that we cannot expect the conjecture to be valid without addressing the following obstacles.

• (1) Strong linear stability. To start with, one needs to show that the solutions to the wave map system system cannot grow out of control. It does not suffice to show that the solutions to the linearized equations are simply bounded; one needs to prove quantitative decay estimates comparable to the known decay estimates for the standard wave equation in the Minkowski space ${\mathbb R}^{1+3}$. Moreover these estimates have to be robust, i.e. the methods used in their derivation can be extended, in principle, to the nonlinear equations.

• (2) Nonlinear stability. Though strong linear stability is an essential ingredient in the proof of nonlinear stability, it is by no means enough. The nonlinear terms of the equation also have to satisfy special structural conditions, such as the null condition.

• (3) Degeneracy on the axis. An additional difficulty is the degeneracy of our system on the axis of symmetry, i.e where A vanishes, see (1.3). Our functional analysis framework, see Definition 1.6, is adapted to handle such a situation.

The first difficulty is the most serious one. The case when the linearized equation is simply $\square _\mathbf{g}\Psi =0$ has now been well understood in full generality, for all $|a|<M$ and under no symmetry assumptions, see [14] and the references therein. Our linearized equations differ significantly, however, from this case. Indeed taking the Fréchet derivative of ${\mathcal F}$ with respect to $\Psi$ we obtain a linear operator with coefficients which depend on $\mathring{\Phi }=(A,B)$ in a non-trivial fashion. The linearized equations are in fact of the form:

$$\begin{aligned} \begin{aligned} 0&= \square \phi +2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \psi -2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi \\ 0&=\square \psi -2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \phi -\frac{\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\psi . \end{aligned} \end{aligned}$$

(1.14)

and cannot be decoupled. It is not apriori clear that such an equation possesses a well defined notion of energy, i.e. a conserved and coercive integral quantity similar to the standard energy quantity for $\square \Psi =0$. Though the existence of such a quantity is by no means enough to prove strong linear stability it is an absolutely necessary first step. Our first result is the following:

Theorem 1.3

The linearized equations (1.14) (for axi-symmmetric solutions $\Psi$) admit an energy-momentum tensor type quantity ${\mathcal Q}_{\mu \nu }={\mathcal Q}[\Psi ]_{\mu \nu }$ and a source ${\mathcal J}_\nu$, both quadratic in $(\Psi , {\partial }\Psi )$, depending also on $(\mathring{\Phi }, {\partial }\mathring{\Phi })$, verifying the following:

• (a) ${\mathcal Q}(X, Y)>0$,    for any future-oriented, timelike, vector-fields X, Y;

• (b) $\mathbf{D}^\nu {\mathcal Q}_{\mu \nu } ={\mathcal J}_{\nu }$.

• (c) $\mathbf{g}(\mathbf{T}, {\mathcal J})=0$.

• (d) ${\mathcal Q}(Z, X) =0$,    for any vector-field X orthogonal to $\mathbf{Z}$.

The underlying reason for the existence of a quantity verifying (b) and (c) is a somewhat less familiar manifestation of Noether’s principle, which we discuss below. The positivity (a), on the other hand, is a consequence of the negative curvature properties of ${\mathbb H}$. The property (d) can be easily derived from the form of ${\mathcal Q}$, displayed below, and the $\mathbf{Z}$-invariance of $\Psi$.

As a consequence of the Theorem we deduce that the current $\mathbf{P}_\mu := \mathbf{Q}_{\mu \nu } \mathbf{T}^\nu$ is conserved, i.e.

$$\begin{aligned} \mathbf{D}^\mu \mathbf{P}_\mu =0. \end{aligned}$$

which leads, by integration on causal domains, to conserved energy type quantities and fluxes. In view of (d) the energy is a coercive quantity in ${\mathcal R}$ with a mild degeneracy on the horizon $r=\,r_{{\mathcal H}}$. Theorem 1.3 is thus a strong first indication of the validity of our conjecture for all values of the Kerr parameters, $|a|<M$. Yet, as alluded above, the bounds provided by the energy are not by themselves enough to even prove the boundedness of solutions to the system (1.14), subject to nice initial conditions.

To actually go beyond the bounds provided by the energy and prove strong linear stability we encounter the same difficulties as for the simpler case of axially symmetric⁶ solutions of the standard wave equation $\square \phi =0$ in the DOC of ${\mathcal K}(a,m)$, i.e. degeneracy of the energy at the horizon, presence of trapped null geodesics and slow decay at null infinity. As it is now well understood, the major ingredient for proving strong linear stability for linear systems on black holes is the derivation of an integrated decay estimate of Morawetz type. Such estimates, which degenerate in the trapping region, i.e. region of ${\mathcal K}(M,a)$ which contain trapped null geodesics, are quite subtle, and difficult to derive.

Fortunately, in the case of axial symmetry, all trapped null geodesics are restricted to the hypersurface at $r=r_*$, the largest root of the polynomial equation in r, $r^3-3 r^2 M+a^2(r+M)=0$. This allows one, in principle, to use a vector-field method approach similar to that used in the derivation of the Morawetz type integrated decay estimate for solutions of the scalar wave equation in Schwarzschild. The main new difficulties are the presence of the source term ${\mathcal J}$ in the divergence equation $\mathbf{Div}{\mathcal Q}={\mathcal J}$, and the degeneracy on the axis. We overcome these difficulties in this paper, for small values of a / M. Inspired by the r-weighted estimates of Dafermos–Rodnianski⁷, see [11], we also prove a stronger version of the Morawetz estimate which provides decay information for an appropriate notion of outgoing energy associated to space-like hypersurfaces.

A precise version of our second theorem requires a space-like $\mathbf{Z}$-invariant foliation $\Sigma _t$ of the entire domain of outer communication, transversal to the horizon and whose leaves are transported by $\mathbf{T}$. In what follows we give a first, informal, version of the theorem, for the linearized equations (1.14) in which we do not specify the foliation. A more precise version will be given later in this section.

To state the theorem we choose a smooth, increasing function $\chi _{\ge 4M}$ supported for $r\ge 4M$, equal to 1 for $r\ge 6M$, and define the outgoing energy density $(e(\phi ),e(\psi ))$,

$$\begin{aligned}&e(\phi )^2:=\frac{(\partial _1\phi )^2}{r^2}+(L\phi )^2+\frac{M^2\big [(\partial _2\phi )^2+(\partial _3\phi )^2\big ]}{r^2}+\frac{\phi ^2}{r^2},\\&e(\psi )^2:=\frac{(\partial _1\psi )^2+\psi ^2(\sin \theta )^{-2}}{r^2}+(L\psi )^2+\frac{M^2\big [(\partial _2\psi )^2+(\partial _3\psi )^2\big ]}{r^2}+\frac{\psi ^2}{r^2}. \end{aligned}$$

where L is the future outgoing vector-field,

$$\begin{aligned} L:=\chi _{\ge 4M}(r) \Big (\partial _r+\frac{r}{r-2M}\partial _t\Big ). \end{aligned}$$

The vector-fields $\partial _1,\partial _2,\partial _3$ are defined precisely in (1.22), as coordinate derivatives in a new set of variables. They agree with the coordinate derivatives $\partial _\theta ,\partial _r,\partial _t$ outside a small neighborhood of the event horizon.

Theorem 1.4

Assume that $(\phi , \psi )$ is an admissible $\mathbf{Z}$-invariant solution of the linear system (1.14). Then, for any $\alpha \in (0,2)$ and any $t_1\le t_2$,

$$\begin{aligned} {\mathcal B}_{\alpha }(t_1, t_2)+\int _{\Sigma _{t_2} } \frac{r^\alpha }{M^{\alpha }}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t\le C_\alpha \int _{\Sigma _{t_1} } \frac{r^\alpha }{M^{\alpha }}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t \end{aligned}$$

with $d\mu _t$ the induced measure on $\Sigma _t$ and ${\mathcal B}_{\alpha }$ the bulk integral,

$$\begin{aligned} \mathcal {B}_\alpha (t_1,t_2):= & {} \int _{\mathcal {D}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }}\Bigg \{\frac{(r-r^*)^2}{r^3}\frac{|\partial _\theta \phi |^2+|\partial _\theta \psi |^2+\psi ^2(\sin \theta )^{-2}}{r^2}\\&+\frac{1}{r}\big [(L\phi )^2+(L\psi )^2\big ]+\frac{1}{r^3}\big (\phi ^2+\psi ^2\big )+\frac{M^{2}}{r^{3}}\big [(\partial _r\phi )^2+(\partial _r\psi )^2\big ]\\&+\frac{M^2(r-r^*)^2}{r^5}\big [(\partial _t \phi )^2+(\partial _t\psi )^2\big ]\Bigg \}\,d\mu . \end{aligned}$$

Note that, as expected the integrand of the bulk integral ${\mathcal B}_{\alpha }$ degenerates at $r=r_*$. Though the presence of the $r^{\alpha }$-weights in our Morawetz type estimate appear to be new even in the particular case of the standard scalar wave equation, they were clearly inspired by the work of Dafermos-Rodnianski [11]. The main new idea in [11] was to observe that one can replace the (t, r) weights of the classical conformal multiplier method, along outgoing null hypersurfaces, by weights which depend only on r, provided that one has already derived a local decay estimate. The new twist in our work is to show that similar estimates can be derived on spacelike hypersurfaces. Unlike in the case of [11], where the proof of r-weighted estimates are can be neatly separated from the main local decay estimate, we are obliged in our work to prove them simultaneously. Proving a simultaneous estimate, on both the space-time integral, requires much more careful choices of the multipliers at infinity.

Proof of Theorem 1.3

In this section we give a first, informal, derivation of Theorem 1.3, based on first principles, which can be easily generalized to other situations. In the next section we shall re-derive the result by a straightforward verification.

Observe first that the linear system (1.14) is derivable from a Lagrangian⁸ ${\mathcal L}[\mathring{\Phi }, \Psi ]$, $\mathring{\Phi }=(\mathring{\Phi }^1, \mathring{\Phi }^2)=(A,B)$, $\Psi =(\Psi ^1, \Psi ^2)=(\phi ,\psi )$, defined as follows:

$$\begin{aligned} {\mathcal L}[\mathring{\Phi }, \Psi ] =\mathbf{g}^{\mu \nu }\big [ \mathring{\mathbf{D}}_\mu \phi \mathring{\mathbf{D}}_\nu \phi + \mathring{\mathbf{D}}_\mu \psi \mathring{\mathbf{D}}_\nu \psi +A^{-2} (\phi {\partial }_\mu B -\psi {\partial }_\mu A)(\phi {\partial }_\nu B -\psi {\partial }_\nu A)\big ] \end{aligned}$$

(1.15)

with,

$$\begin{aligned} \mathring{\mathbf{D}}_\mu \phi= & {} {\partial }_\mu \phi + A^{-1} {\partial }_\mu B \, \psi \qquad \mathring{\mathbf{D}}_\mu \psi ={\partial }_\mu \psi - A^{-1} {\partial }_\mu B \, \phi . \end{aligned}$$

We then define, as ususal, the energy momentum tensor of the linearized field equation to be the quantity,

$$\begin{aligned} {\mathcal Q}[\mathring{\Phi }, \Psi ]_{\mu \nu }:= & {} \frac{{\partial }{\mathcal L}}{{\partial }g^{\mu \nu }}-\frac{1}{2} \mathbf{g}_{\mu \nu }{\mathcal L}. \end{aligned}$$

(1.16)

We also define the source:

$$\begin{aligned} {\mathcal J}[\mathring{\Phi }, \Psi ]_\mu :=2\frac{{\partial }{\mathcal L}[\psi ]}{{\partial }\mathring{\Phi }^c}{\partial }_\mu \mathring{\Phi }^c,\qquad c=1,2. \end{aligned}$$

(1.17)

Note that, in view of the stationarity of $\mathring{\Phi }$,

$$\begin{aligned} \mathbf{T}^\mu {\mathcal J}_\mu =2\frac{{\partial }{\mathcal L}[\psi ]}{{\partial }\mathring{\Phi }^c} \mathbf{T}^\mu {\partial }_\mu \mathring{\Phi }^c=0. \end{aligned}$$

Lemma 1.5

We have the local conservation law:

$$\begin{aligned} \mathbf{D}^\nu {\mathcal Q}_{\mu \nu }= {\mathcal J}_\mu . \end{aligned}$$

Proof

Let $\chi _s$ be the one-parameter group of local diffeomorphisms generated by a given vector-field X. We shall use the flow $\chi$ to vary the fields $\Psi$ according to

$$\begin{aligned} \mathbf{g}_s = (\chi _s)_{*} \mathbf{g},\quad \psi _s= (\chi _s)_{*}\Psi , \quad \phi _s=(\chi _s)_{*}\mathring{\Phi }. \end{aligned}$$

From the invariance of the action integral under diffeomorphisms,   ${\mathcal S}(s) = {\mathcal S}[\Psi _s,\mathbf{g}_s,\mathring{\Phi }_s] = {\mathcal S}[\Psi ,\mathbf{g};\mathring{\Phi }].$ Therefore,

$$\begin{aligned} 0= & {} \frac{d}{ds}{\mathcal S}(s)\Bigm |_{s=0} =\int \frac{{\partial }{\mathcal L}}{{\partial }\Psi ^a} X(\Psi ^a)dv_{\mathbf{g}} \\&+\int \bigg (\frac{{\partial }{\mathcal L}}{\mathbf{g}^{\mu \nu }}-\frac{1}{2} \mathbf{g}_{\mu \nu }\, ^{\text{(lin) }} {\mathcal L}\bigg )\dot{\mathbf{g}}_{\mu \nu } dv_\mathbf{g}+\int \frac{{\partial }{\mathcal L}}{{\partial }\mathring{\Phi }^a} \dot{\mathring{\Phi }}^a dv_{\mathbf{g}} \\= & {} \int {\mathcal Q}^{\mu \nu } (\mathbf{D}_\mu X_\nu +\mathbf{D}_\nu X_\mu ) dv_{\mathbf{g}} +2\int {\mathcal J}^\mu X_\mu dv_\mathbf{g}\\= & {} -2\int \mathbf{D}_\nu {\mathcal Q}^{\mu \nu } X_\mu dv_\mathbf{g}+2\int {\mathcal J}^\mu X_\mu dv_\mathbf{g}\end{aligned}$$

Since the vector-field $X^\mu$ is arbitrary we deduce,

$$\begin{aligned} -\mathbf{D}_\nu {\mathcal Q}^{\mu \nu }+ {\mathcal J}^\mu =0 \end{aligned}$$

as desired. $\square$

In view of the definitions of ${\mathcal Q}$ and ${\mathcal L}$ we can write

$$\begin{aligned} {\mathcal Q}_{\mu \nu }={\mathcal T}_{\mu \nu }-\frac{1}{2} \mathbf{g}_{\mu \nu } ( \text{ tr }_\mathbf{g}{\mathcal T}), \qquad {\mathcal T}_{\mu \nu }:=E_\mu E_\nu +F_\mu F_\nu +M_\mu M_\nu \end{aligned}$$

(1.18)

where,

$$\begin{aligned}&E_\mu := \mathring{\mathbf{D}}_\mu \phi = {\partial }_\mu \phi +\psi A^{-1}{\partial }_\mu B,\,\,\, F_\mu :=\mathring{\mathbf{D}}_\mu \psi ={\partial }_\mu \psi -\phi A^{-1}{\partial }_\mu B,\,\,\,\\&M_\mu := A^{-1}(\phi {\partial }_\mu B-\psi {\partial }_\mu A). \end{aligned}$$

The positivity property (a) is now an immediate consequence of the structure (1.18) of the energy momentum tensor. Indeed, it is well known that at every point of the ergoregion where $r>r_H$, there exists a linear combination of $\mathbf{T}$ and $\mathbf{Z}$, $\mathbf{T}+c \mathbf{Z}$, which is timelike. Therefore, since $\mathbf{T}\cdot E=\mathbf{T}\cdot F=\mathbf{T}\cdot M=0$,

$$\begin{aligned} 0<{\mathcal T}(\mathbf{T}+c\mathbf{Z}, X)={\mathcal T}(\mathbf{T}, X). \end{aligned}$$

On the other hand, since X is orthogonal to $\mathbf{Z}$,

$$\begin{aligned} \mathbf{g}(\mathbf{T}+cZ, X)=\mathbf{g}(\mathbf{T}, X). \end{aligned}$$

Hence,

$$\begin{aligned} 0<{\mathcal Q}(\mathbf{T}+c\mathbf{Z}, X)={\mathcal Q}(\mathbf{T}, X), \end{aligned}$$

as desired.

New Coordinates

As well known the Boyer-Lindquist coordinates, are singular near the horizon and as such are not appropriate for our task. To avoid this difficulty it has become standard to define a new set of variables which are well behaved across the horizon and coincide with the Boyer-Lindquist coordinates away from it.

We fix first a smooth function $\chi :\mathbb {R}\rightarrow [0,1]$ supported in the interval $(-\infty ,5M/2]$ and equal to 1 in the interval $(-\infty ,9M/4]$, and define $g_1,g_2:({r_{\mathcal {H}}},\infty )\rightarrow \mathbb {R}$ such that

$$\begin{aligned} g_1'(r)=\chi (r)\frac{2Mr}{\Delta },\qquad g_2'(r)=\chi (r)\frac{a}{\Delta }. \end{aligned}$$

(1.19)

We define the functions

$$\begin{aligned} t_+:=t+g_1(r),\qquad \phi _+:=\phi +g_2(r). \end{aligned}$$

(1.20)

Therefore

$$\begin{aligned} dt_+=dt+\chi (r)\frac{2Mr}{\Delta }dr,\qquad d\phi _+=d\phi +\chi (r)\frac{a}{\Delta }dr. \end{aligned}$$

In $(\theta ,r,t_+,\phi _+)$ coordinates, the metric $\mathbf{g}$ becomes⁹

$$\begin{aligned} \mathbf{g}= & {} q^2(d\theta )^2+\Big [\frac{q^2}{\Delta }(1-\chi ^2(r))+\frac{2Mr+q^2}{q^2}\chi ^2(r)\Big ](dr)^2\nonumber \\&+\,2\chi (r)\frac{2Mr}{q^2}drdt_+-2\chi (r)\frac{a(\sin \theta )^2(q^2+2Mr)}{q^2}drd\phi _+\nonumber \\&+\frac{2Mr-q^2}{q^2}(dt_+)^2-\frac{4aMr(\sin \theta )^2}{q^2}dt_+d\phi _++\frac{\Sigma ^2(\sin \theta )^2}{q^2}(d\phi _+)^2. \end{aligned}$$

(1.21)

Let

$$\begin{aligned} \partial _1=\partial _\theta =\frac{d}{d\theta },\qquad \partial _2= \partial _r=\frac{d}{dr},\qquad \partial _3=\partial _t=\frac{d}{dt_+}= \mathbf{T},\qquad \partial _4=\partial _\phi =\frac{d}{d\phi _+}=\mathbf{Z}. \end{aligned}$$

(1.22)

The nontrivial components of the metric $\mathbf{g}$ are

$$\begin{aligned} \begin{aligned}&\mathbf{g}_{11}=q^2,\qquad \mathbf{g}_{33}=\frac{2Mr-q^2}{q^2},\qquad \mathbf{g}_{34}=-\frac{2aMr(\sin \theta )^2}{q^2},\qquad \mathbf{g}_{44}=\frac{\Sigma ^2(\sin \theta )^2}{q^2},\\&\mathbf{g}_{22}=\frac{q^2}{\Delta }(1-\chi ^2(r))+\frac{2Mr+q^2}{q^2}\chi ^2(r),\\&\mathbf{g}_{23}=\chi (r)\frac{2Mr}{q^2},\qquad \mathbf{g}_{24}=-\chi (r)\frac{a(\sin \theta )^2(q^2+2Mr)}{q^2}. \end{aligned} \end{aligned}$$

(1.23)

The metric $\mathbf{g}$ extends smoothly to the larger open set

$$\begin{aligned} \widetilde{{\mathcal R}}=\{(\theta ,r,t_+,\phi _+)\in (-\pi ,\pi )\times (0,\infty ) \times \mathbb {R}\times \mathbb {S}^1\}. \end{aligned}$$

For $t\in \mathbb {R}$ and $c\in (0,\infty )$ let

$$\begin{aligned} \Sigma ^c_t:=\{(\theta ,r,t_+,\phi _+)\in \widetilde{R}:t_+=t\text { and }r>c\}. \end{aligned}$$

(1.24)

The surfaces $\Sigma _t^{{r_{\mathcal {H}}}}$, $t\in \mathbb {R}$, form a $\mathbf{Z}$-invariant foliation of spacelike surfaces of the domain of outer communications of the Kerr spacetime $\mathcal {K}(M,a)$. Moreover, the foliation is compatible with the Killing vector-field $\mathbf{T}$, i. e. $\Phi _{t_1}(\Sigma ^c_{t_2})=\Sigma ^c_{t_1+t_2}$ for any $t_1,t_2\in \mathbb {R}$, where $\Phi _t$ denotes the flow associated to $\mathbf{T}$.

As mentioned earlier we are interested in solutions of the form (1.12), i.e., $\Phi =(A',B')=(A,B)+\varepsilon (A\phi ,A\psi )$ of the wave-map equation (1.10), in causal domains of the form

$$\begin{aligned} \mathcal {D}^{c}_I:=\cup _{t\in I}\Sigma ^{c}_t=\{(\theta ,r,t_+,\phi _+)\in \widetilde{{\mathcal R}}:t_+\in I\text { and }r>c\}, \end{aligned}$$

(1.25)

where $I\subseteq \mathbb {R}$ is an interval and $c<{r_{\mathcal {H}}}$. Notice that if $c<{r_{\mathcal {H}}}$ then $\mathcal {D}^{c}_\mathbb {R}$ contains a small neighborhood of the future event horizon $\mathcal {H}^+$ as well as the entire domain of outer communication. For any $c\in (0,\infty )$ and any interval $I\subseteq \mathbb {R}$ let

$$\begin{aligned} \mathcal {N}^{c}_I:=\{(\theta ,r,t_+,\phi _+)\in \widetilde{R}:t_+\in I\text { and }r=c\}. \end{aligned}$$

(1.26)

Notice that the hypersurfaces $\mathcal {N}^{c}_I$ are spacelike if $c<{r_{\mathcal {H}}}$, null (and contained in the future event horizon $\mathcal {H}^+$) if $c={r_{\mathcal {H}}}$, and timelike if $c>{r_{\mathcal {H}}}$.

Precise Version of Our Second Theorem

We define now our main function spaces:

Definition 1.6

For any $m\in \mathbb {Z}^+$, $c\in (0,\infty )$, and $t\in \mathbb {R}$ let $H^m(\Sigma _t^c)$ denote the usual $L^2$-based Sobolev space of functions on the hypersurface $\Sigma _t^c$, with respect to the induced Kerr metric (see (1.23)). Let

$$\begin{aligned} \widetilde{H}^m(\Sigma _t^c):= & {} \Bigg \{f:\Sigma _t^c\rightarrow \mathbb {R}: \,\Vert f\Vert _{\widetilde{H}^m(\Sigma _t^c)}:=\Vert f\Vert _{H^m(\Sigma _t^c)}\nonumber \\&+ \sum _{m'+m''=1}^m\Vert (\widetilde{\partial }_1/r)^{m'}\widetilde{\partial }_2^{m''} f\Vert _{L^2(\Sigma _t^c)}<\infty \Bigg \}, \end{aligned}$$

(1.27)

where, by definition,

$$\begin{aligned} \widetilde{\partial }_1g:=\Big (\partial _1-\frac{2\cos \theta }{\sin \theta }\Big )g,\qquad \widetilde{\partial }_2g:=\partial _2g. \end{aligned}$$

(1.28)

For any $g\in C^1(\Sigma _t^c)$ satisfying $\mathbf{Z}(g)=0$ let

$$\begin{aligned} \nabla g:=(\partial _1g/r,\partial _2g),\qquad \widetilde{\nabla } g:=(\widetilde{\partial }_1g/r,\widetilde{\partial }_2g). \end{aligned}$$

(1.29)

Finally, let

$$\begin{aligned} \mathbf {H}^m(\Sigma _t^c):= & {} \{(\phi ,\psi ):\Sigma _t^c\rightarrow \mathbb {R}\times \mathbb {R}:\,\Vert (\phi ,\psi )\Vert _{\mathbf {H}^m(\Sigma _t^c)}\nonumber \\:= & {} \Vert \phi \Vert _{H^m(\Sigma _t^c)}+ \Vert \psi \Vert _{\widetilde{H}^m(\Sigma _t^c)}<\infty \}. \end{aligned}$$

(1.30)

For any $R\ge 33M/16$ let $\chi _{\ge R}:[0,\infty )\rightarrow [0,1]$ denote a smooth increasing function supported in $[R,\infty )$, equal to 1 in $[2R-2M,\infty )$, and satisfying the natural differential inequalities. Let

$$\begin{aligned} L:=\chi _{\ge 4M}(r)\Big (\partial _2+\frac{r}{r-2M}\partial _3\Big ), \end{aligned}$$

(1.31)

For any $t\in \mathbb {R}$ and $(\phi ,\psi )\in \mathbf {H}^1(\Sigma _t^c)$ we define the outgoing energy density $(e(\phi ),e(\psi ))$,

$$\begin{aligned} \begin{aligned}&e(\phi )^2:=\frac{(\partial _1\phi )^2}{r^2}+(L\phi )^2+\frac{M^2\big [(\partial _2\phi )^2+(\partial _3\phi )^2\big ]}{r^2}+\frac{\phi ^2}{r^2},\\&e(\psi )^2:=\frac{(\partial _1\psi )^2+\psi ^2(\sin \theta )^{-2}}{r^2}+(L\psi )^2+\frac{M^2\big [(\partial _2\psi )^2+(\partial _3\psi )^2\big ]}{r^2}+\frac{\psi ^2}{r^2}. \end{aligned} \end{aligned}$$

(1.32)

We work in the axially symmetric case, therefore the relevant trapped null geodesics are still confined to a codimension 1 set. Assuming that $a\ll M$, it is easy to see that the equation

$$\begin{aligned} r^3-3Mr^2+a^2r+Ma^2=0 \end{aligned}$$

has a unique solution $r^*\in (M,\infty )$. Moreover, $r^*\in [3M-a^2/M,3M]$.

Theorem 1.7

Assume that $M\in (0,\infty )$, $N_0:=4$, $a\in [0,\overline{\varepsilon }M]$ and $c_0\in [{r_{\mathcal {H}}}-\overline{\varepsilon }M,{r_{\mathcal {H}}}]$, where $\overline{\varepsilon }\in (0,1]$ is a sufficiently small constant. Assume that $T\ge 0$, and $(\phi ,\psi )\in C^k([0,T]:\mathbf {H}^{N_0-k}(\Sigma _t^{c_0}))$, $k\in [0,N_0]$, is a solution of the system

$$\begin{aligned} \begin{aligned} \square \phi +2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \psi -2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi&=\mathcal {N}_\phi ,\\ \square \psi -2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \phi -\frac{\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\psi&=\mathcal {N}_\psi , \end{aligned} \end{aligned}$$

(1.33)

satisfying

$$\begin{aligned} \mathbf{Z}(\phi ,\psi )=0\qquad \text { in }\mathcal {D}^{c_0}_{[0,T]}. \end{aligned}$$

(1.34)

Then, for any $\alpha \in (0,2)$ and any $t_1\le t_2\in [0,T]$,

$$\begin{aligned}&\mathcal {B}_\alpha ^{c_0}(t_1,t_2)+\int _{\Sigma _{t_2}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t\nonumber \\&\quad \le \overline{C}_\alpha \int _{\Sigma _{t_1}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t\nonumber \\&\qquad +\,\overline{C}_\alpha \int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }} \big [e(\phi ,\mathcal {N}_\phi )+e(\psi ,\mathcal {N}_\psi )\big ]\,d\mu , \end{aligned}$$

(1.35)

where $\overline{C}_\alpha$ is a large constant that may depend on $\alpha$,

$$\begin{aligned} \mathcal {B}_\alpha ^{c_0}(t_1,t_2):= & {} \int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }}\Bigg \{\frac{(r-r^*)^2}{r^3}\frac{|\partial _1\phi |^2+|\partial _1\psi |^2+\psi ^2(\sin \theta )^{-2}}{r^2}\nonumber \\&+\frac{1}{r}\big [(L\phi )^2+(L\psi )^2\big ]+\frac{1}{r^3}\big (\phi ^2+\psi ^2\big )+\frac{M^{2}}{r^{3}}\big [(\partial _2\phi )^2+(\partial _2\psi )^2\big ]\nonumber \\&+\frac{M^2(r-r^*)^2}{r^5}\big [(\partial _3\phi )^2+(\partial _3\psi )^2\big ]\Bigg \}\,d\mu , \end{aligned}$$

(1.36)

and, for $f\in \{\phi ,\psi \}$,

$$\begin{aligned} e(f,\mathcal {N}_f):=|\mathcal {N}_f|\Big [(Lf)^2+\frac{M^2\big [(\partial _2f)^2+(\partial _3f)^2\big ]+f^2}{r^2}\Big ]^{1/2}. \end{aligned}$$

(1.37)

The point of proving an energy estimate such as (1.35) involving outgoing energies is that it leads directly to decay estimates. For example, we have the following corrolary:

Corollary 1.8

Assume that $N_1=8$ and $(\phi ,\psi )\in C^k([0,T]:\mathbf {H}^{N_1-k}(\Sigma _t^{c_0}))$, $k\in [0,N_1]$, is a solution of the system (1.33) with $\mathcal {N}_\phi =\mathcal {N}_\psi =0$. Then, for any $t\in [0,T]$ and $\beta <2$,

$$\begin{aligned}&\int _{\Sigma _{t}^{c_0}}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t\nonumber \\&\quad \lesssim _{\beta } (1+t/M)^{-\beta }\sum _{k=0}^4M^{2k}\int _{\Sigma _{0}^{c_0}}\frac{r^2}{M^2}\big [e(\mathbf{T}^k\phi )^2+e(\mathbf{T}^k\psi )^2\big ]\,d\mu _t. \end{aligned}$$

(1.38)

The point of the corollary is the almost $(1+t/M)^{-2}$ decay of the outgoing energy on the hypersurface $\Sigma _{t}^{c_0}$, in terms of initial data; such a decay is not possible, of course, for the standard energy. One can further commute the equation with the vector-field $\partial _r$ and use elliptic estimates to prove control decay of higher order outgoing energies as well. Such estimates, with improved decay, can then be combined, in principle, with a bootstrap argument to analyze globally the full semilinear system and prove the Partial Stability Conjecture in the case $a\ll M$. Note that the precise form of the system is given in Proposition 2.1; the nonlinearities $\mathcal {N}_\phi ^\varepsilon$ and $\mathcal {N}_\psi ^\varepsilon$ are quadratic and appear to satisfy suitable null conditions which are needed to prove global existence.

The explicit loss of derivatives of the estimate (1.38) can be improved; however some loss is unavoidable due to the degeneracy of the bulk integral at $r=r^*$ in (1.35). We note that the analogous decay estimate for the standard wave equation in Minkowski space follows, with ${\beta }=2$ and without the loss of derivatives, from the conservation of the conformal energy (see, for example, section 3 in [17]).

Conclusions

The estimates presented in this paper offer convincing evidence for the validity of our conjecture. Further work is needed to remove the smallness condition for a / M, provide sufficiently strong pointwise decay estimate in the wave zone region and implement the standard approach for proving global existence results for nonlinear wave equations which satisfy the null condition¹⁰.

