# Quasilinear wave equations on asymptotically flat spacetimes
with applications to Kerr black holes

Mihalis Dafermos  Gustav Holzegel  Igor Rodnianski  Martin Taylor

(28 December 2022)

###### Abstract

We prove global existence and decay for small-data solutions to a class of
quasilinear wave equations on a wide variety of asymptotically flat spacetime
backgrounds, allowing in particular for the presence of horizons, ergoregions
and trapped null geodesics, and including as a special case the Schwarzschild
and very slowly rotating $|a|\ll M$ Kerr family of black holes in general
relativity. There are two distinguishing aspects of our approach. The first
aspect is its dyadically localised nature: The nontrivial part of the analysis
is reduced entirely to time-translation invariant $r^{p}$-weighted estimates,
in the spirit of [DR09], to be applied on dyadic time-slabs which for large
$r$ are outgoing. Global existence and decay then both immediately follow by
elementary iteration on consecutive such time-slabs, without further global
bootstrap. The second, and more fundamental, aspect is our direct use of a
“black box” linear inhomogeneous energy estimate on exactly stationary
metrics, together with a novel but elementary physical space top order
identity that need not capture the structure of trapping and is robust to
perturbation. In the specific example of Kerr black holes, the required linear
inhomogeneous estimate can then be quoted directly from the literature
[DRSR16], while the additional top order physical space identity can be shown
easily in many cases (we include in the Appendix a proof for the Kerr case
$|a|\ll M$, which can in fact be understood in this context simply as a
perturbation of Schwarzschild). In particular, the approach circumvents the
need either for producing a purely physical space identity capturing trapping
or for a careful analysis of the commutation properties of frequency
projections with the wave operator of time-dependent metrics.

###### Contents

  1. 1 Introduction
    1. 1.1 Assumptions on $(\mathcal{M},g_{0})$: “black box” estimates and physical space identities
    2. 1.2 Assumptions on the nonlinearities $g(\psi,x)$ and $N(\partial\psi,\psi,x)$
    3. 1.3 The dyadically localised approach to global existence: sketch of the proof of the main theorem
    4. 1.4 Discussion
    5. 1.5 Outline of the paper
    6. 1.6 Acknowledgements
  2. 2 Geometric assumptions on the background spacetime
    1. 2.1 Underlying differential structure and the positive function $r$
    2. 2.2 The Lorentzian metric $g_{0}$, time orientation and the causality of the boundary
    3. 2.3 The stationary Killing field $T$
    4. 2.4 The Killing horizon $\mathcal{H}^{+}$
    5. 2.5 Foliations, subregions and volume forms
    6. 2.6 Other vector fields
    7. 2.7 Examples: Minkowski, Schwarzschild and Kerr
    8. 2.8 Table of $r$-parameters
  3. 3 Assumed identities and estimates for $\Box_{g_{0}}\psi=F$
    1. 3.1 Constants and parameters
    2. 3.2 Basic degenerate integrated local energy estimate
    3. 3.3 Physical space identities on a general Lorentzian metric
    4. 3.4 Assumed unweighted first order physical space identities: cases (i)–(iii)
    5. 3.5 The $r^{p}$ hierarchy
    6. 3.6 Higher order estimates
  4. 4 Quasilinear equations: preliminaries, the null condition and local existence
    1. 4.1 The class of equations
    2. 4.2 Smallness parameters
    3. 4.3 Energy currents for $\Box_{g(\psi,x)}$ and the stability of coercivity properties
    4. 4.4 Higher order estimates for the quasilinear equation
    5. 4.5 Summed norm notation and the lower order smallness assumption
    6. 4.6 Estimates on the nonlinear terms in the near region
    7. 4.7 Assumptions for the nonlinearity in the far region: the “null” condition
    8. 4.8 The final estimates
    9. 4.9 Local well posedness
  5. 5 The main theorem: global existence and stability
  6. 6 The estimate hierarchies and global existence and decay
    1. 6.1 Case (i)
    2. 6.2 Case (ii)
    3. 6.3 Case (iii)
  7. A The physical space identity on very slowly rotating Kerr
    1. A.1 The Kerr metric in Boyer–Lindquist coordinates
    2. A.2 The twisted energy momentum tensor
    3. A.3 A Morawetz current vanishing identically in a neighbourhood of trapping
    4. A.4 Adding the redshift and stationary currents and completing the proof
    5. A.5 Computation of the $X$-deformation tensor
  8. B Energy currents in Minkowski space
    1. B.1 The $J^{V,w,q,\varpi}$ current
    2. B.2 The $J_{\rm far}$ current
  9. C Verifying the null condition assumption
    1. C.1 Two model nonlinearities on Minkowski space
    2. C.2 More general nonlinearities and Kerr

##  1 Introduction

We consider here quasilinear equations of the form

$\Box_{g(\psi,x)}\psi=N(\partial\psi,\psi,x)$ (1.1)

on a $4$-dimensional manifold $\mathcal{M}$, where $g(\psi,x)$ and
$N(\partial\psi,\psi,x)$ are appropriate nonlinear terms and where
$g(0,x)=g_{0}(x)$ defines a stationary asymptotically flat Lorentzian metric
on $\mathcal{M}$ foliated by a suitable family of hypersurfaces
$\Sigma(\tau)$, for $\tau\geq 0$, which for large $r$ are outgoing null. In
this paper, we prove the following:

######  Theorem.

For equations (1.1), under appropriate assumptions on the background spacetime
$(\mathcal{M},g_{0})$ and on the nonlinearities $g(\psi,x)$ and
$N(\partial\psi,\psi,x)$ to be described below, we have

  * •

Global existence of small data solutions. Solutions $\psi$ arising from smooth
initial data on $\Sigma_{0}$, sufficiently small as measured in a suitable
weighted Sobolev energy, exist globally on $\mathcal{M}$.

  * •

Orbital stability. Under the above assumptions, the above weighted energy flux
through $\Sigma(\tau)$ is uniformly bounded by a constant times its initial
value.

  * •

Asymptotic stability. Under the above assumptions, a suitable lower order
unweighted energy flux through $\Sigma(\tau)$ decays inverse polynomially in
$\tau$ (implying also pointwise inverse polynomial decay for $\psi$).

One possible motivation for studying (1.1) is as an illustrative model problem
for issues relating to the nonlinear stability of the spacetimes
$(\mathcal{M},g_{0})$, when these themselves are solutions to the celebrated
Einstein equations of general relativity. An important example of such
spacetimes $(\mathcal{M},g_{0})$ allowed by our theorem is provided by the
Schwarzschild family of metrics for $M>0$, and the more general Kerr family
(parametrised by $a$ and $M$) in the very slowly rotating regime $|a|\ll M$.
For a discussion of these spacetimes and the stability problem in general
relativity, see [DHRT21].

Among other features, the Schwarzschild and Kerr spacetimes exhibit _trapped
null geodesics_ , along which energy can concentrate over large time-scales.
Moreover, these are _asymptotically flat_ spacetimes, meaning that linear
waves $\psi$ decay like $\sim r^{-1}$ along outgoing null cones. This then
constrains the decay of $\psi$ in the near region to also be at best only
inverse polynomial. (In the above black hole examples, this slow decay in the
near region is already a linear effect due to scattering off of far away
curvature associated to the nontrivial spacetime mass at infinity; in
Minkowski space, this slow decay in the near region arises due to purely
nonlinear scattering effects, even for compactly supported initial data.) It
is the combination of these two specific analytical difficulties—trapped null
geodesics and the relatively slow decay in the near region necessitated by
asymptotic flatness—of the black hole stability problem (and related problems)
which we wish to capture with our assumptions in the present paper. It is for
this reason that we include both the nonlinear term implicit in the expression
$\Box_{g(\psi,x)}\psi$ on the left hand side of (1.1) (the “quasilinearity”),
as this is the most dangerous term in the vicinity of trapping, as well as the
nonlinear term $N(\partial\psi,\psi,x)$ on the right hand side (the
“semilinearity”), as this term models the true null structure at infinity of
the Einstein equations, when the latter are written in appropriate geometric
gauges. To avoid inessential complications, we will in fact assume that
$g(\psi,x)=g_{0}$ for large $r$ and that $N(\partial\psi,\psi,x)$ satisfies a
generalised version of the null condition [Kla86].

In the case where $(\mathcal{M},g_{0})$ is Minkowski space, a version of the
above theorem follows from classical work of [Kla86, Chr86], and there have
been many further amplifications over the years, particularly in the context
of the obstacle problem, see e.g. [MS05]. Concerning specifically the
Schwarzschild and very slowly rotating $|a|\ll M$ Kerr setting, versions of
the above theorem have been shown previously in various special cases; see
already [Luk13] in the semilinear and [LT16, LT20] in the quasilinear case, as
well as the recent [LT22] (the latter three works concerning slightly
different classes of equations, but satisfying the weak null condition
[LR03]). For axisymmetric solutions to certain semilinear equations on Kerr in
the full subextremal range $|a|<M$, see [Sto17]. See also [Pas19] for a
related physically motivated quasilinear problem and [HV16, Mav21] for the
study of (1.1) on the non-asymptotically flat Schwarzschild–de Sitter and
Kerr–de Sitter black hole backgrounds, where the cosmological constant
$\Lambda$ is positive and decay of $\psi$ is in fact exponential. For work
specifically connected to the related problem of stability of these spacetimes
themselves under the Einstein equations, see already Section 1.4.4.

In comparison to previous related work, there are two distinguishing features
of the present approach.

The first distinguishing feature is our purely dyadic framework: Rather than
relying on a global bootstrap based on time-weighted norms, the argument is
entirely reduced to $r^{p}$-weighted but _time-translation invariant_
estimates to be applied in spacetime slabs of time length $L$ which are
outgoing null for large $r$. Global existence (and decay) is then inferred by
proceeding iteratively in time in consecutive slabs of length $L=2^{i}$, in
the spirit of [DR09]. No bootstrap is necessary for the iteration itself, and
to estimate the $i$’th slab, no information is necessary to remember about the
past other than information on the data of the slab itself. In this sense, the
argument is truly dyadically localised. This yields a more streamlined proof
even restricted to the semilinear case. (In fact, for both the semilinear and
quasilinear cases, if $(\mathcal{M},g_{0})$ is a suitably small perturbation
of Minkowski space, the time-translation invariant estimate can be applied
directly without iteration, and the approach proves global existence without
explicit reference to time decay, cf. the recent [FM22]. In our formalism we
shall always assume $g_{0}$ to be stationary, though we note that results can
also be obtained on non-stationary perturbations of Minkowski space [Yan13].)

The second, and more fundamental, feature is our direct use of a “black box”
linear inhomogeneous energy estimate (which in applications captures both
complicated trapping phenomena as well as low frequency obstructions) on the
exactly stationary background, together with a physical-space top-order
identity which may be applied directly to the quasilinear equation (1.1) and
is in fact completely insensitive to trapping (and, when it holds, robust to
perturbation of the metric $g_{0}$). In the example of Kerr, in fact for the
full subextremal case $|a|<M$, our black box assumption follows directly from
[DRSR16], while we show explicitly how to retrieve the additional top-order
physical space estimate in the case of $|a|\ll M$ (which in this context, can
in fact be thought of simply as a perturbation of Schwarzschild) in Appendix
A. The correct formulation of the companion estimate to be used in connection
with [DRSR16] for the full subextremal case $|a|<M$ will appear elsewhere. (We
emphasise that it is the phenomenon of _superradiance_ which is the primary
difficulty in producing this identity, not the complicated structure of
trapping per se; for instance, the necessary top order physical space identity
can be shown for general stationary spacetimes without horizons or
ergospheres, irrespective of the structure of trapping.) The approach thus
does not depend on whether or not there exists a purely physical-space based
identity capturing trapping (something very fragile!) nor does it require a
careful analysis of the commutation properties of frequency projections with
the wave operator $\Box_{g(\psi,x)}$ of the time-dependent metric $g(\psi,x)$
corresponding to the actual solution $\psi$ (something quite technical, which
was successfully done, however, for instance in [LT20]).

We emphasise that the role of the companion physical-space estimate is purely
in order to deal with quasilinear terms. When our method is restricted to the
semilinear case, i.e. when $g(\psi,x)=g_{0}$ identically in (1.1), then the
black box linear inhomogeneous energy estimate can be applied on its own. In
particular, in view of [DRSR16], our main theorem immediately applies to
semilinear equations on Kerr in the full subextremal range $|a|<M$.

We note that the direct appeal to a linear “black box” statement is similar in
spirit to [MS05] for instance, where results on quasilinear equations outside
obstacles in Minkowski space were studied under an exponential decay
assumption concerning the linear problem.

The remainder of this introduction is structured as follows: In Section 1.1
below, we will discuss in more detail the assumptions we make on the
background $(\mathcal{M},g_{0})$, introducing both the black box linear
inhomogeneous estimate and the additional physical space identity, followed by
the assumptions on the nonlinearity in Section 1.2. We will then sketch our
proof of the above theorem in Section 1.3, introducing our purely dyadic
approach. Finally, we shall end in Section 1.4 with a general discussion,
giving various extensions of the method, describing in more detail the
relation with the nonlinear stability problem for black holes, and comparing
in particular with the case of backgrounds modelled on Schwarzschild–de Sitter
and Kerr–de Sitter spacetimes and with the recent work [Mav21].

###  1.1 Assumptions on $(\mathcal{M},g_{0})$: “black box” estimates and
physical space identities

We will make assumptions directly on the geometry of $(\mathcal{M},g_{0})$ and
on properties of the linear inhomogeneous wave equation

$\Box_{g_{0}}\psi=F$ (1.2)

on the fixed background $g_{0}$.

####  1.1.1 Geometric assumptions on $(\mathcal{M},g_{0})$

The purely geometric assumptions on $(\mathcal{M},g_{0})$ will include
assumptions concerning a suitable notion of asymptotic flatness, that of
stationarity (existence of a Killing field $T$ which is timlike near
infinity), a smooth $T$-invariant strictly positive function $r\geq r_{0}$,
which behaves like the Euclidean area-radius as $r\to\infty$, and the
existence of a $T$-translation invariant foliation of $\mathcal{M}$ by
$\Sigma(\tau)$ for $\tau\geq 0$ hypersurfaces which are spacelike for $r\leq
R$ and “outgoing” null for $r\geq R$. (In the case of Minkowski space, which
will be the most basic example, we note that the function $r$ will only
coincide with the usual radial coordinate for large values.)

To encompass the black hole cases of interest, we will allow for a possibly
empty _spacelike_ future boundary $\mathcal{S}=\\{r=r_{0}\\}$ of
$\mathcal{M}$, in which case we will also assume the presence of a Killing
horizon $\mathcal{H}^{+}$ at $r=r_{\rm Killing}$ for an $r_{0}<r_{\rm
Killing}$, whose surface gravity will be required to be positive and whose
Killing generator $Z$ need not coincide with $T$ (in which case however $Z$
will be required to lie in the span of $T$ and an additional assumed Killing
vector field $\Omega_{1}$). We will furthermore require that for $r>r_{\rm
Killing}$, $T$ (or more generally the span of $T$ and $\Omega_{1}$, if the
latter is assumed Killing) is timelike. This will incorporate Schwarzschild
and Kerr black holes in the full subextremal range $|a|<M$.

With minor modifications, we could also allow $\mathcal{S}$ to be a timelike
boundary along which suitable boundary conditions are imposed, and this would
allow one to consider waves outside of obstacles, cf. [MS05].

We will denote by $\mathcal{R}(\tau_{0},\tau_{1})$ spacetime “slabs”
$\mathcal{R}(\tau_{0},\tau_{1})=\cup_{\tau_{0}\leq\tau\leq\tau_{1}}\Sigma(\tau)=J^{-}(\Sigma(\tau_{1}))\cap
J^{+}(\Sigma(\tau_{0}))$. The region $r\geq R$ will also be foliated by
$T$-translation invariant “ingoing” null hypersurfaces $\underline{C}_{v}$.

$\mathcal{S}$
---

$\underline{C}_{v}$
---

$\Sigma(0)$
---

$\Sigma(\tau_{0})$
---

$\Sigma(\tau_{1})$
---

$\mathcal{R}(\tau_{0},\tau_{1})$
---

$r=R$
---

$\mathcal{H}^{+}$
---

Figure 1: The underlying spacetime $(\mathcal{M},g_{0})$ in the case
$\mathcal{S}\neq\emptyset$ with hypersurfaces and regions depicted

Refer to Figure 1 and already to Section 2 for a detailed discussion.

####  1.1.2 Assumptions on $\Box_{g_{0}}\psi=F$

In this section, we discuss the assumptions which we make at the level of the
equation (1.2).

##### Basic degenerate integrated local energy estimate.

Our fundamental “black box” assumption at the level of solutions of the
inhomogeneous linear equation (1.2) will be the validity of an integrated
local energy estimate

$\mathcal{F}(v),\quad\mathcal{E}(\tau)+c\int_{\tau_{0}}^{\tau_{1}}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+c\int_{\tau_{0}}^{\tau_{1}}\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}\leq\lambda\mathcal{E}(\tau_{0})+C\int_{\mathcal{R}(\tau_{0},\tau_{1})}\left|\left(V^{\mu}_{0}\partial_{\mu}\psi+w_{0}\psi\right)F\right|+C\int_{\mathcal{R}(\tau_{0},\tau_{1})}F^{2},$
(1.3)

where $\mathcal{E}(\tau)$ denotes an energy flux through the hypersurface
$\Sigma(\tau)$, $\mathcal{F}(v)$ denotes an energy flux through
$\underline{C}_{v}\cap\mathcal{R}(\tau_{0},\tau_{1})$, $\lambda\geq 1$, $C>0$,
$c>0$ are constants, and $V_{0}$, $w$ are a fixed vector field and function,
respectively, and $\tau_{0}\leq\tau\leq\tau_{1}$ are arbitrary. In Minkowski
space, such estimates go back to Morawetz [Mor68]. The energy flux
$\mathcal{E}(\tau)$ will control all first order derivatives of $\psi$ on the
spacelike part of $\Sigma(\tau)$ while it will only control all tangential
derivatives on the null parts of $\Sigma(\tau)$; similarly, $\mathcal{F}(v)$
will control all tangential derivatives on $\underline{C}_{v}$. The fluxes
will also control a zeroth order term with weight $r^{-2}$. The two additional
terms on the left hand side are as follows: The quantity
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})$
denotes an integral over $\Sigma(\tau^{\prime})$ controlling all first order
derivatives of $\psi$, whose density is multiplied by $\chi r^{-1-\delta}$,
where $0\leq\chi\leq 1$ denotes a function which is allowed to degenerate but
must satisfy $\chi=1$ in a region to be discussed below, and where $\delta>0$
can be taken to be an arbitrarily small constant which will be fixed
throughout. The quantity
$\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})$
denotes simply the zeroth order quantity

$\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau):=\int_{\Sigma(\tau)}r^{-3-\delta}\psi^{2}.$

The subscript $-1$ indicates that it is of order $1$ less in differentiability
than the other energies considered. We recall that in our setup, what we
define to be $r$ satisfies $r\geq r_{0}>0$, and thus $r$-weights are only
relevant as $r\to\infty$.

In the case where $\mathcal{M}$ has no spacelike boundary, i.e.
$\mathcal{S}=\emptyset$, and $T$ is globally timelike, our only assumption on
$\chi$ will be that $\chi=1$ for large $r$. In the case where
$\mathcal{S}\neq\emptyset$, then we will require that $\chi=1$ for $r\leq
r_{2}$ for some $r_{2}>r_{\rm Killing}$. See already Section 3.2 for the
detailed assumptions.

Let us note that estimate (1.3), with $\chi$ satisfying the above properties
has indeed been shown in [DRSR16] in the Kerr case for the full subextremal
range of parameters $|a|<M$. One essential feature in deriving (1.3) in Kerr
is the fact that the trapped null geodesics of $g_{0}$, corresponding to
photons in bound orbits around the black hole, are unstable in a suitable
sense. We note that one can show in general that estimate (1.3) in the above
form _cannot_ hold in the presence of stable trapping. (Thus, (1.3) in
particular constrains properties of geodesic flow which can be expressed
purely geometrically in terms of $g_{0}$.) Similarly, the possibility of an
estimate (1.3) with $\chi=1$ in a neighbourhood of $\mathcal{H}^{+}$ is
related to the non-degenerate property of the horizon of subextremal Kerr and
its celebrated local _red-shift effect_. See [DR13].

##### Physical space energy identities.

One way to try to prove (1.3) is through a physical space energy _identity_
arising from integrating the divergence identity

$\nabla^{\mu}J^{V,w,q,\varpi}_{\mu}[\psi]=K^{V,w,q}[\psi]+(V^{\mu}\partial_{\mu}\psi)F+w\psi
F$ (1.4)

for well chosen currents $J^{V,w,q,\varpi}[\psi,g_{0}]$ (associated to a
vector field $V$, a scalar function $w$, a one-form $q$ and a two-form
$\varpi$) with (degenerate) coercivity properties for the arising bulk and
boundary terms. In this case, if one defines

$\mathfrak{E}(\tau)=\int_{\Sigma(\tau)}J_{\mu}^{V,w,q,\varpi}[\psi]n^{\mu}_{\Sigma(\tau)},\qquad\mathfrak{F}(v)=\int_{\underline{C}_{v}\cap\mathcal{R}(\tau_{0},\tau_{1})}J_{\mu}^{V,w,q,\varpi}[\psi]n^{\mu}_{\underline{C}_{v}},$

then one has $\mathfrak{E}(\tau)\sim\mathcal{E}(\tau)$,
$\mathfrak{F}(v)\sim\mathcal{F}(v)$, and one can reexpress (1.3) with
$\mathfrak{E}(\tau)$, $\mathfrak{E}(\tau_{0})$ replacing $\mathcal{E}(\tau)$,
$\mathcal{E}(\tau_{0})$, respectively, but with $\lambda$ in (1.3) now equal
to $1$. This is precisely the situation in the Schwarzschild case, where
appropriate coercive purely physical space identities of the form (1.4) can be
deduced from [DR07a, MMTT10].

If (1.3) indeed holds as a consequence of the coercivity properties of a
divergence identity (1.4), then this situation appears very advantageous for
nonlinear applications. The advantage of such coercive identities (1.4) is
that they are robust and easily amenable to _direct_ application to the
quasilinear (1.1). This is because the latter can be viewed as an
inhomogeneous wave equation _with respect to the wave operator
$\Box_{g(\psi,x)}$ corresponding to the solution itself_, and both the
existence of energy identities like (1.4) and their coercivity properties are
stable to passing from $g_{0}$ to $g(\psi,x)$, i.e. still hold for
$J^{V,w,q,\varpi}[g(\psi,x),\psi]$.

If one only has estimate (1.3) as a “black box”, i.e. not proven via a
physical-space identity (1.4), one may still of course apply the estimate to
(1.1) by rewriting (1.1) as an inhomogeneous equation with respect to
$\Box_{g_{0}}$:

$\Box_{g_{0}}\psi=(\Box_{g_{0}}-\Box_{g(\psi,x)})\psi+N(\partial\psi,\psi,x).$
(1.5)

Applied in this manner, however, the resulting estimate _loses derivatives_ ,
in view of the quasilinear second order terms on the right hand side of (1.5).
(Note in contrast that in the purely semilinear case, where $g(\psi,x)=g_{0}$,
there is no such loss in estimating (1.5), and we may base our argument
entirely on (1.3). See already Section 1.4.1.) At first glance, this loss
would appear to present a fundamental difficulty.

##### An auxiliary physical space estimate.

Here we come to the key observation of the present approach: To avoid the loss
of derivatives described above, one does not need that (1.3) be proven via a
physical space identity (1.4); one only needs a much weaker (from the
coercivity point of view) estimate arising from (1.4) which need not imply
(1.3) but can be used _in conjunction_ with (1.3).

Specifically, our main assumption is that, in addition to (1.3), one has a
physical space identity (1.4) with (a) coercive boundary terms and with (b)
bulk terms which are only assumed to be nonnegative _at highest order_ , i.e.
up to lower order “error” terms. Importantly, however, these allowed lower
order error terms must moreover be supported entirely in the region where the
degeneration function $\chi$ of (1.3) satisfies $\chi=1$, i.e. where the
estimate of (1.3) is in fact non-degenerate. The identity (1.4) applied to
(1.2), on its own, gives rise thus to an estimate of the form

$\mathfrak{F}(v),\quad\mathfrak{E}(\tau^{\prime})+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}\leq\mathfrak{E}(\tau_{0})+A\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\int_{\mathcal{R}(\tau_{0},\tau^{\prime})}\left|\left(V^{\mu}\partial_{\mu}\psi+w\psi\right)F\right|,$
(1.6)

where
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}$
is defined similarly to
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}$
but with degeneration function $\rho$ which in general vanishes on a bigger
set than $\chi$. The zeroth order term
${}^{\rho}\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}$
on the left hand side also degenerates, unlike the analogous term in (1.3).
The presence of the term multiplying the new constant $A$ on the right hand
side is necessary in view of the fact that nonnegativity is only assumed for
the highest order terms. Here,
$\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}}^{\prime}$
is a zeroth order energy whose density is supported only in the support of a
function $\xi(r)$

$\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau):=\int_{\Sigma(\tau)}\xi(r)\psi^{2}.$

Importantly, in view of our comments above, we will require that $\xi$
vanishes identically where $\chi$ of (1.3) degenerates, i.e. $\xi=0$ where
$\chi\neq 1$. We will also require $\xi$ to vanish for large $r$, and, in the
case where $\mathcal{S}\neq\emptyset$, in the region $r\leq r_{2}$.