Organization

The rest of the paper is organized as follows. In section 2 we derive the main identities in the paper, including the precise form of the system and the divergence identities; this provides an alternative explicit proof of Theorem 1.3. In section 3 we give an outline of the proof of the main theorem in the simplified case (1.3). In sections 4 and 5 we give a complete proof of the main Theorem 1.7, first in the case of the pure wave equation on the Schwarzschild space, and then for the full system on the Kerr spaces. In section 6 we provide a proof of Corollary 1.8, using Theorem 1.7 and an elliptic estimate. Finally, the appendix contains several explicit calculations in Kerr spaces, some Hardy inequalities, and some properties of the modified Sobolev spaces $\widetilde{H}^m$.

Derivation of the Main Algebraic Identities. Theorem 1.3 Revisited

Assume that $(A',B')=(A,B)+(\varepsilon A\phi ,\varepsilon A\psi )$ is a solution of the wave-map equation (1.10) on some interval I, where $(\phi ,\psi )\in C^k(I:\mathbf {H}^{N_1-k}(\Sigma _t^{c_0}))$, $k=0,\ldots ,N_1$. The functions $(\phi ,\psi )$ satisfy the system

$$\begin{aligned}&A^2\square \phi +2A\mathbf{D}^\mu B\mathbf{D}_\mu \psi -2\mathbf{D}^\mu B\mathbf{D}_\mu B\phi +2\mathbf{D}^\mu B\mathbf{D}_\mu A\psi \\&\quad +\,\varepsilon \big [A\phi \square (A\phi )-\mathbf{D}^\mu (A\phi )\mathbf{D}_\mu (A\phi )+\mathbf{D}^\mu (A\psi )\mathbf{D}_\mu (A\psi )\big ]=0, \end{aligned}$$

and

$$\begin{aligned}&A^2\square \psi -2A\mathbf{D}^\mu B\mathbf{D}_\mu \phi -(\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B)\psi \\&\quad +\,\varepsilon \big [A\phi \square (A\psi )-2\mathbf{D}^\mu (A\phi )\mathbf{D}_\mu (A\psi )]=0. \end{aligned}$$

Using the formulas (1.10) these equations become

$$\begin{aligned}&A^2(1+\varepsilon \phi )\square \phi +2A\mathbf{D}^\mu B\mathbf{D}_\mu \psi -2\mathbf{D}^\mu B\mathbf{D}_\mu B\phi +2\mathbf{D}^\mu B\mathbf{D}_\mu A\psi \\&\quad +\,\varepsilon \big [A^2\mathbf{D}^\mu \psi \mathbf{D}_\mu \psi +2A\psi \mathbf{D}^\mu A\mathbf{D}_\mu \psi +\mathbf{D}^\mu A\mathbf{D}_\mu A\psi ^2\\&\quad -\,A^2\mathbf{D}^\mu \phi \mathbf{D}_\mu \phi -\mathbf{D}^\mu B\mathbf{D}_\mu B\phi ^2\big ]=0, \end{aligned}$$

and

$$\begin{aligned}&A^2(1+\varepsilon \phi )\square \psi -2A\mathbf{D}^\mu B\mathbf{D}_\mu \phi -(\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B)\psi \\&\quad +\,\varepsilon \big [-2A^2\mathbf{D}^\mu \phi \mathbf{D}_\mu \psi -\mathbf{D}^\mu A\mathbf{D}_\mu A\phi \psi -\mathbf{D}^\mu B\mathbf{D}_\mu B\phi \psi -2A\psi \mathbf{D}^\mu A\mathbf{D}_\mu \phi ]=0. \end{aligned}$$

We divide the equations by $A^2(1+\varepsilon \phi )$ to conclude that

$$\begin{aligned} \begin{aligned}&\square \phi +2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \psi -2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi =\varepsilon \mathcal {N}_\phi ^\varepsilon ,\\&\square \psi -2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \phi -\frac{\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\psi =\varepsilon \mathcal {N}_\psi ^\varepsilon , \end{aligned} \end{aligned}$$

(2.1)

where

$$\begin{aligned} \mathcal {N}_\phi ^\varepsilon= & {} \frac{A^2\mathbf{D}^\mu \phi \mathbf{D}_\mu \phi -A^2\mathbf{D}^\mu \psi \mathbf{D}_\mu \psi -2A\psi \mathbf{D}^\mu A\mathbf{D}_\mu \psi +\mathbf{D}^\mu B\mathbf{D}_\mu B\phi ^2 -\mathbf{D}^\mu A\mathbf{D}_\mu A\psi ^2}{A^2(1+\varepsilon \phi )}\\&+\frac{\phi }{A^2(1+\varepsilon \phi )}[2A\mathbf{D}^\mu B\mathbf{D}_\mu \psi -2\mathbf{D}^\mu B\mathbf{D}_\mu B\phi +2\mathbf{D}^\mu B\mathbf{D}_\mu A\psi ], \end{aligned}$$

and

$$\begin{aligned} \mathcal {N}_\psi ^\varepsilon= & {} \frac{2A^2\mathbf{D}^\mu \phi \mathbf{D}_\mu \psi +(\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B)\phi \psi +2A\psi \mathbf{D}^\mu A\mathbf{D}_\mu \phi }{A^2(1+\varepsilon \phi )}\\&-\frac{\phi }{A^2(1+\varepsilon \phi )}[2A\mathbf{D}^\mu B\mathbf{D}_\mu \phi +(\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B)\psi ]. \end{aligned}$$

The formulas for the nonlinear terms $\mathcal {N}_\phi ^\varepsilon$ and $\mathcal {N}_\psi ^\varepsilon$ can be simplified, and the calculations can be reversed. To summarize, we have proved the following:

Proposition 2.1

Assume $I\subseteq \mathbb {R}$ is an interval, $\varepsilon >0$, and $(\phi ,\psi )\in C^k(I:\mathbf {H}^{N_1-k}(\Sigma _t^{c_0}))$, $k=0,\ldots ,N_1$. Then $(A',B')=(A,B)+(\varepsilon A\phi ,\varepsilon A\psi )$ is a solution of the wave-map equation (1.10) on the interval I if and only if $(\phi ,\psi )$ satisfy the nonlinear system

$$\begin{aligned} \begin{aligned}&\square \phi +2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \psi -2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi =\varepsilon \mathcal {N}_\phi ^\varepsilon ,\\&\square \psi -2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \phi -\frac{\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\psi =\varepsilon \mathcal {N}_\psi ^\varepsilon , \end{aligned} \end{aligned}$$

(2.2)

where

$$\begin{aligned} \begin{aligned}&\mathcal {N}_\phi ^\varepsilon =\frac{A^2(\mathbf{D}^\mu \phi \mathbf{D}_\mu \phi -\mathbf{D}^\mu \psi \mathbf{D}_\mu \psi )+(\phi \mathbf{D}^\mu B-\psi \mathbf{D}^\mu A) (2A\mathbf{D}_\mu \psi -\phi \mathbf{D}_\mu B+\psi \mathbf{D}_\mu A)}{A^2(1+\varepsilon \phi )},\\&\mathcal {N}_\psi ^\varepsilon =\frac{2A^2\mathbf{D}^\mu \phi \mathbf{D}_\mu \psi +2A(\psi \mathbf{D}^\mu A-\phi \mathbf{D}^\mu B)\mathbf{D}_\mu \phi }{A^2(1+\varepsilon \phi )}. \end{aligned} \end{aligned}$$

(2.3)

The Energy–Momentum Tensor

We study now solutions of the system

$$\begin{aligned} \begin{aligned} \square \phi +2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \psi -2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi&=\mathcal {N}_\phi ,\\ \square \psi -2\frac{\mathbf{D}^\mu B}{A}\mathbf{D}_\mu \phi -\frac{\mathbf{D}^\mu A\mathbf{D}_\mu A+\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\psi&=\mathcal {N}_\psi . \end{aligned} \end{aligned}$$

(2.4)

Our main goal is to construct a suitable energy-momentum tensor that verifies a good divergence equation. More precisely, let

$$\begin{aligned} E_\mu :=\mathbf{D}_\mu \phi +\psi A^{-1}\mathbf{D}_\mu B,\quad F_\mu :=\mathbf{D}_\mu \psi -\phi A^{-1}\mathbf{D}_\mu B,\quad M_\mu :=\frac{\phi \mathbf{D}_\mu B-\psi \mathbf{D}_\mu A}{A}. \end{aligned}$$

(2.5)

Using the formulas

$$\begin{aligned} A\mathbf{D}_\mu \phi =AE_\mu -\psi \mathbf{D}_\mu B,\quad A\mathbf{D}_\mu \psi =AF_\mu +\phi \mathbf{D}_\mu B, \end{aligned}$$

(2.6)

the identities (2.4) and (1.10) show that

$$\begin{aligned} \begin{aligned}&\mathbf{D}^\mu E_\mu +\frac{\mathbf{D}^\mu BF_\mu }{A}-\frac{\mathbf{D}^\mu BM_\mu }{A}=\mathcal {N}_\phi ,\\&\mathbf{D}^\mu F_\mu -\frac{\mathbf{D}^\mu BE_\mu }{A}+\frac{\mathbf{D}^\mu AM_\mu }{A}=\mathcal {N}_\psi ,\\&\mathbf{D}^\mu M_\mu -\frac{\mathbf{D}^\mu BE_\mu }{A}+\frac{\mathbf{D}^\mu AF_\mu }{A}=0. \end{aligned} \end{aligned}$$

(2.7)

We also calculate

$$\begin{aligned} \begin{aligned} \mathbf{D}_\mu E_\nu -\mathbf{D}_\nu E_\mu&=\frac{F_\mu \mathbf{D}_\nu B-F_\nu \mathbf{D}_\mu B}{A}+\frac{M_\mu \mathbf{D}_\nu B-M_\nu \mathbf{D}_\mu B}{A},\\ \mathbf{D}_\mu F_\nu -\mathbf{D}_\nu F_\mu&=-\frac{E_\mu \mathbf{D}_\nu B-E_\nu \mathbf{D}_\mu B}{A}-\frac{M_\mu \mathbf{D}_\nu A-M_\nu \mathbf{D}_\mu A}{A},\\ \mathbf{D}_\mu M_\nu -\mathbf{D}_\nu M_\mu&=\frac{E_\mu \mathbf{D}_\nu B-E_\nu \mathbf{D}_\mu B}{A}-\frac{F_\mu \mathbf{D}_\nu A-F_\nu \mathbf{D}_\mu A}{A}.\\ \end{aligned} \end{aligned}$$

(2.8)

Let

$$\begin{aligned} \begin{aligned}&T_{\mu \nu }:=E_\mu E_\nu +F_\mu F_\nu +M_\mu M_\nu ,\\&Q_{\mu \nu }:=T_{\mu \nu }+\mathbf{g}_{\mu \nu }\mathcal {L},\\&\mathcal {L}:=-(1/2)\mathbf{g}^{\alpha {\beta }}T_{\alpha {\beta }}=-(1/2)(E_\alpha E^\alpha +F_\alpha F^\alpha +M_\alpha M^\alpha ). \end{aligned} \end{aligned}$$

(2.9)

We calculate the divergence

$$\begin{aligned} \mathbf{D}^\mu Q_{\mu \nu }= & {} E_\nu \mathbf{D}^\mu E_\mu +E^\mu (\mathbf{D}_\mu E_\nu -\mathbf{D}_\nu E_\mu )\\&+\,F_\nu \mathbf{D}^\mu F_\mu +F^\mu (\mathbf{D}_\mu F_\nu -\mathbf{D}_\nu F_\mu )\\&+\,M_\nu \mathbf{D}^\mu M_\mu +M^\mu (\mathbf{D}_\mu M_\nu -\mathbf{D}_\nu M_\mu ), \end{aligned}$$

Using (2.7) and (2.8) we calculate

$$\begin{aligned}&E_\nu \mathbf{D}^\mu E_\mu +E^\mu (\mathbf{D}_\mu E_\nu -\mathbf{D}_\nu E_\mu )\\&\quad =\frac{E_\nu (\mathbf{D}^\mu BM_\mu -\mathbf{D}^\mu BF_\mu )-F_\nu E^\mu \mathbf{D}_\mu B-M_\nu E^\mu \mathbf{D}_\mu B}{A}\\&\qquad +\frac{\mathbf{D}_\nu B(E^\mu F_\mu +E^\mu M_\mu )}{A}+\mathcal {N}_\phi E_\nu ,\\&F_\nu \mathbf{D}^\mu F_\mu +F^\mu (\mathbf{D}_\mu F_\nu -\mathbf{D}_\nu F_\mu )\\&\quad =\frac{E_\nu F^\mu \mathbf{D}_\mu B+F_\nu (\mathbf{D}^\mu BE_\mu -\mathbf{D}^\mu AM_\mu )+M_\nu F^\mu \mathbf{D}_\mu A }{A}\\&\qquad +\frac{-\mathbf{D}_\nu B E^\mu F_\mu -\mathbf{D}_\nu AF^\mu M_\mu }{A}+\mathcal {N}_\psi F_\nu , \end{aligned}$$

and

$$\begin{aligned}&M_\nu \mathbf{D}^\mu M_\mu +M^\mu (\mathbf{D}_\mu M_\nu -\mathbf{D}_\nu M_\mu )\\&\quad =\frac{-E_\nu M^\mu \mathbf{D}_\mu B+F_\nu M^\mu \mathbf{D}_\mu A +M_\nu (\mathbf{D}^\mu BE_\mu -\mathbf{D}^\mu AF_\mu ) }{A}\\&\qquad +\frac{\mathbf{D}_\nu BM^\mu E_\mu -\mathbf{D}_\nu AM^\mu F_\mu }{A}. \end{aligned}$$

Therefore

$$\begin{aligned} \mathbf{D}^\mu Q_{\mu \nu }=\frac{2\mathbf{D}_\nu BM^\mu E_\mu -2\mathbf{D}_\nu AM^\mu F_\mu }{A}+\mathcal {N}_\phi E_\nu +\mathcal {N}_\psi F_\nu . \end{aligned}$$

(2.10)

Divergence Identities

Given a vector-field X, a function w, and 1-forms $m,m'$ we define the form

$$\begin{aligned} P_\mu= & {} P_\mu [X,w,m,m']:=Q_{\mu \nu }X^\nu +\frac{1}{2}w(\phi E_\mu +\psi F_\mu )\nonumber \\&-\frac{1}{4}\mathbf{D}_\mu w(\phi ^2+\psi ^2)+\frac{1}{4}(m_\mu \phi ^2+m'_\mu \psi ^2). \end{aligned}$$

(2.11)

Then, using (2.5)–(2.7) we calculate the divergence

$$\begin{aligned} \mathbf{D}^\mu P_\mu= & {} X^\nu {\mathcal J}_\nu +\frac{1}{2}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }+\frac{1}{2}\mathbf{D}^\mu w(\phi E_\mu +\psi F_\mu )+\frac{1}{2}w(\mathbf{D}^\mu \phi E_\mu +\mathbf{D}^\mu \psi F_\mu )\\&+\,\frac{1}{2}w(\phi \mathbf{D}^\mu E_\mu +\psi \mathbf{D}^\mu F_\mu )-\frac{1}{4}\square w(\phi ^2+\psi ^2)-\frac{1}{2}\mathbf{D}^\mu w(\phi \mathbf{D}_\mu \phi +\psi \mathbf{D}_\mu \psi )\\&+\,\frac{1}{4}(\phi ^2\mathbf{D}^\mu m_\mu +\psi ^2\mathbf{D}^\mu m'_\mu )+\frac{1}{2}(\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi )\\= & {} X^\nu {\mathcal J}_\nu +\frac{1}{2}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }-\frac{1}{4}\square w(\phi ^2+\psi ^2)\\&+\,\frac{1}{4}(\phi ^2\mathbf{D}^\mu m_\mu +\psi ^2\mathbf{D}^\mu m'_\mu )+\frac{1}{2}(\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi )+E', \end{aligned}$$

where

$$\begin{aligned} E'= & {} \frac{1}{2}\mathbf{D}^\mu w(\phi E_\mu +\psi F_\mu -\phi \mathbf{D}_\mu \phi -\psi \mathbf{D}_\mu \psi )\\&+\,\frac{1}{2}w(\mathbf{D}^\mu \phi E_\mu +\mathbf{D}^\mu \psi F_\mu +\phi \mathbf{D}^\mu E_\mu +\psi \mathbf{D}^\mu F_\mu )\\= & {} 0+\frac{1}{2}w(E^\mu E_\mu +F^\mu F_\mu +M^\mu M_\mu +\phi \mathcal {N}_\phi +\psi \mathcal {N}_\psi ). \end{aligned}$$

Therefore

$$\begin{aligned} \mathbf{D}^\mu P_\mu= & {} X^\nu {\mathcal J}_\nu +\frac{1}{2}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }-\frac{1}{4}\square w(\phi ^2+\psi ^2)-w\mathcal {L}\nonumber \\&+\,\frac{1}{4}(\phi ^2\mathbf{D}^\mu m_\mu +\psi ^2\mathbf{D}^\mu m'_\mu )+\frac{1}{2}(\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi )\nonumber \\&+\,\frac{1}{2}w(\phi \mathcal {N}_\phi +\psi \mathcal {N}_\psi ). \end{aligned}$$

Summary

We summarize the results of the section in the following:

Proposition 2.2

(i) Assume that $(\phi ,\psi )\in C^k(I:\mathbf {H}^{N_0-k}(\Sigma _t^{c_0}))$, $k=0,\ldots ,N_0$ satisfy the system (2.4). Let

$$\begin{aligned} \begin{aligned}&E_\mu :=\mathbf{D}_\mu \phi +\psi A^{-1}\mathbf{D}_\mu B,\quad F_\mu :=\mathbf{D}_\mu \psi -\phi A^{-1}\mathbf{D}_\mu B,\quad M_\mu :=\frac{\phi \mathbf{D}_\mu B-\psi \mathbf{D}_\mu A}{A},\\&Q_{\mu \nu }:=E_\mu E_\nu +F_\mu F_\nu +M_\mu M_\nu +\mathbf{g}_{\mu \nu }\mathcal {L},\\&\mathcal {L}:=-\frac{1}{2}(E_\alpha E^\alpha +F_\alpha F^\alpha +M_\alpha M^\alpha ). \end{aligned} \end{aligned}$$

(2.12)

Then

$$\begin{aligned} \mathbf{D}^\mu Q_{\mu \nu }=:J_\nu =\frac{2\mathbf{D}_\nu BM^\mu E_\mu -2\mathbf{D}_\nu AM^\mu F_\mu }{A}+\mathcal {N}_\phi E_\nu +\mathcal {N}_\psi F_\nu . \end{aligned}$$

(2.13)

(ii) Let

$$\begin{aligned} P_\mu= & {} P_\mu [X,w,m,m']:=Q_{\mu \nu }X^\nu +\frac{1}{2}w(\phi E_\mu +\psi F_\mu )\nonumber \\&-\frac{1}{4}\mathbf{D}_\mu w(\phi ^2+\psi ^2)+\frac{1}{4}(m_\mu \phi ^2+m'_\mu \psi ^2), \end{aligned}$$

(2.14)

where X is a smooth vector-field, w is a smooth function, and $m,m'$ are smooth 1-forms. Then

$$\begin{aligned} 2\mathbf{D}^\mu P_\mu= & {} 2X^\nu J_\nu +Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }-2w\mathcal {L}+(\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi )\nonumber \\&+\frac{1}{2}\phi ^2(\mathbf{D}^\mu m_\mu -\square w)+\frac{1}{2}\psi ^2(\mathbf{D}^\mu m'_\mu -\square w)+w(\phi \mathcal {N}_\phi +\psi \mathcal {N}_\psi ). \end{aligned}$$

(2.15)

Note that theorem 1.3 is an immediate consequence of the first part of the proposition. Indeed, assuming that $({\mathcal N}_\phi , {\mathcal N}_\psi )=0$ it is immediate that J is orthogonal to $\mathbf{T}$. The positivity of the energy momentum tensor ${\mathcal Q}$ is an immediate consequence of its form (2.12).

Main Ideas in the Proof of Theorem 1.7

In this section we provide main ideas and motivation for the various choices we need to make in the proof of theorem 1.7. Our proof follows the well established pattern of proving integrated local energy decay estimates on black holes, such as Schwarzschild, for which the ergoregion is trivial and the trapped region is contained to a level surface $r=r^*>\,r_{{\mathcal H}}$. It is quite fortunate that our axially symmetric linearized system can be treated in the same manner. Though our treatment follows the clear and efficient approach of [19], we should point out that many of the ideas go back to other authors such as [5, 6, 10]. An essential ingredient in the proof is to take into account the red shift effects of the horizon, idea which goes back to [10].

In our problem we need to make two important modifications. Most importantly, to get any estimate at all, we need to account for the source term ${\mathcal J}$. This requires, in particular, a serious modification of the current $P_\mu$ in (2.15), modification which adds considerably to the complexity of the proof.

The second important modification has to do with the presence of weights in our main estimate. Typically, integrated decay estimates are designed to deal with the region close to the black hole, most importantly the trapping region. They are then complemented by weighted estimates in the asymptotic region. Thus, for example, J. Luk (see [18]), relies on an integrated local decay estimate (proved earlier by Dafermos-Rodnianski (see [12]) for small a / M), which he combines with weighted estimates in the asymptotic region based on a straightforward adaptation of the classical conformal method. The use of conformal method, however, is quite awkward in the black hole region, because the weights involved in the conformal method lead to errors which grow linearly in t. This problem was later fixed by a different method of Dafermos-Rodnianski in [11], who replace the conformal method by r-weighted estimates. The new method allows one to prove decay estimate for the energy associated to hypersurfaces which are spacelike near the black hole region but become null in the asymptotic region. This depends, however, on having first derived an integrated local decay estimate¹¹. In our work here we refine the analysis significantly by deriving r-weighted estimates for the outgoing energy across spacelike hypersurfaces, simultaneously with the integrated local decay estimates.

Outline of the Proof

We discuss now the main ideas in the proof. For simplicity, we consider only the equation for $\psi$ in the Schwarzschild case $a=0$, which carries most of the conceptual difficulties of the problem. In this case $B=0$, $A=r^2(\sin \theta )^2$, and the equation is

$$\begin{aligned} \square \psi -\frac{4-8(M/r)(\sin \theta )^2}{r^2(\sin \theta )^2}\psi =0. \end{aligned}$$

(3.1)

As in (2.2) we define

$$\begin{aligned} F_\mu :=\mathbf{D}_\mu \psi ,\quad M_\mu :=\frac{-\psi \mathbf{D}_\mu A}{A},\quad Q_{\mu \nu }:=F_\mu F_\nu +M_\mu M_\nu -\frac{1}{2}\mathbf{g}_{\mu \nu }(F_\alpha F^\alpha +M_\alpha M^\alpha ). \end{aligned}$$

(3.2)

For suitable triplets $(X,w,m')$ we define

$$\begin{aligned} \widetilde{P}_\mu =\widetilde{P}_\mu [X,w,m']:=Q_{\mu \nu }X^\nu +\frac{w}{2}\psi F_\mu -\frac{\psi ^2}{4}\mathbf{D}_\mu w+\frac{\psi ^2}{4}m'_\mu -\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi ^2. \end{aligned}$$

(3.3)

Notice the correction $-\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi ^2$, compared to the definition of P in (2.14), which is needed to partially compensate for the source term J. Then we have the divergence identity

$$\begin{aligned} 2\mathbf{D}^\mu \widetilde{P}_\mu =\sum _{j=1}^5 L^j, \end{aligned}$$

(3.4)

where

$$\begin{aligned} \begin{aligned}&L^1=L^1[X,w,m']:=Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }+w(F_\alpha F^\alpha +M_\alpha M^\alpha ),\\&L^2=L^2[X,w,m']:=\psi {m'}^\mu \mathbf{D}_\mu \psi ,\\&L^3=L^3[X,w,m']:=\frac{1}{2}\psi ^2(\mathbf{D}^\mu m'_\mu -\square w),\\&L^4=L^4[X,w,m']:=-2\mathbf{D}^\mu \Big [\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\Big ]\psi ^2. \end{aligned} \end{aligned}$$

(3.5)

The divergence identity gives

$$\begin{aligned} \int _{\Sigma _{t_1}^c} \widetilde{P}_\mu n_0^\mu \,d\mu _{t_1}=\int _{\Sigma _{t_2}^c}\widetilde{P}_\mu n_0^\mu \,d\mu _{t_2}+\int _{\mathcal {N}^c_{[t_1,t_2]}}\widetilde{P}_\mu k_0^\mu \,d\mu _c+\int _{\mathcal {D}^c_{[t_1,t_2]}}\mathbf{D}^\mu \widetilde{P}_\mu \,d\mu , \end{aligned}$$

(3.6)

where $t_1,t_2\in [0,T]$, $c\in (c_0,2M]$, $n_0:=n/|\mathbf{g}^{33}|^{1/2}$, $k_0:=k/|\mathbf{g}^{22}|^{1/2}$, and the integration is with respect to the natural measures induced by the metric $\mathbf{g}$. To prove the main theorem we need to choose a suitable multiplier triplet $(X,w,m')$ in a such a way that all the terms in the identity above are nonnegative. This is the method of simultaneous inequalities of Marzuola–Metcalfe–Tataru–Tohaneanu [19].

To accomplish our task we need to superimpose four different choices of multiplier triplets $(X, w, m')$, denoted $(X_{(k)},w_{(k)},m'_{(k)} )$, $k\in \{1,2,3,4\}$. The first multiplier ($k=1$) is important in a neighborhood of the trapped set $\{r=3M\}$; the second multiplier ($k=2$) is important in a neighborhood of the horizon $\{r=\,r_{{\mathcal H}}\}$; the third multiplier ($k=3$) is important at infinity, in the construction of outgoing energies at infinity; the fourth multiplier is important to control the term $L^4$, which is connected to the presence of the nontrivial potential in (3.1).

The Multipliers $(X_{(1)},w_{(1)},m'_{(1)} )$ and $(X_{(2)},w_{(2)},m'_{(2)} )$.

The first two multipliers are similar to the multipliers used in [19] in the case of the homogeneous wave equation. Set

$$\begin{aligned}&X_{(1)}:=f_1(r)\partial _2+g_1(r)\partial _3,\qquad f_1(r):=\frac{a_1(r)\Delta }{r^2},\qquad g_1(r):=\frac{a_1(r)\chi (r)2M}{r}+1,\\&w_{(1)}(r,\theta ):=f'_1(r)+f_1(r)\partial _r\log \big (r^4/\Delta )-\epsilon _1\widetilde{w}(r),\\&\widetilde{w}(r):=M^2(r-33M/16)^3(r-r^*)^2r^{-8}\mathbf {1}_{[33M/16,\infty )}(r),\\&m'_{(1)}:=0, \end{aligned}$$

where $r^*=3M$, $\epsilon _1\in (0,1]$ is a small constant, and $a_1:(0,\infty )\rightarrow \mathbb {R}$ is a smooth function. The important function $a_1$, which vanishes on the trapped region $\{r=r^*\}$, is defined by

$$\begin{aligned}&R(r):=(r-r^*)(r+2M)+6M^2\log \Big (\frac{r-2M}{r^*-2M}\Big ),\\&a_1(r):=r^{-2}\delta ^{-1}\kappa (\delta R(r))+\Big [\frac{r^*-2M}{r}-\frac{6M^2}{r^2}\log \Big (\frac{r-{r_{\mathcal {H}}}}{r^*-{r_{\mathcal {H}}}}\Big )\Big ]\chi _{\ge DM}(r), \end{aligned}$$

where D is a sufficiently large constant, $\delta =\epsilon _2^2M^{-2}$ for a small positive constant $\epsilon _2$, and $\kappa :\mathbb {R}\rightarrow \mathbb {R}$ is an increasing smooth function satisfying $\kappa (y)=y$ on $[-1,\infty )$ and $\kappa (y)=-2$ on $(-\infty ,-3]$. This is essentially the choice of [19], except for the correction at infinity, containing the cutoff function $\chi _{\ge DM}$; this correction is needed in order to match properly with the third multiplier at infinity to produce outgoing energies.

In a small neighborhood of the horizon we need to use the redshift effect. We define the second multiplier

$$\begin{aligned} \begin{aligned}&X_{(2)}:=f_2(r)\partial _2+g_2(r)\partial _3,\qquad f_2(r):=-\epsilon _2a_2(r),\qquad g_2(r):=\epsilon _2 a_2(r)(1-\epsilon _2),\\&w_{(2)}(r):=-2\epsilon _2 a_2(r)/r,\qquad m'_{(2)2}=m'_{(2)3}:=\epsilon _2M^{-2}\gamma (r),\qquad m'_{(2)1}=m'_{(2)4}:=0, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} a_2(r):={\left\{ \begin{array}{ll} M^{-3}(9M/4-r)^3&{}\text { if }r\le 9M/4,\\ 0&{}\text { if }r\ge 9M/4, \end{array}\right. } \end{aligned}$$

and $\gamma :[c_0,\infty )\rightarrow [0,1]$ is a function supported in $[c_0,17M/8]$, satisfying $\gamma (2M)=1/2$ and the more technical property (4.38). As in [19], the multipliers $(X_{(1)},w_{(1)},m'_{(1)} )$ and $(X_{(2)},w_{(2)},m'_{(2)} )$ cooperate well to generate mostly positive bulk contributions. More precisely, the constants $\epsilon _1, \epsilon _2$ can be chosen such that, for some absolute constant $\epsilon _3>0$,

$$\begin{aligned} \sum _{j=1}^4\big (L^j_{(1)}+L^j_{(2)}\big )\ge & {} \epsilon _3\sum _{Y\in \{F,M\}}\Big [\frac{(r-r^*)^2}{r^3} (Y_1/r)^2+\frac{M^2}{r^3}(Y_2)^2+\frac{M^2(r-r^*)^2}{r^5}(Y_3)^2\Big ]\nonumber \\&+\,\epsilon _3\frac{M}{r^4}\psi ^2-\epsilon _3^{-1}\frac{M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\psi ^2+\widetilde{L}, \end{aligned}$$

(3.7)

where

$$\begin{aligned} \widetilde{L}:= & {} \frac{8\Delta (r^2-4Mr)}{r^7}a_1(r) \psi ^2+(1-2C_1\epsilon _1)\mathbf {1}_{[r^*,\infty )}(r) \left\{ \frac{M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )\psi ^2\right. \nonumber \\&\left. +\frac{8a_1(r)(r-r^*)}{r^4}\frac{(\cos \theta )^2}{(\sin \theta )^2}\psi ^2+\frac{2a_1(r)(r-r^*)}{r^4}(F_1)^2+2a'_1(r)\frac{\Delta ^2}{r^4}(F_2)^2\right\} . \end{aligned}$$

(3.8)

Moreover, letting $\widetilde{P}_{(j)}:=\widetilde{P}_\mu [X_{(j)},w_{(j)},m'_{(j)}]$, $j=1,2$, we have

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) k^\mu\ge & {} \epsilon _3\sum _{Y\in \{F,M\}}\big [(Y_1/r)^2+(Y_2)^2(2-c/M)\big ]\nonumber \\&+\,\epsilon _3M^{-2}\psi ^2-\epsilon _3^{-1}(F_3)^2, \end{aligned}$$

(3.9)

along $\mathcal {N}^c_{[t_1,t_2]}$. Also, with $p=2M/r$,

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) n^\mu\ge & {} -\epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)(F_3)^2\big \}\nonumber \\&-\,\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\psi ^2)+\epsilon _3(F_2)^2\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(3.10)

and

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) n^\mu\le & {} \epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)(F_3)^2\big \}\nonumber \\&-\,\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\psi ^2)+\epsilon _3^{-1}(F_2)^2\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(3.11)

where

$$\begin{aligned} \widetilde{e}_0=\frac{(F_1)^2+(M_1)^2}{r^2}+(L\psi )^2+\frac{M^2|r-2M|}{r^3}(F_2)^2+ \frac{M^2}{r^2}(F_3)^2+\frac{1}{r^2}\psi ^2. \end{aligned}$$

Notice that the bulk terms in (3.7) are mostly positive, with the exception of the term $\widetilde{L}$. The terms along $\mathcal {N}^c_{[t_1,t_2]}$ are also mostly positive. On the other hand, the bounds (3.10) and (3.11) we have so far on the integrals along the hypersurfaces $\Sigma _t^{c}$ are very weak; these bounds will be improved by choosing a suitable multiplier $(X_{(3)},w_{(3)},m'_{(3)})$ at infinity.