In addition, we will make on $\rho$ the same asymptotic non-vanishing
assumption that we we made previously on $\chi$, namely that $\rho=1$ for
large $r$, and, in the presence of a nonempty spacelike boundary
$\mathcal{S}\neq\emptyset$, that $\rho=1$ in $r\leq r_{2}$. See already
Section 3.4.3 for the detailed assumptions and Figure 2 for a depiction of the
supports of $\chi$, $\rho$ and $\xi$.

We note that in the case where $\mathcal{S}=\emptyset$ and $T$ is globally
timelike, one can in fact easily construct a current satisfying (a) and (b)
and leading to (1.6), _irrespective of the structure of trapping and the
validity of an estimate of the form ( 1.3)_. We will show in Appendix A that
for the very slowly rotating Kerr case $|a|\ll M$, although there is no
globally timelike $T$, there is a similar elementary construction, based
essentially only on the fact that superradiance is effectively governed by a
small parameter and the ergoregion is confined to a part of spacetime which
can be understood using only the red-shift effect (cf. the original treatment
of the boundedness problem on Kerr [DR11]). In general, however, the main
difficulty in proving (1.6) is ensuring the boundary coercivity (a).

Though on its own, estimate (1.6) is clearly weaker than (1.3), its robustness
to direct application to (1.1) allows one to avoid the above loss of
derivatives, when (1.6) is used in conjunction with (1.3) applied to (1.5).
Before turning to describe how this is done in the context of the proof, we
will in fact have to slightly strengthen our assumed estimates (1.3) and (1.6)
as follows.

##### Extending to $r^{p}$ weighted estimates.

First of all, we will need to extend (1.3) and (1.6) to $r^{p}$-weighted
estimates, for $0\leq p<2$, of the type first introduced in [DR09]:

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{E}}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{F}}(v)+\int_{\tau_{0}}^{\tau_{1}}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$-1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{E}}(\tau_{0})+\text{\rm
inhomogeneous terms},$ (1.7)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{F}}(v),\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{E}}(\tau)+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{E}}(\tau_{0})+A\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\text{\rm
inhomogeneous terms}.$ (1.8)

In the above, $\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{E}}$,
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\mathcal{E}}^{\prime}$,
etc., are $r^{p}$, respectively $r^{p-1}$, etc., weighted versions of
$\mathcal{E}$,
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}$,
etc, while $\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{E}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{F}}$ satisfy
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{E}}\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{E}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{F}}\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{F}}$
but can moreover be represented as the flux term of a current

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{E}}(\tau)=\int_{\Sigma(\tau)}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{J}_{\mu}^{V,w,q,\varpi}[\psi]n^{\mu},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathfrak{F}}(v)=\int_{\underline{C}_{v}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{J}_{\mu}^{V,w,q,\varpi}[\psi]n^{\mu},$

and we in fact assume that (1.8), just as (1.6), is the result of a pointwise
coercive energy identity (1.4) associated to the current
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{J}$.

To obtain (1.7) and (1.8), given (1.3) and (1.6), it suffices to assume the
existence of currents defined only in the region $r\geq\tilde{R}$ with
suitable far away coercivity properties. For maximal generality, we will here
simply directly postulate the existence of such currents as an additional
assumption. See already Section 3.5 for the precise formulation. This
assumption can be shown to follow from suitable pointwise asymptotically flat
assumptions on the metric $g_{0}$ and holds of course in all our examples of
interest, where the currents can easily be explicitly constructed. See
Appendix B, and also [Mos16] for an even more general setting.

##### Extending to higher order estimates.

Secondly, we will need, through suitable commutations, to extend (1.3), (1.6),
(1.7) and (1.8) to higher order statements. We will distinguish two energies

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}$

at $k$’th order.

The energy
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}$
is the sum of the usual energy, with $r^{p}$ weights $0\leq p<2$, and with
commutation vector fields $\widetilde{\mathfrak{D}}^{k}$ up to order $k$,
_spanning the entire tangent space_. The set will include the vector fields
$\Omega_{i}$, $i=1,\ldots,3$, which for large $r$ correspond to the usual
rotational vector fields. (Note that the latter will be the only commutation
vector fields which are $r$-weighted.) This is a fundamental energy with
respect to which initial data can be measured and with respect to which
suitable Sobolev inequalities hold.

The energy
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}$,
on the other hand, denotes the precise energy flux of the current leading to
(1.4) applied moreover with a new set $\mathfrak{D}^{k}$ of commutation vector
fields, which will include $T$ (and $\Omega_{1}$, if this is assumed to be
Killing), but for which _all non-Killing vectors are cut off to vanish in a
suitable region of finite $r$_. (See already Sections 3.6.1–3.6.4 for the
definition of these commutation vector fields and energies.) Nonetheless, by
our geometric assumptions and elliptic estimates, we will have that the
energies

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}\sim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}$
(1.9)

are equivalent for solutions of $\Box_{g_{0}}\psi=0$.

To extend our estimates to higher order, we will need some properties of the
commutation errors arising from $[\Box_{g_{0}},\mathfrak{D}^{k}]$. Let us note
that in the case where $\mathcal{S}=\emptyset$ and $T$ is globally timelike,
these errors are only supported in the asymptotic region of large $r$ and can
be controlled if the metric is suitably asymptotically flat. For maximum
generality, we will formulate the relevant asymptotic assumption directly in
terms of pointwise decay bounds for these commutators. This will include all
examples of interest. See already Section 3.6.2.

In the case where $\mathcal{S}\neq\emptyset$, there will be additional
commutation errors $[\Box_{g_{0}},\mathfrak{D}^{k}]$ arising in a
neighbourhood of the boundary $\mathcal{S}$ including the Killing horizon
$\mathcal{H}^{+}$. Key to their control is the assumption of positive surface
gravity of $\mathcal{H}^{+}$, discussed in Section 1.1.1, and the inclusion
among the collection $\mathfrak{D}$ of a well-chosen vector field $Y$ which is
null and transversal to $\mathcal{H}^{+}$. As first shown in [DR13],
commutation by $Y$ generates a term with a good sign at $\mathcal{H}^{+}$ (see
already Proposition 3.6.1 of Section 3.6.3), and this fact, together with
elliptic estimates in $r>r_{\rm Killing}$, an enhanced red-shift estimate near
$\mathcal{H}^{+}$ and enhanced positivity in the black hole interior up to
$\mathcal{S}$ (see already Propositions 3.4.1 and 3.4.2 of Section 3.4.5), can
be used to absorb the errors arising from commutation and to extend the
estimates to higher order. We note that the above-mentioned good sign is yet
another manifestation of the local red-shift at the horizon $\mathcal{H}^{+}$
associated to the positivity of the surface gravity. To understand how all
commutation errors can indeed be absorbed, see already Section 3.6.6.

The final extended estimates on (1.2) resulting from (1.3) and (1.6) take the
form

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}+\int_{\tau_{0}}^{\tau_{1}}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\mathcal{E}}(\tau_{0})+\text{\rm
inhomogeneous terms},$ (1.10)
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau)+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+c\int_{\tau_{0}}^{\tau_{1}}{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+A\int_{\tau_{0}}^{\tau_{1}}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}+\text{\rm
inhomogeneous terms},$ (1.11)

where again (1.11) derives from the pointwise coercivity properties of the
energy identity (1.4) associated to a current
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{J}}$,
whose flux terms are precisely given by
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}$.
See already Section 3.6.8 for the precise form of the estimates and
inhomogeneous terms.

###  1.2 Assumptions on the nonlinearities $g(\psi,x)$ and
$N(\partial\psi,\psi,x)$

We now turn to the nonlinear equation (1.1). In formulating assumptions on the
allowed nonlinearities, we are here largely motivated by geometric
formulations of the Einstein equations, as in [DHRT21], which produce “almost
decoupled” equations similar to (1.1), which satisfy an appropriate form of
the null condition. We emphasise, however, that the resulting equations in
such a formulation do not constitute a pure system of wave equations in the
form (1.1), and the quasilinear structure is mitigated through coupling with
transport equations. Thus, in this context, one should really only think of
(1.1) as an indicative model equation. For more specific discussion of the
Einstein equations, see already Section 1.4.4.

In addressing the combined difficulties of the interaction of quasilinear
terms, trapped null geodesics, and merely polynomial decay at the level of
model scalar equations of the form (1.1), the most natural simplest setting is
to require that the _quasilinear_ term $g(\psi,x)-g_{0}(x)$ be supported
entirely in the region $r\leq R$, and to impose on the _semilinear_ term
$N(\partial\psi,\psi,x)$ a generalised version of the null condition [Kla86].
This also allows us to use the exact null hypersurfaces of the background
$(\mathcal{M},g_{0})$ without additional complications, making the null
hierarchical structure of the estimates clearer. (In the case of the Einstein
equations, in our framework, these would be replaced by null hypersurfaces of
the actual dynamic spacetime $(\mathcal{M},g)$, for instance associated to a
double null gauge.) We note, however, that use of exact null hypersurfaces is
in no way fundamental; one can always replace these with suitable
hyperboloidal hypersurfaces, for instance, using the setup of [Mos16].

For maximum generality, rather than attempt a general algebraic definition of
the null condition for the semilinear terms $N(\partial\psi,\psi,x)$ of (1.1),
we will formalise a “null condition assumption” in the form of a time-
translation translation invariant $r^{p}$ weighted estimate for the
inhomogeneous terms of (1.10)–(1.11) when applied to (1.1), restricted
entirely to the asymptotic region $r\geq R$.

To describe the condition, it will be convenient to introduce “master”
energies

${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1}):=\sup_{v}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v)+\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\int_{\tau_{0}}^{\tau_{1}}\left({}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)\right)d\tau\,\,$
(1.12)

for all $\delta\leq p\leq 2-\delta$. In the case $p=0$, which is anomalous, we
will define

$\displaystyle{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=\sup_{v}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v)+\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\int_{\tau_{0}}^{\tau_{1}}\left({}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau)+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau)\right)d\tau\,\,,$
(1.13)
$\displaystyle{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})$
$\displaystyle:=\sup_{v}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}(v)+\sup_{\tau_{0}\leq\tau\leq\tau_{1}}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\int_{\tau_{0}}^{\tau_{1}}\left({}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(-3\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})\right)d\tau\,\,.$

We may define similar master energies
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$,
etc., with $\chi$ replacing $\rho$, and where extra non-degenerate lower order
terms
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}$,
$\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}$
are added to the integrands on the right hand side of (1.12) and (1.13),
respectively, and finally
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}$
where $\rho$ is removed. Let us note that

${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}.$

Specifically, our null condition assumption is that in any spacetime slab
$\mathcal{R}(\tau_{0},\tau_{1})$, the far away (i.e. $r\geq R$) contribution
of the inhomogeneous terms arising from $N(\partial\psi,\psi,x)$ in the
estimates (1.10), (1.11) can be bounded by the expressions

${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})}+\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})}\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})},\qquad{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau_{1})}$
(1.14)

for $\delta\leq p\leq 2-\delta$ and $p=0$, respectively, and where all
energies may in fact be restricted to the region $r\geq 8R/9$, where $\rho=1$.
(Note how the $p=0$ weighted estimate is anomalous, in that the nonlinear
terms require the boundedness of a bulk term associated to the energy identity
of a higher ($p>0$) weight so as to be estimated.) See already Section 4.7 for
the precise formulation of the assumption.

The assumption that one can bound the relevant inhomogeneous terms arising
from $N(\partial\psi,\psi,x)$ by (1.14) isolates precisely that aspect of the
usual null condition which is relevant for global existence in our method. We
will then show in Appendix C that our assumption indeed holds in particular
for the usual nonlinearities allowed by the classical null condition [Kla86]
and includes also the class of semilinear terms $N(\partial\psi,\psi,x)$ on
Kerr studied previously in [Luk13].

###  1.3 The dyadically localised approach to global existence: sketch of the
proof of the main theorem

We now turn to the proof of the main theorem and the other novel aspect of the
present work, namely the replacement of a global bootstrap built on global
decay-in-time estimates with dyadic iteration in consecutive spacetime slabs
based entirely on dyadically localised, $r^{p}$ weighted—but time-translation
invariant—estimates.