The Multiplier $(X_{(4)},w_{(4)},m'_{(4)} )$

Our next goal is to control the term $\widetilde{L}$ in (3.8). This is a new term, when compared to solutions of the homogeneous wave equation, connected to the nontrivial potential in (3.1) and the bulk term $L^4$ in (3.5). Since $a'_1(r)\ge 0$ and $a_1(r)(r-r^*)\ge 0$, this term can only be problematic in the region $\{r\in [r^*,4M]\}$. We define

$$\begin{aligned} \begin{aligned}&X_{(4)}:=0,\qquad w_{(4)}:=0,\\&\widetilde{m}'_{(4)1}(r,\theta ):=-(1-2C_1\epsilon _1) \frac{8(r-r^*)a_1(r)\chi _{\le 6R}(r)}{r^2}\frac{\cos \theta }{\sin \theta }\mathbf {1}_{[r^*,\infty )}(r),\\&\widetilde{m}'_{(4)2}(r):=(1-2C_1\epsilon _1)\frac{2b(r)}{\Delta },\qquad \widetilde{m}'_{(4)3}:=0,\qquad \widetilde{m}'_{(4)4}:=0, \end{aligned} \end{aligned}$$

for a suitable function b supported in $[r^*,4M]$. Careful estimates, as in Lemma 5.3, and completion of squares show that one can choose the function b in such a way that

$$\begin{aligned} L^1_{(4)}=L^4_{(4)}=0,\qquad \widetilde{L}+L^2_{(4)}+L^3_{(4)}\ge -C_2|2M-c_0|r^{-4}\psi ^2 \end{aligned}$$

(3.12)

for some constant $C_2$ sufficiently large, and

$$\begin{aligned} \big |2\widetilde{P}_{(4)\mu }n^\mu \big |\lesssim \epsilon _3^{-1}\psi ^2/r^2\qquad \text { and }\qquad 2\widetilde{P}_{(4)\mu }k^\mu =0\,\,\text { along }\,\,\mathcal {N}^c_{[t_1,t_2]}. \end{aligned}$$

(3.13)

These two bounds can be combined with (3.7)–(3.11) to effectively remove the contribution of the term $\widetilde{L}$.

The Multiplier $(X_{(3)},w_{(3)},m'_{(3)} )$

Finally, we are ready to define the multiplier at infinity and close the estimate. First of all, to obtain any simultaneous estimate at all, we need to make sure that the contributions of the integrals of $2\widetilde{P}_\mu n^\mu$ on the hypersurfaces $\Sigma _t^c$ are positive. So far, these integrals are far from positive, in view of the estimates (3.10), (3.11), and (3.13).

The formula (A.16) shows that

$$\begin{aligned} 2n^\mu \widetilde{P}_\mu [K\partial _3,0,0]= & {} 2n^\mu Q_{\mu \nu }(K\partial _3)^\nu \\= & {} K\sum _{Y\in \{F,M\}}\big [\mathbf{g}^{11}(Y_1)^2+\mathbf{g}^{22}(Y_2)^2+(-\mathbf{g}^{33})(Y_3)^2\big ]. \end{aligned}$$

Therefore, one could make the integrals of $2\widetilde{P}_\mu n^\mu$ along the hypersurfaces $\Sigma _t^c$ positive by adding a multiplier of the form $(K\partial _3,0,0)$, for some positive constant K sufficiently large, and using a Hardy estimate to control the integral of the 0’s order term in terms of the first order terms. Notice that such a multiplier does not affect the bulk integrals. This is precisely the argument used in [19] to close the simultaneous estimate for the standard energy for the wave equation.

In our case, however, we are looking to prove stronger estimates involving outgoing energies. A multiplier of the form $(K\partial _3,0,0)$ is not allowed, since this would create contributions at infinity of the form $(F_2)^2+(F_3)^2$, which are unacceptable in view of the definition (1.32). Instead, we choose the last multiplier of the form

$$\begin{aligned} \begin{aligned}&X_{(3)}:=f_3\partial _2+\Big (\frac{f_3}{1-p}+g_3\Big )\partial _3,\qquad w_{(3)}:=\frac{2f_3}{r},\\&m'_{(3)1}:=m'_{(3)4}:=0,\qquad m'_{(3)2}:=\frac{2h_3}{r(1-p)},\qquad m'_{(3)3}:=-\frac{2h_3}{r}, \end{aligned} \end{aligned}$$

(3.14)

for some suitable functions $f_3,g_3,h_3$. The function $f_3$ should behave like $(r/M)^{\alpha }$ for large r, in order to produce the desired power in the outgoing energy. To make sure that it does not interfere with the crucial trapping region we have to choose it to vanish for $r\le 8M$. The role of the function $g_3$ is to match, to some extent, the role played by the multiplier $K\mathbf{T}$ in the boundary estimate discussed earlier. Thus we choose $g_3$ to be a very large constant when $r\le C_4M$, for some large constant $C_4$, but we choose it to decay as $r\rightarrow \infty$, at the rate $r^{\alpha -2}$, such that it does not interfere with the outgoing energy. Precise choices are provided in (5.59)–(5.61),

$$\begin{aligned} f_3(r):=\epsilon _4\chi _{\ge 8M}(r)e^{\beta (r)},\qquad g_3(r):=\int _{r}^\infty \Big [\rho (s)+\frac{\epsilon _4M^2}{s^3}f_3(s)\Big ]\,ds, \end{aligned}$$

where

$$\begin{aligned} \beta (8M):=0,\qquad \beta '(r):=\Big (\frac{4M}{r^2}+\frac{1}{r}\Big )\big (1-\chi _{\ge C_4^4M}(r)\big )+\frac{\alpha }{r}\chi _{\ge C_4^4M}(r), \end{aligned}$$

and

$$\begin{aligned} \rho (r):=\delta M^{-1}\Big [\chi _{\ge C_4M}(r)+\chi _{\ge 4C_4^4M}(r)\Big (C_4^7e^{\beta (r)}\frac{M^3}{r^3}-1\Big )\Big ]. \end{aligned}$$

The constants $\epsilon _4,C_4$ satisfy $\epsilon _4=\epsilon _3^2$ and $C_4\ge \epsilon _4^{-4}\alpha ^{-1}(2-\alpha )^{-1}$, while $\delta \in [10^{-4}C_4^{-3},10^4C_4^{-3}]$ is such that $\int _{C_4M}^\infty \rho (s)\,ds=C_4$.

The function $h_3$ can be chosen explicitly in terms of $f_3$ and $g_3$, in such a way to complete squares and create positive 0’s order contributions. The positivity of the bulk terms in (3.7) and (3.12), together with the choice $\epsilon _4\ll \epsilon _3$, is used to show positivity of the total bulk contribution in the transition region. Overall, we derive the desired lower bound on the bulk term,

$$\begin{aligned}&\sum _{j=1}^4\big (L^j_{(1)}+L^j_{(2)}+L^j_{(4)}+L^j_{(3)}\big )\gtrsim _\alpha e^{\beta }\left\{ \frac{(r-r^*)^2}{r^2}\frac{(\partial _1\psi )^2+(\psi /\sin \theta )^2}{r^3}\right. \nonumber \\&\quad \left. +\frac{M^2}{r^3}(\partial _2\psi )^2+\frac{M^2(r-r^*)^2}{r^5}(\partial _3\phi )^2+\frac{\psi ^2}{r^3}+\frac{(L\psi )^2}{r}\right\} . \end{aligned}$$

(3.15)

At the same time one can estimate precisely the size of the term $2\widetilde{P}_{(3)\mu }n^\mu$ at infinity, and use positivity of the function $g_3$ in the transition region to absorb the contributions of the other terms $2\widetilde{P}_{(j)\mu }n^\mu$, $j\in \{1,2,4\}$. Overall, we find that

$$\begin{aligned} \int _{\Sigma _{t}^c}2\big [\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(3)\mu }+\widetilde{P}_{(4)\mu } \big ]n_0^\mu \,d\mu _{t}\approx _{\alpha } \int _{\Sigma _{t}^c}e^\beta \big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t. \end{aligned}$$

(3.16)

Finally we find that

$$\begin{aligned} 2\big [\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(3)\mu }+ \widetilde{P}_{(4)\mu }\big ]k^\mu \ge 0\qquad \text { along }\mathcal {N}^c_{[0,T]}. \end{aligned}$$

(3.17)

The theorem follows from (3.15)–(3.17), and the divergence identity (3.6).

The Wave Equation in the Schwarzschild Spacetime

We show first how to prove Theorem 1.7 in the simplest case: $a=0$ (the Schwarzschild spacetime) and $\psi =0$. In this case $B=0$ and we are simply considering $\mathbf{Z}$-invariant solutions of the wave equation

$$\begin{aligned} \square \phi =0. \end{aligned}$$

In the rest of this section we use the coordinates $(\theta ,r,u_+,\phi _+)$ and the induced vector-fields $\partial _1=\partial _\theta ,\,\partial _2=\partial _r,\,\partial _3=\partial _t, \,\partial _4=\partial _{\phi }$, see (1.21)–(1.22). For simplicity of notation, we identify functions that depend on r (or on some of the other variables) with the corresponding functions defined on the spacetime.

Notice that

$$\begin{aligned} q^2=r^2,\qquad p=\frac{2M}{r}, \end{aligned}$$

(4.1)

with p introduced in (1.7). The nontrivial components of the metric are

$$\begin{aligned}&\mathbf{g}^{11}=r^{-2},\nonumber \\&\mathbf{g}^{22}=1-p,\nonumber \\&\mathbf{g}^{23}=p\chi ,\\&\mathbf{g}^{33}=\frac{-1+p^2\chi ^2}{1-p},\nonumber \\&\mathbf{g}^{44}=\frac{1}{r^2(\sin \theta )^2}.\nonumber \end{aligned}$$

(4.2)

Given a function H that depends only on r, the formula (A.9) shows that

$$\begin{aligned} \square H=\mathbf{g}^{22}\partial _2^2H+D^2\partial _2H=\frac{r-2M}{r}\partial _2^2H+\frac{2r-2M}{r^2}\partial _2H. \end{aligned}$$

(4.3)

Similarly, if m is a 1-form with $m_4=0,\mathcal {L}_\mathbf{T}m=0,\mathcal {L}_\mathbf{Z}m=0$, then

$$\begin{aligned} \mathbf{D}^\mu m_\mu= & {} \mathbf{g}^{\alpha {\beta }}\partial _\alpha m_{\beta }+\big [\partial _\mu \mathbf{g}^{\mu \nu }+(1/2)\mathbf{g}^{\mu \nu }\partial _\mu \log |r^4(\sin \theta )^2|\big ]m_\nu \nonumber \\= & {} \frac{1}{r^2}\Big [\partial _1 m_1+\frac{\cos \theta }{\sin \theta }m_1\Big ]+\Big [(1-p)\partial _2m_2+\frac{2r-2M}{r^2}m_2\Big ]\nonumber \\&+\,p\Big [\chi \partial _2m_3+(\chi '+\chi /r)m_3\Big ]. \end{aligned}$$

(4.4)

Therefore, given a vector-field

$$\begin{aligned} X=f(r)\partial _2+g(r)\partial _3, \end{aligned}$$

(4.5)

as in (A.12), and a 1-form Y with $Y_4=0$, and letting

$$\begin{aligned} {}^{(Y)}Q_{\mu \nu }=Y_\mu Y_\nu -(1/2)\mathbf{g}_{\mu \nu }(Y_\rho Y^\rho ), \end{aligned}$$

we have, see (A.15)–(A.17),

$$\begin{aligned} {}^{(Y)}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }= & {} (Y_1)^2\frac{-f'(r)}{r^2}+ (Y_2)^2\frac{-f(r)(2r-2M)+f'(r)(r^2-2Mr)}{r^2}\nonumber \\&+\,(Y_3)^2\Big [-f(r)\partial _2\mathbf{g}^{33}+2g'(r)\mathbf{g}^{23}-f'(r)\mathbf{g}^{33}-\frac{2rf(r)\mathbf{g}^{33}}{r^2}\Big ]\nonumber \\&+\,2Y_2Y_3\frac{-2Mrf(r)\chi '(r)-2Mf(r)\chi (r)+g'(r)(r^2-2Mr)}{r^2}, \end{aligned}$$

(4.6)

$$\begin{aligned} 2{}^{(Y)}Q(n,X)= & {} (Y_1)^2\frac{g(r)}{r^2}+(Y_2)^2[g(r)(1-p)-2f(r)\mathbf{g}^{23}]\nonumber \\&+\,(Y_3)^2[-g(r)\mathbf{g}^{33}]+2Y_2Y_3[-f(r)\mathbf{g}^{33}], \end{aligned}$$

(4.7)

and

$$\begin{aligned} 2{}^{(Y)}Q(k,X)= & {} (Y_1)^2\frac{-f(r)}{r^2}+(Y_2)^2[f(r)(1-p)]\nonumber \\&+\,(Y_3)^2[-f(r)\mathbf{g}^{33}+2g(r)\mathbf{g}^{23}]+2Y_2Y_3[g(r)(1-p)]. \end{aligned}$$

(4.8)

Here $f'$ and $g'$ denote the derivatives with respect to r of the functions f and g, and

$$\begin{aligned} n=-\mathbf{g}^{3\mu }\partial _\mu =-\mathbf{g}^{23}\partial _2-\mathbf{g}^{33}\partial _3,\qquad k=\mathbf{g}^{2\mu }\partial _\mu =(1-p)\partial _2+\mathbf{g}^{23}\partial _3. \end{aligned}$$

(4.9)

In this section we prove the following:

Theorem 4.1

Assume that $M\in (0,\infty )$, $N_0=4$, $a=0$ and $c_0:=2M-\overline{\varepsilon }M$, where $\overline{\varepsilon }\in [0,1)$ is a sufficiently small constant. Assume that $T\ge 0$, and $\phi \in C^k([0,T]:H^{N_0-k}(\Sigma _t^{c_0}))$, $k\in [0,N_0]$, is a $\mathbf{Z}$-invariant real-valued solution of the wave equation

$$\begin{aligned} \square \phi =0. \end{aligned}$$

(4.10)

Then, for any $\alpha \in (0,2)$ and any $t_1\le t_2\in [0,T]$,

$$\begin{aligned} \mathcal {E}^{c_0}_\alpha (t_2)+\mathcal {B}_\alpha ^{c_0}(t_1,t_2)\le \overline{C}_\alpha \mathcal {E}^{c_0}_\alpha (t_1), \end{aligned}$$

(4.11)

where $\overline{C}_\alpha$ is a large constant that may depend on $\alpha$,

$$\begin{aligned}&E_\mu :=\mathbf{D}_\mu \phi ,\qquad L\phi :=\chi _{\ge 4M}(r)\Big (\partial _2+\frac{1}{1-p}\partial _3\Big )\phi =\chi _{\ge 4M}(r)\Big (E_2+\frac{1}{1-p}E_3\Big ), \end{aligned}$$

(4.12)

$$\begin{aligned}&\mathcal {E}^{c_0}_\alpha (t):=\int _{\Sigma _t^{c_0}}\frac{r^\alpha }{M^{\alpha }} \Big [(E_1/r)^2+(L\phi )^2+M^2r^{-2}\big [(E_2)^2+(E_3^2)\big ]+r^{-2}\phi ^2\Big ]\,d\mu _t, \nonumber \\ \end{aligned}$$

(4.13)

$$\begin{aligned} \mathcal {B}_\alpha ^{c_0}(t_1,t_2):= & {} \int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }}\Big \{\frac{(r-3M)^2}{r^3}\frac{(E_1)^2}{r^2}+\frac{1}{r}(L\phi )^2\nonumber \\&+\frac{1}{r^3}\phi ^2+\frac{M^{2}}{r^{3}}\Big [(E_2)^2+\frac{(r-3M)^2}{r^2}(E_3)^2\Big ]\Big \}\,d\mu . \end{aligned}$$

(4.14)

The rest of the section is concerned with the proof of Theorem 4.1. Let

$$\begin{aligned} Q_{\mu \nu }:=E_\mu E_\nu -(1/2)\mathbf{g}_{\mu \nu }(E_\rho E^\rho ),\qquad J_\nu :=\mathbf{D}^\mu Q_{\mu \nu }=\mathcal {N}E_\nu . \end{aligned}$$

(4.15)

For any vector-field X, real scalar function w, and 1-form m we define

$$\begin{aligned} P_\mu =P_\mu [X,w,m]:=Q_{\mu \nu }X^\nu +\frac{1}{2}w\phi E_\mu -\frac{1}{4}\phi ^2\mathbf{D}_\mu w+\frac{1}{4}m_\mu \phi ^2. \end{aligned}$$

(4.16)

The formula (2.15) becomes

$$\begin{aligned} 2\mathbf{D}^\mu P_\mu =\mathcal {T}[X,w,m]:= & {} {}^{(X)}\pi ^{\mu \nu }Q_{\mu \nu } +wE^\mu E_\mu +\phi m^\mu E_\mu \nonumber \\&+\,\frac{1}{2}\phi ^2\big (\mathbf{D}^\mu m_\mu -\square w\big ). \end{aligned}$$

(4.17)

We use the divergence identity

$$\begin{aligned} \int _{\Sigma _{t_1}^c}P_\mu n_0^\mu \,d\mu _{t_1}=\int _{\Sigma _{t_2}^c}P_\mu n_0^\mu \,d\mu _{t_2}+\int _{\mathcal {N}^c_{[t_1,t_2]}}P_\mu k_0^\mu \,d\mu _c+\int _{\mathcal {D}^c_{[t_1,t_2]}}\mathbf{D}^\mu P_\mu \,d\mu , \end{aligned}$$

(4.18)

where $t_1,t_2\in [0,T]$, $c\in (c_0,2M]$, $n_0:=n/|\mathbf{g}^{33}|^{1/2}$, $k_0:=k/|\mathbf{g}^{22}|^{1/2}$, and the integration is with respect to the natural measures induced by the metric $\mathbf{g}$. We would like to find multipliers (X, w, m) in such a way that the contributions of the integrals in (4.18) are all nonnegative.

The Multipliers $(X_{(k)},w_{(k)},m_{(k)})$, $k\in \{1,2\}$

In this subsection we define three multipliers $(X_{(k)},w_{(k)},m_{(k)})$, $k\in \{1,2,3\}$, which are used to generate positive terms in the divergence identity (4.18). The first multiplier $(X_{(1)},w_{(1)},m_{(1)})$ is relevant in a neighborhood of the trapped set $\{r=3M\}$ and the second multiplier $(X_{(2)},w_{(2)},m_{(2)})$ is relevant in a neighborhood of the horizon $\{r=2M\}$. The third multiplier $(X_{(3)},w_{(3)},m_{(3)})$ generates outgoing energies at infinity; at the same time it contains a large multiple of the vector-field $\partial _3$ which helps with the positivity of the boundary integrals $P_\mu n^\mu$.

Analysis Around the Trapped Set $r=3M$

This is similar to the construction in [19]. We define the first multiplier $(X_{(1)},w_{(1)}, m_{(1)})$ by the formulas

$$\begin{aligned} \begin{aligned}&X_{(1)}:=f_1(r)\partial _2+g_1(r)\partial _3,\qquad f_1(r):=\frac{a_1(r)\Delta }{r^2},\qquad g_1(r):=\frac{a_1(r)\chi (r)2M}{r}+1,\\&w_{(1)}(r):=f'_1(r)+f_1(r)\partial _r\log \big (r^4/\Delta )-\epsilon _1\widetilde{w}(r),\qquad m_{(1)}\equiv 0,\\&\widetilde{w}(r):=M^2(r-33M/16)^3(r-3M)^2r^{-8}\mathbf {1}_{[33M/16,\infty )}(r), \end{aligned} \end{aligned}$$

(4.19)

where $a_1:(0,\infty )\rightarrow \mathbb {R}$ is a smooth increasing function to be fixed, $\lim _{r\rightarrow \infty }a_1(r)=1$, and $\epsilon _1\in (0,1]$ is a small constant. Using (4.6),

$$\begin{aligned} Q_{\mu \nu }{}^{(X_{(1)})}\pi ^{\mu \nu }+w_{(1)}E_\mu E^\mu =\big [K_{(1)}^{11}(E_1)^2+ K_{(1)}^{22}(E_2)^2+K_{(1)}^{33}(E_3)^2+2K_{(1)}^{23}E_2E_3\big ], \end{aligned}$$

where

$$\begin{aligned} K_{(1)}^{11}= & {} \frac{-f'_1(r)}{r^2}+w_{(1)}(r)\mathbf{g}^{11}=\frac{2(r-3M)}{r^4}a_1-\epsilon _1\widetilde{w}\mathbf{g}^{11},\\ K_{(1)}^{22}= & {} \frac{-f_1(r)(2r-2M)+f'_1(r)\Delta }{r^2}+w_{(1)}(r)\mathbf{g}^{22}=\frac{2\Delta ^2}{r^4}a'_1-\epsilon _1\widetilde{w}\mathbf{g}^{22},\\ K_{(1)}^{33}= & {} -f_1(r)\partial _2\mathbf{g}^{33}+2g'_1(r)\mathbf{g}^{23}-f'_1(r)\mathbf{g}^{33}-\frac{2rf_1(r)\mathbf{g}^{33}}{r^2}+w_{(1)}(r)\mathbf{g}^{33}\\= & {} \frac{8M^2\chi ^2}{r^2}a'_1-\epsilon _1\widetilde{w}\mathbf{g}^{33},\\ K_{(1)}^{23}= & {} \frac{-2Mrf_1(r)\chi '(r)-2Mf_1(r)\chi (r)+g'_1(r)\Delta }{r^2}+w_{(1)}(r)\mathbf{g}^{23}\\= & {} \frac{4M\Delta \chi }{r^3}a'_1-\epsilon _1\widetilde{w}\mathbf{g}^{23}. \end{aligned}$$

where $a'_1$ denotes the r derivative of the function $a_1$. Therefore

$$\begin{aligned} Q_{\mu \nu }{}^{(X_{(1)})}\pi ^{\mu \nu }+w_{(1)}E_\mu E^\mu= & {} \frac{2(r-3M)a_1-\epsilon _1\widetilde{w}r^2}{r^4}(E_1)^2\nonumber \\&+\,\Big (2a'_1-\frac{\epsilon _1\widetilde{w}}{1-p}\Big )\Big (\frac{\Delta }{r^2}E_2+ \frac{2M\chi }{r}E_3\Big )^2+\frac{\epsilon _1\widetilde{w}}{1-p}(E_3)^2. \end{aligned}$$

(4.20)

Moreover

$$\begin{aligned} \phi m^\mu _{(1)} E_\mu +\frac{1}{2}\phi ^2\big (\mathbf{D}^\mu m_{(1)\mu }-\square w_{(1)}\big )=-\frac{1}{2}\square w_{(1)}\phi ^2. \end{aligned}$$

(4.21)

We define now the important function $a_1(r)$. Assume $\kappa :\mathbb {R}\rightarrow \mathbb {R}$ is an increasing smooth function satisfying $\kappa (y)=y$ on $[-1,\infty )$ and $\kappa (y)=-2$ on $(-\infty ,-3]$. We set

$$\begin{aligned} \begin{aligned}&R(r):=(r-3M)(r+2M)+6M^2\log \Big [\frac{r-2M}{M}\Big ],\\&a_1(r):=r^{-2}\delta ^{-1}\kappa (\delta R(r))+\Big [\frac{M}{r}-\frac{6M^2}{r^2}\log \Big (\frac{r-2M}{M}\Big )\Big ]\chi _{\ge DM}(r), \end{aligned} \end{aligned}$$

(4.22)

where $\delta :=\epsilon _2^2M^{-2}$ is a small constant and $D\gg 1$ is a large constant. The function $a_1$ is well defined, using the formula above, for $r>2M$. Clearly $a_1(r)=-2r^{-2}\delta ^{-1}$ for r sufficiently close to 2M. Therefore $a_1$ can be extended smoothly by this formula to the full interval $r\in (c_0,\infty )$.

Clearly

$$\begin{aligned} R'(r)=2r-M+\frac{6M^2}{r-2M}. \end{aligned}$$

(4.23)

The function R is increasing on $(2M,\infty )$. Let $r_\delta$ denote the unique number in $(2M,\infty )$ with the property that $R(r_\delta )=-1/\delta$, and notice that

$$\begin{aligned} \frac{r_\delta -2M}{M}\approx e^{-(6\delta M^2)^{-1}}. \end{aligned}$$

Clearly $a_1(3M)=0$,

$$\begin{aligned} a'_1(r)=r^{-2}\Big [R'(r)\kappa '(\delta R(r))-\frac{2\kappa (\delta R(r))}{\delta r}\Big ] \end{aligned}$$

(4.24)

if $r\le DM$, and

$$\begin{aligned} a'_1(r)= & {} \frac{12M^2}{r^3}+\Big [\frac{M}{r}-\frac{6M^2}{r^2}\log \Big (\frac{r-2M}{M}\Big )\Big ]\chi '_{\ge DM}(r)\nonumber \\&+\,\Big [\frac{M}{r^2}-\frac{12M^2}{r^3}\log \Big (\frac{r-2M}{M}\Big )+\frac{6M^2}{r^2(r-2M)}\Big ](1-\chi _{\ge DM}(r)) \end{aligned}$$

(4.25)

if $r\ge r_\delta$. In view of (4.24), if $r\in (c_0,r_\delta ]$ then $a'_1(r)\ge 2\delta ^{-1}r^{-3}$. On the other hand, if $r\in [r_\delta ,\infty )$ then $a'_1(r)\ge 12M^2r^{-3}$. Therefore

$$\begin{aligned} a_1(3M)=0\qquad \text { and }\qquad a'_1(r)\ge 12 M^2r^{-3}\qquad \text { for }r\in (c_0,\infty ), \end{aligned}$$

(4.26)

provided that $\delta \le (10M)^{-2}$.

Let

$$\begin{aligned} h_1(r):=f'_1(r)+f_1(r)\partial _r\log \big (r^4/\Delta )=\frac{r-2M}{r^3} \partial _r\big (r^2a_1(r)\big ). \end{aligned}$$

(4.27)

We calculate, as before,

$$\begin{aligned} h_1(r)=\frac{r-2M}{r^3}R'(r)\kappa '(\delta R(r)) \end{aligned}$$

(4.28)

if $r\le DM$, and

$$\begin{aligned} h_1(r)= & {} \frac{r-2M}{r^3}\Big \{2r-\Big [M-\frac{6M^2}{r-2M}\Big ](1-\chi _{\ge DM}(r))\nonumber \\&+\,\Big [Mr-6M^2\log \Big (\frac{r-2M}{M}\Big )\Big ]\chi '_{\ge DM}(r)\Big \} \end{aligned}$$

(4.29)

if $r\ge r_\delta$. Letting

$$\begin{aligned} \widetilde{R}(r):=\frac{r-2M}{r^3} R'(r)=\frac{2}{r}-\frac{5M}{r^2}+\frac{8M^2}{r^3}, \end{aligned}$$

we have

$$\begin{aligned} (\square h_1)(r)= & {} \frac{\partial _2(\Delta \cdot \partial _2h_1)}{r^2}\\= & {} r^{-2}\Big \{\kappa '(\delta R(r))\partial _r[\Delta \widetilde{R}'(r)]+\delta ^2\kappa '''(\delta R(r))r^7\widetilde{R}(r)^3(r-2M)^{-1}\\&+\,\delta \kappa ''(\delta R(r))[3r^4\widetilde{R}(r)\widetilde{R}'(r)+4r^3\widetilde{R}(r)^2]\Big \} \end{aligned}$$

if $r\le DM$, and

$$\begin{aligned} (\square h_1)(r)= & {} \frac{\partial _2(\Delta \cdot \partial _2h_1)}{r^2}\\= & {} r^{-2}\partial _r[\Delta \widetilde{R}'(r)]+O(Mr^{-4})\mathbf {1}_{[DM,\infty )}(r)\\= & {} -\frac{2M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )+O(Mr^{-4})\mathbf {1}_{[DM,\infty )}(r). \end{aligned}$$

if $r\ge r_\delta$. Therefore, the last two identities show that

$$\begin{aligned} (\square h_1)(r)= & {} -\frac{2M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )+O(Mr^{-4})\mathbf {1}_{[DM,\infty )}(r)\nonumber \\&+\,M^{-3}O(1)\mathbf {1}_{(c_0,r_\delta ]}(r)+O\Big (\frac{\delta ^2M^2}{r-2M}\Big )\mathbf {1}_{[r'_\delta ,r_\delta ]}(r), \end{aligned}$$

(4.30)

where $r'_\delta$ denotes the unique number in $(2M,\infty )$ with the property that $R(r'_\delta )=-2/\delta$. Notice that

$$\begin{aligned} 7-\frac{44M}{r}+\frac{72 M^2}{r^2}\ge 1/10\qquad \text { for any }r\ge M. \end{aligned}$$

(4.31)

Therefore, since $w_{(1)}=h_1-\epsilon _1\widetilde{w}$, it follows that

$$\begin{aligned} -\frac{1}{2}(\square w_{(1)})(r)\ge \frac{M}{10r^4}-\frac{C_1M}{r^4}\mathbf {1}_{[DM,\infty )}(r)-\frac{C_1}{M^3}\mathbf {1}_{(c_0,r_\delta ]}(r)-\frac{C_1\delta ^2M^2}{r-2M}\mathbf {1}_{[r'_\delta ,r_\delta ]}(r), \end{aligned}$$

(4.32)

for a sufficiently large constant $C_1$, provided that the constant $\epsilon _1$ is sufficiently small. Using also (4.20)–(4.21) and (4.26),

$$\begin{aligned}&\mathcal {T}[X_{(1)},w_{(1)},m_{(1)}]\ge \frac{(2-C_1\epsilon _1)(r-3M)a_1(r)}{r^4}(E_1)^2\nonumber \\&\quad +\,(2-C_1\epsilon _1)a'_1(r)\big ((1-p)E_2+p\chi (r)E_3\big )^2+\epsilon _1\widetilde{w}(r)(E_3)^2+\frac{M}{10r^4}\phi ^2\nonumber \\&\quad -\frac{C_1M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\phi ^2-\frac{C_1}{M^3}\mathbf {1}_{(c_0,r_\delta ]}(r)\phi ^2-\frac{C_1\delta ^2M^2}{r-2M}\mathbf {1}_{[r'_\delta ,r_\delta ]}(r)\phi ^2, \end{aligned}$$

(4.33)

for a sufficiently large constant $C_1$, provided that the constant $\epsilon _1$ is sufficiently small.

The remaining contributions $2P_\mu n_0^\mu$ and $2P_\mu k_0^\mu$ in the divergence identity (4.18) cannot be estimated effectively at this time. We will prove partial estimates for these terms in Lemma 4.2 below, after we construct the second multiplier $(X_{(2)},w_{(2)},m_{(2)})$ and show how to fix some of the parameters.

Analysis in a Neighborhood of the Horizon

In a small neighborhood of the horizon we need to use the redshift effect. For this we define the second multiplier $(X_{(2)},w_{(2)},m_{(2)})$ by the formulas

$$\begin{aligned} \begin{aligned}&X_{(2)}:=f_2(r)\partial _2+g_2(r)\partial _3,\qquad f_2(r):=-\epsilon _2a_2(r),\qquad g_2(r):=\epsilon _2 a_2(r)(1-\epsilon _2),\\&w_{(2)}(r):=-2\epsilon _2 a_2(r)/r,\qquad m_{(2)2}:=\epsilon _2M^{-2}\gamma (r),\qquad m_{(2)3}:=\epsilon _2M^{-2}\gamma (r) \end{aligned} \end{aligned}$$

(4.34)

where $\epsilon _2$ is a small positive constant (recall that $\delta =\epsilon _2^2M^{-2}$),

$$\begin{aligned} a_2(r):={\left\{ \begin{array}{ll} M^{-3}(9M/4-r)^3&{}\text { if }r\le 9M/4,\\ 0&{}\text { if }r\ge 9M/4, \end{array}\right. } \end{aligned}$$

(4.35)

and $\gamma :[c_0,\infty )\rightarrow [0,1]$ is a suitable function (to be fixed later) supported in $[c_0,17M/8]$ and satisfying $\gamma (2M)=1/2$.

Notice that $\chi =1$ in the support of the functions $a_2$ and $\gamma$. As before, we calculate

$$\begin{aligned} Q_{\mu \nu }{}^{(X_{(2)})}\pi ^{\mu \nu }+w_{(2)}E_\mu E^\mu =\big [K_{(2)}^{11}(E_1)^2+ K_{(2)}^{22}(E_2)^2+K_{(2)}^{33}(E_3)^2+2K_{(2)}^{23}E_2E_3\big ], \end{aligned}$$

where

$$\begin{aligned} K_{(2)}^{11}= & {} \frac{-f'_2(r)}{r^2}+w_{(2)}(r)\mathbf{g}^{11}=\epsilon _2\frac{ra'_2-2a_2}{r^3},\\ K_{(2)}^{22}= & {} \frac{-f_2(r)(2r-2M)+f'_2(r)\Delta }{r^2}+w_2(r)\mathbf{g}^{22}=\epsilon _2\Big [-\frac{r-2M}{r}a'_2+\frac{2M}{r^2}a_2\Big ],\\ K_{(2)}^{33}= & {} -f_2(r)\partial _2\mathbf{g}^{33}+2g'_2(r)\mathbf{g}^{23}-f'_2(r)\mathbf{g}^{33}-\frac{2rf_2(r)\mathbf{g}^{33}}{r^2}+w_2(r)\mathbf{g}^{33}\\= & {} \epsilon _2\Big [-\frac{r-2M+4\epsilon _2M}{r}a'_2+\frac{2M}{r^2}a_2\Big ],\\ K_{(2)}^{23}= & {} \frac{-2Mrf_2(r)\chi '(r)-2Mf_2(r)\chi (r)+g'_2(r)\Delta }{r^2}+w_2(r)\mathbf{g}^{23}\\= & {} -\epsilon _2\Big [-\frac{(1-\epsilon _2)(r-2M)}{r}a'_2(r)+\frac{2M}{r^2}a_2(r)\Big ]. \end{aligned}$$

Using the explicit formula (4.35), it is easy to see that

$$\begin{aligned}&Q_{\mu \nu }{}^{(X_{(2)})}\pi ^{\mu \nu }+w_{(2)}E_\mu E^\mu \ge \mathbf {1}_{(c_0,9M/4)}(r)(9M/4-r)^2M^{-3}\nonumber \\&\quad \times \,\Big [C_2^{-1}\epsilon _2(E_2-E_3)^2+C_2^{-1}\epsilon _2^2(E_3)^2-C_2\epsilon _2(E_1)^2/r^2\Big ], \end{aligned}$$

(4.36)

for a sufficiently large constant $C_2$, provided that $\epsilon _2$ is sufficiently small and $c_0$ is sufficiently close to 2M. Moreover, using the definitions and (4.3)–(4.4),

$$\begin{aligned}&\phi m_{(2)}^\mu E_\mu +\frac{1}{2}\phi ^2\big (\mathbf{D}^\mu m_{(2)\mu }-\square w_{(2)}\big )\\&\quad =\frac{\epsilon _2\gamma }{M^2}\phi (E_2-E_3)+\frac{\epsilon _2}{2}\phi ^2\Big (\frac{1}{M^2}\gamma '+\frac{2}{rM^2}\gamma +2\square (a_2/r)\Big ). \end{aligned}$$

Therefore, recalling also that $\gamma \in [0,1]$ and completing the square,

$$\begin{aligned} \mathcal {T}[X_{(2)},w_{(2)},m_{(2)}]\ge & {} \frac{\epsilon _2}{2M^2}\phi ^2\gamma '+M^{-1}\epsilon _2^4\mathbf {1}_{(c_0,17M/8)}(r)\big [(E_2)^2+(E_3)^2\big ]\nonumber \\&-\,C_2\epsilon _2\mathbf {1}_{(c_0,9M/4)}(r)\big [M^{-1}(E_1)^2/r^2+M^{-3}\phi ^2\big ], \end{aligned}$$

(4.37)

provided that $\epsilon _2$ is sufficiently small and $c_0$ is sufficiently close to 2M.

We examine now (4.33) and (4.37) and fix the constant $\epsilon _1,\epsilon _2$ and the function $\gamma$ such that the sum $\mathcal {T}[X_{(1)},w_{(1)},m_{(1)}]+\mathcal {T}[X_{(2)},w_{(2)},m_{(2)}]$ is nonnegative when $r\in (c_0,DM]$. For the positivity of the zero order term we need that

$$\begin{aligned} \frac{M}{20r^4}+\frac{\epsilon _2\gamma '(r)}{2M^2}\ge \frac{C_1}{M^3}\mathbf {1}_{(c_0,r_\delta ]}(r)+\frac{C_1\delta ^2M^2}{r-2M}\mathbf {1}_{[r'_\delta ,r_\delta ]}(r)+\frac{C_2\epsilon _2}{M^3}\mathbf {1}_{(c_0,9M/4)}(r). \end{aligned}$$

(4.38)

Recall that $\delta =\epsilon _2^2M^{-2}$. The point is that

$$\begin{aligned} \int _{c_0}^\infty \frac{C_1}{M^3}\mathbf {1}_{(c_0,r_\delta ]}(r)+\frac{C_1\delta ^2M^2}{r-2M}\mathbf {1}_{[r'_\delta ,r_\delta ]}(r)\,dr\le \frac{C_1^2\epsilon _2^2}{M^2}, \end{aligned}$$

provided that $2M-c_0\le \epsilon _2^2$. This is easy to see if one recalls the definitions of $r_\delta$ and $r'_\delta$. Therefore, assuming that $\epsilon _2$ is sufficiently small, one can fix the function $\gamma$ to achieve the inequality (4.38), while still preserving the other properties of $\gamma$, namely

$$\begin{aligned} \gamma :[c_0,\infty )\rightarrow [0,1]\text { is supported in }[c_0,17M/8]\text { and satisfies }\gamma (2M)=1/2. \end{aligned}$$

(4.39)

Indeed, the function $\gamma$ can be chosen to increase on the interval $(c_0,r_\delta ]$ and then decrease for $r\ge 2r_\delta -2M$ in a way to satisfy both (4.38) and (4.39).

Notice that the sum of the first order terms in $\mathcal {T}[X_{(1)},w_{(1)},m_{(1)}]+\mathcal {T}[X_{(2)},w_{(2)},m_{(2)}]$ is nonnegative and nondegenerate if we simply have $\epsilon _1,\epsilon _2>0$ sufficiently small. Therefore, one can fix the parameters $\epsilon _1,\epsilon _2$ and the function $\gamma$ in such a way that

$$\begin{aligned}&\mathcal {T}[X_{(1)},w_{(1)},m_{(1)}]+\mathcal {T}[X_{(2)},w_{(2)},m_{(2)}]\nonumber \\&\quad \ge \epsilon _3\Big [\frac{(r-3M)^2}{r^3}(E_1/r)^2+\frac{M^2}{r^3}(E_2)^2+\frac{M^2(r-3M)^2}{r^5}(E_3)^2+\frac{M}{r^4}\phi ^2\Big ]\nonumber \\&\qquad -\,\epsilon _3^{-1}\frac{M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\phi ^2, \end{aligned}$$

(4.40)

for a constant $\epsilon _3>0$ sufficiently small (relative to $\epsilon _1$ and $\epsilon _2$). The parameter D will be fixed later, sufficiently large depending on $\epsilon _3$.

We can prove now some partial bounds on the remaining terms

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu })n_0^\mu ,\qquad 2(P_{(1)\mu }+P_{(2)\mu }) k_0^\mu , \end{aligned}$$

in the divergence identity (4.18), where $P_{(k)}:=P[X_{(k)},w_{(k)},m_{(k)}]$, $k\in \{1,2\}$.

Lemma 4.2

There is a sufficiently small absolute constant $\epsilon _3$ such that

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) k^\mu \ge \epsilon _3\big [(E_1/r)^2+(E_2)^2(2-c/M)+M^{-2}\phi ^2\big ]-\epsilon _3^{-1}(E_3)^2. \end{aligned}$$

(4.41)

along $\mathcal {N}^c_{[t_1,t_2]}$. Also

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) n^\mu\ge & {} -\epsilon _3^{-1}\big [\widetilde{F}_0+\mathbf {1}_{[8M,2DM]}(r)(E_3)^2\big ]\nonumber \\&-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2)+\epsilon _3(E_2)^2\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(4.42)

and

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) n^\mu\le & {} \epsilon _3^{-1}\big [\widetilde{F}_0+\mathbf {1}_{[8M,2DM]}(r)(E_3)^2+(E_2)^2\mathbf {1}_{(c_0,17M/8]}(r)\big ]\nonumber \\&-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2), \end{aligned}$$

(4.43)

where

$$\begin{aligned} \widetilde{F}_0=(E_1/r)^2+(L\phi )^2+M^2r^{-2}\big [(E_2)^2|1-p|+ (E_3^2)\big ]+r^{-2}\phi ^2. \end{aligned}$$

(4.44)

Proof

We start with the term $2(P_{(1)\mu }+P_{(2)\mu }) k^\mu$,

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) k^\mu= & {} 2k^\mu Q_{\mu \nu }\big (X^\nu _{(1)}+X^\nu _{(2)}\big )+(w_{(1)}+w_{(2)})\phi E_\mu k^\mu \\&-\frac{1}{2}\phi ^2k^\mu (\mathbf{D}_\mu w_{(1)}+\mathbf{D}_\mu w_{(2)})+\frac{1}{2}k^\mu m_{(2)\mu }\phi ^2. \end{aligned}$$

When $r=c\in (c_0,2M]$ and assuming that $2M-c$ is sufficiently small, we use the definitions, the identity $m_{(2)3}(2M)\ge \epsilon _2/(2M^2)$, and the identities (4.8). We have

$$\begin{aligned} 2k^\mu Q_{\mu \nu }\big (X^\nu _{(1)}+X^\nu _{(2)}\big )\ge & {} \epsilon _3\big [(E_1/r)^2+(E_2)^2(2-c/M)\big ]\\&-\,\epsilon _3^{-1}(E_3)^2-\epsilon _3^{-1}|E_2E_3|(2-c/M),\\ \big |(w_{(1)}+w_{(2)})\phi E_\mu k^\mu \big |\le & {} \epsilon _3^{-1}M^{-1}|\phi |\big [(2-c/M)|E_2|+|E_3|\big ], \end{aligned}$$

and

$$\begin{aligned} -\frac{1}{2}\phi ^2k^\mu (\mathbf{D}_\mu w_{(1)}+\mathbf{D}_\mu w_{(2)})+\frac{1}{2}k^\mu m_{(2)\mu }\phi ^2\ge \epsilon _3M^{-2}\phi ^2, \end{aligned}$$

provided that $\epsilon _3$ is sufficiently small. The bound (4.41) follows by further reducing $\epsilon _3$ and assuming that $2M-c$ is sufficiently small.

We consider now the term $2(P_{(1)\mu }+P_{(2)\mu }) n^\mu$,

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) n^\mu= & {} 2n^\mu Q_{\mu \nu }\big (X^\nu _{(1)}+X^\nu _{(2)}\big )+(w_{(1)}+w_{(2)})\phi E_\mu n^\mu \\&-\frac{1}{2}\phi ^2n^\mu (\mathbf{D}_\mu w_{(1)}+\mathbf{D}_\mu w_{(2)})+\frac{1}{2}n^\mu m_{(2)\mu }\phi ^2. \end{aligned}$$

Using the definitions and the identities (4.7) we estimate

$$\begin{aligned} \big |\phi ^2n^\mu (\mathbf{D}_\mu w_{(1)}+\mathbf{D}_\mu w_{(2)})\big |+\big |n^\mu m_{(2)\mu }\phi ^2\big |\le \epsilon _3^{-1}M^2r^{-4}\phi ^2. \end{aligned}$$

(4.45)

Moreover, with $\widetilde{F}_0$ as in (4.44), we write

$$\begin{aligned}&2n^\mu Q_{\mu \nu }X^\nu _{(1)}+w_{(1)}\phi E_\mu n^\mu \\&\quad =(E_1)^2\frac{g_1(r)}{r^2}+(E_2)^2[g_1(r)(1-p)-2f_1(r)\mathbf{g}^{23}]\\&\qquad +(E_3)^2[-g_1(r)\mathbf{g}^{33}]+2E_2E_3[-f_1(r)\mathbf{g}^{33}]+w_1\phi (-\mathbf{g}^{33}E_3-\mathbf{g}^{23}E_2)\\&\quad \ge -(10\epsilon _3)^{-1}\widetilde{F}_0+\big [(E_2)^2(1-p)+(E_3)^2(1-p)^{-1}\\&\qquad +2E_2E_3a_1+\phi E_3 w_1(1-p)^{-1}\big ]\chi _{\ge 8M}(r). \end{aligned}$$

Using the definitions and the formula (4.29),

$$\begin{aligned} |a_1-1|\chi _{\ge 8M}(r)\lesssim & {} Mr^{-1}\mathbf {1}_{[8M,2DM]}(r)+M^2r^{-2}\mathbf {1}_{[8M,\infty )}(r),\\ |w_1-2(1-p)/r|\chi _{\ge 8M}(r)\lesssim & {} Mr^{-2}\mathbf {1}_{[8M,2DM]}(r)+M^2r^{-3}\mathbf {1}_{[8M,\infty )}(r). \end{aligned}$$

Therefore

$$\begin{aligned}&2n^\mu Q_{\mu \nu }X^\nu _{(1)}+w_{(1)}\phi E_\mu n^\mu \\&\quad \ge -(9\epsilon _3)^{-1}\big [\widetilde{F}_0+\mathbf {1}_{[8M,2DM]}(r)(E_3)^2\big ]-\Big [\frac{\phi ^2}{r^2}+\frac{2\phi }{r}E_2\Big ]\chi _{\ge 8M}(r)(1-p)\\&\quad \ge -(8\epsilon _3)^{-1}\big [\widetilde{F}_0+\mathbf {1}_{[8M,2DM]}(r)(E_3)^2\big ]-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2). \end{aligned}$$

Similarly, using also the observation that $-f_2(2M)\gtrsim 1$,

$$\begin{aligned} 2n^\mu Q_{\mu \nu }X^\nu _{(2)}+w_{(2)}\phi E_\mu n^\mu \ge -(8\epsilon _3)^{-1}\widetilde{F}_0+\epsilon _3(E_2)^2\mathbf {1}_{(c_0,17M/8]}(r). \end{aligned}$$

The bound (4.42) follows using the last two inequalities and (4.45).

The proof of the upper bound (4.43) follows in a similar way. $\square$

Remark 4.3

At this point one can recover the energy estimate of Marzuola–Metcalfe–Tataru–Tohaneanu [19, Theorem1.2],

$$\begin{aligned} \mathcal {E}^{c_0}(t_2)+\mathcal {B}^{c_0}(t_1,t_2)\le \overline{C}\mathcal {E}^{c_0}(t_1), \end{aligned}$$

where

$$\begin{aligned} \mathcal {E}^{c_0}(t):= & {} \int _{\Sigma _t^{c_0}}\Big [(E_1/r)^2+(E_2)^2+(E_3^2)\Big ]\,d\mu _t,\\ \mathcal {B}^{c_0}(t_1,t_2):= & {} \int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\Big [\frac{(r-3M)^2}{r^3}(E_1/r)^2+\frac{M^2}{r^3}(E_2)^2\\&+\frac{M^2(r-3M)^2}{r^5}(E_3)^2+\frac{M}{r^4}\phi ^2\Big ]\,d\mu . \end{aligned}$$

To see this, we simply set $D:=\infty$ and add in a very large multiple of the Killing vector-field $\partial _3$. The spacetime integral $\mathcal {B}^{c_0}(t_1,t_2)$ is generated by the right-hand side of (4.40) (some of the powers of r in the spacetime integral could in fact be improved by reexamining the proof). The formulas for the nondegenerate energies $\mathcal {E}^{c_0}(t)$ follow from the bounds (4.42) and (4.43), the identity (4.7), and the Hardy inequality in Lemma A.1 (i). The contribution of $P_\mu k^\mu$ along $\mathcal {N}^c_{[t_1,t_2]}$ becomes nonnegative, in view of (4.8), and can be neglected.

Outgoing Energies

To prove the stronger estimates in Theorem 4.1 we consider now a multiplier $(X_{(3)},w_{(3)},m_{(3)})$ of the form

$$\begin{aligned} \begin{aligned}&X_{(3)}=f_3\partial _2+\Big (\frac{f_3}{1-p}+g_3\Big )\partial _3,\qquad w_{(3)}=\frac{2f_3}{r},\\&m_{(3)1}=m_{(3)4}=0,\qquad m_{(3)2}=\frac{2h_3}{r(1-p)},\qquad m_{(3)3}=-\frac{2h_3}{r}, \end{aligned} \end{aligned}$$

(4.46)

where $f_3,g'_3,h_3$ are smooth functions supported in $\{r\ge 8M\}$, which depend only on r. The function $g_3$ is not supported in $\{r\ge 8M\}$, it is in fact a very large constant in the region $r\in [c,10M]$.

As before, using (4.6), we calculate

$$\begin{aligned} Q_{\mu \nu }{}^{(X_{(3)})}\pi ^{\mu \nu }+w_{(3)}E_\mu E^\mu =\big [K^{11}_{(3)}(E_1)^2+ K^{22}_{(3)}(E_2)^2+K^{33}_{(3)}(E_3)^2+2K^{23}_{(3)}E_2E_3\big ], \end{aligned}$$

where

$$\begin{aligned}&K^{11}_{(3)}=\frac{-f_3'(r)}{r^2}+w_{(3)}(r)\mathbf{g}^{11}=\frac{2f_3-rf'_3}{r^3},\\&K^{22}_{(3)}=\frac{-f_3(r)(2r-2M)+f'_3(r)\Delta }{r^2}+w_{(3)}(r)\mathbf{g}^{22}=(1-p)f'_3-\frac{2Mf_3}{r^2},\\&K^{33}_{(3)}=-f_3(r)\partial _2\mathbf{g}^{33}-f'_3(r)\mathbf{g}^{33} -\frac{2f_3(r)\mathbf{g}^{33}}{r}+w_{(3)}(r)\mathbf{g}^{33} =\frac{f'_3}{1-p}-\frac{2Mf_3}{r^2(1-p)^2},\\&K^{23}_{(3)}=(1-p)\Big (\frac{f_3}{1-p}+g_3\Big )'=f'_3-\frac{2Mf_3}{r^2(1-p)}+(1-p)g'_3. \end{aligned}$$

Moreover

$$\begin{aligned}&\phi m_{(3)}^\mu E_\mu +\frac{1}{2}\phi ^2\big (\mathbf{D}^\mu m_{(3)\mu }-\square w_{(3)}\big )\\&\quad =2h_3\frac{\phi }{r}\Big (E_2+\frac{E_3}{1-p}\Big )+\phi ^2\Big [\frac{h'_3}{r}+\frac{h_3}{r^2}-\frac{(1-p)f''_3}{r}-\frac{2Mf'_3}{r^3}+\frac{2Mf_3}{r^4}\Big ]. \end{aligned}$$

Set

$$\begin{aligned} \begin{aligned}&H_3:=(1-p)f'_3-\frac{2Mf_3}{r^2}+(1-p)^2g'_3,\\&h_3:=H_3\cdot (1-\widetilde{\alpha }), \end{aligned} \end{aligned}$$

(4.47)

where $\widetilde{\alpha }=(2-\alpha )/10>0$. The identities above show that

$$\begin{aligned} \mathcal {T}[X_{(3)},w_{(3)},m_{(3)}]= & {} \frac{(E_1)^2}{r^2}\frac{2f_3-rf'_3}{r}+H_3\Big (E_2+\frac{E_3}{1-p}\Big )^2\\&-\,(1-p)^2g'_3\Big [(E_2)^2+\frac{(E_3)^2}{(1-p)^2}\Big ]\\&+\,2h_3\frac{\phi }{r}\Big (E_2+\frac{E_3}{1-p}\Big )\\&+\,\phi ^2\Big [\frac{h_3}{r^2}+\frac{h'_3}{r}-\frac{(1-p)f''_3}{r}-\frac{2Mf'_3}{r^3}+\frac{2Mf_3}{r^4}\Big ]. \end{aligned}$$

After completing the square this becomes

$$\begin{aligned} \mathcal {T}[X_{(3)},w_{(3)},m_{(3)}]= & {} \frac{(E_1)^2}{r^2}\frac{2f_3-rf'_3}{r}+H_3\Big (L\phi +\frac{(1-\widetilde{\alpha })\phi }{r}\Big )^2\nonumber \\&-\,(1-p)^2g'_3\Big [(E_2)^2+\frac{(E_3)^2}{(1-p)^2}\Big ]\nonumber \\&+\,\phi ^2\Big [\frac{(\widetilde{\alpha }-\widetilde{\alpha }^2)H_3-\widetilde{\alpha }rH'_3}{r^2}+\frac{6Mf_3}{r^4}\nonumber \\&-\,\frac{2Mf'_3}{r^3}+\frac{(1-p)^2g''_3}{r}+\frac{4M(1-p)g'_3}{r^3}\Big ]. \end{aligned}$$

(4.48)

Using (4.7) we calculate

$$\begin{aligned} 2P_{(3)\mu } n^\mu= & {} 2Q_{\mu \nu }X^\nu _{(3)} n^\mu +w_{(3)}\phi E_\mu n^\mu -\frac{1}{2}\phi ^2n^\mu \mathbf{D}_\mu w_{(3)}+\frac{1}{2}n^\mu m_{(3)\mu }\phi ^2\\= & {} \frac{(E_1)^2}{r^2}\Big [\frac{f_3}{1-p}+g_3\Big ]+(E_2)^2\big [f_3+g_3(1-p)\big ]\\&+\,(E_3)^2\Big [\frac{f_3}{(1-p)^2}+\frac{g_3(1-p^2\chi ^2)}{1-p}\Big ]\\&+\,2E_2E_3\frac{f_3}{1-p}+\frac{2f_3}{r(1-p)}\phi E_3+\frac{m_{(3)3}}{2(1-p)}\phi ^2\\= & {} \frac{(E_1)^2}{r^2}\Big [\frac{f_3}{1-p}+g_3\Big ]+f_3\Big [E_2+\frac{E_3}{1-p}+\frac{\phi }{r}\Big ]^2-f_3\frac{\phi ^2}{r^2}\\&-\,2f_3E_2\frac{\phi }{r}+g_3(1-p)\Big [(E_2)^2+\frac{(E_3)^2(1-p^2\chi ^2)}{(1-p)^2}\Big ]-\frac{h_3}{r(1-p)}\phi ^2. \end{aligned}$$

Therefore

$$\begin{aligned} 2P_{(3)\mu } n^\mu= & {} \frac{(E_1)^2}{r^2}\Big [\frac{f_3}{1-p}+g_3\Big ]+f_3\Big [L\phi +\frac{\phi }{r}\Big ]^2\nonumber \\&+\,g_3(1-p)\Big [(E_2)^2+\frac{(E_3)^2(1-p^2\chi ^2)}{(1-p)^2}\Big ]\nonumber \\&-\frac{1}{r^2}\partial _2\big [f_3r\phi ^2\big ]+\phi ^2\Big [\frac{\widetilde{\alpha }H_3}{r(1-p)}+\frac{2Mf_3}{r^3(1-p)}-\frac{(1-p)g'_3}{r}\Big ]. \end{aligned}$$

(4.49)

Proof of the Theorem 4.1

We compare now the expressions (4.48) and (4.49) with the lower bounds in (4.40) and (4.42). We would like to fix the functions $f_3$ and $g_3$ and the constant D in such a way that the sum of the corresponding expressions is bounded from below. More precisely, the sum of the spacetime integrals is pointwise bounded from below, while the sum of the integrals on the surfaces $\Sigma _t^c$ is bounded from below after integration by parts and the use of a simple Hardy-type inequality.

One should think of the functions $f_3$ and $g_3$ in the following way: the function $f_3$ vanishes when $r\le 8M$ and behaves like $r^\alpha$ as $r\rightarrow \infty$. On the other hand the function $g_3$ is an extremely large constant when $r\le C_4M$, for some large constant $C_4$ but vanishes as $r\rightarrow \infty$ at a rate of $r^{\alpha -2}$. More precisely, we are looking for functions $f_3,g_3$ of the form

$$\begin{aligned} f_3(r)=\epsilon _4\chi _{\ge 8M}(r)e^{\beta (r)},\qquad g_3(r)=\int _{r}^\infty \rho (s)\,ds, \end{aligned}$$

(4.50)

where $\epsilon _4=\epsilon _3^2$ is a small constant, $C_4=C_4(\alpha )\ge \epsilon _4^{-4}\alpha ^{-1}(2-\alpha )^{-1}$ is a large constant (to be fixed), and $\beta ,\rho :(c,\infty )\rightarrow [0,\infty )$ are smooth functions satisfying

$$\begin{aligned} \begin{aligned} \beta (r)\in [-10,0]\text { and }M\beta '(r)\in [1/10,10]&\qquad \text { if }r\in (c,8M],\\ \max \Big (\frac{\alpha }{100r},\frac{4M}{r^2}+\frac{1}{r}\mathbf {1}_{[8M,C_4M]}(r)\Big )\le \beta '(r)\le \frac{2}{r}&\qquad \text{ if } r\in [8M,\infty ),\\ \rho (r)=0\text { and }g_3(r)\in [C_4/2,2C_4]&\qquad \text { if }r\le C_4M,\\ \rho (r)\le \frac{\epsilon _4}{100}\beta '(r)e^{\beta (r)}\text { and }\rho '(r)\le \frac{\epsilon _4M}{100r^3}e^{\beta (r)}&\qquad \text { if }r\ge C_4M,\\ \frac{e^\beta M^2}{r^2}\le g_3(r)\le \frac{C_4^{10}e^\beta M^2}{r^2}&\qquad \text { if }r\ge C_4M,\\ (1-2\widetilde{\alpha })H_3(r)-rH'_3(r)\ge 0&\qquad \text { if }r\in [16M,\infty ). \end{aligned} \end{aligned}$$

(4.51)

A specific choice satisfying these conditions is given in (4.58)–(4.59). As a result of these conditions, we clearly have

$$\begin{aligned}&g'_3=-\rho ,\nonumber \\&H_3\ge \frac{\epsilon _4}{100}e^{\beta }\beta '\chi _{\ge 8M},\\&e^{\beta (r)}\in [r/(100M),r^2/M^2]\qquad \text { for }r\in (c,C_4M].\nonumber \end{aligned}$$

(4.52)

Let

$$\begin{aligned} (X,w,m):= & {} (X_{(1)},w_{(1)},m_{(1)})+(X_{(2)},w_{(2)},m_{(2)})+(X_{(3)},w_{(3)},m_{(3)}),\\ \mathcal {T}[X,w,m]:= & {} \mathcal {T}[X_{(1)},w_{(1)},m_{(1)}]+\mathcal {T}[X_{(2)},w_{(2)},m_{(2)}]+\mathcal {T}[X_{(3)},w_{(3)},m_{(3)}],\\ P_{\mu }:= & {} P_{(1)\mu }+P_{(2)\mu }+P_{(3)\mu }. \end{aligned}$$

Our next lemma contains the main bounds on the terms in the divergence identity (4.18).