We sketch the argument here. For details, see already Section 6. In our
discussion below, we will focus directly on our main case of interest,
referred to later in the paper as case (iii), where the black box
inhomogeneous estimate (1.3) has nontrivial degeneration and indeed does not
arise from a physical space identity, and thus must be used in conjunction
with the auxiliary identity (1.6). We emphasise, however, that the dyadic
approach described here is already novel in the case, referred to later in the
paper as case (ii), where the black box estimate (1.3) again has degeneration
but is actually a consequence of a physical space identity (as in
Schwarzschild, see the discussion following (1.4)). Finally, in the case where
the black box estimate (1.3) derives from a physical space identity and is
moreover non-degenerate (i.e. $\chi=1$ identically), referred to in the paper
as case (i), then dyadic iteration is in fact unnecessary, and the approach
reduces to a completely elementary time-translation invariant estimate. (This
latter case applies for instance when $g_{0}$ is a small stationary
perturbation of Minkowski space.) _In the paper, we will provide self-
contained treatments of all three cases so the reader can compare with the
more traditional approach._

####  1.3.1 The $r^{p}$ weighted estimate hierarchy on a spacetime slab of
time-length $L$

In the dyadic approach, the key element is an appropriate time-translation
invariant $r^{p}$ weighted hierarchy on a slab of time length $L$. The
hierarchy will be formed by combining the $r^{p}$ and higher $k$ order
estimates associated to (1.3) and (1.6) with various choices of $p$ weights
and differentiability order $k$. The estimates will depend only on a time-
translation invariant bootstrap assumption, to be retrieved within the slab,
concerning only lower order energy quantities.

Specifically, from the assumptions of Section 1.1 and 1.2, on any spacetime
slab $\mathcal{R}(\tau_{0},\tau_{0}+L)$ of time-length $L\geq 1$, then, we
obtain for $p=2-\delta$ and $p=1$, and for $0<\delta<\frac{1}{10}$, the
following hierarchy of estimates:

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}(v),\quad\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau),\quad{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)$
$\displaystyle\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})+\boxed{A\int_{\tau_{0}}^{\tau}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}}+C\,{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}+C\,\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}$
$\displaystyle\qquad+\boxed{C\sup_{\tau_{0}\leq\tau^{\prime}\leq\tau}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{L}},$
(1.15)
$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)$
$\displaystyle\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}(\tau_{0})+{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}+\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{{}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}$
(1.16)
$\displaystyle\qquad+\boxed{\sup_{\tau_{0}\leq\tau^{\prime}\leq\tau}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau^{\prime})}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}\sqrt{L}.$

The estimates are in fact contingent on an appropriate time-translation
invariant bootstrap assumption on the lower order energy

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)\lesssim\varepsilon$
(1.17)

for $p=0$. The hierarchy (1.15)–(1.16) descends also to $p=0$, but where
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{X}}}(\tau_{0},\tau)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}$
and
${}^{\rho}\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau)\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0+)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)}$
replace the sum of the third to last and second to last terms of (1.15) and
(1.16), respectively. (This anomaly, referred to already immediately after
(1.14), is a fundamental aspect of the estimates.) See already Section 6.3.1.

Estimate (1.15) arises from applying the physical space identity associated to
the current
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{J}}$,
which in the linear case led to (1.11), _directly_ to the $k$ times commuted
equation (1.1), i.e. it arises from the divergence identity (1.4) associated
to the current
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{J}}[g(\psi,x),\psi]$,
where $g(\psi,x)$ replaces $g_{0}$ covariantly in the definition of
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{J}}$.
(The energies
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}$
and
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{F}}}$
in (1.15) now in fact denote the exact fluxes arising from these currents; in
view of the bootstrap assumption (1.17), one can again show the equivalence
(1.9) for solutions of (1.1).) As discussed already in Section 1.1.2, it
follows that (1.15) does not “lose derivatives”, as is clear from examining
the order of differentiability of the terms on the right hand side. We remark
that the first boxed term in (1.15) reflects the analogous term in (1.11). On
the right hand side of (1.15), we recognise the bound for the far away
contribution of the nonlinear term (1.14) from our assumption capturing the
null condition, while the second boxed term in (1.15) is necessary to control
the nonlinear terms on the set where $\rho$ degenerates, for there, this boxed
term cannot be absorbed in the spacetime bulk term on the left hand side. (The
bad explicit dependence on the length $L$ arises because one must take the
supremum over $\tau$ for the energy of the highest order terms there.)

Estimate (1.16), on the other hand, arises from applying the “black box”
estimate (1.10) to the nonlinear equation written in the form (1.5), i.e. now
thought of simply as an inhomogeneous equation with respect to the background
$g_{0}$. The boxed term in (1.16), which is $k$’th order, reflects the loss of
derivatives due to the quasilinearity discussed already in Section 1.1.2.

The estimates can in principle close because, in view of our restrictions on
the support of $\xi$, we have the fundamental relation

$\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}+\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}$
(1.18)

and thus we may bound the term

$A\int_{\tau_{0}}^{\tau}\accentset{\scalebox{0.8}{\mbox{\tiny$\xi$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{E}}}^{\prime}(\tau^{\prime})d\tau^{\prime}\lesssim{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu1$}}}{\mathcal{X}}}(\tau_{0},\tau),$

which is in turn bounded by the term on the left hand side of (1.16) for any
$p$.

####  1.3.2 Global existence in a slab

The first task is to show global existence in a given slab
$\mathcal{R}(\tau_{0},\tau_{0}+L)$ for arbitrary $L\geq 1$, given suitable
($L$-dependent) smallness assumptions at $\Sigma(\tau_{0})$.

We first note that the estimate (1.15) alone can be used to easily show
_local_ existence, by which we mean existence in an entire smaller slab of the
form $\mathcal{R}(\tau_{0},\tau_{0}+\epsilon)$, provided that some
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}\lesssim\varepsilon_{0}$
for sufficiently high $k$. This is in fact true also for $p=0$. (Note, that
this “local” result already uses non-trivially the null condition assumption
of Section 1.2. This is because our foliation $\Sigma(\tau)$ is outgoing null
for $r\geq R$. Equations (1.1) not satisfying some version of the null
condition can fail to yield solutions in
$\mathcal{R}(\tau_{0},\tau_{0}+\epsilon)$ for any $\epsilon>0$, no matter how
small the data are.) Moreover, we show that smallness of a suitably high
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}$
norm defines a continuation criterion.

In view of the above, for global existence, it suffices to show that, under
suitable assumptions at $\tau=\tau_{0}$, the quantity
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}$,
for some $p\in\\{0\\}\cup[\delta,2-\delta]$ and suitably high $k$ remains
globally small in the slab. Examining the hierarchy, for global existence in a
slab $\mathcal{R}(\tau_{0},\tau_{0}+L)$ of length $L$, it seems necessary to
have at least

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{X}}}(\tau_{0},\tau)\lesssim\varepsilon
L^{-1}.$ (1.19)

Indeed, in view of the nonlinear terms, the $L^{-1}$ factor in (1.19)
represents a natural threshold, as $s=1$ represents the minimum value of $s$
for which $\sqrt{L^{-s}}\sqrt{L}\lesssim 1$, allowing the quantities in a
fixed slab to be easily bounded by their initial values.

Assuming

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})\lesssim\varepsilon_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau_{0})\lesssim
L^{-1}\varepsilon_{0},$ (1.20)

which of course imply, for $\varepsilon_{0}\ll\varepsilon$, both (1.17) (for
$p=1$) and (1.19) at $\tau=\tau_{0}$, an easy continuity argument, with (1.17)
and (1.19) taken as bootstrap assumptions, shows that if estimates (1.20) hold
for $\tau=\tau_{0}$, then the solution indeed exists globally in the entire
slab $\mathcal{R}(\tau_{0},\tau_{0}+L)$.

See already Section 6.3.2 for the details of the proof. Note that our above
appeal to (1.17) and (1.19) is in fact the only instance that a bootstrap
assumption must be used in the proof of our main theorem.

####  1.3.3 Improved estimates and the pigeonhole principle

Once global existence within a slab has been established, one can improve a
posteriori the estimates given stronger assumptions on initial data.
Anticipating dyadic iteration, one seeks a set of initial estimates at
$\tau=\tau_{0}$ with the property that they are _exactly recoverable_ at the
top of the slab $\tau_{0}+L$, with suitable redefinition of $\varepsilon_{0}$
and with $L$ replaced by $2L$.

Re-examining the hierarchy (1.15)–(1.16), we show that for an appropriate
large constant $\alpha$, and any $L\geq 1$, $\tau_{0}\geq 1$, then for
$\hat{\epsilon}_{0}$ sufficiently small, given for example the stronger
estimates

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha
L^{-1+\delta},\qquad\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu6$}}}{\mathfrak{E}}}(\tau_{0})\leq\hat{\varepsilon}_{0}\alpha^{2}L^{-2+\delta},$
(1.21)

then (1.21) holds also at the final time $\tau_{0}+L$ where $\tau_{0}$, $L$
and $\hat{\varepsilon}_{0}$ are now however replaced by

$\tau_{0}+L,\qquad 2L,\qquad\hat{\varepsilon}_{0}(1+\alpha L^{-\frac{1}{4}}),$
(1.22)

respectively. Note that the statement that the last three inequalities of
(1.21) hold at $\tau_{0}+L$ with $2L$ in place of $L$ is a _stronger_
smallness assumption than that assumed for these quantities at $\tau_{0}$,
signifying that these quantities have in fact decayed, and relies on a
pigeonhole argument as in [DR09], based on the fundamental relation that, for
$p-1\geq 0$, the _bulk_ integrand
$\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}^{\prime}$
of the $p$-weighted identity (1.16) at order $k-1$ (when integrated, included
in the
${}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{X}}}$
master energy) controls the _boundary_ term of the $p-1$ weighted identity at
arbitrary order $k$, denoted without the prime:

$\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\gtrsim\accentset{\scalebox{0.6}{\mbox{\tiny$(p\mkern-6.0mu-\mkern-6.0mu1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}.$
(1.23)

Briefly, the pigeonhole principle is applied as follows: The integral of the
left hand side of (1.23) for $p=1$ and $k$ replaced by $k-2$ can be shown to
be bounded $\lesssim\hat{\varepsilon}_{0}$, whence we obtain the existence of
a $\tau=\tau^{\prime\prime}$ slice for which the right hand side
$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau^{\prime\prime})$
is bounded $\lesssim\hat{\varepsilon}_{0}L^{-1}$, where the $L^{-1}$ factor
arises from the length of the interval. We can then propagate this to
$\tau_{0}+L$, and use that $\alpha\gg 1$ to arrange that we have the precise
new version of the third inequality of (1.21) at $\tau_{0}+L$, i.e. for
$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathfrak{E}}}(\tau_{0}+L)$.
Similarly, the integral of the left hand side of (1.23) for $p=2-\delta$ and
$k$ replaced by $k-4$ can be shown to be bounded
$\lesssim\hat{\varepsilon}_{0}$, whence we obtain the existence of a
$\tau=\tau^{\prime\prime}$ slice for which the right hand side
$\accentset{\scalebox{0.6}{\mbox{\tiny$(1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathcal{E}}}(\tau^{\prime\prime})$
is bounded $\lesssim\hat{\varepsilon}_{0}L^{-1}$, the $L^{-1}$ factor again
arising from the length of the interval. Interpolating with the boundedness
statement
$\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathcal{E}}}(\tau^{\prime\prime})\lesssim\hat{\varepsilon}_{0}$
we obtain that on this slice
$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathcal{E}}}(\tau^{\prime\prime})\lesssim\hat{\varepsilon}_{0}L^{-1+\delta}$.
Again, we can then propagate this to $\tau_{0}+L$, and use that $\alpha\gg 1$
to arrange that we have the precise new version of the fourth inequality of
(1.21) at $\tau_{0}+L$, i.e. for
$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu4$}}}{\mathfrak{E}}}(\tau_{0}+L)$.
Finally, a similar argument, applying (1.23) for $p=1$ and $k-6$, yields the
new version of the last inequality of (1.21).

Note that to obtain the new versions of the first two inequalities of (1.21)
at $\tau_{0}+L$, which refer to quantities that do not decay with respect to
their initial values, it is important that the constant appearing in the first
term on the right hand side of (1.15) is indeed $1$.

####  1.3.4 Iteration over consecutive spacetime slabs of dyadic time-length
$L_{i}=2^{i}$

Let us assume that our initial data on $\tau_{0}:=1$ satisfy

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau_{0})+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau_{0})\leq\varepsilon_{0}$

for sufficiently small $\varepsilon_{0}$. It follows that (1.21) is satisfied
for $\tau_{0}=1$, $L:=L_{0}=1$ and appropriate $\hat{\varepsilon}_{0}$.

Examining closely (1.21) and (1.22), we see that we have now obtained a series
of estimates which, in addition to being sufficient to obtain existence in
$\mathcal{R}(\tau_{0},\tau_{0}+L)$, are moreover preserved under dyadic
iteration. We simply iterate the above statement on consecutive spacetime
slabs of dyadic time-length $L_{i}=2^{i}$, defining
$\tau_{i+1}=\tau_{i}+L_{i}=2^{i}$, and
$\hat{\varepsilon}_{i+1}=\hat{\varepsilon}_{i}(1+\alpha
L_{i}^{-\frac{1}{4}})$. Note that
$\hat{\varepsilon}_{i+1}\lesssim\hat{\varepsilon}_{0}\lesssim\varepsilon_{0}$
for all $i$.