Lemma 4.4

Assume that the conditions (4.51) hold and that $C_4$ sufficiently large (depending on $\epsilon _4$). Then there is an absolute constant $\epsilon _5=\epsilon _5(\alpha )>0$ sufficiently small such that

$$\begin{aligned} \mathcal {T}[X,w,m]\ge & {} \epsilon _5\Big [\Big (\frac{2}{r}e^\beta -\beta 'e^\beta +\frac{100}{r}\Big )\frac{(r-3M)^2}{r^2}\frac{(E_1)^2}{r^2}+e^{\beta }\beta '(L\phi )^2\nonumber \\&+\,\Big (\rho +\frac{M^2}{r^3}\Big )(E_2)^2+\Big (\rho +\frac{M^2(r-3M)^2}{r^5}\Big )(E_3)^2+\frac{e^\beta \beta '}{r^2}\phi ^2\Big ]. \end{aligned}$$

(4.53)

Moreover, for any $t\in [0,T]$,

$$\begin{aligned} \int _{\Sigma _{t}^c}2P_\mu n_0^\mu \,d\mu _{t}\ge \epsilon _5\int _{\Sigma _{t}^c}e^\beta \frac{(E_1)^2}{r^2}+e^\beta (L\phi )^2+g_3\big [(E_2)^2+(E_3)^2\big ]+\frac{e^\beta \beta '}{r}\phi ^2\,d\mu _t \end{aligned}$$

(4.54)

and

$$\begin{aligned} \int _{\Sigma _{t}^c}2P_\mu n_0^\mu \,d\mu _{t}\le \epsilon _5^{-1}\int _{\Sigma _{t}^c}e^\beta \frac{(E_1)^2}{r^2} +e^\beta (L\phi )^2+g_3\big [(E_2)^2 +(E_3)^2\big ]+\frac{e^\beta \beta '}{r}\phi ^2\,d\mu _t. \end{aligned}$$

(4.55)

Finally,

$$\begin{aligned} 2P_\mu k^\mu \ge \epsilon _5\Big [\frac{(E_1)^2}{M^2}+(E_2)^2\frac{2M-c}{M}+(E_3)^2+\frac{\phi ^2}{M^2}\Big ]\qquad \text { along }\mathcal {N}^c_{[t_1,t_2]}, \end{aligned}$$

(4.56)

Proof

We start with the proof of (4.53). Using the definitions we have

$$\begin{aligned} \begin{aligned}&\frac{2f_3-rf'_3}{r}=\epsilon _4 e^{\beta }\big [(2/r-\beta ')\chi _{\ge 8M}-\chi '_{\ge 8M}\big ],\\&\frac{6Mf_3}{r^4}-\frac{2Mf'_3}{r^3} +\frac{(1-p)^2g''_3}{r}+\frac{4M(1-p)g'_3}{r^3} \ge \frac{\epsilon _4 M}{100r^4}e^\beta \chi _{\ge 8M}-\frac{2\epsilon _4 M}{r^3}e^\beta \chi '_{\ge 8M}. \end{aligned} \end{aligned}$$

(4.57)

We combine the formulas (4.40) and (4.48) to estimate

$$\begin{aligned} \mathcal {T}[X,w,m]\ge I_1+I_2+I'_2+I_3, \end{aligned}$$

where

$$\begin{aligned} I_1:= & {} \frac{(E_1)^2}{r^2}\frac{2f_3-rf'_3}{r}+\epsilon _3\frac{(E_1)^2}{r^2}\frac{(r-3M)^2}{r^3},\\ I_2:= & {} H_3\Big (L\phi +\frac{(1-\widetilde{\alpha })\phi }{r}\Big )^2,\\ I'_2:= & {} -(1-p)^2g'_3\Big [(E_2)^2+\frac{(E_3)^2}{(1-p)^2}\Big ]+\epsilon _3\Big [\frac{M^2}{r^3}(E_2)^2+\frac{M^2(r-3M)^2}{r^5}(E_3)^2\Big ],\\ I_3:= & {} \phi ^2\Big [\frac{(\widetilde{\alpha }-\widetilde{\alpha }^2)H_3-\widetilde{\alpha }rH'_3}{r^2}+\frac{\epsilon _4 M}{100r^4}e^\beta \chi _{\ge 8M}\\&-\frac{2\epsilon _4 M}{r^3}e^\beta \chi '_{\ge 8M}+\frac{\epsilon _3M}{r^4}-\epsilon _3^{-1}\frac{M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\Big ]. \end{aligned}$$

Using (4.51), (4.52), and (4.57) it is easy to see that, for some sufficiently small constant $\epsilon _5$ (which may depend on $\alpha$),

$$\begin{aligned} I_1\ge & {} \epsilon _5\Big [e^\beta \Big (\frac{2}{r}-\beta '\Big )\chi _{\ge 8M}+\frac{(r-3M)^2}{r^3}\Big ]\frac{(E_1)^2}{r^2},\\ I_2+I'_2\ge & {} \epsilon _5e^{\beta }\beta '\chi _{\ge 8M}\Big (L\phi +\frac{(1-\widetilde{\alpha })\phi }{r}\Big )^2+\epsilon _5\Big (\rho +\frac{M^2}{r^3}\Big )(E_2)^2\\&+\,\epsilon _5\Big (\rho +\frac{M^2(r-3M)^2}{r^5}\Big )(E_3)^2,\\ I_3\ge & {} \epsilon _5\Big (\frac{M}{r^4}e^{\beta }\chi _{\ge 8M}+\frac{e^{\beta }\beta '}{r^2}\Big )\phi ^2, \end{aligned}$$

provided that $\epsilon _4$ is fixed (sufficiently small relative to $\epsilon _3$), and D is sufficiently large depending on $\epsilon _4$ such that $e^{\beta (DM)}\ge \epsilon _4^{-4}$). The bound (4.53) follows.

To prove (4.54) we combine now the formulas (4.49), (4.42), and (4.44) to estimate

$$\begin{aligned} 2P_{\mu } n^\mu \ge I_4+I_5+I_6-\frac{1}{r^2}\partial _2\big [f_3r\phi ^2\big ]-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2), \end{aligned}$$

where

$$\begin{aligned} I_4:= & {} \frac{(E_1)^2}{r^2}\Big [\frac{f_3}{1-p}+g_3\Big ]-\epsilon _3^{-1}\frac{(E_1)^2}{r^2},\\ I_5:= & {} f_3\Big (L\phi +\frac{\phi }{r}\Big )^2+g_3(1-p)\Big [(E_2)^2+\frac{(E_3)^2(1-p^2\chi ^2)}{(1-p)^2}\Big ]\\&+\,\epsilon _3(E_2)^2\mathbf {1}_{(c_0,17M/8]}(r)-\epsilon _3^{-1}\Big [(L\phi )^2+\frac{M^2}{r^{2}}\big [(E_2)^2|1-p|+(E_3^2)\big ]\Big ],\\ I_6:= & {} \phi ^2\Big [\frac{\widetilde{\alpha }F_3}{r(1-p)}+\frac{2Mf_3}{r^3(1-p)}-\frac{(1-p)g'_3}{r}\Big ]-\epsilon _3^{-1}\frac{1}{r^2}\phi ^2. \end{aligned}$$

Using (4.51), (4.52), and (4.57) it follows that

$$\begin{aligned} I_4\ge & {} \epsilon _5e^\beta \frac{(E_1)^2}{r^2},\\ I_5+I_6\ge & {} \epsilon _5e^\beta \Big (L\phi +\frac{\phi }{r}\Big )^2+[g_3(1-p)+\epsilon _3\mathbf {1}_{(c_0,17M/8]}(r)]\frac{(E_2)^2}{2}+\epsilon _5 g_3(E_3)^2\\&+\,\epsilon _4\frac{\widetilde{\alpha }e^\beta \beta '}{1000r}\phi ^2-10\epsilon _3^{-1}\frac{1}{r^2}\phi ^2, \end{aligned}$$

provided that $C_4$ is sufficiently large (relative to $\epsilon _4$) and $|c_0-2M|\le C_4^{-10}$ is sufficiently small. Using the Hardy inequalities in Lemma A.1 (i) and (ii) it is easy to see that the integral on the negative term $-10\epsilon _3^{-1}r^{-2}\phi ^2$ in $I_6$ along $\Sigma _t^c$ can be absorbed by the integrals of the positive terms $\epsilon _4\frac{\widetilde{\alpha }e^\beta \beta '}{1000r}\phi ^2$ and $g_3(1-p)\frac{(E_2)^2}{2}$, provided that the constant $C_4$ is sufficiently large.

Moreover, notice that for any $t\in [0,T]$

$$\begin{aligned} \int _{\Sigma _t^c}2P_\mu n_0^\mu \,d\mu _{t}=C\int _{\mathbb {S}^2}\int _{c}^\infty 2P_\mu n^\mu r^2(\sin \theta )\,drd\theta . \end{aligned}$$

After integration by parts in r it follows that

$$\begin{aligned}&\Big |\int _{\mathbb {S}^2} \int _{c}^\infty \frac{1}{r^2}\partial _2\big [f_3r\phi ^2\big ]r^2(\sin \theta )\,drd\theta \Big |\\&\quad +\Big |\int _{\mathbb {S}^2} \int _{c}^\infty \frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2)r^2(\sin \theta )\,drd\theta \Big |\le \epsilon _4^{-1}\int _{\Sigma _t^c}\frac{1}{r^2}\phi ^2\,d\mu _t, \end{aligned}$$

so these terms can also be absorbed. The desired bound (4.54) follows.

The proof of (4.55) is similar, starting from the inequality (4.43) and the identity (4.49). To prove (4.56) we start from the bound (4.41),

$$\begin{aligned} 2(P_{(1)\mu }+P_{(2)\mu }) k^\mu \ge \epsilon _3\big [(E_1/r)^2+(E_2)^2(2-c/M)+M^{-2}\phi ^2\big ]-\epsilon _3^{-1}(E_3)^2. \end{aligned}$$

The identity (4.8) shows that

$$\begin{aligned} 2P_{(3)\mu } k^\mu =2k^\mu Q_{\mu \nu }X^\nu _{(3)}=2g_3(c)p(E_3)^2+2g_3(c)(1-p)E_2E_3. \end{aligned}$$

The lower bound (4.56) follows since $g_3(c)\in [C_4/2,2C_4]$, provided that $C_4$ is sufficiently large and $|c-2M|/M$ is sufficiently small. $\square$

Proof of Theorem 4.1

We can now complete the proof of Theorem 4.1, using Lemma 4.4 and the divergence identity. We have to fix functions $\beta$ and $\rho$ satisfying (4.51). With $\alpha$ as in the statement of the theorem, we define first the smooth function $\beta$ by setting $\beta (8M)=0$ and

$$\begin{aligned} \beta '(r)=\Big (\frac{4M}{r^2}+\frac{1}{r}\Big )\big (1-\chi _{\ge C_4^4M}(r)\big )+\frac{\alpha }{r}\chi _{\ge C_4^4M}(r). \end{aligned}$$

(4.58)

This choice clearly satisfies the first two conditions in (4.51). Then we define

$$\begin{aligned} \rho (r)=\delta M^{-1}\Big [\chi _{\ge C_4M}(r)+\chi _{\ge 4C_4^4M}(r)\Big (C_4^7e^{\beta (r)}\frac{M^3}{r^3}-1\Big )\Big ], \end{aligned}$$

(4.59)

where $\delta \in [10^{-4}C_4^{-3},10^4C_4^{-3}]$ is such that $\int _{C_4M}^\infty \rho (s)\,ds=C_4$.

Notice that

$$\begin{aligned} e^{\beta (r)}\approx \frac{r}{M}\text { if }r\le 10 C_4^4M\quad \text { and }\quad e^{\beta (r)}\approx C_4^4\Big (\frac{r}{C_4^4M}\Big )^\alpha \text { if }r\ge (1/10) C_4^4M. \end{aligned}$$

(4.60)

The other bounds in (4.51) follow easily. Moreover, the definitions show that

$$\begin{aligned}&e^{\beta (r)}\approx _{C_5}\frac{r^\alpha }{M^\alpha },\qquad \beta '(r)\approx _{C_5}\frac{1}{r},\qquad \Big (\frac{2}{r}-\beta '(r)\Big )\approx _{C_5}\frac{1}{r},\\&\rho (r)\approx _{C_5}\chi _{\ge C_4M}(r)\frac{M^{2-\alpha }}{r^{3-\alpha }}, \qquad g_3(r)\approx _{C_5}\frac{r^{\alpha -2}}{M^{\alpha -2}}. \end{aligned}$$

for some large constant $C_5$, where $A\approx _{C_5}B$ means $A\in [C_5^{-1}B,C_5B]$. The desired conclusion of the theorem follows from Lemma 4.4 and the divergence identity. $\square$

Proof of Theorem 1.7

In this section we prove Theorem 1.7. We still use some of the ideas from the previous section. We use the more complicated divergence identities (2.14) and (2.15),

$$\begin{aligned} 2\mathbf{D}^\mu P_\mu= & {} 2X^\nu J_\nu +Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu } +w(E_\alpha E^\alpha +F_\alpha F^\alpha +M_\alpha M^\alpha )\nonumber \\&+\,(\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi )+\frac{1}{2}\phi ^2(\mathbf{D}^\mu m_\mu -\square w)\nonumber \\&+\frac{1}{2}\psi ^2(\mathbf{D}^\mu m'_\mu -\square w)+w(\phi \mathcal {N}_\phi +\psi \mathcal {N}_\psi ), \end{aligned}$$

(5.1)

where

$$\begin{aligned} E_\mu= & {} \mathbf{D}_\mu \phi +\psi A^{-1}\mathbf{D}_\mu B,\quad F_\mu =\mathbf{D}_\mu \psi -\phi A^{-1}\mathbf{D}_\mu B,\quad M_\mu =\frac{\phi \mathbf{D}_\mu B-\psi \mathbf{D}_\mu A}{A}, \end{aligned}$$

(5.2)

$$\begin{aligned} Q_{\mu \nu }:= & {} E_\mu E_\nu +F_\mu F_\nu +M_\mu M_\nu -(1/2)\mathbf{g}_{\mu \nu }(E_\alpha E^\alpha +F_\alpha F^\alpha +M_\alpha M^\alpha ), \end{aligned}$$

(5.3)

$$\begin{aligned} P_\mu= & {} P_\mu [X,w,m,m']=Q_{\mu \nu }X^\nu +\frac{1}{2}w(\phi E_\mu +\psi F_\mu )-\frac{1}{4}\mathbf{D}_\mu w(\phi ^2+\psi ^2)\nonumber \\&+\,\frac{1}{4}(m_\mu \phi ^2+m'_\mu \psi ^2), \end{aligned}$$

(5.4)

and

$$\begin{aligned} J_\nu =\frac{2\mathbf{D}_\nu BM^\mu E_\mu -2\mathbf{D}_\nu AM^\mu F_\mu }{A}+\mathcal {N}_\phi E_\nu +\mathcal {N}_\psi F_\nu . \end{aligned}$$

(5.5)

Recall (see (1.9)) that

$$\begin{aligned} A=\frac{\Sigma ^2(\sin \theta )^2}{q^2},\quad B=-\Big [2aM(3\cos \theta -(\cos \theta )^3)+\frac{2a^3M(\sin \theta )^4\cos \theta }{q^2}\Big ]. \end{aligned}$$

(5.6)

These formulas show that

$$\begin{aligned} \begin{aligned}&A^{-1}\mathbf{D}_1B=\frac{6aM q^2\sin \theta }{\Sigma ^2}-\frac{2a^3M[4\sin \theta q^2-5(\sin \theta )^3q^2+2a^2(\sin \theta )^3(\cos \theta )^2]}{\Sigma ^2q^2},\\&A^{-1}\mathbf{D}_2B=\frac{4ra^3M(\sin \theta )^2\cos \theta }{q^2\Sigma ^2},\\&A^{-1}\mathbf{D}_1A=\frac{2\cos \theta }{\sin \theta }-\frac{2a^2\Delta \sin \theta \cos \theta }{\Sigma ^2}-\frac{2a^2\sin \theta \cos \theta }{q^2},\\&A^{-1}\mathbf{D}_2A=\frac{4r(r^2+a^2)-a^2(\sin \theta )^2(2r-2M)}{\Sigma ^2}-\frac{2r}{q^2}. \end{aligned} \end{aligned}$$

(5.7)

Notice that

$$\begin{aligned} r^{-1}\Big |\frac{\mathbf{D}_1 B}{A}\Big |+\Big |\frac{\mathbf{D}_2 B}{A}\Big |+r^{-1}\Big |\frac{\mathbf{D}_1 A}{A}-\frac{2\cos \theta }{\sin \theta }\Big |+\Big |\frac{\mathbf{D}_2 A}{A}-\frac{2}{r}\Big |\lesssim aMr^{-3}. \end{aligned}$$

(5.8)

and

$$\begin{aligned} \begin{aligned} \frac{|E_1-\mathbf{D}_1\phi |}{r}+|E_2-\mathbf{D}_2\phi |+|E_3-\mathbf{D}_3\phi |&\lesssim aMr^{-3}\big (|\phi |+|\psi |\big ),\\ \frac{|F_1-\mathbf{D}_1\psi |}{r}+|F_2-\mathbf{D}_2\psi |+|F_3-\mathbf{D}_3\psi |&\lesssim aMr^{-3}\big (|\phi |+|\psi |\big ),\\ \Big |\frac{M_1}{r}+\frac{2\cos \theta }{r\sin \theta }\psi \Big |+\Big |M_2+\frac{2\psi }{r}\Big |+|M_3|&\lesssim aMr^{-3}\big (|\phi |+|\psi |\big ). \end{aligned} \end{aligned}$$

(5.9)

Letting ${}^{0}\mathbf {g}^{\alpha {\beta }}$ denote the Schwarzschild components of the metric, see (4.2), and $\mathbf{g}^{\alpha {\beta }}$ the Kerr components, we notice that

$$\begin{aligned} \begin{aligned}&\mathbf{g}^{11}={}^{0}\mathbf {g}^{11}+O(a^2r^{-4}),\qquad \quad \mathbf{g}^{22}={}^{0}\mathbf {g}^{22}+O(a^2r^{-2}),\\&\mathbf{g}^{23}={}^{0}\mathbf {g}^{23}+O(a^2M^2r^{-4}),\qquad \mathbf{g}^{33}={}^{0}\mathbf {g}^{33}+O(a^2r^{-2}). \end{aligned} \end{aligned}$$

(5.10)

We notice that the term $J_1$ in (5.5) is singular when $\theta =0$, due to the fraction $\mathbf{D}_1A/A$. To eliminate this singularity we work with a modification of the 1-form P, namely

$$\begin{aligned} \widetilde{P}_\mu =\widetilde{P}_\mu [X,w,m,m']:=P_\mu -\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi ^2. \end{aligned}$$

(5.11)

Then

$$\begin{aligned} 2\mathbf{D}^\mu \widetilde{P}_\mu =2\mathbf{D}^\mu P_\mu -4\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi \mathbf{D}^\mu \psi - 2\mathbf{D}^\mu \Big [\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\Big ]\psi ^2=\sum _{j=1}^5 L^j, \end{aligned}$$

(5.12)

where

$$\begin{aligned} \begin{aligned}&L^1=L^1[X,w,m,m']:=Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }+w(E_\alpha E^\alpha +F_\alpha F^\alpha +M_\alpha M^\alpha ),\\&L^2=L^2[X,w,m,m']:=\phi m^\mu \mathbf{D}_\mu \phi +\psi {m'}^\mu \mathbf{D}_\mu \psi ,\\&L^3=L^3[X,w,m,m']:=\frac{1}{2}\phi ^2(\mathbf{D}^\mu m_\mu -\square w)+\frac{1}{2}\psi ^2(\mathbf{D}^\mu m'_\mu -\square w),\\&L^4=L^4[X,w,m,m']:=-2\mathbf{D}^\mu \Big [\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\Big ]\psi ^2,\\&L^5=L^5[X,w,m,m']:=2X^\nu J_\nu -4\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi \mathbf{D}^\mu \psi +w(\phi \mathcal {N}_\phi +\psi \mathcal {N}_\psi ). \end{aligned} \end{aligned}$$

(5.13)

The terms $L^1,L^2,L^3$ are similar to the corresponding terms we estimated in the proof of Theorem 4.1. The main new terms are $L^4$ and the quadratic part of $L^5$. We describe these terms below.

Lemma 5.1

Assuming that $X=f\partial _2+g\partial _3$, where f may depend only on r, we have

$$\begin{aligned} L^4=-8\frac{\mathbf{g}^{22}}{r}\partial _2\big [r^{-1}f\big ]\psi ^2+O(a^2r^{-5}) \big [|f|+r|f'|\big ]\psi ^2 \end{aligned}$$

(5.14)

and

$$\begin{aligned} |L^5|\lesssim & {} \frac{aM}{r^4}|f|\big (|\phi |+|\psi |\big )\Big \{\sum _{Y\in \{E,F\}}\Big (\frac{|Y_1|}{r}+\frac{M}{r}|Y_2|+\frac{M}{r}|Y_3|\Big )+\frac{1}{r}\big (|\phi |+|\psi |\big )\Big \}\nonumber \\&+\,|\mathcal {N}_\phi |\big |2fE_2+2gE_3+w\phi \big |+|\mathcal {N}_\psi |\big |2fF_2+2gF_3+w\psi \big |. \end{aligned}$$

(5.15)

Proof

We rewrite

$$\begin{aligned} L^4=-2\mathbf{D}^\mu \Big [\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\Big ] \psi ^2=-2\mathbf{D}^\mu \Big [\frac{f\mathbf{D}_2 A}{A}\frac{\mathbf{D}_\mu A}{A}\Big ]\psi ^2. \end{aligned}$$

In view of (1.10) and (5.7),

$$\begin{aligned} \Big |\frac{f\mathbf{D}_2 A}{A}\mathbf{D}^\mu \Big [\frac{\mathbf{D}_\mu A}{A}\Big ]\Big |= \Big |\frac{f\mathbf{D}_2 A}{A}\frac{\mathbf{D}_\mu B\mathbf{D}^\mu B}{A^2}\Big |\lesssim \frac{a^2M^2}{r^7}|f|. \end{aligned}$$

Also

$$\begin{aligned} \Big |\mathbf{g}^{11}\partial _1\Big [\frac{f\partial _2 A}{A}\Big ]\frac{\partial _1 A}{A}\Big |\lesssim \frac{a^2}{r^5}|f|. \end{aligned}$$

and

$$\begin{aligned} \Big |\mathbf{g}^{22}\partial _2\Big [\frac{f\partial _2 A}{A}\Big ]\frac{\partial _2 A}{A}-\mathbf{g}^{22}\partial _2\Big [\frac{2f}{r}\Big ]\frac{2}{r}\Big |\lesssim \frac{a^2}{r^5}|f|+\frac{a^2}{r^4}|f'|. \end{aligned}$$

The desired formula (5.14) follows.

We estimate now the term $L^5$. We start by rewriting

$$\begin{aligned} L^5= & {} 2X^\nu \Big [\frac{2\mathbf{D}_\nu BM^\mu E_\mu -2\mathbf{D}_\nu AM^\mu F_\mu }{A}\Big ]-4\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi \mathbf{D}^\mu \psi \\&+\,\mathcal {N}_\phi \big (2X^\nu E_\nu +w\phi \big )+\mathcal {N}_\psi \big (2X^\nu F_\nu +w\psi \big ). \end{aligned}$$

Using (5.7) and (5.2), we estimate

$$\begin{aligned} \Big |2X^\nu \frac{2\mathbf{D}_\nu BM^\mu E_\mu }{A}\Big |\lesssim \frac{a^2M}{r^5}|f|\big (|\phi |+|\psi |\big )\big [|E_1/r|+Mr^{-1}|E_2|+Mr^{-1}|E_3|\big ], \end{aligned}$$

and

$$\begin{aligned}&\Big |2X^\nu \Big [\frac{-2\mathbf{D}_\nu AM^\mu F_\mu }{A}\Big ]-4\frac{X^\nu \mathbf{D}_\nu A}{A}\frac{\mathbf{D}_\mu A}{A}\psi \mathbf{D}^\mu \psi \Big |\\&\qquad \lesssim \frac{aM}{r^4}|f||\phi |\Big [|F_1/r|+\frac{M}{r}|F_2|+\frac{M}{r}|F_3|+\frac{1}{r}|\psi |\Big ]. \end{aligned}$$

The desired formula (5.15) follows. $\square$

As in the proof of Theorem 4.1, our goal is to choose suitable multipliers $(X,w,m,m')$ in a such a way that the quadratic terms in the divergence formula

$$\begin{aligned} \int _{\Sigma _{t_1}^c}\widetilde{P}_\mu n_0^\mu \,d\mu _{t_1}= \int _{\Sigma _{t_2}^c}\widetilde{P}_\mu n_0^\mu \,d\mu _{t_2}+ \int _{\mathcal {N}^c_{[t_1,t_2]}}\widetilde{P}_\mu k_0^\mu \,d\mu _c+ \int _{\mathcal {D}^c_{[t_1,t_2]}}\mathbf{D}^\mu \widetilde{P}_\mu \,d\mu \end{aligned}$$

(5.16)

are nonegative, where $t_1,t_2\in [0,T]$, $c\in (c_0,{r_{\mathcal {H}}}]$, $n_0:=n/|\mathbf{g}^{33}|^{1/2}$, $k_0:=k/|\mathbf{g}^{22}|^{1/2}$, and the integration is with respect to the natural measures induced by the metric $\mathbf{g}$.

The Multipliers $(X_{(k)},w_{(k)},m_{(k)},m'_{(k)})$, $k\in \{1,2,3,4\}$

In this subsection we introduce the main multipliers. The multipliers $(X_{(k)},w_{(k)}, m_{(k)},m'_{(k)})$, $k\in \{1,2,3\}$ are analogous to the multipliers $(X_{(k)},w_{(k)},m_{(k)})$, $k\in \{1,2,3\}$, used in the analysis of the wave equation in Schwarzschild spacetime in the previous section. On the other hand, the multiplier $(X_{(4)},w_{(4)},m_{(4)},m'_{(4)})$, which is supported in a small region close to the trapped set, is new and is used mostly to control the contribution of the new term $L^4$ in (5.13).

Analysis Around the Trapped Set

As in the previous section, we start by constructing the multiplier $(X_{(1)},w_{(1)},m_{(1)}, m'_{(1)})$, which is relevant in a neighborhood of the trapped set. For now our main concern is the positivity of the spacetime integral $\mathbf{D}^\mu \widetilde{P}_\mu$; as in the proof of Theorem 4.1, the positivity of the surfaces integrals along $\Sigma _{t}^c$ and $\mathcal {N}^c_{[t_1,t_2]}$ can only be addressed after the other multipliers are introduced.

It is important to recall that we are in the axially symmetric case. Therefore the relevant trapped null geodesics are still confined to a codimension 1 set. More precisely, recalling that $a\ll M$, it is easy to see that the equation $r^3-3Mr^2+a^2r+Ma^2=0$ has a unique solution $r^*\in (c_0,\infty )$. Moreover, $r^*\in [3M-a^2/M,3M]$ and

$$\begin{aligned} \big |r^3-3Mr^2+a^2r+Ma^2-(r-r^*)r^2\big |\lesssim (a^2/M)r|r-r^*|\qquad \,\text { if }r\in (c_0,\infty ). \end{aligned}$$

(5.17)

We start by setting, as before,

$$\begin{aligned} \begin{aligned}&X_{(1)}:=f_1(r)\partial _2+g_1(r)\partial _3,\qquad f_1(r):=\frac{a_1(r)\Delta }{r^2},\qquad g_1(r):=\frac{a_1(r)\chi (r)2M}{r}+1,\\&w_{(1)}(r,\theta ):=f'_1(r)+f_1(r)\partial _r\log \big (\Sigma ^2/\Delta )-\epsilon _1\widetilde{w}(r),\\&\widetilde{w}(r):=M^2(r-33M/16)^3(r-r^*)^2r^{-8}\mathbf {1}_{[33M/16,\infty )}(r),\\&m_{(1)}=m'_{(1)}:=0,\\ \end{aligned} \end{aligned}$$

(5.18)

where $a_1:(0,\infty )\rightarrow \mathbb {R}$ is a smooth function to be fixed, $\lim _{r\rightarrow \infty }a_1(r)=1$, $\epsilon _1\in (0,1]$ is a small constant and $\Sigma ^2=(r^2+a^2)^2-a^2(\sin \theta )^2\Delta$ is as in (1.5).

Let

$$\begin{aligned} L^j_{(1)}:=L^j[X_{(1)},w_{(1)},m_{(1)},m'_{(1)}], \end{aligned}$$

for $j\in \{1,2,3,4,5\}$, see (5.13). Notice that

$$\begin{aligned} L^2_{(1)}=0,\qquad L^3_{(1)}=-\frac{1}{2}\square w_{(1)}(\phi ^2+\psi ^2). \end{aligned}$$

(5.19)

Using (A.15),

$$\begin{aligned} L^1_{(1)}=\sum _{Y\in \{E,F,M\}}\big [K_{(1)}^{11}(Y_1)^2+ K_{(1)}^{22}(Y_2)^2+K_{(1)}^{33}(Y_3)^2+2K_{(1)}^{23}Y_2Y_3\big ], \end{aligned}$$

where

$$\begin{aligned}&K_{(1)}^{11}=\frac{-f'_1(r)}{q^2}+w_{(1)}(r,\theta )\mathbf{g}^{11},\\&K_{(1)}^{22}=\frac{-f_1(r)(2r-2M)+f'_1(r)\Delta }{q^2}+w_{(1)}(r,\theta )\mathbf{g}^{22},\\&K_{(1)}^{33}=-f_1(r)\partial _2\mathbf{g}^{33}+2g'_1(r)\mathbf{g}^{23}-f'_1(r)\mathbf{g}^{33}-\frac{2rf_1(r)\mathbf{g}^{33}}{q^2}+w_{(1)}(r,\theta )\mathbf{g}^{33},\\&K_{(1)}^{23}=\frac{-2Mrf_1(r)\chi '(r)-2Mf_1(r)\chi (r)+g'_1(r)\Delta }{q^2}+w_{(1)}(r,\theta )\mathbf{g}^{23}. \end{aligned}$$

Simple calculations, using also (A.6), show that

$$\begin{aligned} \begin{aligned}&\partial _r\log \big (\Sigma ^2/\Delta )=\frac{\Delta \partial _r\Sigma ^2-\Sigma ^2\partial _r\Delta }{\Delta \Sigma ^2}=\frac{2(r^2+a^2)(r^3-3Mr^2+a^2r+Ma^2)}{\Delta \Sigma ^2},\\&\mathbf{g}^{33}=-\frac{\Sigma ^2}{q^2\Delta }+\frac{4M^2r^2}{q^2\Delta }\chi (r)^2. \end{aligned} \end{aligned}$$

(5.20)

Using also the formulas (4.19) and (A.6) we calculate

$$\begin{aligned} K_{(1)}^{11}= & {} a_1(r)\frac{2(r^2+a^2)(r^3-3Mr^2+a^2r+Ma^2)}{r^2q^2\Sigma ^2}-\epsilon _1\widetilde{w}(r)\mathbf{g}^{11},\\ K_{(1)}^{22}= & {} \frac{2\Delta ^2}{q^2r^2}\Big [a'_1(r)+a_1(r)\frac{-2a^2(r^2+a^2)+a^2(\sin \theta )^2(r^2-3Mr+2a^2)}{\Sigma ^2r}\Big ]\\&-\,\epsilon _1\widetilde{w}(r)\mathbf{g}^{22},\\ K_{(1)}^{33}= & {} \frac{8M^2\chi (r)^2}{q^2}\Big [a'_1(r)+a_1(r)\frac{-2a^2(r^2+a^2)+a^2(\sin \theta )^2(r^2-3Mr+2a^2)}{\Sigma ^2r}\Big ]\\&-\,\epsilon _1\widetilde{w}(r)\mathbf{g}^{33},\\ K_{(1)}^{23}= & {} \frac{4M\Delta \chi (r)}{q^2r}\Big [a'_1(r)+a_1(r)\frac{-2a^2(r^2+a^2)+a^2(\sin \theta )^2(r^2-3Mr+2a^2)}{\Sigma ^2r}\Big ]\\&-\,\epsilon _1\widetilde{w}(r)\mathbf{g}^{23}. \end{aligned}$$

Therefore

$$\begin{aligned} L^1_{(1)}\ge & {} \sum _{Y\in \{E,F,M\}}\Big \{\frac{(2-a/M)a_1(r)(r-r^*)-\epsilon _1r^4\widetilde{w}(r)q^{-2}}{r^4}(Y_1)^2\nonumber \\&+\,\Big [(2-a/M)a'_1(r)-\epsilon _1\widetilde{w}(r)\frac{r^4}{q^2\Delta }\Big ]\Big (\frac{\Delta }{r^2}Y_2+\frac{2M\chi (r)}{r}Y_3\Big )^2\nonumber \\&+\,\epsilon _1\widetilde{w}(r)\frac{\Sigma ^2}{q^2\Delta }(Y_3)^2\Big \}, \end{aligned}$$

(5.21)

provided that a is sufficiently small and

$$\begin{aligned} a_1(r^*)=0\qquad \text { and }\qquad a'_1(r)\ge a^{1/2}M^{3/2}r^{-3}|a_1(r)|\text { for }r\in (c_0,\infty ). \end{aligned}$$

(5.22)

This condition is clearly satisfied by the function $a_1$ defined below.

The important function $a_1$ is defined as in the proof of Theorem 4.1, see (4.22),

$$\begin{aligned} \begin{aligned}&R(r):=(r-r^*)(r+2M)+6M^2\log \Big (\frac{r-{r_{\mathcal {H}}}}{r^*-{r_{\mathcal {H}}}}\Big ),\\&a_1(r):=r^{-2}\delta ^{-1}\kappa (\delta R(r))+\Big [\frac{r^*-2M}{r}-\frac{6M^2}{r^2}\log \Big (\frac{r-{r_{\mathcal {H}}}}{r^*-{r_{\mathcal {H}}}}\Big )\Big ]\chi _{\ge DM}(r), \end{aligned} \end{aligned}$$

(5.23)

where $\delta :=\epsilon _2^2M^{-2}$ is a small constant and $D\gg 1$ is a large constant. This function can be analyzed as in section 4, see (4.23)–(4.32), once we observe that

$$\begin{aligned} {r_{\mathcal {H}}}=2M+O(a^2/M),\qquad r^*=3M+O(a^2/M),\qquad \Sigma ^2=r^4+O(a^2r^2). \end{aligned}$$

Recalling also the identities (5.10) and defining

$$\begin{aligned} h_1(r,\theta ):=f'_1(r)+f_1(r)\partial _r\log \big (\Sigma ^2/\Delta )=\frac{\Delta }{\Sigma ^2}\partial _r\big [a_1(r)\Sigma ^2r^{-2}\big ], \end{aligned}$$

(5.24)

we estimate, as in (4.30),

$$\begin{aligned} (\square h_1)(r,\theta )= & {} -\frac{2M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )+O(ar^{-4})+O(Mr^{-4})\mathbf {1}_{[DM,\infty )}(r)\nonumber \\&+\,M^{-3}O(1)\mathbf {1}_{(c_0,r_\delta ]}(r)+O\Big (\frac{\delta ^2M^2}{r-{r_{\mathcal {H}}}}\Big )\mathbf {1}_{[r'_\delta ,r_\delta ]}(r), \end{aligned}$$

(5.25)

where $r_\delta$ and $r'_\delta$ denote the unique numbers in $({r_{\mathcal {H}}},\infty )$ with the property that $R(r_\delta )=-1/\delta$ and $R(r'_\delta )=-2/\delta$. We also have, compare with (4.26),

$$\begin{aligned} a_1(r^*)=0\qquad \text { and }\qquad a'_1(r)\ge 10 M^2r^{-3}\qquad \text { for }r\in (c_0,\infty ), \end{aligned}$$

(5.26)

if $\delta$ is sufficiently small. In particular, this implies (5.22) if a is sufficiently small relative to $\epsilon _2$.

The bound (5.21) shows that

$$\begin{aligned} L_{(1)}^1\ge & {} \sum _{Y\in \{E,F,M\}}\Big \{\frac{(2-C_1\epsilon _1)a_1(r)(r-r^*)}{r^4}(Y_1)^2+\epsilon _1\widetilde{w}(r)(Y_3)^2\nonumber \\&\qquad \qquad \qquad +\,(2-C_1\epsilon _1)a'_1(r)\Big (\frac{\Delta }{r^2}Y_2+\frac{2M\chi (r)}{r}Y_3\Big )^2\Big \}, \end{aligned}$$

(5.27)

for a sufficiently large constant $C_1$, provided that the constant $\epsilon _1$ is sufficiently small and $a/M\le \epsilon _1$. Moreover, the identities (5.19) and (5.25) show that

$$\begin{aligned} L^3_{(1)}\ge & {} \frac{M(1-C_1\epsilon _1)}{r^4} \Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big ) (\phi ^2+\psi ^2)-\frac{C_1M}{r^4}\mathbf {1}_{[DM,\infty )}(r)(\phi ^2+\psi ^2)\nonumber \\&-\frac{C_1}{M^3}\mathbf {1}_{(c_0,r_\delta ]}(r)(\phi ^2+\psi ^2)-\frac{C_1\delta ^2M^2}{r-{r_{\mathcal {H}}}}\mathbf {1}_{[r'_\delta ,r_\delta ]}(r)(\phi ^2+\psi )^2. \end{aligned}$$

(5.28)

The bounds (5.14) and (5.15) and the definitions show that

$$\begin{aligned} L^2_{(1)}= & {} 0, \end{aligned}$$

(5.29)

$$\begin{aligned} L^4_{(1)}= & {} -8\frac{\mathbf{g}^{22}}{r}\partial _2\big [r^{-1}f_1\big ]\psi ^2+O(a^2r^{-5})\big [|f_1|+r|f'_1|\big ]\psi ^2\nonumber \\= & {} \Big [-\frac{8\Delta ^2}{q^2r^4}a'_1(r)+\frac{8\Delta (r^2-4Mr)}{q^2r^5}a_1(r)\Big ] \psi ^2\nonumber \\&+\,O(a^2r^{-5})\big [|a_1|+|r-{r_{\mathcal {H}}}||a'_1|\big ]\psi ^2. \end{aligned}$$

(5.30)

and

$$\begin{aligned} |L^5_{(1)}|\lesssim & {} \frac{aM}{r^4}|f|\big (|\phi |+|\psi |\big )\Big \{\sum _{Y\in \{E,F\}}\Big (\frac{|Y_1|}{r}+\frac{M}{r}|Y_2|+\frac{M}{r}|Y_3|\Big )+\frac{1}{r}\big (|\phi |+|\psi |\big )\Big \}\nonumber \\&+\,|\mathcal {N}_\phi |\big |2f_1E_2+2g_1E_3+w_1\phi \big |+|\mathcal {N}_\psi |\big |2f_1F_2+2g_1F_3+w_1\psi \big |. \end{aligned}$$

(5.31)

Using (5.25) and (5.30), together with the inequalities in the last line of (5.9), after possibly increasing the constant $C_1$ we have

$$\begin{aligned} L_{(1)}^1+L_{(1)}^4\ge & {} \sum _{Y\in \{E,F\}}\Big \{\frac{(2-C_1\epsilon _1)a_1(r)(r-r^*)}{r^4}(Y_1)^2+\epsilon _1\widetilde{w}(r)(Y_3)^2\nonumber \\&+\,(2-C_1\epsilon _1)a'_1(r)\Big (\frac{\Delta }{r^2}Y_2 +\frac{2M\chi (r)}{r}Y_3\Big )^2\Big \}+\frac{8\Delta (r^2-4Mr)}{r^7}a_1(r)\psi ^2\nonumber \\&+\,\frac{(2-C_1\epsilon _1)a_1(r)(r-r^*)}{r^4}\frac{4(\cos \theta )^2\psi ^2}{(\sin \theta )^2}\nonumber \\&-\,C_1\frac{a^2|a_1(r)|+\epsilon _1r^2|r-{r_{\mathcal {H}}}|a'_1(r)}{r^5}(\phi ^2+\psi ^2). \end{aligned}$$

(5.32)

Analysis in a Neighborhood of the Horizon

In a small neighborhood of the horizon we need to use the redshift effect. As in subsection 4.1, we define

$$\begin{aligned} \begin{aligned}&X_{(2)}:=f_2(r)\partial _2+g_2(r)\partial _3,\qquad f_2(r):=-\epsilon _2a_2(r),\qquad g_2(r):=\epsilon _2 a_2(r)(1-\epsilon _2),\\&w_{(2)}(r):=-2\epsilon _2 a_2(r)/r,\qquad m_{(2)2}=m_{(2)3}=m'_{(2)2}=m'_{(2)3}:=\epsilon _2M^{-2}\gamma (r),\\&m_{(2)1}=m_{(2)4}=m'_{(2)1}=m'_{(2)4}:=0, \end{aligned} \end{aligned}$$

(5.33)

where $\epsilon _2$ is a small positive constant (recall that $\delta =\epsilon _2^2M^{-2}$),

$$\begin{aligned} a_2(r):={\left\{ \begin{array}{ll} M^{-3}(9M/4-r)^3\qquad &{}\text { if }\, r\le 9M/4,\\ 0\qquad &{}\text { if }\, r\ge 9M/4, \end{array}\right. } \end{aligned}$$

(5.34)

and $\gamma :[c_0,\infty )\rightarrow [0,1]$ is a function supported in $[c_0,17M/8]$, and satisfying $\gamma ({r_{\mathcal {H}}})=1/2$ and a property similar to (4.38).

Let $L^j_{(2)}:=L^j[X_{(2)},w_{(2)},m_{(2)},m'_{(2)}]$, $j\in \{1,2,3,4,5\}$. As in the proof of Theorem 4.1, see Lemma 4.2 and (4.40), the multipliers $(X_{(1)},w_{(1)},m_{(1)},m'_{(1)})$ and $(X_{(2)},w_{(2)},m_{(2)},m'_{(2)})$ can be combined to prove the following:

Lemma 5.2

The constants $\epsilon _1,\epsilon _2$ can be fixed small enough such that there is a sufficiently small absolute constant $\epsilon _3>0$ with the property that

$$\begin{aligned}&\sum _{j=1}^4\big (L^j_{(1)}+L^j_{(2)}\big )\nonumber \\&\quad \ge \epsilon _3\sum _{Y\in \{E,F,M\}}\Big [\frac{(r-r^*)^2}{r^3}(Y_1/r)^2+\frac{M^2}{r^3}(Y_2)^2+\frac{M^2(r-r^*)^2}{r^5}(Y_3)^2\Big ]\nonumber \\&\qquad +\,\epsilon _3\frac{M}{r^4}\big (\phi ^2+\psi ^2\big )-\epsilon _3^{-1}\frac{M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\big (\phi ^2+\psi ^2\big )+\widetilde{L}, \end{aligned}$$

(5.35)

where

$$\begin{aligned} \widetilde{L}:= & {} \frac{8\Delta (r^2-4Mr)}{r^7}a_1(r)\psi ^2+(1-2C_1\epsilon _1)\mathbf {1}_{[r^*,\infty )}(r)\Big \{\frac{M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )\psi ^2\nonumber \\&+\frac{8a_1(r)(r-r^*)}{r^4}\frac{(\cos \theta )^2}{(\sin \theta )^2}\psi ^2+\frac{2a_1(r)(r-r^*)}{r^4}(F_1)^2+2a'_1(r)\frac{\Delta ^2}{r^4}(F_2)^2\Big \}, \end{aligned}$$

(5.36)

provided that a / M and $({r_{\mathcal {H}}}-c_0)/M$ are very small relative to $\epsilon _3$. Moreover

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) k^\mu\ge & {} \epsilon _3\sum _{Y\in \{E,F,M\}} \big [(Y_1/r)^2+(Y_2)^2({r_{\mathcal {H}}}-c)/M\big ]+\epsilon _3M^{-2}\big (\phi ^2+\psi ^2\big )\nonumber \\&-\,\epsilon _3^{-1}\big [(E_3)^2+(F_3)^2\big ], \end{aligned}$$

(5.37)

along $\mathcal {N}^c_{[t_1,t_2]}$. Also

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) n^\mu\ge & {} -\epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)\big [(E_3)^2+(F_3)^2\big ]\big \}\nonumber \\&-\,\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2+r\psi ^2)\nonumber \\&+\,\epsilon _3\big [(E_2)^2+(F_2)^2\big ]\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(5.38)

and

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }) n^\mu\le & {} \epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)\big [(E_3)^2+(F_3)^2\big ]\big \}\nonumber \\&-\,\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2+r\psi ^2)\nonumber \\&+\,\epsilon _3^{-1}\big [(E_2)^2+(F_2)^2\big ]\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(5.39)

where

$$\begin{aligned} \widetilde{e}_0= & {} \frac{(E_1)^2+(F_1)^2+(M_1)^2}{r^2}+(L\phi )^2+(L\psi )^2\nonumber \\&+\,\frac{M^2|r-{r_{\mathcal {H}}}|}{r^3}\big [(E_2)^2+(F_2)^2\big ]+\frac{M^2}{r^2}\big [(E_3^2)+(F_3)^2\big ]+\frac{1}{r^2}(\phi ^2+\psi ^2). \end{aligned}$$

(5.40)

Finally,

$$\begin{aligned} \big |L^5_{(1)}\big |+\big |L^5_{(2)}\big |\le & {} \frac{\epsilon _3^{-1}aM|r-r^*|}{r^5}\big (|\phi |+|\psi |\big )\nonumber \\&\times \,\Big \{\sum _{Y\in \{E,F\}}\Big (\frac{|Y_1|}{r}+\frac{M(|Y_2|+|Y_3|)}{r}\Big )+\frac{1}{r}\big (|\phi |+|\psi |\big )\Big \}\nonumber \\&+\,\epsilon _3^{-1}\big [e(\phi ,\mathcal {N}_\phi )+e(\psi ,\mathcal {N}_\psi )\big ]. \end{aligned}$$

(5.41)

Proof

The order of the constants to keep in mind is

$$\begin{aligned} \max \big (a/M,({r_{\mathcal {H}}}-c_0)/M\big )\ll \epsilon _3\ll \min (\epsilon _1,\epsilon _2)\le \max (\epsilon _1,\epsilon _2)\ll C_1^{-1}\ll 1. \end{aligned}$$

(5.42)

Most of the proof follows in the same way as in Lemma 4.2, using the identities/inequalities (A.16), (A.17), (5.25), (5.31), and (5.32)

The term $\widetilde{L}$ is new, when compared to the corresponding inequality (4.40) in the case of the pure wave equation. It is necessary to have this term because of the term $L_{(1)}^4$ in (5.30), which leads to the term

$$\begin{aligned} \frac{8\Delta (r^2-4Mr)}{r^7}a_1(r)\psi ^2 \end{aligned}$$

in (5.32). This term is clearly nonnegative if $r\le r^*$ or $r\ge 4M$; however, for $r\in [r^*,4M]$ we need an additional multiplier to control this term. The other terms in (5.36) are coming from corresponding terms in (5.32) and (5.25), and their role is to help $\widetilde{L}$ become positive. We show how to control this term below. $\square$

The New Multiplier $(X_{(4)},w_{(4)},m_{(4)},m'_{(4)})$

We define, with $a_1$ as in (5.23),

$$\begin{aligned} \begin{aligned}&X_{(4)}:=0,\qquad w_{(4)}=0,\qquad m_{(4)}=0,\\&\widetilde{m}'_{(4)1}(r,\theta ):=-(1-2C_1\epsilon _1) \frac{8(r-r^*)a_1(r)\chi _{\le 6R}(r)}{r^2}\frac{\cos \theta }{\sin \theta }\mathbf {1}_{[r^*,\infty )}(r),\\&\widetilde{m}'_{(4)2}(r):=(1-2C_1\epsilon _1)\frac{2b(r)}{\Delta },\qquad \widetilde{m}'_{(4)3}:=0,\qquad \widetilde{m}'_{(4)4}:=0, \end{aligned} \end{aligned}$$

(5.43)

for some function b supported in $[r^*,4M]$ to be fixed. We prove the following:

Lemma 5.3

Letting $L^j_{(4)}:=L^j[X_{(4)},w_{(4)},m_{(4)},m'_{(4)}]$, $j\in \{1,2,3,4,5\}$, we have

$$\begin{aligned} L^1_{(4)}=L^4_{(4)}=L^5_{(4)}=0 \end{aligned}$$

(5.44)

and, for some constant $C_2$ sufficiently large,

$$\begin{aligned} \widetilde{L}+L^2_{(4)}+L^3_{(4)}\ge -C_2(a+|{r_{\mathcal {H}}}-c_0|)r^{-4}(\phi ^2+\psi ^2). \end{aligned}$$

(5.45)

Moreover,

$$\begin{aligned} \big |2\widetilde{P}_{(4)\mu }n^\mu \big |\lesssim \epsilon _3^{-1}\psi ^2/r^2\qquad \text { and }\qquad 2\widetilde{P}_{(4)\mu }k^\mu =0\,\,\text { along }\,\,\mathcal {N}^c_{[t_1,t_2]}. \end{aligned}$$

(5.46)

Proof

The identities in (5.44) are clear. The inequality in (5.45) is also clear in the regions $\{r\le r^*\}$ and $\{r\ge 12M\}$.

Using the formula (A.10) we calculate, in the region $\{r\in [r^*,12M]\}$,

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\mathbf{D}^\mu \widetilde{m}'_{(4)\mu }=(1-2C_1\epsilon _1)\Big [\frac{4(r-r^*)a_1(r)\chi _{\le 6R}(r)}{q^2r^2}+\frac{b'(r)}{q^2}\Big ],\\&\psi \widetilde{m}'^\mu _{(4)}\mathbf{D}_\mu \psi =(1-2C_1\epsilon _1)\Big [-\frac{8(r-r^*)a_1(r)\chi _{\le 6R}(r)}{q^2r^2}\frac{\cos \theta }{\sin \theta }\psi \mathbf{D}_1\psi +\frac{2b(r)}{q^2}\psi \mathbf{D}_2\psi \Big ]. \end{aligned} \end{aligned}$$

Therefore, in the region $\{r\in [r^*,12M]\}$,

$$\begin{aligned}&L^2_{(4)}+L^3_{(4)}+\widetilde{L}\nonumber \\&\quad =\frac{8\Delta (r^2-4Mr)}{r^7}a_1(r)\psi ^2+(1-2C_1\epsilon _1)\Big \{\frac{M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )\psi ^2\\&\qquad +\frac{8a_1(r)(r-r^*)}{r^4}\frac{(\cos \theta )^2}{(\sin \theta )^2}\psi ^2+\frac{2a_1(r)(r-r^*)}{r^4}(F_1)^2+2a'_1(r)\frac{\Delta ^2}{r^4}F^2_2\Big \}\\&\qquad +\,(1-2C_1\epsilon _1)\Big [\frac{4(r-r^*)a_1(r)\chi _{\le 6R}(r))}{q^2r^2}+\frac{b'(r)}{q^2}\Big ]\psi ^2\\&\qquad +\,(1-2C_1\epsilon _1)\Big [-\frac{8(r-r^*)a_1(r)\chi _{\le 6R}(r)}{q^2r^2}\frac{\cos \theta }{\sin \theta }\psi \mathbf{D}_1\psi +\frac{2b(r)}{q^2}\psi \mathbf{D}_2\psi \Big ]. \end{aligned}$$

Recalling (5.9), we may replace $\mathbf{D}_1\psi$ and $\mathbf{D}_2\psi$ with $F_1$ and $F_2$, up to acceptable errors. Then we divide by $(1-2C_1\epsilon _1)$ and complete squares. For (5.45) it suffices to prove that

$$\begin{aligned} -C_2a\le & {} \frac{8\Delta (r^2-4Mr)a_1(r)}{r^7(1-2C_1\epsilon _1)}+\frac{M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )\\&+\,\Big [\frac{4(r-r^*)a_1(r)\chi _{\le 6R}(r))}{r^4}+\frac{b'(r)}{r^2}\Big ]-\frac{b(r)^2}{2\Delta ^2a'_1(r)}, \end{aligned}$$

for any $r\in [r^*,12M]$, for some function b supported in $[r^*,4M]$ to be fixed. After algebraic simplifications, it suffices to prove that, for any $r\in [r^*,4M]$,

$$\begin{aligned} 0\le & {} \frac{M}{r^4}\Big (7-\frac{44M}{r}+\frac{72M^2}{r^2}\Big )+\frac{8\Delta (r-4M)a_1(r)}{r^6(1-2C_1\epsilon _1)}\nonumber \\&+\,\Big [\frac{4(r-r^*)a_1(r)}{r^4}+\frac{b'(r)}{r^2}\Big ]-\frac{b(r)^2}{2a'_1(r)\Delta ^2}. \end{aligned}$$

(5.47)

We multiply both sides of (5.47) by $r^6/M^3$. It suffices to find a function b supported in $[r^*,4M]$ such that, for $r\in [r^*,4M]$,

$$\begin{aligned} 1\lesssim & {} \frac{r^4b'(r)}{M^3}+\Big (\frac{7r^2}{M^2}-\frac{44r}{M}+72\Big )-\frac{r^4b(r)^2}{2M^3a'_1(r)(r-2M)^2}\nonumber \\&+\,4a_1(r)\Big (\frac{3r^3}{M^3}-\frac{15r^2}{M^2}+\frac{16r}{M}\Big ). \end{aligned}$$

(5.48)

Let

$$\begin{aligned} r=(3+s)M,\qquad \widetilde{b}(s):=b((3+s)M). \end{aligned}$$

Notice also that, for $s\in [0,1]$,

$$\begin{aligned} \big |a_1((3+s)M)-\widetilde{a}_1(s)\big |+\big |Ma'_1((3+s)M)-\widetilde{a}'_1(s)\big |\lesssim a, \end{aligned}$$

where

$$\begin{aligned} \widetilde{a}_1(s):=\frac{5s+s^2+6\log (1+s)}{(3+s)^2},\qquad \widetilde{a}'_1(s):=\frac{33+s-12\log (1+s)-12\frac{s}{s+1}}{(3+s)^3}. \end{aligned}$$

(5.49)

For (5.48) it suffices to prove that, for $s\in [0,1]$,

$$\begin{aligned} 1\lesssim \widetilde{b}'(s)-\frac{\widetilde{b}(s)^2}{2\widetilde{a}'_1(s)(1+s)^2}+\frac{7s^2-2s+3}{(3+s)^4}+4\widetilde{a}_1(s)\frac{3s^2+3s-2}{(3+s)^3}. \end{aligned}$$

(5.50)

Notice that $\widetilde{a}'_1(s)(1+s)^2\ge 1$ for any $s\in [0,1]$. Indeed, using (5.49),

$$\begin{aligned} (3+s)^3\big [\widetilde{a}'_1(s)(1+s)^2 -1]= & {} (1+s)[33+10s-11s^2+12(1+s)(s-\log (1+s))]\\&-(3+s)^3\\= & {} 12(1+s)^2(s-\log (1+s))+6+16s-10s^2-12s^3\\\ge & {} 0. \end{aligned}$$

Therefore, for (5.50) it suffices to prove that, for $s\in [0,1]$,

$$\begin{aligned} 1\lesssim \widetilde{b}'(s)-\frac{\widetilde{b}(s)^2}{2}+\frac{7s^2-2s+3}{(3+s)^4}+4\widetilde{a}_1(s)\frac{3s^2+3s-2}{(3+s)^3}. \end{aligned}$$

(5.51)

Moreover, for $s\in [0,1]$,

$$\begin{aligned}&\frac{7s^2-2s+3}{(3+s)^4}+4\widetilde{a}_1(s)\frac{3s^2+3s-2}{(3+s)^3}=\frac{9-91s+167s^2+115s^3-24s^4}{(3+s)^5}\\&\qquad +\frac{24(-2+3s+3s^2)[\log (1+s)-s+s^2/2]}{(3+s)^5}\\&\quad \ge \frac{9-91s+167s^2+91s^3}{(3+s)^5}\\&\quad \ge \frac{9(1-10s+18s^2)}{(3+s)^5}\mathbf {1}_{[1/10,1]}(s)+\frac{4s^2}{(3+s)^5}+10^{-10}. \end{aligned}$$

Therefore, to prove (5.51) it suffices to find a function $\widetilde{b}$ supported in [1 / 10, 1] such that

$$\begin{aligned} \widetilde{b}'(s)+\frac{9(1-10s+18s^2)}{(3+s)^5}\ge 0\quad \text { and }\quad |\widetilde{b}(s)|\le \frac{\sqrt{2}s}{16} \end{aligned}$$

(5.52)

for any $s\in [1/10,1]$.

Notice that $1-10s+18s^2=18(s-s_1)(s-s_2)$ where $s_1=(5-\sqrt{7})/18$, $s_2=(5+\sqrt{7})/18$. We define $\widetilde{b}(s)=0$ for $s\le s_1$ and

$$\begin{aligned} \widetilde{b}(s):=\int _{s_1}^s\frac{9(10\rho -1-18\rho ^2)}{3^5}\,d\rho \end{aligned}$$

for $s\in [s_1,s_2]$. The desired inequalities (5.52) are easy to verify for $s\in [1/10,s_2]$, and, moreover, $\widetilde{b}(s_2)=7^{3/2}9^{-4}\le 3\cdot 10^{-3}$.

On the other hand, for $s\ge s_2$, we would like to define the function $\widetilde{b}$ decreasing, still satisfying (5.52), and vanishing for $s\ge 1$. The only condition for this to be possible is the inequality

$$\begin{aligned} \int _{s_2}^1\frac{9(1-10\rho +18\rho ^2)}{4^5}\,d\rho \ge \widetilde{b}(s_2), \end{aligned}$$

which is easy to verify. This completes the proof of the main inequality (5.45).

The identity and the inequality in (5.46) follow from definitions. $\square$

As a consequence of Lemma 5.2 and Lemma 5.3 we have:

Corollary 5.4

There is a sufficiently small absolute constant $\epsilon _3>0$ with the property that

$$\begin{aligned}&\sum _{j=1}^4\big (L^j_{(1)}+L^j_{(2)}+L^j_{(4)}\big )\nonumber \\&\quad \ge \epsilon _3\sum _{Y\in \{E,F,M\}}\Big [\frac{(r-r^*)^2}{r^3}(Y_1/r)^2+\frac{M^2}{r^3}(Y_2)^2+\frac{M^2(r-r^*)^2}{r^5}(Y_3)^2\Big ]\nonumber \\&\qquad +\,\epsilon _3\frac{M}{r^4}\big (\phi ^2+\psi ^2\big )-\epsilon _3^{-1}\frac{M}{r^4}\mathbf {1}_{[DM,\infty )}(r)\big (\phi ^2+\psi ^2\big ), \end{aligned}$$

(5.53)

and

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(4)\mu }) k^\mu\ge & {} \epsilon _3\sum _{Y\in \{E,F,M\}}\big [(Y_1/r)^2+(Y_2)^2({r_{\mathcal {H}}}-c)/M\big ]\nonumber \\&+\,\epsilon _3M^{-2}\big (\phi ^2+\psi ^2\big )-\epsilon _3^{-1}\big [(E_3)^2+(F_3)^2\big ], \end{aligned}$$

(5.54)

along $\mathcal {N}^c_{[t_1,t_2]}$. Moreover, with $\widetilde{e}_0$ as in (5.40),

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(4)\mu }) n^\mu\ge & {} -\epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)\big [(E_3)^2+(F_3)^2\big ]\big \}\nonumber \\&-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2+r\psi ^2)\nonumber \\&+\,\epsilon _3\big [(E_2)^2+(F_2)^2\big ]\mathbf {1}_{(c_0,17M/8]}(r), \end{aligned}$$

(5.55)

and

$$\begin{aligned} 2(\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(4)\mu }) n^\mu\le & {} \epsilon _3^{-1}\big \{\widetilde{e}_0+\mathbf {1}_{[8M,2DM]}(r)\big [(E_3)^2+(F_3)^2\big ]\big \}\nonumber \\&-\frac{\chi _{\ge 8M}(r)(1-p)}{r^2}\partial _2(r\phi ^2+r\psi ^2)\nonumber \\&+\,\epsilon _3^{-1}\big [(E_2)^2+(F_2)^2\big ]\mathbf {1}_{(c_0,17M/8]}(r). \end{aligned}$$

(5.56)

Finally,

$$\begin{aligned} \big |L^5_{(1)}\big |+\big |L^5_{(2)}\big |+\big |L^5_{(4)}\big |\le & {} \frac{\epsilon _3^{-1}aM|r-r^*|}{r^5}\big (|\phi |+|\psi |\big )\nonumber \\&\times \,\Big \{\sum _{Y\in \{E,F\}}\Big (\frac{|Y_1|}{r}+\frac{M(|Y_2|+|Y_3|)}{r}\Big )+\frac{1}{r}\big (|\phi |+|\psi |\big )\Big \}\nonumber \\&+\,\epsilon _3^{-1}\big [e(\phi ,\mathcal {N}_\phi )+e(\psi ,\mathcal {N}_\psi )\big ]. \end{aligned}$$

(5.57)

These inequalities should be compared with the inequalities (4.40) and the corresponding inequalities in Lemma 4.2.

Outgoing Energies

Finally, as in subsection 4.2, we define $(X_{(3)},w_{(3)},m_{(3)},m'_{(3)})$ by

$$\begin{aligned} \begin{aligned}&X_{(3)}:=f_3\partial _2+\Big (\frac{f_3}{1-\widetilde{p}}+g_3\Big )\partial _3,\qquad w_{(3)}:=\frac{2f_3}{r},\qquad m'_{(3)}:=m_{(3)},\\&m_{(3)1}:=m_{(3)4}:=0,\quad m_{(3)2}:=\frac{2h_3}{r(1-\widetilde{p})},\quad m_{(3)3}:=-\frac{2h_3}{r}, \end{aligned} \end{aligned}$$

(5.58)

where $\widetilde{p}:=2M/r$, and $f_3,g_3$ are defined by

$$\begin{aligned} f_3(r):=\epsilon _4\chi _{\ge 8M}(r)e^{\beta (r)},\qquad g_3(r):=\int _{r}^\infty \Big [\rho (s)+\frac{\epsilon _4M^2}{s^3}f_3(s)\Big ]\,ds, \end{aligned}$$

(5.59)

where

$$\begin{aligned} \beta (8M):=0,\qquad \beta '(r):=\Big (\frac{4M}{r^2}+\frac{1}{r}\Big )\big (1-\chi _{\ge C_4^4M}(r)\big )+\frac{\alpha }{r}\chi _{\ge C_4^4M}(r), \end{aligned}$$

(5.60)

and

$$\begin{aligned} \rho (r):=\delta M^{-1}\Big [\chi _{\ge C_4M}(r)+\chi _{\ge 4C_4^4M}(r)\Big (C_4^7e^{\beta (r)}\frac{M^3}{r^3}-1\Big )\Big ]. \end{aligned}$$

(5.61)

The constants $\epsilon _4,C_4$ satisfy $\epsilon _4=\epsilon _3^2$ and $C_4\ge \epsilon _4^{-4}\alpha ^{-1}(2-\alpha )^{-1}$, while $\delta \in [10^{-4}C_4^{-3},10^4C_4^{-3}]$ is such that $\int _{C_4M}^\infty \rho (s)\,ds=C_4$. Recall (4.60),

$$\begin{aligned} e^{\beta (r)}\approx \frac{r}{M}\text { if }r\le 10 C_4^4M\quad \text { and }\quad e^{\beta (r)}\approx C_4^4\Big (\frac{r}{C_4^4M}\Big )^\alpha \text { if }r\ge (1/10) C_4^4M. \end{aligned}$$

(5.62)

Notice the additional term $M^2s^{-3}f_3(s)$ in the definition of the function $g_3$; this term is needed in order to be able to estimate the contributions of the new terms containing the small coefficient a, in a way that is uniform as $\alpha \rightarrow 0$ or $\alpha \rightarrow 2$.