This immediately yields global existence in
$\mathcal{R}(\tau_{0},\infty)=\cup_{i\geq 0}\mathcal{R}(\tau_{i},\tau_{i+1})$,
the top order boundedness statement

$\accentset{\scalebox{0.6}{\mbox{\tiny$(1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}(\tau)+\accentset{\scalebox{0.6}{\mbox{\tiny$(2-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu2$}}}{\mathcal{E}}}(\tau)\lesssim\varepsilon_{0}$
(1.24)

and the decay estimate

$\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k\mkern-6.0mu-\mkern-6.0mu6$}}}{\mathcal{E}}}(\tau_{i})\lesssim\varepsilon_{0}\tau^{-2+\delta}.$

Thus, our result is a true orbital stability statement at highest order, in
addition to yielding asymptotic stability.

###  1.4 Discussion

We end with some additional discussion.

####  1.4.1 The semilinear case

We have already remarked that in the semilinear case, i.e. the case where
$g(\psi,x)=g_{0}$ identically and the equation takes the form

$\Box_{g_{0}}\psi=N(\partial\psi,\psi,x),$ (1.25)

there is no loss of derivatives when estimating (1.25) as a linear
inhomogeneous equation. Thus, one can base the entire argument directly on the
black box estimate (1.3), without the need for the auxiliary physical space
based estimate (1.6), provided of course that the appropriate asymptotic
flatness and commutation assumptions are made so that (1.3) can be extended to
the higher order weighted (1.10). See already Remark 5.1 for the modified
statement appropriate in this case and a guide to its proof. In particular,
our theorem applies directly to the purely semilinear version of (1.1) on Kerr
in the full subextremal range $|a|<M$.

####  1.4.2 Additional examples and extensions

We collect here some additional examples to which our results apply to, as
well as various natural future extensions.

##### Other spacetimes $(\mathcal{M},g_{0})$ satisfying our assumptions.

We have already specifically mentioned the examples where
$(\mathcal{M},g_{0})$ is Minkowski space, Schwarzschild and Kerr, but let us
remark that our black box assumption (1.3) holds in fact for the full
subextremal Kerr–Newman family [Civ15]. We also note that combining the use of
an energy current construction similar to Appendix A with [WZ11], one can show
that (1.3) holds for _arbitrary_ stationary perturbations
$(\mathcal{M},g_{0})$ of Schwarzschild, provided that they are close to
Schwarzschild in a suitably regular norm. Since we have already remarked that
the physical space based estimate (1.6) holds also for such spacetimes, in
fact under much weaker regularity assumptions, this means that our main
theorem indeed applies in this case. Thus, for very slowly rotating $|a|\ll M$
Kerr spacetimes, one sees that the special additional structure of Kerr (like
the various manifestations of Carter separability, the Killing tensor, etc.)
has no significance for the problem, and the spacetimes can just be understood
as a very small stationary perturbation of Schwarzschild with a Killing
horizon. _It is only in the case $|a|\sim M$ where true Kerr properties are of
any relevance._ A treatment of this case, reducing directly to the black box
estimate proven in [DRSR16], will appear elsewhere.

##### The axisymmetric case.

In the case where we assume that the metric $g_{0}$ admits an additional
Killing field $\Omega_{1}$, one may also restrict to nonlinear equations of
the form (1.1) which moreover preserve this symmetry, i.e. with the property
that if the data are preserved by $\Omega_{1}$, then so is the solution. Let
us call such equations and solutions axisymmetric. Under the geometric
assumptions of this paper, for axisymmetric equations (1.1) and restricted to
axisymmetric solutions, one may in fact easily derive the existence of the
necessary auxiliary physical space identity yielding (1.6), just as in
Schwarzschild. This in particular applies to Kerr–Newman in the full
subextremal case. Thus, in view of [DRSR16, Civ15] which obtain (1.3), the
main theorem of the paper applies also in these settings. We leave the details
as an exercise for the reader. Note that in this case, if one prefers, one may
quote the original [DR10] for (1.3) for axisymmetric solutions, in place of
the more general (1.3) obtained for all solutions in [DRSR16].

##### Potentials.

Our method applies also if a suitably decaying $T$-independent potential
$\mathcal{V}(x)$ is added to (1.1), i.e. for

$\Box_{g(\psi,x)}\psi-\mathcal{V}(x)\psi=N(\partial\psi,\psi,x),$

provided that both the black box assumption (1.3) and the physical space based
(1.6) hold for solutions of the inhomogeneous version of the new linearised
equation equation $\Box_{g_{0}}\psi-\mathcal{V}(x)\psi=F$ in place of (1.2).
We note that in the case where $T$ is globally timelike and $\mathcal{V}$
decays suitably and is nonnegative, then, one can always obtain also the
analogue of (1.6). The same statement applies for any such potential in the
slowly rotating Kerr case $|a|\ll M$.

##### Systems.

For notational convenience, we have restricted here to scalar equations. The
considerations generalise readily to systems, for which a generalised null
condition can again be formulated in terms of bounds of the nonlinear terms by
(1.14); this now includes many examples. See also the discussion of the weak
null condition in Section 1.4.5 below.

##### The obstacle problem.

We have already remarked that, with a mild adaptation of the setup, one can
also consider the case where the boundary $\mathcal{S}$ is in fact timelike,
imposing now also suitable boundary conditions on $\mathcal{S}$. This is for
instance the setting for the classical obstacle problem. The analogue of (1.3)
has indeed been proven in the nontrapping case (with $\chi=1$ identically,
i.e. without degeneration), but also, with suitable degeneration functions
$\chi$, for many examples with nontrivial hyperbolic trapping (see for example
[Laf22]). We note again that the additional physical space estimate (1.6) can
always be retrieved in this case. Thus, this situation can also be
incorporated in our framework.

##### Extremal black holes.

Extremal black holes do not satisfy our assumptions on $(\mathcal{M},g_{0})$,
already because the stationary vector field $T$ will not be tangential to the
boundary $\mathcal{S}$, but more seriously, because $\chi$ must degenerate at
the black hole horizon [Sbi15], where $T$ will not be timelike. Nonetheless, a
version of (1.3) has been proven in the extremal Reissner–Nordström case
[Are11] (and in the extremal Kerr case, under the assumption of axisymmetry
[Are12]), with an additional hierarchy of degeneration associated to the
horizon, and nonlinear stability for semilinear equations has been proven
[AAG20], where, however, an additional null structural condition is required
at the horizon, in full analogy with the situation at null infinity. It would
be interesting to reformulate this latter work in the dyadic setup used here.
This is related to the nonlinear stability problem of extremal black holes.
See the discussion in [DHRT21] and Conjecture IV.2 therein.

##### The asymptotically AdS case.

Finally, although our assumptions are modelled on the asymptotically flat
setting, one could also try to reformulate things in the asymptotically anti-
de Sitter case, appropriate for solutions of the Einstein equations with
negative cosmological constant $\Lambda<0$ (for a discussion of the
$\Lambda>0$ case see already Section 1.4.6 below). Here, one must also impose
boundary conditions _at infinity_ , since infinity itself can be thought of as
an asymptotic _timelike_ boundary. We note that under reflecting boundary
conditions, there exist periodic (and thus non-decaying) solutions of the wave
equation on pure AdS, while general solutions on the Schwarzschild–anti de
Sitter and Kerr–anti de Sitter case [HS13, HS14], again under such boundary
conditions, decay only inverse logarithmically. (This is due to stable photon
orbits repeatedly reflecting off of infinity only to return later having been
repelled centrifugally by the black hole.) Based on this lack of decay, pure
AdS has in fact been proven to be nonlinearly unstable under reflecting
boundary conditions as a solution to various Einstein–matter systems, where
the problem can be studied already under spherical symmetry [Mos22]. Thus, if
one hopes to show nonlinear stability for an equation of the type (1.1) on
asymptotically AdS backgrounds, it is more natural to consider _dissipative_
boundary conditions like those considered in [HLSW20]. Note that on Kerr–AdS,
one expects to indeed satisfy the analogue of (1.6) with the help of the
Hawking–Reall Killing vector field, provided that the parameters satisfy the
Hawking–Reall bound. Thus, given a version of (1.3), nonlinear stability for a
suitable class of equations (1.1) should be tractable for all such Kerr–AdS
parameters.

####  1.4.3 The dyadic approach vs. the traditional approach

The dyadic approach followed here is of course closely related to the more
traditional approach. Our $L$-weighted smallness translates easily into
$t$-decay assumptions, and if one wishes, one can rewrite the argument using a
bootstrap with $t$-decaying norms. We believe, however, that the dyadic
localisation of the argument both makes the proof more modular and may serve
to better identify possible future refinements with respect to regularity and
decay. One sees clearly, for instance, that the $L^{-1}$ smallness assumptions
necessary for global existence _within a slab_ are manifestly weaker than the
$L^{-2+\delta}$ smallness necessary to improve and iterate (in fact one may
weaken this to $L^{-1-\epsilon}$). Since bootstrap is only used to show
existence _within a slab_ , this already simplifies the argument considerably.
We also note that the $L^{-1-\epsilon}$ threshold corresponds to pointwise
decay $t^{-\frac{1}{2}-\frac{\epsilon}{2}}$, considerably weaker than the
$t^{-1-\epsilon}$ “improved decay” which is often invoked for global existence
and stability for (1.1). Proving sharper decay is of course an extremely
important problem in itself (see [AAG16, Hin22, AAG21, Keh22]) but is not
necessary (or even particularly helpful) for nonlinear stability.

Let us also point out that for problems with gauge invariance, like the
Einstein equations, dyadic localisation also provides a convenient opportunity
to refresh the gauge. This suggests an alternative approach to the global
teleological normalisations of gauge done in [DHR, DHRT21].

####  1.4.4 Applications to the Teukolsky equation and derived equations and
to the nonlinear stability of black holes

As discussed in Section 1.2, one motivation for the precise assumptions on the
nonlinearities of (1.1) made here is viewing these as a model for
understanding the Einstein equations, when the latter are considered under
suitable geometrically defined gauges. Such gauges were first exploited
analytically in the monumental proof of the nonlinear stability of Minkowski
space [CK93].

The geometric-gauge approach to black hole stability was taken up in [DHR],
where the problem was studied in double null gauge, and the full linear
stability of Schwarzschild was first proven. (See [Chr09] for an introduction
to double null gauge, including a discussion of the characteristic initial
value problem and a derivation of all equations.) The setup is of course
intimately connected to the Newman–Penrose formalism [NP62]. In the system
resulting from linearising the reduced Einstein equations in double null gauge
around Schwarzschild, the gauge invariant quantities are determined by the
extremal curvature components $\alpha$ and $\underline{\alpha}$, which each
satisfy the Teukolsky equation, first derived in this setting by [BP73] (and
generalised to Kerr in [Teu73]). The transport equations satisfied by the
residual gauge dependent quantities, on the other hand, are always already
_linearly_ coupled to the gauge invariant ones. To analyse first the gauge
invariant quantities, [DHR] introduced a pair of novel physical space
quantities $P$ and $\underline{P}$, related to $\alpha$ and
$\underline{\alpha}$ by second order weighted null differential operators, but
satisfying the more tractable Regge–Wheeler equation (an equation first
derived in a different context in [RW57]), which, unlike the Teukolsky
equation, could be understood by the exact same methods as the linear wave
equation. In particular, an analogue of (1.3) was proven, via a physical space
identity, for $P$ and $\underline{P}$, and this led to boundedness and decay
through the hierarchical structure of the system, first for the quantities $P$
and $\underline{P}$ themselves, then for the original gauge-independent
quantities $\alpha$ and $\underline{\alpha}$, and then, after teleological
normalisation of the double null gauge, of all residual gauge dependent
quantities as well. (For a complete scattering theory for this system, see
[Mas22a, Mas22b]. For other approaches to the gauge in Schwarzschild under
linear theory, see [Ben22] and the references discussed in Section 1.4.5
below.) The origin of this relation between Teukolsky and Regge–Wheeler goes
back to the fixed frequency transformation theory of [Cha75]. On the basis of
this linear theory, the full nonlinear stability of the Schwarzschild family,
without symmetry, was proven in our [DHRT21]. (For previous nonlinear results
for Schwarzschild under symmetry assumptions, see [Chr87, DR05] in the case of
the Einstein-scalar field system under spherical symmetry, and [Hol10, KS20]
for vacuum, the former in the higher dimensional case under biaxial Bianchi
symmetry, and the latter under polarised axisymmetry, reducing to $2+1$
dimensions, the first such result beyond $1+1$ dimensional reductions.) Note
that since Schwarzschild is a co-dimension $3$ subfamily of Kerr, nonlinear
asymptotic stability of Schwarzschild refers to constructing the full
(teleologically defined) codimension $3$ set of initial data which asymptote
to Schwarzschild in the future.