Also let

$$\begin{aligned} H_3:=(1-\widetilde{p})f'_3-\frac{2Mf_3}{r^2}-(1-\widetilde{p})^2\rho -\frac{\epsilon _4M^2f_3}{r^3}(1-\widetilde{p})^2,\quad h_3:=H_3\cdot (1-\widetilde{\alpha }), \end{aligned}$$

(5.63)

where $\widetilde{\alpha }:=(2-\alpha )/10$. Recall the bounds (4.51) and (4.52),

$$\begin{aligned}&\begin{aligned} \beta (r)\in [-10,0]\text { and }M\beta '(r)\in [1/10,10]&\qquad \text { if }r\in (c,8M],\\ \max \Big (\frac{\alpha }{100r},\frac{4M}{r^2}+\frac{1}{r}\mathbf {1}_{[8M,C_4M]}(r)\Big )\le \beta '(r)\le \frac{2}{r}&\qquad \text{ if } r\in [8M,\infty ),\\ \rho (r)=0\text { and }g_3(r)\in [C_4/2,2C_4]&\qquad \text { if }r\le C_4M,\\ \rho (r)\le \frac{\epsilon _4}{100}\beta '(r)e^{\beta (r)}\text { and }\rho '(r)\le \frac{\epsilon _4M}{100r^3}e^{\beta (r)}&\qquad \text { if }r\ge C_4M,\\ \frac{e^\beta M^2}{r^2}\le g_3(r)\le \frac{C_4^{10}e^\beta M^2}{r^2}&\qquad \text { if }r\ge C_4M,\\ (1-2\widetilde{\alpha })H_3(r)-rH'_3(r)\ge 0&\qquad \text { if }r\in [16M,\infty ), \end{aligned} \end{aligned}$$

(5.64)

$$\begin{aligned}&\begin{aligned}&g'_3=-\rho -\epsilon _4M^2r^{-3}f_3,\\&\big |H_3-(1-\widetilde{p})f'_3\big |\le \frac{(2+\epsilon _4)Mf_3}{r^2}+\rho ,\\&e^{\beta (r)}\in [r/(100M),r^2/M^2]\qquad \text { for }r\in (c,C_4M], \end{aligned} \end{aligned}$$

(5.65)

and

$$\begin{aligned} \begin{aligned}&\frac{2f_3-rf'_3}{r}=\epsilon _4 e^{\beta }\big [(2/r-\beta ')\chi _{\ge 8M}-\chi '_{\ge 8M}\big ],\\&\frac{6Mf_3}{r^4}-\frac{2Mf'_3}{r^3} +\frac{(1-\widetilde{p})^2g''_3}{r} +\frac{4M(1-\widetilde{p})g'_3}{r^3} \ge \frac{\epsilon _4 M}{100r^4}e^\beta \chi _{\ge 8M}-\frac{2\epsilon _4 M}{r^3}e^\beta \chi '_{\ge 8M}. \end{aligned} \end{aligned}$$

(5.66)

Notice that

$$\begin{aligned} \mathbf{g}^{33}=-\frac{r^2+a^2}{\Delta }+O(a^2Mr^{-3})\qquad \text { if }r\ge 5M/2. \end{aligned}$$

(5.67)

Proof of Theorem 1.7

Let $L^j_{(3)}:=L^j[X_{(3)},w_{(3)},m_{(3)},m'_{(3)}]$, $j\in \{1,2,3,4,5\}$. As in the proof of (4.48), we have

$$\begin{aligned} L^1_{(3)}=\sum _{Y\in \{E,F,M\}}\big [K^{11}_{(3)}(Y_1)^2+K^{22}_{(3)}(Y_2)^2+K^{33}_{(3)}(Y_3)^2+2K^{23}_{(3)}Y_2Y_3\big ], \end{aligned}$$

where, with $O':=O[a^2r^{-2}(f_3/r+f'_3)]$,

$$\begin{aligned} K_{(3)}^{11}= & {} \frac{-f'_3(r)}{q^2}+w_{(3)}(r)\mathbf{g}^{11}=\frac{2f_3-rf'_3}{rq^2},\\ K_{(3)}^{22}= & {} \frac{-f_3(r)(2r-2M)+f'_3(r)\Delta }{q^2}+w_{(3)}(r)\mathbf{g}^{22}=(1-\widetilde{p})f'_3-\frac{2Mf_3}{r^2}+O',\\ K_{(3)}^{33}= & {} -f_3(r)\partial _2\mathbf{g}^{33} -f'_3(r)\mathbf{g}^{33}-\frac{2rf_3(r)\mathbf{g}^{33}}{q^2} +w_{(3)}(r)\mathbf{g}^{33}= \frac{f'_3}{1 -\widetilde{p}}-\frac{2Mf_3}{r^2(1 -\widetilde{p})^2}\\&+\,O',\\ K_{(3)}^{23}= & {} \Big (\frac{f_3}{1-\widetilde{p}}+g_3\Big )'\frac{\Delta }{q^2}=f'_3-\frac{2Mf_3}{r^2(1-\widetilde{p})}-\rho (1-\widetilde{p})-\frac{\epsilon _4M^2 f_3}{r^3}(1-\widetilde{p})+O'. \end{aligned}$$

Therefore

$$\begin{aligned} L^1_{(3)}\ge & {} \sum _{Y\in \{E,F,M\}}\Big \{\frac{2f_3-rf'_3}{rq^2}(Y_1)^2+H_3\Big [Y_2+\frac{Y_3}{1-\widetilde{p}}\Big ]^2\\&+\,\Big [\rho +\frac{\epsilon _4M^2f_3}{2r^3}\Big ][(1-\widetilde{p})^2(Y_2)^2+(Y_3)^2]\Big \}\\&-\,aMr^{-3}e^{\beta (r)}\chi _{\ge 5M}(r)[(Y_2)^2+(Y_3)^2]. \end{aligned}$$

Also, using also (A.9), (A.10) the definitions (5.13), and Lemma 5.1,

$$\begin{aligned} L^2_{(3)}\ge & {} \frac{2h_3}{r}\phi \Big [\mathbf{D}_2\phi +\frac{\mathbf{D}_3\phi }{1-\widetilde{p}}\Big ]+\frac{2h_3}{r}\psi \Big [\mathbf{D}_2\psi +\frac{\mathbf{D}_3\psi }{1-\widetilde{p}}\Big ]\\&-\,aMr^{-4}e^{\beta (r)}\chi _{\ge 5M}(r)\big [|\phi ||\mathbf{D}_2\phi |+|\phi ||\mathbf{D}_3\phi |+|\psi ||\mathbf{D}_2\psi |+|\psi ||\mathbf{D}_3\psi |\big ],\\ L^3_{(3)}\ge & {} (\phi ^2+\psi ^2)\Big [\frac{h_3}{r^2}+\frac{h'_3}{r}+\frac{2Mf_3}{r^4}-\frac{2Mf'_3}{r^3}-\frac{(1-\widetilde{p})f''_3}{r}\Big ]\\&-\,(\phi ^2+\psi ^2)ar^{-4}e^{\beta (r)}\chi _{\ge 5M}(r), \end{aligned}$$

and

$$\begin{aligned} L^4_{(3)}\ge \frac{8(1-\widetilde{p})}{r^3}(f_3-rf'_3)\psi ^2-\psi ^2ar^{-4}e^{\beta (r)}\chi _{\ge 5M}(r). \end{aligned}$$

We combine now the $M_2^2$ term in the right-hand side of $L_{(3)}^1$ and $L^4_{(3)}$. Recalling also the definition and (5.9) we have $(M_2)^2\ge 4r^{-2}\psi ^2-(\phi ^2+\psi ^2)aMr^{-4}$. Therefore,

$$\begin{aligned} H_3(M_2)^2+L^4_{(3)}\ge -(\phi ^2+\psi ^2)ar^{-4}e^{\beta (r)}\chi _{\ge 5M}(r), \end{aligned}$$

using the second inequality in (5.65) and the definitions.

We add up the estimates above and complete the square to conclude that

$$\begin{aligned}&L^1_{(3)}+L^2_{(3)}+L^3_{(3)}+L^4_{(3)}\\&\quad \ge \sum _{Y\in \{E,F,M\}}\frac{2f_3-rf'_3}{2r^3}(Y_1)^2\\&\quad \quad +\,H_3\Big (E_2+\frac{E_3}{1-\widetilde{p}}+\frac{(1-\widetilde{\alpha })\phi }{r}\Big )^2+H_3\Big (F_2+\frac{F_3}{1-\widetilde{p}}+\frac{(1-\widetilde{\alpha })\psi }{r}\Big )^2\\&\quad \quad +\,\Big [\rho +\frac{\epsilon _4M^2f_3}{2r^3}\Big ][(1-\widetilde{p})^2(E_2)^2+(E_3)^2+(1-\widetilde{p})^2(F_2)^2+(F_3)^2]\\&\quad \quad +\,(\phi ^2+\psi ^2)\Big [\frac{(\widetilde{\alpha }-\widetilde{\alpha }^2)H_3-\widetilde{\alpha }rH'_3}{r^2}+\frac{H'_3}{r}+\frac{2Mf_3}{r^4}-\frac{2Mf'_3}{r^3}-\frac{(1-\widetilde{p})f''_3}{r}\Big ]\\&\quad \quad -\,\epsilon _3^{-1}ar^{-4}e^{\beta (r)}\chi _{\ge 5M}(r)(\phi ^2+\psi ^2). \end{aligned}$$

Combining this with (5.53) and estimating as in the proof of Lemma 4.4 we conclude that

$$\begin{aligned}&\sum _{j=1}^4\big (L^j_{(1)}+L^j_{(2)}+L^j_{(4)}+L^j_{(3)}\big )\nonumber \\&\quad \ge \sum _{Y\in \{E,F,M\}}\epsilon _4^2\Big (\frac{e^{\beta }(2-r\beta ')}{r}+\frac{100}{r}\Big )\frac{(r-r^*)^2}{r^2}\frac{(Y_1)^2}{r^2}\nonumber \\&\qquad +\,\epsilon _4^2\Big (\frac{\widetilde{\alpha }^2e^{\beta }\beta '}{r^2} +\frac{M e^{\beta }}{r^4}\Big )\big (\phi ^2 +\psi ^2\big )+\sum _{Y\in \{E,F\}}\epsilon _4^2\frac{M^2e^{\beta }}{100r^3} \Big [(Y_2)^2+\frac{(r-r^*)^2}{r^2}(Y_3)^2\Big ]\nonumber \\&\qquad +\,\epsilon _4^2e^{\beta }\beta '\Big [\Big (E_2 +\frac{E_3}{1-\widetilde{p}} +\frac{(1-\widetilde{\alpha })\phi }{r}\Big )^2 +\Big (F_2+\frac{F_3}{1-\widetilde{p}} +\frac{(1-\widetilde{\alpha })\psi }{r}\Big )^2\Big ], \end{aligned}$$

(5.68)

provided that D is taken large enough and $\epsilon _4$ is sufficiently small.

Moreover, using Lemma 5.1,

$$\begin{aligned} |L^5_{(3)}|\le & {} \frac{aM}{r^4}\epsilon _4e^{\beta }\chi _{\ge 8M}\big (|\phi |+|\psi |\big )\Big \{\sum _{Y\in \{E,F\}}\frac{|Y_1|+M|Y_2| +M|Y_3|}{r}+\frac{1}{r}\big (|\phi |+|\psi |\big )\Big \}\\&+\,e^{\beta }e(\phi ,\mathcal {N}_\phi )+e^{\beta }e(\psi ,\mathcal {N}_{\psi }). \end{aligned}$$

Combining this with (5.57), (5.68), and (5.9), we obtain the final lower bound on the space-time term, for some small constant $\epsilon _5=\epsilon _5(\alpha )$,

$$\begin{aligned}&\sum _{j=1}^5\big (L^j_{(1)}+L^j_{(2)}+L^j_{(4)}+L^j_{(3)}\big )\nonumber \\&\quad \ge \epsilon _5e^{\beta }\Big \{\frac{(r-r^*)^2}{r^2}\frac{(\partial _1\phi )^2+(\partial _1\psi )^2+(\psi /\sin \theta )^2}{r^3}\nonumber \\&\qquad +\frac{M^2}{r^3}\big [(\partial _2\phi )^2+(\partial _2\psi )^2\big ]+\frac{M^2(r-r^*)^2}{r^5}\big [(\partial _3\phi )^2+(\partial _3\psi )^2\big ]\nonumber \\&\qquad +\frac{\phi ^2+\psi ^2}{r^3}+\frac{(L\phi )^2+(L\psi )^2}{r}\Big \}-e^{\beta }\big [e(\phi ,\mathcal {N}_\phi )+e(\psi ,\mathcal {N}_{\psi })\big ]. \end{aligned}$$

(5.69)

We consider now the contribution of $\widetilde{P}_{(3)\mu }n^\mu$. Using (A.16) and the definitions we write

$$\begin{aligned} 2\widetilde{P}_{(3)\mu }n^\mu= & {} 2Q_{\mu \nu }X_{(3)}^\nu n^\mu +w_{(3)}(\phi E_\mu +\psi F_\mu )n^\mu -\frac{n^\mu \mathbf{D}_\mu w_{(3)}}{2}(\phi ^2+\psi ^2)\\&+\frac{n^\mu }{2}(m_{(3)\mu }\phi ^2+m'_{(3)\mu }\psi ^2)-2\frac{X_{(3)}^\nu \mathbf{D}_\nu A}{A}\frac{n^\mu \mathbf{D}_\mu A}{A}\psi ^2\\= & {} \frac{m_{(3)3}(-\mathbf{g}^{33})}{2}(\phi ^2+\psi ^2)\\&+\sum _{Y\in \{E,F,M\}}\Big [\frac{(Y_1)^2}{q^2}\Big (\frac{f_3}{1-\widetilde{p}}+g_3\Big )+\frac{\Delta (Y_2)^2}{q^2}\Big (\frac{f_3}{1-\widetilde{p}}+g_3\Big )\\&+\,(Y_3)^2(-\mathbf{g}^{33})\Big (\frac{f_3}{1-\widetilde{p}}+g_3\Big )+2Y_2Y_3(-\mathbf{g}^{33})f_3\Big ]\nonumber \\&+\frac{2f_3}{r}(-\mathbf{g}^{33})(\phi E_3+\psi F_3). \end{aligned}$$

As before, the main point is that the function $g_3$ is extremely large when r is small. We can combine this last identity with the bounds (5.55) and (5.56), as in the proof of Lemma 4.4 to conclude that, for any $t\in [0,T]$,

$$\begin{aligned} \int _{\Sigma _{t}^c}2\big [\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(3)\mu }+\widetilde{P}_{(4)\mu } \big ]n_0^\mu \,d\mu _{t}\approx _{\alpha } \int _{\Sigma _{t}^c} e^\beta \big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _t. \end{aligned}$$

(5.70)

Finally, using (A.17), the contribution of $\widetilde{P}_{(3)\mu }k^\mu$ along $\mathcal {N}^c_{[0,T]}$ is

$$\begin{aligned} 2\widetilde{P}_{(3)\mu }k^\mu= & {} 2Q_{\mu \nu }X_{(3)}^\nu k^\mu +w_{(3)}(\phi E_\mu +\psi F_\mu )k^\mu -\frac{k^\mu \mathbf{D}_\mu w_{(3)}}{2}(\phi ^2+\psi ^2)\\&+\frac{k^\mu }{2}(m_{(3)\mu }\phi ^2+m'_{(3)\mu }\psi ^2)-2\frac{X_{(3)}^\nu \mathbf{D}_\nu A}{A}\frac{k^\mu \mathbf{D}_\mu A}{A}\psi ^2\\= & {} \sum _{Y\in \{E,F\}}\big [2g_3(c)\mathbf{g}^{23}(Y_3)^2+2Y_2Y_3g_3(c)\mathbf{g}^{22}\big ]. \end{aligned}$$

Combining with (5.54) we obtain

$$\begin{aligned} 2\big [\widetilde{P}_{(1)\mu }+\widetilde{P}_{(2)\mu }+\widetilde{P}_{(3)\mu }+\widetilde{P}_{(4)\mu }\big ]k^\mu \ge 0\qquad \text { along }\mathcal {N}^c_{[0,T]}. \end{aligned}$$

(5.71)

The theorem follows from (5.69), (5.70), (5.71), and the divergence identity (5.16). $\square$

Proof of Corollary 1.8

In this section we provide a proof of Corollary 1.8. The main issue is the degeneracy of the weights in the bulk term at $r=r^*$. We compensate for this by losing derivatives. More precisely:

Lemma 6.1

Assume that $(\phi ,\psi )\in C^k([0,T]:\mathbf {H}^{6-k}(\Sigma _t^{c_0}))$, $k\in [0,6]$, is a solution of the system (1.33) with $\mathcal {N}_\phi =\mathcal {N}_\psi =0$. Then

$$\begin{aligned}&\mathcal {BB}_\alpha ^{c_0}(t_1,t_2)+\sum _{k=0}^2\int _{\Sigma _{t_2}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _t\nonumber \\&\quad \lesssim _\alpha \sum _{k=0}^2\int _{\Sigma _{t_1}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _t, \end{aligned}$$

(6.1)

for any $\alpha \in (0,2)$ and any $t_1\le t_2\in [0,T]$, where $\phi _k:=M^k\mathbf{T}^k\phi$, $\psi _k:=M^k\mathbf{T}^k\psi$, and

$$\begin{aligned} \mathcal {BB}_\alpha ^{c_0}(t_1,t_2):= & {} \int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }}\Big \{\frac{|\partial _1\phi |^2+|\partial _1\psi |^2+\psi ^2(\sin \theta )^{-2}}{r^3}+\frac{1}{r}\big [(L\phi )^2+(L\psi )^2\big ]\nonumber \\&+\frac{1}{r^3}\big (\phi ^2+\psi ^2\big )+\frac{M^{2}}{r^{3}}\big [(\partial _2\phi )^2+(\partial _2\psi )^2+(\partial _3\phi )^2+(\partial _3\psi )^2\big ]\Big \}\,d\mu . \end{aligned}$$

(6.2)

Assuming Lemma 6.1, it is not hard to complete the proof of Corollary 1.8

Proof of Corollary 1.8

We prove the estimate in two steps. Notice first that the inequality (6.2) is equivalent to

$$\begin{aligned}&\int _{t_1}^{t_2}\Big (\int _{\Sigma _{s}^{c_0}}\frac{r^{\alpha -1}}{M^{\alpha }}\big [e(\phi )^2+e(\psi )^2\big ]\,d\mu _s\Big )\,ds+\sum _{k=0}^2\int _{\Sigma _{t_2}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _t\\&\quad \lesssim _\alpha \sum _{k=0}^2\int _{\Sigma _{t_1}^{c_0}}\frac{r^\alpha }{M^{\alpha }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _t, \end{aligned}$$

for any $t_1\le t_2\in [0,T]$ and $\alpha \in (0,2)$. Let

$$\begin{aligned} I_{\beta ,l}(s):=\sum _{k=0}^l\int _{\Sigma _{s}^{c_0}}\frac{r^\beta }{M^{\beta }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _s. \end{aligned}$$

(6.3)

Therefore, for any $\alpha \in (0,2)$, $l\in \{0,1,2\}$, and $t_1\le t_2\in [0,T]$, we have

$$\begin{aligned} I_{\alpha ,l+2}(t_2)+\int _{t_1}^{t_2}\frac{1}{M}I_{\alpha -1,l}(s)\,ds\lesssim _\alpha I_{\alpha ,l+2}(t_1). \end{aligned}$$

(6.4)

We apply (6.4) first with $\alpha$ close to 2 and $l=2,4$; the result is

$$\begin{aligned} \int _{0}^{T}\frac{1}{M}I_{\alpha -1,2}(s)\,ds\lesssim _\alpha I_{\alpha ,4}(0)\quad \text { and }\quad I_{\alpha -1,2}(s')\lesssim _\alpha I_{\alpha -1,2}(s)\quad \text { if }s\le s'. \end{aligned}$$

These inequalities show easily that

$$\begin{aligned} I_{\alpha -1,2}(s)\lesssim _\alpha I_{\alpha ,4}(0)\frac{M}{M+s}\quad \text { for any }\quad s\in [0,T]\quad \text { and }\quad \alpha \in (0,2). \end{aligned}$$

(6.5)

To apply this argument again we need to improve slightly on (6.5). More precisely, we’d like to show that

$$\begin{aligned} I_{1+\epsilon ,2}(s)\lesssim _\epsilon I_{2,4}(0)\frac{M^{1-2\epsilon }}{(M+s)^{1-2\epsilon }}\quad \text { for any }\quad s\in [0,T]\quad \text { and }\quad \epsilon \in (0,1/10]. \end{aligned}$$

(6.6)

Indeed, we estimate

$$\begin{aligned} I_{1+\epsilon ,2}(s)\lesssim II(s)+III(s), \end{aligned}$$

where, using (6.5) and (6.4),

$$\begin{aligned} {\textit{II}}(s):= & {} \sum _{k=0}^l\int _{\Sigma _{s}^{c_0},\,r\le M+s}\frac{r^{1+\epsilon }}{M^{1+\epsilon }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _s\\\lesssim & {} I_{1-\epsilon /2,2}\frac{(M+s)^{7\epsilon /4}}{M^{7\epsilon /4}}\lesssim _\epsilon I_{2,4}(0)\frac{M^{1-2\epsilon }}{(M+s)^{1-2\epsilon }} \end{aligned}$$

and

$$\begin{aligned} {\textit{III}}(s):= & {} \sum _{k=0}^l\int _{\Sigma _{s}^{c_0},\,r\ge M+s}\frac{r^{1+\epsilon }}{M^{1+\epsilon }}\big [e(\phi _k)^2+e(\psi _k)^2\big ]\,d\mu _s\\\lesssim & {} I_{2-\epsilon /2,2}\frac{M^{1-3\epsilon /2}}{(M+s)^{1-3\epsilon /2}}\lesssim _\epsilon I_{2,2}(0)\frac{M^{1-3\epsilon /2}}{(M+s)^{1-3\epsilon /2}}. \end{aligned}$$

The bound (6.6) follows.

We can now repeat the argument at the beginning of the proof, starting from the bounds,

$$\begin{aligned} \int _{t_1}^{T}\frac{1}{M}I_{\epsilon ,0}(s)\,ds\lesssim _\alpha I_{1+\epsilon ,2}(t_1)\quad \text { and }\quad I_{\epsilon ,0}(s')\lesssim _\alpha I_{\epsilon ,0}(s)\quad \text { if }s\le s', \end{aligned}$$

which follow from (6.4) and Theorem 1.7. Using now (6.6) it follows easily that

$$\begin{aligned} I_{\epsilon ,0}(s)\lesssim _\epsilon I_{2,4}(0)\frac{M^{2-2\epsilon }}{(M+s)^{2-2\epsilon }}\quad \text { for any }\quad s\in [0,T]\quad \text { and }\quad \epsilon \in (0,1/10], \end{aligned}$$

which gives the conclusion of Corollary 1.8. $\square$

We turn now to the proof of Lemma 6.1.

Proof of Lemma 6.1

In view of Theorem 1.7, with the notation (6.3), we know that

$$\begin{aligned}&I_{\alpha ,2}(t_2)+\sum _{k=0}^2\int _{\mathcal {D}^{c_0}_{[t_1,t_2]}}\frac{r^\alpha }{M^{\alpha }}\Big \{\frac{(r-r^*)^2}{r^3}\frac{(\partial _1\phi _k)^2+(\partial _1\psi _k)^2+\psi _k^2(\sin \theta )^{-2}}{r^2}\nonumber \\&\quad +\frac{1}{r^3}\big (\phi _k^2+\psi _k^2\big )+\frac{M^{2}}{r^{3}}\big [(\partial _2\phi _k)^2+(\partial _2\psi _k)^2\big ]\Big \}\,d\mu \lesssim _\alpha I_{\alpha ,2}(t_1), \end{aligned}$$

(6.7)

for any $t_1\le t_2\in [0,T]$ and $\alpha \in (0,2)$. It suffices to prove that

$$\begin{aligned} \int _{\mathcal {D}^{c_0}_{[t_1,T]}}\frac{r^\alpha }{M^{\alpha }}\widetilde{\chi }(r)\frac{(\partial _1\phi )^2+(\partial _1\psi )^2+\psi ^2(\sin \theta )^{-2}}{r^3}\,d\mu \lesssim _\alpha I_{\alpha ,2}(t_1), \end{aligned}$$

(6.8)

where $\widetilde{\chi }:=\chi _{\ge 9M/4}-\chi _{\ge 4M}$. For this we use elliptic estimate and (6.7).

The equation for $\phi$ and the formula (A.9) show that

$$\begin{aligned} \mathbf{g}^{11}\Big [\partial _1^2\phi +\frac{\cos \theta }{\sin \theta }\partial _1\phi \Big ]+\mathbf{g}^{22}\partial _2^2\phi +\frac{2\mathbf{g}^{11}\mathbf{D}_1B}{A}\partial _1\psi =-F_\phi , \end{aligned}$$

(6.9)

where

$$\begin{aligned} F_{\phi }:= & {} \mathbf{g}^{33}\partial _3^2\phi +2\mathbf{g}^{23}\partial _2\partial _3\phi +D^2\partial _2\phi +D^3\partial _3\phi \\&+\,2\frac{\mathbf{D}^2B\mathbf{D}_2\psi +\mathbf{D}^3B\mathbf{D}_3\psi }{A}-2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu B}{A^2}\phi +2\frac{\mathbf{D}^\mu B\mathbf{D}_\mu A}{A^2}\psi . \end{aligned}$$

If follows from (6.7) that

$$\begin{aligned} \int _{\mathcal {D}^{c_0}_{[t_1,T]}}\frac{M^4}{r^3}|F_{\phi }|^2\,d\mu \lesssim _\alpha I_{\alpha ,2}(t_1). \end{aligned}$$

(6.10)

Using then integration by parts and (6.9), we have

$$\begin{aligned}&\int _{\mathcal {D}^{c_0}_{[t_1,T]}}\frac{r^\alpha }{M^{\alpha }}\widetilde{\chi }(r)\frac{(\partial _1\phi )^2}{r^3}\,d\mu \\&\quad \lesssim \int _{[t_1,T]\times (0,\pi )\times (c_0,\infty )}\widetilde{\chi }(r)\frac{(\partial _1\phi )^2}{M^3}r^2(\sin \theta )\,drd\theta dt\\&\quad \lesssim \Big |\int _{[t_1,T]\times (0,\pi )\times (c_0,\infty )}\widetilde{\chi }(r)\phi \cdot \Big [\partial _1^2\phi +\frac{\cos \theta }{\sin \theta }\partial _1\phi \Big ]\frac{r^2}{M^3}(\sin \theta )\,drd\theta dt\Big |\\&\quad \lesssim \Big |\int _{[t_1,T]\times (0,\pi )\times (c_0,\infty )}\widetilde{\chi }(r)\phi \cdot \Big [\Delta \partial _2^2\phi +\frac{2\mathbf{D}_1B}{A}\partial _1\psi +\frac{F_\phi }{\mathbf{g}^{11}}\Big ]\frac{r^2}{M^3}(\sin \theta )\,drd\theta dt\Big |. \end{aligned}$$

Using (6.7), (6.10), and integration by parts it follows that

$$\begin{aligned}&\int _{\mathcal {D}^{c_0}_{[t_1,T]}}\frac{r^\alpha }{M^{\alpha }}\widetilde{\chi }(r)\frac{(\partial _1\phi )^2}{r^3}\,d\mu \lesssim _\alpha I_{\alpha ,2}(t_1)+[I_{\alpha ,2}(t_1)]^{1/2}\Big (\int _{\mathcal {D}^{c_0}_{[t_1,T]}}\widetilde{\chi }(r)\frac{(\partial _1\psi )^2}{r^3}\,d\mu \Big )^{1/2}\!\!.\nonumber \\ \end{aligned}$$

(6.11)

Similarly, the equation for $\psi$ and the formula (A.9) show that

$$\begin{aligned} \mathbf{g}^{11}\Big [\partial _1^2\psi +\frac{\cos \theta }{\sin \theta } \partial _1\psi -\frac{4(\cos \theta )^2}{(\sin \theta )^2}\psi \Big ]+ \mathbf{g}^{22}\partial _2^2\psi -\frac{2\mathbf{g}^{11}\mathbf{D}_1B}{A}\partial _1\phi =-F_\psi , \end{aligned}$$

where $F_{\psi }$ satisfies the same bound (6.10) as $F_{\phi }$, and the additional term in the left-hand side comes from the fraction $\frac{2\cos \theta }{\sin \theta }$ in $A^{-1}\mathbf{D}_1A$ (see (5.7)). Integrating by parts as before we have

$$\begin{aligned}&\int _{\mathcal {D}^{c_0}_{[t_1,T]}}\frac{r^\alpha }{M^{\alpha }}\widetilde{\chi }(r)\frac{(\partial _1\psi )^2+\psi ^2(\sin \theta )^{-2}}{r^3}\,d\mu \\&\quad \lesssim _\alpha I_{\alpha ,2}(t_1)+[I_{\alpha ,2}(t_1)]^{1/2}\Big (\int _{\mathcal {D}^{c_0}_{[t_1,T]}}\widetilde{\chi }(r)\frac{(\partial _1\phi )^2}{r^3}\,d\mu \Big )^{1/2}. \end{aligned}$$

The desired bound (6.8) follows using also (6.11). $\square$

Appendix 1: Explicit Formulas in Kerr Spaces

Recall the Kerr spacetimes $\mathcal {K}(m,a)$, in standard Boyer–Lindquist coordinates,

$$\begin{aligned} \mathbf{g}=-\frac{q^2\Delta }{\Sigma ^2}(dt)^2+\frac{\Sigma ^2(\sin \theta )^2}{q^2}\Big (d\phi -\frac{2aMr}{\Sigma ^2}dt\Big )^2 +\frac{q^2}{\Delta }(dr)^2+q^2(d\theta )^2, \end{aligned}$$

(A.1)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta =r^2+a^2-2Mr;\\ q^2=r^2+a^2(\cos \theta )^2;\\ \Sigma ^2=(r^2+a^2)q^2+2Mra^2(\sin \theta )^2=(r^2+a^2)^2-a^2(\sin \theta )^2\Delta . \end{array}\right. } \end{aligned}$$

(A.2)

Observe that

$$\begin{aligned} (2Mr-q^2)\Sigma ^2=-q^4\Delta +4a^2M^2r^2(\sin \theta )^2. \end{aligned}$$

(A.3)

Recall the change of variables (1.19)–(1.20) and let

$$\begin{aligned} p:=\frac{2Mr}{q^2}. \end{aligned}$$

Therefore

$$\begin{aligned} \frac{\Sigma ^2}{q^2}=q^2+(p+1)a^2(\sin \theta )^2,\qquad \Delta =q^2(1-p)+a^2(\sin \theta )^2. \end{aligned}$$

Recall that

$$\begin{aligned} \partial _1=\partial _\theta =\frac{d}{d\theta }, \quad \partial _2=\partial _r=\frac{d}{dr}, \quad \partial _3=\partial _t=\frac{d}{dt_+}=\mathbf{T},\quad \partial _4=\partial _\phi =\frac{d}{d\phi _+}=\mathbf{Z}. \end{aligned}$$

(A.4)

The nontrivial components of the metric $\mathbf{g}$ become

$$\begin{aligned} \begin{aligned}&\mathbf{g}_{11}=q^2,\quad \mathbf{g}_{33}=p-1,\quad \mathbf{g}_{34}=-a(\sin \theta )^2p, \quad \mathbf{g}_{44}=q^2(\sin \theta )^2+(p+1)a^2(\sin \theta )^4,\\&\mathbf{g}_{22}=\frac{q^2}{\Delta }(1-\chi ^2)+(p+1)\chi ^2,\quad \mathbf{g}_{23}=p\chi ,\quad \mathbf{g}_{24}=-a(\sin \theta )^2(p+1)\chi , \end{aligned} \end{aligned}$$

(A.5)

and, letting $\mathrm {Det}:=-q^2(\sin \theta )^2$,

$$\begin{aligned} \begin{aligned}&\mathbf{g}^{11}=\frac{1}{\mathbf{g}_{11}}=\frac{1}{q^2},\\&\mathbf{g}^{22}=\frac{\mathbf{g}_{33}\mathbf{g}_{44}-\mathbf{g}_{34}^2}{\mathrm {Det}}=\frac{\Delta }{q^2},\\&\mathbf{g}^{23}=\frac{\mathbf{g}_{24}\mathbf{g}_{34}-\mathbf{g}_{23}\mathbf{g}_{44}}{\mathrm {Det}}=p\chi ,\\&\mathbf{g}^{24}=\frac{\mathbf{g}_{23}\mathbf{g}_{34}-\mathbf{g}_{33}\mathbf{g}_{24}}{\mathrm {Det}}=\frac{a\chi }{q^2},\\&\mathbf{g}^{33}=\frac{\mathbf{g}_{22}\mathbf{g}_{44}-\mathbf{g}_{24}^2}{\mathrm {Det}}=-(p+1)\chi ^2-\frac{q^2+(p+1)a^2(\sin \theta )^2}{\Delta }(1-\chi ^2),\\&\mathbf{g}^{34}=\frac{\mathbf{g}_{24}\mathbf{g}_{23}-\mathbf{g}_{22}\mathbf{g}_{34}}{\mathrm {Det}}=\frac{-ap}{\Delta }(1-\chi ^2),\\&\mathbf{g}^{44}=\frac{\mathbf{g}_{22}\mathbf{g}_{33}-\mathbf{g}_{23}^2}{\mathrm {Det}}=\frac{\Delta -a^2(\sin \theta )^2(1-\chi ^2)}{q^2\Delta (\sin \theta )^2}. \end{aligned} \end{aligned}$$

(A.6)

The metric $\mathbf{g}$ extends to the larger open set

$$\begin{aligned} \widetilde{R}=\{(\theta ,r,t_+,\phi _+)\in (-\pi ,\pi )\times (0,\infty )\times \mathbb {R}\times \mathbb {S}^1\}. \end{aligned}$$

Recall also the sets, see (1.24)–(1.26),

$$\begin{aligned} \begin{aligned}&\mathcal {D}^{c}_I=\{(\theta ,r,t_+,\phi _+)\in \widetilde{R}:t_+\in I\text { and }r>c\},\\&\Sigma ^c_t:=\{(\theta ,r,t_+,\phi _+)\in \widetilde{R}:t_+=t\text { and }r>c\},\\&\mathcal {N}^{c}_I:=\{(\theta ,r,t_+,\phi _+)\in \widetilde{R}:t_+\in I\text { and }r=c\}, \end{aligned} \end{aligned}$$

(A.7)

defined for $c\in (0,\infty )$, $t\in \mathbb {R}$, and intervals $I\subseteq \mathbb {R}$.