A generalisation of the quantities $P$ and $\underline{P}$ of [DHR] to the
slowly rotating Kerr case $|a|\ll M$ was given independently by [DHR19, Ma20].
The equations satisfied by these quantities are now however (weakly) coupled
to the quantities satisfying the Teukolsky equation. The works [DHR19, Ma20]
both show an analogue of (1.3) for this system, not via a physical space
identity however but based on the framework for frequency localised estimates
on Kerr introduced in [DR10]. These results were followed by full linear
stability statements for the gauge dependent quantities in various gauges
[ABBM19, HHV21] (the work [HHV21] considers in fact harmonic gauge, cf. the
discussion in Section 1.4.5). The estimates of [DHR19, Ma20] have recently
been reformulated by [GKS22] in the language of the higher order physical
space commutation by the Carter tensor first introduced in [AB15]. The work
[GKS22] then uses this, among other ingredients, in the context of the
formalism [GKS20] to obtain nonlinear stability results for the very slowly
rotating Kerr case $|a|\ll M$, culminating an impressive series of preprints.
For generalisations to the Einstein–Maxwell system see [Gio20, Gio21, Ape22].

The full subextremal Kerr case $|a|<M$ is more subtle, as it cannot be
understood as a small perturbation of Schwarzschild. Remarkably, however, an
analogue of (1.3) for this system has been obtained in the full subextremal
case [SRTdC20], using the already highly non-trivial mode stability results
[Whi89, SR15, TdC20] (see also [AMPW17]), but, also, non-trivial new high
frequency structure with no apparent precise analogue in the context of the
wave equation. To exploit frequency analysis, the proof of [SRTdC20] uses a
version of the frequency localisation framework of [DRSR16].

In view of the method introduced in the present paper, it should be clear that
there is absolutely nothing to fear in these type of frequency localisations
for nonlinear applications, provided that they are indeed used to prove a
spacetime localised statement which can be expressed in the form (1.3). The
results of [DHR19, Ma20] and [SRTdC20] can thus in principle be used
_directly_ for the nonlinear problem, in fact, as “black box” results. Thus,
in our view, given the breakthrough of [SRTdC20], not only is there no real
conceptual difficulty in extending the nonlinear Schwarzschild analysis of
[DHRT21] to the full subextremal case $|a|<M$ of Kerr, but the technical
complications are not as severe as one might naively have thought.

####  1.4.5 Harmonic gauge and the weak null condition

In our discussion in Section 1.4.4 we have focussed above primarily on
“geometric” gauges, but as is well known, another way to relate (1.1) to the
Einstein equations is via the harmonic gauge condition (see for instance
[LR10], where the nonlinear stability of Minkowski space is proven in this
gauge). The resulting reduced equations (for
$\psi^{\mu\nu}=g^{\mu\nu}-g^{\mu\nu}_{0}$) produce additional complicated
linear terms, and moreover fail to satisfy the null condition, although they
do satisfy the so-called _weak null condition_ introduced in [LR03]. Note that
a linear stability result has been given for this system in the Schwarzschild
case [Joh19] (see also [Hun18]), and as mentioned earlier, also on very slowly
rotating Kerr $|a|\ll M$ in [HHV21].

In order to model the Einstein equations in harmonic gauge, classes of
equations (1.1) satisfying the weak null condition have been studied in the
recent [LT16, LT20, LT22]. Though we do not here implement our method in this
latter setting, nonetheless, we emphasise that our analysis can in principle
be extended to equations or systems satisfying the weak null condition,
following [Kei18], under an appropriate black box assumption for the
linearisation.

####  1.4.6 Comparison with the $\Lambda>0$ case

Finally, it is interesting to compare with the $\Lambda>0$ case. Here, the
analogue of the Schwarzschild and Kerr black holes are the Schwarzschild–de
Sitter and Kerr–de Sitter family. See [DR13] for a discussion of their
geometry. The study of decay for semilinear and quasilinear equations of the
type (1.1) on Kerr–de Sitter was pioneered by Hintz and Vasy [HV16],
eventually leading to their groundbreaking proof of the nonlinear stability of
very slowly rotating ($|a|\ll M,\Lambda$) Kerr–de Sitter in harmonic gauge.
The proof appeals to extensive machinery from microlocal analysis, with an
elaborate compactification of spacetime, and with a Nash–Moser iteration. For
a more recent approach replacing Nash–Moser iteration with a global bootstrap,
see [Fan21].

Returning to the model equation (1.1), an elementary new method was recently
introduced by Mavrogiannis [Mav21] to treat nonlinear stability on
Schwarzschild–de Sitter and Kerr–de Sitter backgrounds. In [Mav21], the entire
analysis is reduced to a “black box” estimate for the linear problem on
timeslabs of some _fixed_ length $L$. The required black box linear estimate,
however, is not just the analogue of the (degenerate) integrated local energy
estimate (1.3) (first proven in this context in [DR07b]), but a higher order
refinement of (1.3) which is _relatively non-degenerate_ , i.e. where the bulk
and boundary term have identical degeneration function and thus again are
comparable. This is possible through a commutation with a globally defined
well chosen vector field orthogonal to the photon sphere and vanishing at the
horizons. This is an analogue of an energy originally constructed in the
Schwarzschild case in [HK20]. (In the Kerr–de Sitter case, both the degenerate
integrated local energy decay statement and the accompanying relatively non-
degenerate statement are now proven using frequency localisation in a
framework similar to [DRSR16].) Exponential decay is then a derived statement
which follows from iterating a suitable estimate on consecutive slabs, each
now of fixed length $L$.

In comparison to the asymptotically flat case studied in the present paper, it
is noteworthy that in the Kerr–de Sitter case, one does _not_ require an
additional non-trivial physical space-based ingredient, analogous to (1.6),
other than this new, relatively non-degenerate version of the black box linear
estimate (1.3). From the time-slab localised point of view of the two works,
the reason for the difference is clear: Since estimates in [Mav21] have been
reduced to a fixed time scale $L$, the role of (1.6) is essentially provided
by _Cauchy stability_ , a completely soft statement.

From this point of view, the difference between the Schwarzschild and Kerr
case on the one hand and their de Sitter analogues on the other is more than
simply one between polynomial and exponential decay, but is one between
_dyadically localised_ and _truly local_. In this approach, it is this
fundamentally local nature of the analysis in the de Sitter case that renders
the _nonlinear_ aspects of stability problems on such de Sitter backgrounds to
be essentially soft.

###  1.5 Outline of the paper

We end with an outline of the remainder of the paper.

In Section 2 we shall describe the geometric assumptions on the background
spacetime, followed in Section 3 by the assumptions on properties of the
linear inhomogeneous equation (1.2). We shall then introduce in Section 4 the
class of nonlinear equations (1.1) that we shall consider, stating in
particular a local well posedness theorem and continuation criterion. We shall
give the precise statement of the main theorem in Section 5. The proof will
then be carried out in Section 6.

The paper also contains three appendices. In Appendix A, we show how to obtain
(1.6) from a physical space identity in the very slowly rotating Kerr case. In
Appendix B, we will show how to define the far away currents encoding the
$r^{p}$ method necessary to extend the estimates (1.3), (1.6) to (1.7), (1.8).
Finally, in Appendix C, we show that our null condition assumption encoded by
the bounds (1.14) indeed includes the classical null condition [Kla86] and
also the more general class of semilinear terms considered on Kerr in [Luk13].
Together, these appendices show that our main theorem holds in particular for
a wide class of equations of the form (1.1) on very slowly rotating $|a|\ll M$
Kerr backgrounds (and, for the semilinear case $g=g_{0}$, in the full
subextremal range $|a|<M$).

###  1.6 Acknowledgements

MD acknowledges support through NSF grant DMS-2005464. GH acknowledges support
by the Alexander von Humboldt Foundation in the framework of the Alexander von
Humboldt Professorship endowed by the Federal Ministry of Education and
Research and ERC Consolidator Grant 772249. IR acknowledges support through
NSF grants DMS-2005464 and a Simons Investigator Award. MT acknowledges
support through Royal Society Tata University Research Fellowship
URF\R1\191409.

##  2 Geometric assumptions on the background spacetime

We consider a manifold $\mathcal{M}$ with a background metric $g_{0}$
satisfying certain assumptions. In this section, we collect the assumptions
which concern directly the geometry of $(\mathcal{M},g_{0})$. The assumptions
here will not be the most general possible but are sufficient to include the
main examples of interest. They can be easily further generalised in various
directions if desired. Some of the assumptions are in principle redundant but
we have not attempted to derive them from the most minimal considerations. We
note already that the assumptions of this section are modelled on (and are
satisfied by—see Section 2.7!) the basic cases of Minkowski space and the
(extended) exterior regions of Schwarzschild and subextremal $|a|<M$ Kerr
spacetimes.

###  2.1 Underlying differential structure and the positive function $r$

For definiteness, we consider the underlying differential structure of
$\mathcal{M}$ to be given by $\mathbb{R}^{4}_{(x^{0},x^{1},x^{2},x^{3})}$ or
alternatively by the manifold with boundary
$\mathbb{R}^{4}\setminus\\{|x^{1}|^{2}+|x^{2}|^{2}+|x^{3}|^{2}<r_{0}^{2}\\}$.
(The black hole examples in fact correspond to the latter case.) In this
latter case, let us define

$r:=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}.$ (2.1)

In the former case where the underlying manifold is $\mathbb{R}^{4}$, let us
pick an arbitrary $r_{0}>0$ and define

$r:=f\left(\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}\right).$ (2.2)

where $f:[0,\infty)\to[r_{0},\infty)$ is a smooth function with
$f^{\prime}(v)>0$ such that $f(z)=\sqrt{r_{0}^{2}+z^{2}}$ near $z=0$ and
$f(z)=z$ for $z\geq 2r_{0}$. Thus, in all cases $r$ is a smooth positive
function on $\mathcal{M}$ satisfying $r\geq r_{0}>0$. Let us also fix a large
$R\geq 20r_{0}$.

We may also define ambient spherical coordinates in the usual way by the
relation

$(x^{1},x^{2},x^{3})=(r\cos\varphi\sin\vartheta,r\sin\varphi\sin\vartheta,r\cos\vartheta).$
(2.3)

In the case of no boundary, we will denote by $\Omega_{1}=\partial_{\varphi}$,
$\Omega_{2}$, and $\Omega_{3}$ the standard rotation vector fields associated
to the ambient spherical coordinates $(x^{0},r,\vartheta,\varphi)$. These are
globally regular vector fields.

In the case where the underlying structure is $\mathbb{R}^{4}$, the above
vector fields would only be regular in $r>r_{0}$. In this case then, we will
use the notation $\Omega_{i}$, $i=1,\ldots,3$ for the above standard vector
fields multiplied by $\omega(r)$, for a smooth cutoff function satisfying
$\omega(r)=1$ for $r\geq R/2$, and $\omega(r)=0$ in a neighbourhood of
$r_{0}$. These are again globally regular vector fields.

###  2.2 The Lorentzian metric $g_{0}$, time orientation and the causality of
the boundary

We assume that $g_{0}$ is a time oriented Lorentzian metric on
$\mathcal{M}_{0}$ such that the coordinate vector field $\partial_{x^{0}}$ is
future directed timelike for $r\geq R/6$.

In the case where $\mathcal{M}$ is a manifold with boundary
$\mathcal{S}=\\{r=r_{0}\\}$, we assume that the boundary $\mathcal{S}$ is
spacelike and the future directed normal points out of the spacetime.

###  2.3 The stationary Killing field $T$

Denoting now by $T$ the coordinate vector field

$T=\partial_{x^{0}}$

with respect to the ambient coordinates $(x^{0},x^{1},x^{2},x^{3})$, we assume
that $T$ is Killing with respect to the metric $g_{0}$. It follows by the
previous assumptions that the ambient function $r$ above is $T$-invariant and
that $T$ is future directed timelike in the region $r\geq R/6$. Let us further
assume that $g(T,T)\to-1$ as $r\to\infty$.

We will denote by $\phi_{\tau}$ the $1$-parameter group of diffeomorphisms
generated by $T$.

In the case where $\mathcal{S}=\emptyset$, it is natural to already assume
that $T$ is globally timelike, in view of the Friedman instability [Mos18] and
its evanescent analogue [Kei20] which applies in the marginal case, both of
which would be incompatible with assumptions we will make later on concerning
the behaviour of waves.