Notice that

$$\begin{aligned} \begin{aligned}&\partial _1(q^2)=-2a^2\sin \theta \cos \theta ,\qquad \partial _2(q^2)=2r,\\&\partial _1p=\frac{4Mra^2\sin \theta \cos \theta }{q^4},\qquad \partial _2p=-\frac{2M(r^2-a^2(\cos \theta )^2)}{q^4}. \end{aligned} \end{aligned}$$

(A.8)

Recall the general formula

$$\begin{aligned} \Gamma _{\mu \alpha {\beta }}=\mathbf{g}(\mathbf{D}_{\partial _{\beta }}\partial _\alpha ,\partial _\mu )= \frac{1}{2}(\partial _\alpha \mathbf{g}_{{\beta }\mu }+\partial _{\beta }\mathbf{g}_{\alpha \mu }-\partial _\mu \mathbf{g}_{\alpha {\beta }}). \end{aligned}$$

In the case of $\mathbf{Z}$-invariant functions f, i.e. if $\mathbf{Z}(f)=0$, we have the general formula

$$\begin{aligned} \square f= & {} \mathbf{g}^{\alpha {\beta }}\partial _\alpha \partial _{\beta }f-\mathbf{g}^{\alpha {\beta }}\mathbf{g}^{\mu \nu }\Gamma _{\mu \alpha {\beta }}\partial _\nu f\nonumber \\= & {} \mathbf{g}^{\alpha {\beta }}\partial _\alpha \partial _{\beta }f+\big [\partial _\mu \mathbf{g}^{\mu \nu }+(1/2)\mathbf{g}^{\mu \nu }\partial _\mu \log \big |q^4(\sin \theta )^2\big |\big ]\partial _\nu f\nonumber \\= & {} \mathbf{g}^{11}\partial _1^2f+\mathbf{g}^{22}\partial _2^2f+\mathbf{g}^{33}\partial _3^2f+2\mathbf{g}^{23}\partial _2\partial _3f\nonumber \\&+\,\big [\partial _1\mathbf{g}^{11}+(1/2)\mathbf{g}^{11}\partial _1\log \big |q^4(\sin \theta )^2\big |\big ]\partial _1 f\nonumber \\&+\,\big [\partial _2\mathbf{g}^{22}+(1/2)\mathbf{g}^{22}\partial _2\log \big |q^4(\sin \theta )^2\big |\big ]\partial _2 f\nonumber \\&+\,\big [\partial _2\mathbf{g}^{23}+(1/2)\mathbf{g}^{23}\partial _2\log \big |q^4(\sin \theta )^2\big |\big ]\partial _3 f\nonumber \\= & {} \mathbf{g}^{11}\Big [\partial _1^2f+\frac{\cos \theta }{\sin \theta }\partial _1f\Big ]+\mathbf{g}^{22}\partial _2^2f+\mathbf{g}^{33}\partial _3^2f\nonumber \\&+\,2\mathbf{g}^{23}\partial _2\partial _3f+D^2\partial _2 f+D^3\partial _3 f, \end{aligned}$$

(A.9)

where

$$\begin{aligned} D^2:=\partial _2\mathbf{g}^{22}+\mathbf{g}^{22}\frac{2r}{q^2}=\frac{2r-2M}{q^2}, \quad D^3:=\partial _2\mathbf{g}^{23}+\mathbf{g}^{23}\frac{2r}{q^2}=\frac{2M\chi (r)+2Mr\chi '(r)}{q^2}. \end{aligned}$$

Also, if m is a 1-form satisfying $m_4=0$ and $\partial _4 m_\alpha =0$, $\alpha \in \{1,2,3,4\}$, then

$$\begin{aligned} \mathbf{D}^\alpha m_\alpha= & {} \mathbf{g}^{\alpha {\beta }}\partial _\alpha m_{\beta }-\mathbf{g}^{\alpha {\beta }}\mathbf{g}^{\mu \nu }\Gamma _{\mu \alpha {\beta }}m_\nu \nonumber \\= & {} \mathbf{g}^{11}\Big [\partial _1m_1+\frac{\cos \theta }{\sin \theta }m_1\Big ]+\mathbf{g}^{22}\partial _2m_2\nonumber \\&+\,\mathbf{g}^{33}\partial _3m_3+\mathbf{g}^{23}(\partial _2m_3+\partial _3m_2)+D^2m_2+D^3m_3. \end{aligned}$$

(A.10)

Vector-fields

Letting

$$\begin{aligned} \pi _{\alpha {\beta }}=(\mathcal {L}_{\partial _2}\mathbf{g})_{\alpha {\beta }}=\Gamma _{\alpha 2 {\beta }}+\Gamma _{{\beta }2 \alpha }, \end{aligned}$$

we calculate

$$\begin{aligned} \begin{aligned}&\pi _{\alpha {\beta }}=\partial _2\mathbf{g}_{\alpha {\beta }},\\&\pi ^{\alpha {\beta }}=\mathbf{g}^{\alpha \mu }\mathbf{g}^{{\beta }\nu }\pi _{\mu \nu }=\mathbf{g}^{\alpha \mu }\mathbf{g}^{{\beta }\nu }\partial _2\mathbf{g}_{\mu \nu }=-\mathbf{g}^{{\beta }\nu }\mathbf{g}_{\mu \nu }\partial _2\mathbf{g}^{\alpha \mu }=-\partial _2\mathbf{g}^{\alpha {\beta }},\\&\pi _{\alpha {\beta }}\mathbf{g}^{\alpha {\beta }}=\partial _2\log |q^4(\sin \theta )^2|=4r/q^2. \end{aligned} \end{aligned}$$

(A.11)

Therefore, for any vector field

$$\begin{aligned} X=f(r)\partial _2+g(r)\partial _3, \end{aligned}$$

(A.12)

we calculate

$$\begin{aligned} {}^{(X)}\pi ^{\mu \nu }:= & {} \mathbf{D}^\mu X^\nu +\mathbf{D}^\nu X^\mu \nonumber \\= & {} f\pi ^{\mu \nu }+(\mathbf{D}^\mu f\delta _2^\nu +\mathbf{D}^\nu f\delta _2^\mu )+(\mathbf{D}^\mu g\delta _3^\nu +\mathbf{D}^\nu g\delta _3^\mu )\nonumber \\= & {} f\pi ^{\mu \nu }+f'(r)(\mathbf{g}^{\mu 2} \delta _2^\nu +\mathbf{g}^{\nu 2} \delta _2^\mu )+g'(r)(\mathbf{g}^{\mu 2} \delta _3^\nu +\mathbf{g}^{\nu 2} \delta _3^\mu ). \end{aligned}$$

(A.13)

For any 1-form Y with $Y_4=0$ let

$$\begin{aligned} {}^{(Y)}Q_{\mu \nu }=Y_\mu Y_\nu -(1/2)\mathbf{g}_{\mu \nu }(Y_\rho Y^\rho ). \end{aligned}$$

(A.14)

We calculate the contraction

$$\begin{aligned} {}^{(Y)}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }= & {} {}^{(X)}\pi ^{\mu \nu }Y_\mu Y_\nu -(1/2)\mathbf{g}_{\mu \nu }(Y_\rho Y^\rho ){}^{(X)}\pi ^{\mu \nu }\\= & {} f(r)\pi ^{\mu \nu }Y_\mu Y_\nu +2f'(r)Y^2Y_2+2g'(r)Y^2Y_3\\&-\,(Y_\rho Y^\rho )[2rf(r)/q^2+f'(r)]\\= & {} (Y_1)^2\Big [f(r)\pi ^{11}-\frac{2rf(r)\mathbf{g}^{11}}{q^2}-f'(r)\mathbf{g}^{11}\Big ]\\&+\,(Y_2)^2\Big [f(r)\pi ^{22}+f'(r)\mathbf{g}^{22}-\frac{2rf(r)\mathbf{g}^{22}}{q^2}\Big ]\\&+\,(Y_3)^2\Big [f(r)\pi ^{33}+2g'(r)\mathbf{g}^{23}-f'(r)\mathbf{g}^{33}-\frac{2rf(r)\mathbf{g}^{33}}{q^2}\Big ]\\&+\,2Y_2Y_3\Big [f(r)\pi ^{23}+g'(r)\mathbf{g}^{22}-\frac{2rf(r)\mathbf{g}^{23}}{q^2}\Big ]. \end{aligned}$$

Using also the formulas (A.6) and (A.11) this simplifies to

$$\begin{aligned} {}^{(Y)}Q_{\mu \nu }{}^{(X)}\pi ^{\mu \nu }= & {} (Y_1)^2\frac{-f'(r)}{q^2}+ (Y_2)^2\frac{-f(r)(2r-2M)+f'(r)\Delta }{q^2}\nonumber \\&+\,(Y_3)^2\Big [-f(r)\partial _2\mathbf{g}^{33}+2g'(r)\mathbf{g}^{23}-f'(r)\mathbf{g}^{33}-\frac{2rf(r)\mathbf{g}^{33}}{q^2}\Big ]\nonumber \\&+\,2Y_2Y_3\frac{-2Mrf(r)\chi '(r)-2Mf(r)\chi (r)+g'(r)\Delta }{q^2}. \end{aligned}$$

(A.15)

Recall the vector-fields $n=-\mathbf{g}^{\mu \nu }\partial _\nu u_+\partial _\mu =-\mathbf{g}^{3\mu }\partial _\mu$ and $k=\mathbf{g}^{\mu \nu }\partial _\nu r\partial _\mu =\mathbf{g}^{2\mu }\partial _\mu$, defined in $\widetilde{R}$, which are normal to the hypersurfaces $\Sigma ^c_t$ and $\mathcal {N}^{c}_I$ respectively. We calculate

$$\begin{aligned}&{}^{(Y)}Q(n,\partial _2)=-\mathbf{g}^{3\mu }Y_\mu Y_2=-\mathbf{g}^{32}(Y_2)^2-\mathbf{g}^{33}Y_2Y_3,\\&{}^{(Y)}Q(n,\partial _3)=-\mathbf{g}^{3\mu }Y_\mu Y_3+(1/2)(Y_\rho Y^\rho )=(1/2)[\mathbf{g}^{11}(Y_1)^2+\mathbf{g}^{22}(Y_2)^2-\mathbf{g}^{33}(Y_3)^2],\\&{}^{(Y)}Q(k,\partial _2)=\mathbf{g}^{2\mu }Y_\mu Y_2-(1/2)(Y_\rho Y^\rho )=(1/2)[-\mathbf{g}^{11}(Y_1)^2+\mathbf{g}^{22}(Y_2)^2-\mathbf{g}^{33}(Y_3)^2], \end{aligned}$$

and

$$\begin{aligned} {}^{(Y)}Q(k,\partial _3)=\mathbf{g}^{2\mu }Y_\mu Y_3=\mathbf{g}^{23}(Y_3)^2+\mathbf{g}^{22}Y_2Y_3. \end{aligned}$$

Therefore, if $X=f(r)\partial _2+g(r)\partial _3$ as in (A.12) then

$$\begin{aligned} 2{}^{(Y)}Q(n,X)= & {} (Y_1)^2[g(r)\mathbf{g}^{11}]+(Y_2)^2[g(r)\mathbf{g}^{22}-2f(r)\mathbf{g}^{23}]\nonumber \\&+\,(Y_3)^2[-g(r)\mathbf{g}^{33}]+2Y_2Y_3[-f(r)\mathbf{g}^{33}] \end{aligned}$$

(A.16)

and

$$\begin{aligned} 2{}^{(Y)}Q(k,X)= & {} (Y_1)^2[-f(r)\mathbf{g}^{11}]+(Y_2)^2[f(r)\mathbf{g}^{22}]\nonumber \\&+\,(Y_3)^2[-f(r)\mathbf{g}^{33}+2g(r)\mathbf{g}^{23}]+2Y_2Y_3[g(r)\mathbf{g}^{22}]. \end{aligned}$$

(A.17)

Hardy Inequalities

In this subsection we prove the following lemma:

Lemma A.1

• (i) If $c\ge c_0$ and $f\in H^1_{\mathrm {loc}}((c,\infty ))$ satisfies $\lim _{D\rightarrow \infty }\int _D^{2D}|f(r)|^2\,dr=0$ then

  $$\begin{aligned} \int _{c}^\infty |f/r|^2\cdot r^2\,dr\lesssim \int _{c}^\infty |f'|^2\cdot r^2\,dr. \end{aligned}$$

  (A.18)

• (ii) If $g\in H^1_{\mathrm {loc}}((0,\pi ))$ and $p\in [0,10]$ then

  $$\begin{aligned} \int _0^\pi |g|^2(\sin \theta )^p\,d\theta \lesssim \int _0^\pi |g'|^2(\sin \theta )^{p+2}\,d\theta +\int _0^\pi |g|^2(\sin \theta )^{p+2}\,d\theta . \end{aligned}$$

  (A.19)

• (iii) If $f\in H^1_{\mathrm {loc}}((0,\pi ))$ then

  $$\begin{aligned}&\int _0^\pi |f'|^2\sin \theta \,d\theta +\int _0^\pi |f|^2(\sin \theta )^{-1}\,d\theta \nonumber \\&\qquad \approx \int _0^\pi \Big |f'-\frac{2\cos \theta }{\sin \theta }f\Big |^2\sin \theta \,d\theta +\int _0^\pi |f|^2\sin \theta \,d\theta . \end{aligned}$$

  (A.20)

• (iv) If $g\in L^2_{\mathrm {loc}}((0,\pi ))$ then

  $$\begin{aligned} \int _0^\pi |g|^2(\sin \theta )^{-1}\,d\theta \lesssim \int _0^\pi \Big |g'+\frac{\cos \theta }{\sin \theta }g\Big |^2\sin \theta \,d\theta +\int _0^\pi |g|^2\sin \theta \,d\theta . \end{aligned}$$

  (A.21)

• (v) If $f\in H^1_{\mathrm {loc}}((0,\pi ))$ then

  $$\begin{aligned}&\int _0^{\pi } |f''|^2\sin \theta +|f'|^2(\sin \theta )^{-1}+|f|^2(\sin \theta )^{-3}\,d\theta \nonumber \\&\quad \lesssim \int _0^\pi \Big |f''+\frac{\cos \theta }{\sin \theta }f'-\frac{4(\cos \theta )^2}{(\sin \theta )^2}f\Big |^2\sin \theta \,d\theta \nonumber \\&\qquad +\int _0^\pi |f'|^2\sin \theta +|f|^2(\sin \theta )^{-1}\,d\theta . \end{aligned}$$

  (A.22)

Proof

The inequalities in this lemma are standard Hardy-type inequalities, and we provide the proofs mostly for sake of completeness.

For (i) we may assume that f is real-valued and

$$\begin{aligned} \int _{c}^\infty |f'(r)|^2r^2\,dr=1. \end{aligned}$$

Given $\delta >0$ small and $D\gg 1$ we fix a smooth function $K=K_{\delta ,D}:\mathbb {R}\rightarrow \mathbb {R}$ supported in the interval $[c+\delta /2,2D]$ with the properties

$$\begin{aligned} \begin{aligned}&K'(r)=1\qquad \text { if }r\in [c+\delta ,D],\\&|K'(r)|\lesssim 1\qquad \text { if }r\in [D,2D],\\&K'\text { is increasing on the interval }[c+\delta /2,c+\delta ]. \end{aligned} \end{aligned}$$

(A.23)

By taking D sufficiently large, we may assume that

$$\begin{aligned} \int _D^{2D}|f(r)|^2\,dr\le 1. \end{aligned}$$

Notice that $K(r)\lesssim rK'(r)^{1/2}$ for any $r\in [c,D]$, which follows easily from (A.23). Then we estimate, using integration by parts

$$\begin{aligned} \Big |\int _{c}^Df(r)^2K'(r)\,dr\Big |\lesssim & {} \Big |\int _{\mathbb {R}}f(r)^2K'(r)\,dr\Big |+\int _D^{2D}|f(r)|^2\,dr\\\lesssim & {} \int _{\mathbb {R}}|f(r)||f'(r)||K(r)|\,dr+1\\\lesssim & {} \int _{c}^D|f(r)||f'(r)||K(r)|\,dr+1\\\lesssim & {} \Big |\int _{c}^D|f(r)|^2K'(r)\,dr\Big |^{1/2}\Big |\int _{c}^D|f'(r)|r^2\,dr\Big |^{1/2}+1. \end{aligned}$$

Therefore

$$\begin{aligned} \Big |\int _{c}^Df(r)^2K'(r)\,dr\Big |\lesssim 1, \end{aligned}$$

and the desired inequality follows by letting $\delta \rightarrow 0$ and $D\rightarrow \infty$.

To prove (ii) we may assume that g is real-valued and

$$\begin{aligned} \int _0^\pi |g'(\theta )|^2(\sin \theta )^{p+2}\,d\theta +\int _0^\pi |g(\theta )|^2(\sin \theta )^{p+2}\,d\theta =1. \end{aligned}$$

As before, given $\delta >0$ small we fix a smooth function $K=K_{\delta }:\mathbb {R}\rightarrow \mathbb {R}$ supported in the interval $[\delta /2,1/2]$ with the properties

$$\begin{aligned} \begin{aligned}&K'(\theta )=(\sin \theta )^p\qquad \text { if }\theta \in [\delta ,1/4],\\&|K'(\theta )|\lesssim 1\qquad \text { if }\theta \in [1/4,1/2],\\&K'\text { is increasing on the interval }[\delta /2,\delta ]. \end{aligned} \end{aligned}$$

(A.24)

As before, we notice that these assumptions imply that $K(\theta )\lesssim (\sin \theta )^{(p+2)/2}K'(\theta )^{1/2}$ for any $\theta \in [0,1/4]$. Then we estimate, using integration by parts,

$$\begin{aligned}&\Big |\int _{0}^{1/4}g(\theta )^2K'(\theta )\,d\theta \Big |\lesssim \Big |\int _{0}^{1/2}g(\theta )^2K'(\theta )\,d\theta \Big |+1\\&\quad \lesssim \int _0^{1/2}|g(\theta )||g'(\theta )||K(\theta )|\,d\theta +1\\&\quad \lesssim \int _0^{1/4}|g(\theta )||g'(\theta )||K(\theta )|\,d\theta +1\\&\quad \lesssim \Big |\int _{0}^{1/4}|g(\theta )|^2K'(\theta )\,d\theta \Big |^{1/2}\Big |\int _{0}^{1/4}|g'(\theta )|^2(\sin \theta )^{p+2}\,d\theta \Big |^{1/2}+1. \end{aligned}$$

Therefore

$$\begin{aligned} \Big |\int _{0}^{1/4}g(\theta )^2K'(\theta )\,d\theta \Big |\lesssim 1. \end{aligned}$$

Letting $\delta \rightarrow 0$ it follows that

$$\begin{aligned} \int _0^{1/4}|g(\theta )|^2(\sin \theta )^p\,d\theta \lesssim 1. \end{aligned}$$

The change of variables $\theta \rightarrow \pi -\theta$ now shows that

$$\begin{aligned} \int _{\pi -1/4}^{\pi }|g(\theta )|^2(\sin \theta )^p\,d\theta \lesssim 1, \end{aligned}$$

and the desired estimate follows.

To prove (iii), we notice first that the right-hand side of (A.20) is clearly dominated by the left-hand side. To prove the reverse inequality, let $f(\theta )=(\sin \theta )^2 g(\theta )$ and notice that

$$\begin{aligned} f'(\theta )-\frac{2\cos \theta }{\sin \theta }f(\theta )=(\sin \theta )^2g'(\theta ). \end{aligned}$$

The desired bound follows from the inequality

$$\begin{aligned} \int _0^\pi |g(\theta )|^2(\sin \theta )^3\,d\theta \lesssim \int _0^\pi |g'(\theta )|^2(\sin \theta )^5\,d\theta +\int _0^\pi |g(\theta )|^2(\sin \theta )^5\,d\theta , \end{aligned}$$

which is a consequence of (A.19).

To prove (iv), we may assume that $g\in H^1_{\mathrm {loc}}((0,\pi ))$ is real-valued and let $g(\theta )=h(\theta )/\sin \theta$. Then

$$\begin{aligned} g'(\theta )+\frac{\cos \theta }{\sin \theta }g(\theta )=\frac{h'(\theta )}{\sin \theta }. \end{aligned}$$

The inequality to prove becomes

$$\begin{aligned} \int _0^\pi \frac{h(\theta )^2}{(\sin \theta )^3}\,d\theta \lesssim \int _0^\pi \frac{h'(\theta )^2}{\sin \theta }\,d\theta +\int _0^\pi \frac{h(\theta )^2}{\sin \theta }\,d\theta . \end{aligned}$$

(A.25)

This is nontrivial only if $h'\in L^2((0,\pi ))$, which shows that $h'\in L^1((0,\pi ))$. Therefore, in proving (A.25) we may assume that h extends to a continuous function on the interval $[0,\pi ]$ and $h(0)=h(\pi )=0$ (otherwise the right-hand side of (A.25) is equal to $\infty$). In particular, for any $\theta \in [0,\pi /2]$,

$$\begin{aligned} h(\theta )=\int _{0}^\theta h'(\mu )\,d\mu . \end{aligned}$$

(A.26)

For $k\le 0$ let $c_k:=2^{-k/2}\Big [\int _{[2^{k-1},2^k]}|h'(\mu )|^2\,d\mu \Big ]^{1/2}$. The formula (A.26) above shows that

$$\begin{aligned} |h(\theta )|\lesssim \sum _{k'\le k}2^{k'}c_{k'}\qquad \text { if }k\le 0\text { and }\theta \in [2^{k-1},2^{k}]. \end{aligned}$$

Therefore

$$\begin{aligned} \int _0^1\frac{h(\theta )^2}{(\sin \theta )^3}\,d\theta \lesssim \sum _{k\le 0}\Big (\sum _{k'\le k}2^{k'-k}c_{k'}\Big )^2\lesssim \sum _{k\le 0}c_k^2\lesssim \int _0^1\frac{h'(\theta )^2}{\sin \theta }\,d\theta . \end{aligned}$$

The change of variables $\theta \rightarrow \pi -\theta$ shows that

$$\begin{aligned} \int _{\pi -1}^\pi \frac{h(\theta )^2}{(\sin \theta )^3}\,d\theta \lesssim \int _{\pi -1}^\pi \frac{h'(\theta )^2}{\sin \theta }\,d\theta +\int _0^\pi \frac{h(\theta )^2}{\sin \theta }\,d\theta , \end{aligned}$$

and the desired bound (A.25) follows.

To prove (v), we may assume that $f\in H^2_{\mathrm {loc}}((0,\pi ))$ is real-valued and let $f(\theta )=g(\theta )(\sin \theta )^2$. Then

$$\begin{aligned}&f''(\theta )+\frac{\cos \theta }{\sin \theta }f'(\theta )-\frac{4(\cos \theta )^2}{(\sin \theta )^2}f(\theta )\\&\quad =(\sin \theta )^2g''(\theta )+3\sin \theta \cos \theta g'(\theta )-2(\sin \theta )^2g(\theta ). \end{aligned}$$

The inequality (A.22) becomes

$$\begin{aligned}&\int _0^{\pi } |g''|^2(\sin \theta )^5+|g'|^2(\sin \theta )^3+|g|^2\sin \theta \,d\theta \\&\quad \lesssim \int _0^\pi |(\sin \theta )^2g''+3\sin \theta \cos \theta g'|^2\sin \theta \,d\theta \\&\qquad +\int _0^\pi |g'|^2(\sin \theta )^5+|g|^2(\sin \theta )^3\,d\theta . \end{aligned}$$

In view of the inequality (A.19) with $p=1$ it suffices to prove that

$$\begin{aligned} \int _0^{\pi } |g'|^2(\sin \theta )^3\,d\theta\lesssim & {} \int _0^\pi |(\sin \theta )^2 g''+3\sin \theta \cos \theta g'|^2\sin \theta \,d\theta \\&+\int _0^\pi |g'|^2(\sin \theta )^5\,d\theta . \end{aligned}$$

Letting $h(\theta )=(\sin \theta )^3g'(\theta )$ this is equivalent to

$$\begin{aligned} \int _0^\pi \frac{h(\theta )^2}{(\sin \theta )^3}\,d\theta \lesssim \int _0^\pi \frac{h'(\theta )^2}{\sin \theta }\,d\theta +\int _0^\pi \frac{h(\theta )^2}{\sin \theta }\,d\theta , \end{aligned}$$

which was proved earlier, see (A.25). $\square$

The Main Function Spaces

We summarize now some of the main properties of the spaces $H^m(\Sigma ^c_t)$ and $\widetilde{H}^m(\Sigma ^c_t)$:

Lemma A.2

Assume $t\in \mathbb {R}$ and $c\ge c_0$.

• (i) If $f\in H^1(\Sigma _t^c)$ satisfies $\mathbf{Z}(f)=0$ then

  $$\begin{aligned} \Vert f\Vert _{\widetilde{H}^1(\Sigma _t^c)}\approx \Vert f\Vert _{H^1(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

  (A.27)

• (ii) If $f\in H^2(\Sigma _t^c)$, satisfies $\mathbf{Z}(f)=0$ then

  $$\begin{aligned} \Vert f\Vert _{\widetilde{H}^2(\Sigma _t^c)}\approx \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}f\Vert _{H^{1}(\Sigma _t^c)}+\Vert (r\sin \theta )^{-2}f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

  (A.28)

Proof of Lemma A.2

The bound (A.27) follows easily from the definitions and (A.20). We prove now part (ii) and the bounds (A.28) for $m=2$. In view of the definition, and using also (A.27),

$$\begin{aligned} \Vert f\Vert _{\widetilde{H}^2(\Sigma _t^c)}\approx & {} \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert \widetilde{\partial }_2f\Vert _{\widetilde{H}^{1}(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}\nonumber \\\approx & {} \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}\partial _2f\Vert _{L^2(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}\nonumber \\&+\,\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

(A.29)

Using (A.18), we have

$$\begin{aligned} \Vert (r\sin \theta )^{-1}\partial _2f\Vert _{L^2(\Sigma _t^c)}\lesssim \Vert \partial _2[(r\sin \theta )^{-1}f]\Vert _{L^2(\Sigma _t^c)}\lesssim \Vert (r\sin \theta )^{-1}f\Vert _{H^1(\Sigma _t^c)}. \end{aligned}$$

Moreover, using the definition,

$$\begin{aligned}&\Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}\\&\quad \lesssim \Vert (\partial _1/r)^2f\Vert _{L^2(\Sigma _t^c)}+\Vert [1+(r\sin \theta )^{-1}](\partial _1/r)f\Vert _{L^2(\Sigma _t^c)}\\&\qquad +\,\Vert [1+(r\sin \theta )^{-2}]f\Vert _{L^2(\Sigma _t^c)}\\&\quad \lesssim \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}f\Vert _{H^1(\Sigma _t^c)}+\Vert (r\sin \theta )^{-2}f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

Using also (A.29), it follows that

$$\begin{aligned} \Vert f\Vert _{\widetilde{H}^2(\Sigma _t^c)}\lesssim \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}f\Vert _{H^1(\Sigma _t^c)}+\Vert (r\sin \theta )^{-2}f\Vert _{L^2(\Sigma _t^c)}, \end{aligned}$$

as desired.

For the reverse inequality, using (A.29), it remains to prove that

$$\begin{aligned}&\Vert (r\sin \theta )^{-2}f\Vert _{L^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}(\partial _1/r)f\Vert _{L^2(\Sigma _t^c)}\nonumber \\&\quad \lesssim \Vert f\Vert _{H^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}\partial _2f\Vert _{L^2(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}\nonumber \\&\qquad +\,\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

(A.30)

Using (A.20),

$$\begin{aligned} \Vert (r\sin \theta )^{-1}(\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}\lesssim \Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}+\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}. \end{aligned}$$

Also, using (A.22) and then (A.27),

$$\begin{aligned}&\Vert (r\sin \theta )^{-2}f\Vert _{L^2(\Sigma _t^c)}\\&\quad \lesssim \Big \Vert r^{-2}\Big [\partial _1^2+\frac{\cos \theta }{\sin \theta }\partial _1-\frac{4(\cos \theta )^2}{(\sin \theta )^2}\Big ]f\Big \Vert _{L^2(\Sigma _t^c)}\\&\qquad +\,\Vert (\partial _1/r)f\Vert _{L^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}f\Vert _{L^2(\Sigma _t^c)}\\&\quad \lesssim \Vert (\widetilde{\partial }_1/r)^2f\Vert _{L^2(\Sigma _t^c)}+\Vert (r\sin \theta )^{-1}(\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}\\&\qquad +\,\Vert (\widetilde{\partial }_1/r)f\Vert _{L^2(\Sigma _t^c)}+\Vert f\Vert _{H^1(\Sigma _t^c)}. \end{aligned}$$

The desired bound (A.30) follows from these two estimates. $\square$