In the case where $\mathcal{S}\neq\emptyset$, we notice that $T$ cannot be
globally timelike as $T$ is tangential to and thus spacelike on
$\mathcal{S}=\\{r=r_{0}\\}$. In this case we will need to assume the existence
of a Killing horizon.

###  2.4 The Killing horizon $\mathcal{H}^{+}$

If $\mathcal{S}\neq\emptyset$ we will assume the following further properties:

We assume that there exists an $r_{\rm Killing}$ such that the hypersurface
$\mathcal{H}^{+}:=\\{r=r_{\rm Killing}\\}$ is a Killing horizon with future
directed null generator $Z=T$, or more generally, with future directed null
generator $Z$ in the span of $T$ and $\Omega_{1}$, in which case we will
assume that $\Omega_{1}$ is globally Killing. We will denote by $Z$ the
globally defined Killing field given by this combination of $T$ and
$\Omega_{1}$.

We will assume furthermore that $\mathcal{H}^{+}$ has strictly positive
surface gravity, i.e. $\nabla_{Z}Z=\kappa(\vartheta,\varphi)Z$ for some
$\kappa>0$ and we will assume that $T$, or more generally the span of $T$ and
$\Omega_{1}$, is timelike for $r>r_{\rm Killing}$.

Finally, in the case $\mathcal{S}\neq\emptyset$ we will assume that the vector
field $\nabla r$ is future directed timelike in the region $r<r_{\rm
Killing}$. Note that $\nabla r$ on $\mathcal{H}^{+}=\\{r=r_{\rm Killing}\\}$
will be null and in the direction of $Z$.

###  2.5 Foliations, subregions and volume forms

####  2.5.1 The foliation $\Sigma(\tau)$

We will assume $\mathcal{M}$ admits a hypersurface $\Sigma_{0}$ with
$2$-dimensional corner at $\Sigma_{0}\cap\\{r=R\\}$, such that
$\Sigma_{0}\cap\\{r\leq R\\}$ is strictly spacelike and
$\Sigma_{0}\cap\\{r\geq R\\}$ is null and $\Sigma_{0}$ is transversal to $T$.

We assume that, for $r^{\prime}\geq R/2$, $\Sigma_{0}$ is transversal to the
hypersurface $\\{r=r^{\prime}\\}$, and that $\Omega_{1}$, $\Omega_{2}$,
$\Omega_{3}$ are tangent to $\Sigma_{0}\cap\\{r\geq R/2\\}$. In particular the
$2$-dimensional space $T_{p}(\Sigma_{0}\cap\\{r\leq R\\})\cap
T_{p}(\Sigma_{0}\cap\\{r\geq R\\})$ should coincide with the span of these
vectors.

If $\mathcal{S}\neq\emptyset$, we assume that $\Sigma_{0}$ is transversal to
the hypersurface $\mathcal{S}$, and $\Sigma_{0}\cap\mathcal{S}$ is
diffeomorphic to the $2$-sphere. We assume finally that the vector field $Z$
of Section 2.4 is orthogonal to $\Sigma_{0}\cap\mathcal{H}^{+}$, while
$\Omega_{1}$, $\Omega_{2}$, $\Omega_{3}$ are tangent to
$\Sigma_{0}\cap\mathcal{H}^{+}$.

We assume that $\Sigma_{0}$ separates $\mathcal{M}$ into two connected
components and that $\Sigma_{0}$ is a past Cauchy hypersurface for
$J^{+}(\Sigma_{0})$.

Denoting by $\Sigma(\tau):=(\phi_{\tau})_{*}(\Sigma_{0})$ we assume

$J^{+}(\Sigma_{0})=\cup_{\tau\geq 0}\Sigma(\tau),$

where $J^{+}$ denotes causal future in $\mathcal{M}$ with respect to the
metric $g_{0}$.

Clearly $\\{\Sigma(\tau)\\}_{\tau\in\mathbb{R}}$ defines a foliation of
$\mathcal{M}$ and thus defines globally on $\mathcal{M}$ a Lipschitz function
$\tau$, which is smooth separately on $r\leq R$ and $r\geq R$.

Note that by our assumptions above, for $r\geq R/2$, we have that
$\tau=\tau(x^{0},r)$ and the triple $(r,\vartheta,\varphi)$ represent a smooth
coordinate system on $\Sigma(\tau)\cap\\{r\geq R\\}$ (modulo the spherical
degeneration).

We will denote by $L$ a smooth choice of future directed null generator of
$\Sigma_{0}\cap\\{r\geq R\\}$ normalised to satisfy the constraint
$g(L,T)\sim-1$. By translation invariance, this extends to a smooth vector
field on all of $\\{r\geq R\\}$ in the direction of the null generator of
$\Sigma(\tau)$. (Note that we also use $L$ to denote a general length
parameter; in practice these two notations will not interfere with one
another.)

####  2.5.2 The regions $\mathcal{R}(\tau_{0},\tau_{1})$

Let us denote

$\mathcal{R}(\tau_{0},\tau_{1}):=\cup_{\tau_{0}\tau\leq\tau_{1}}\Sigma(\tau).$

We shall refer to such regions as spacetime slabs. We will also use the
notation $\mathcal{R}(\tau_{0},\infty):=\cup_{\tau\geq\tau_{0}}\Sigma(\tau)$.
Note that $\mathcal{R}(\tau_{0},\infty)=J^{+}(\Sigma_{0})$.

We shall denote

$\mathcal{S}(\tau_{0},\tau_{1}):=\mathcal{S}\cap\mathcal{R}(\tau_{0},\tau_{1}).$

####  2.5.3 The ingoing cones $\underline{C}_{v}$ and truncated regions

We also assume that the region $r\geq R$ is foliated by translation invariant
“ingoing” null cones $\underline{C}_{v}$ parameterised by a smooth function
$v$ defined on $r\geq R$, increasing towards the future, which may moreover be
expressed as $v(x^{0},r)$. In particular, the vector fields $\Omega_{i}$ are
tangent to $\underline{C}_{v}$. We again define a smooth future directed null
generator $\underline{L}$ of $\underline{C}_{v}$ normalised by the constraint
$g(\underline{L},T)\sim-1$, and translation invariant; this defines a smooth
vector field on $r\geq R$.

Let us assume moreover that $g(\underline{L},L)\sim-1$ globally in $r\geq R$.

Note that, under the above assumptions, $r$ is constant on
$\underline{C}_{v}\cap\Sigma(\tau)$, and the future boundary of
$\underline{C}_{v}$ is the set $\\{r=R\\}\cap\Sigma(\tau(v))$, where this
relation defined $\tau(v)$.

If $\tau_{0}\leq\tau_{1}\leq\tau(v)$, we shall denote

$\mathcal{R}(\tau_{0},\tau_{1},v):=\mathcal{R}(\tau_{0},\tau_{1})\setminus\bigcup_{\tilde{v}>v}\underline{C}_{\tilde{v}}$

and

$\Sigma(\tau,v):=\Sigma(\tau)\setminus\bigcup_{\tilde{v}>v}\underline{C}_{\tilde{v}}.$

The spacetime region $\mathcal{R}(\tau_{0},\tau_{1},v)$ is a compact subset of
spacetime with past boundary $\Sigma(\tau_{0},v)$ and future boundary
$\mathcal{S}(\tau_{0},\tau_{1})\cup\Sigma(\tau_{1},v)\cup\underline{C}_{v}$.

####  2.5.4 Comparison of volume forms

We will assume that in the region $r\geq R$, writing the volume form of
$(\mathcal{M},g)$ as

$dV_{\mathcal{M}}=\upsilon(r,\vartheta,\varphi)d\tau\,r^{2}dr\sin\vartheta
d\vartheta d\varphi$

and, for $\Sigma(\tau)\cap\\{r\geq R\\}$, with the choice $L$ for the normal,
as

$dV_{\Sigma(\tau)\cap\\{r\geq
R\\}}:=\tilde{\upsilon}(r,\vartheta,\varphi)r^{2}dr\sin\vartheta d\vartheta
d\varphi$

and for $\underline{C}_{v}$, with the choice $\underline{L}$ for the normal,
as

$dV_{\underline{C}_{v}}:=\tilde{\tilde{\upsilon}}(r,\vartheta,\varphi)r^{2}dr\sin\vartheta
d\vartheta d\varphi,$

then

$\upsilon\sim\tilde{\upsilon}\sim\tilde{\tilde{\upsilon}}\sim 1.$

With this assumption, it follows by the coarea formula and compactness that
globally, the volume form of $(\mathcal{M},g_{0})$ is related to the volume
form of $\Sigma(\tau)$

$dV_{\mathcal{M}}\sim d\tau\,dV_{\Sigma(\tau)},$

where $\sim$ is interpreted for $4$-forms in the obvious sense.

Note that when volume forms are omitted from integrals, the above induced
volume forms from the metric $g_{0}$ will be understood, unless otherwise
noted.

###  2.6 Other vector fields

It will be useful to extend the vector fields defined above to a global set of
vector fields which span the tangent space $T_{x}\mathcal{M}$ for all
$x\in\mathcal{M}$.

####  2.6.1 The global extensions of the vector fields $L$, $\underline{L}$
and $\Omega_{i}$

For notational convenience, in the case where $\mathcal{S}=\emptyset$, let us
define $\Omega_{4}=(1-\omega(r))\partial_{x^{1}}$,
$\Omega_{5}=(1-\omega(r))\partial_{x^{2}}$,
$\Omega_{6}=(1-\omega(r))\partial_{x^{3}}$, where $\omega$ is as in Section
2.1, and let us define $L$ and $\underline{L}$ to be translation invariant
extensions of the vector fields defined already with the property that $L$,
$\underline{L}$ and $\Omega_{1},\ldots,\Omega_{6}$ span the tangent space
globally. Note that this is easy to satisfy in general, and moreover we are
not requiring that $L$ and $\underline{L}$ be null, and the $\Omega_{i}$,
$i=1,\ldots,6$ are not required to be tangential to the ambient spheres for
$r\leq R/2$. (See already Section 2.7.1.)

In the case where $\mathcal{S}\neq\emptyset$ we will define $L$ and
$\underline{L}$ to be smooth $\phi_{\tau}$-invariant extensions of $L$ and
$\underline{L}$ to $r_{0}\leq r\leq R$, with the property that $L$,
$\underline{L}$ everywhere span the same plane spanned by the coordinate
vector fields $\partial_{x^{0}}$ and $\partial_{r}$ of ambient spherical
coordinates $(x^{0},r,\vartheta,\varphi)$. This again can easily seen to be
possible. Note again that we are not requiring these vector fields to be
globally null.

####  2.6.2 The notation $|\not{\nabla}\psi|^{2}$

We define the notation

$|\not{\nabla}\psi|^{2}:=\sum_{i}r^{-2}|\Omega_{i}\psi|^{2}.$

We note that under our assumptions, the above expression is comparable to the
induced gradient squared on the spacelike spheres
$\Sigma(\tau)\cap\\{r=r^{\prime}\\}$ in the region $r\geq R/2$.

In view, however, of our spanning assumptions, in both the case
$\mathcal{S}=\emptyset$ and $\mathcal{S}\neq\emptyset$, the expression

$|L\psi|^{2}+|\underline{L}\psi|^{2}+|\not{\nabla}\psi|^{2}$

will always be a translation invariant coercive expression on first
derivatives of $\psi$, similar (by compactness) to any other such coercive
expression in $r\leq R$, while in the region $r\geq R$ it will have the right
weights in $r$ so as to be a suitable reference point for natural energy
expressions.

####  2.6.3 The one-forms $\varrho^{L}$, $\varrho^{\underline{L}}$,
$\varrho^{1}$, $\varrho^{2}$ and $\varrho^{3}$

We will need to introduce a set of weighted spanning $1$-forms on the sphere
so as to properly measure weighted boundedness of forms. Define one forms
$\varrho^{L}$, $\varrho^{\underline{L}}$, and $\varrho^{1}$, $\varrho^{2}$,
$\varrho^{3}$ on the far region $\mathcal{M}\cap\\{r\geq R\\}$ as follows.

The region $\mathcal{M}\cap\\{r\geq R\\}$ can be written as the product
manifold $\mathcal{M}\cap\\{r\geq R\\}=\mathbb{R}\times[R,\infty)\times
S^{2}$. Let $\pi\colon\mathcal{M}\cap\\{r\geq
R\\}\to\mathbb{R}\times[R,\infty)$ denote the canonical projection. Let
$\varrho^{L}$, $\varrho^{\underline{L}}$ be defined by
$\varrho^{L}=\pi^{*}\tilde{\varrho}^{L}$ and
$\varrho^{\underline{L}}=\pi^{*}\tilde{\varrho}^{\underline{L}}$, where
$\tilde{\varrho}^{L},\tilde{\varrho}^{\underline{L}}$ is the dual coframe of
$\pi_{*}L,\pi_{*}\underline{L}$ on $\mathbb{R}\times[R,\infty)$. It follows
that

$\varrho^{L}(L)=\varrho^{\underline{L}}(\underline{L})=1,\qquad\varrho^{L}(\underline{L})=\varrho^{\underline{L}}(L)=\varrho^{L}(\Omega_{i})=\varrho^{\underline{L}}(\Omega_{i})=0,\qquad
i=1,2,3.$

For all smooth functions $\psi$, one can then write

$d\psi=L\psi\,\varrho^{L}+\underline{L}\psi\,\varrho^{\underline{L}}+\not{d}\psi,$

where $\not{d}\psi(L)=\not{d}\psi(\underline{L})=0$.

For each $i=1,2,3$, there is a polar coordinate system
$(\vartheta^{i},\varphi^{i})$ on $S^{2}$, associated to the corresponding
rotation vector field $\Omega_{i}$, with the property that, in the
$(x^{0},r,\vartheta^{i},\varphi^{i})$ coordinate system for
$\mathcal{M}\cap\\{r\geq R\\}$, one has $\partial_{\varphi^{i}}=\Omega_{i}$.
Such a $(\vartheta^{i},\varphi^{i})$ coordinate system on $S^{2}$ is unique up
to a choice of meridian. Define then, for $i=1,2,3$,

$\varrho^{1}=r\sin\vartheta^{1}d\varphi^{1},\qquad\varrho^{2}=r\sin\vartheta^{2}d\varphi^{2},\qquad\varrho^{3}=r\sin\vartheta^{3}d\varphi^{3}.$

There exist (non-unique, but universal) bounded functions
${a^{i}}_{j}(\vartheta,\varphi)$ for $i,j=1,2,3$, such that, for any smooth
function $\psi\colon\mathcal{M}\cap\\{r\geq R\\}\to\mathbb{R}$,

$d\psi=L\psi\,\varrho^{L}+\underline{L}\psi\,\varrho^{\underline{L}}+r^{-1}{a^{i}}_{j}(\vartheta,\varphi)\,\Omega_{i}\psi\,\varrho^{j}.$
(2.4)

####  2.6.4 The vector field $Y$

In the case $\mathcal{S}\neq\emptyset$, we may define $Y$ to be a
$\phi_{\tau}$ invariant vector field such that $Y$ is future directed null on
$\mathcal{H}^{+}$ and transversal to $\mathcal{H}^{+}$ and orthogonal to
$\mathcal{H}^{+}\cap\Sigma_{0}$, $Y$ is supported in $r\leq
r_{1}+(r_{2}-r_{1})/2$, for some $r_{2}>r_{1}>r_{\rm Killing}$, and such that
$Z$, $Y$, $\Omega_{i}$ span the tangent space in $r\leq
r_{1}+(r_{2}-r_{1})/4$.

The existence of such a vector field in a neighbourhood of a Killing horizon
follows from [DR13].

In the case where $\mathcal{S}=\emptyset$, since we are assuming that $T$ is
globally timelike, we may simply set $Y=0$.

###  2.7 Examples: Minkowski, Schwarzschild and Kerr

We note that Schwarzschild and Kerr in the full subextremal black hole range
of parameters $|a|<M$ satisfy the assumptions of this section with an
appropriate definition of the underlying differential structure.

####  2.7.1 Minkowski

For the Minkowski case, we consider the underlying manifold to be
$\mathbb{R}^{4}$, i.e. without boundary, and we define the metric to be the
familiar expression

$g_{0}=-(dx^{0})^{2}+(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}.$

We define $r_{0}:=1$ and this determines the function $r$. Let us distinguish
this from $\tilde{r}:=\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$ which we may
think of as a function $\tilde{r}(r)$. Fixing any $R\geq 20$, we note that
$u=t-\tilde{r}$, $v=t+\tilde{r}$ are null coordinates. We may define
$L=\partial_{u}$, $\underline{L}=\partial_{v}$, with respect to coordinates
$(u,v,\vartheta,\varphi)$, in the region $r\geq 2$.

We may take $\Sigma_{0}=(\\{t=0\\}\cap\\{r\leq
R\\})\cup(\\{v=\tilde{r}(R)\\}\cap\\{r\geq R\\})$ and $\underline{C}_{v}$
defined by the level sets of $v$.

Note that $r/\tilde{r}\sim 1$ for $r\geq r_{0}$ and that the spacetime volume
forms is $\tilde{r}^{2}\sin\vartheta d\tilde{r}d\tau d\vartheta d\varphi$,
while the volume form on $\Sigma_{0}\cap\\{r\geq R\\}$ and $\underline{C}_{v}$
is $\tilde{r}^{2}\sin\vartheta d\tilde{r}d\vartheta d\varphi$ with our choice
of $L$ and $\underline{L}$.

Note also in this case, we define $\Omega_{4}=\eta(r)\partial_{x^{1}}$,
$\Omega_{5}=\eta(r)\partial_{x^{2}}$ and $\Omega_{6}=\eta(r)\partial_{x^{3}}$
where $\eta$ is a cutoff vanishing with $\eta=1$ for $r\geq\frac{1}{4}$ and
$\eta=0$ for $r\geq\frac{1}{2}$, and we can extend $L$ and $\underline{L}$
simply by $L=(1-\eta)\partial_{u}+\eta\partial_{v}$,
$\underline{L}=(1-\eta)\partial_{v}$. (We may then define the cutoff $\omega$
of Section 2.1 so that $\omega=1$ for $r\geq\frac{1}{4}$ and $\omega=0$ for
$r\leq\frac{1}{8}$.) We have of course $Y=0$ in this case.

####  2.7.2 Schwarzschild

For the Schwarzschild case, given a real parameter $M>0$, we may define
$r_{0}:=(2-\delta_{1})M$ for sufficiently small $0<\delta_{1}\ll M$, denote
$x^{0}$ by $t^{*}$, let $r$ be defined by (1.1) and standard spherical
coordinates $(\vartheta,\varphi)$ on $\mathbb{R}^{3}$ by (2.3) and define the
metric $g_{M}$ to be

$-(1-2M/r)(dt^{*})^{2}+4M/rdrdt^{*}+(1+2M/r)dr^{2}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2})$

We define the vector fields $\Omega_{i}$ to be the standard Killing vector
fields associated to spherical symmetry and we define $T=\partial_{t^{*}}$ to
be the coordinate vector field with respect to the above coordinates.

Note that $r=2M$ is a Killing horizon with null generator $T$ and positive
surface gravity, and the hypersurfaces $r=r^{\prime}$ for
$r^{\prime}\in[r_{0},2M)$ are indeed spacelike.

We may define the function $t$ in the region $r>2M$ by $t=t^{*}-2M\log(r-2M)$.
We note that in the coordinates $(t,r,\theta,\phi)$, the metric takes the
familiar form

$-(1-2M/r)dt^{2}+(1-2M/r)^{-1}dr^{2}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}).$

We may define $r^{*}=r+2M\log(r-2M)$ and we may define coordinates $u$ and $v$
in $r>2M$ by $u=t-r^{*}$, $v=t+r^{*}$.

We may fix $R\geq 20r_{0}$ and define now $L=\partial_{v}$ and
$\underline{L}=\partial_{u}$ to be the coordinate vector fields with respect
to $(u,v,\theta,\phi)$ coordinates in $r\geq R$. We may extend these to be
globally defined linearly independent translation invariant vector fields in
the span of the coordinate vector fields $\partial_{t*}$ and $\partial_{r}$.

We then may define $\Sigma_{0}=(\\{t^{*}=0\\}\cap\\{r\leq
R\\})\cup(\\{v=R\\}\cap\\{r\geq R\\})$ and this satisfies all the required
transversality properties, etc.

Finally, we may define for instance $Y$ to be a translation invariant vector
field in the span of $\partial_{t*}$ and $\partial_{r}$ which is null and
future directed and satisfies $g(\partial_{t*},Y)=-2$ at $r=2M$, such that $Y$
is supported entirely in $r\leq r_{1}+(r_{2}-r_{1})/2$, with
$r_{1}:=(2+\delta_{1})M$.

####  2.7.3 Kerr

To put Kerr in our preferred form, one uses a combination of Kerr star
coordinates (based on Boyer–Lindquist coordinates) and double null
coordinates. We leave the details to the reader, but note the following.

Given subextremal parameters $|a|<M$, and defining
$r_{+}=M+\sqrt{M^{2}-a^{2}}$, we set $r_{0}=r_{+}-\delta_{1}$ for a small
$\delta_{1}$.

One may define the ambient differential structure so that the $r$ of (2.1)
will coincide with the Boyer–Lindquist $r$ of Appendix A in the region $r\leq
R/3$, while for $r\geq R/2$ it will coincide with the coordinate $r_{*}$ of
Appendix C.2. The coordinate $v$ coincides with the double null $v$ of
Appendix C.2, and $\Sigma(\tau)\cap\\{r\geq R\\}$ will be a hypersurface of
constant $u$, where again $u$ is as defined in Appendix C.2.

Further, one may set things up so that $\Omega_{1}=\partial_{\varphi}$ is the
axisymmetric Killing field.

Let us note finally that extremal Kerr (corresponding to $|a|=M$) cannot in
fact be put into our preferred form already because of our requirement that
$T$ be tangential to $\mathcal{S}$ and thus spacelike. Note also that the
Killing horizon of extremal Kerr has zero surface gravity, which would also
contradict our assumptions of Section 2.4.

###  2.8 Table of $r$-parameters

We collect finally a list of important $r$-values in increasing order. Some
will only be introduced later in the paper:

$r_{0}$ |  $r\geq r_{0}$; moreover if $\mathcal{S}\neq\emptyset$, then $\mathcal{S}=\\{r=r_{0}\\}$
---|---
$r_{\rm Killing}$ |  $\mathcal{H}^{+}=\\{r=r_{\rm Killing}\\}$ a Killing horizon with positive surface gravity;
|  span of $T$ and $\Omega_{1}$ is timelike for $r>r_{\rm Killing}$
$r_{1}$ |  parameter related to the vector field $Y$ if $\mathcal{S}\neq\emptyset$
$r_{1}+(r_{2}-r_{1})/4$ |  $Z$, $Y$, $\Omega_{i}$ span the tangent space for $r_{0}\leq r_{1}+(r_{2}-r_{1})/4$
$r_{1}+(r_{2}-r_{1})/2$ |  commutation vector fields $\mathfrak{D}$ all Killing for $r_{1}+(r_{2}-r_{1})/2\leq r\leq R/2$
$r_{2}$ |  $\rho=1$, $\chi=1$ for $r\leq r_{2}$
$R/6$ |  $T$ timelike for $r\geq R/6$
$R/4$ |  $\rho=1$, $\chi=1$ for $r\geq R/4$
$R/2$ |  commutation vector fields $\mathfrak{D}$ all Killing for $r_{1}+(r_{2}-r_{1})/2\leq r\leq R/2$ ;
|  $g=g_{0}$ for $r\geq R/2$
$8R/9$ |  the generalised null condition assumption concerns $r\geq 8R/9$
$R$ |  $\Sigma(\tau)\cap\\{r\geq R\\}$ is null, $\underline{C}_{v}\subset\\{r\geq R\\}$
$\tilde{R}$ | parameter related to positivity properties of far-away currents

The parameters $r_{\rm Killing}$, $r_{1}$, $r_{2}$ only occur if
$\mathcal{S}\neq\emptyset$.

##  3 Assumed identities and estimates for $\Box_{g_{0}}\psi=F$

Our fundamental assumptions in this paper are connected with the behaviour of
solutions of the linear inhomogeneous equation (1.2) on the exactly stationary
background $g_{0}$.

###  3.1 Constants and parameters

Before stating assumptions, we make a remark concerning constants and
parameters.

Given a spacetime $(\mathcal{M},g_{0})$ satisfying the assumptions of Section
2, we will consider the parameters of Section 2.8 as fixed. Let us also fix
once and for all a

$0<\delta<\frac{1}{10}.$ (3.1)

We will use $k$ to denote integers $\geq 0$ which will parametrise number of
derivatives.

In inequalities, we will denote by $C$ and $c$ generic positive constants
depending only on (a) $(\mathcal{M},g_{0})$ (with the choice of
$r$-parameters), (b) the above choice of $\delta$, and (c) if there is
$k$-dependence in the relevant statement, also on $k$. (We use $C$ for large
constants and $c$ for small constants.)

For nonnegative quantities $A$ and $B$, the notation

$A\lesssim B$

means $A\leq CB$, while

$A\sim B$

means $cB\leq A\leq CB$.

The reader should be aware to distinguish between $\leq$ and $\lesssim$, as
both will appear!

For a discussion of additional smallness parameters depending also on the
nonlinearity, see already Section 4.2.

###  3.2 Basic degenerate integrated local energy estimate

As discussed in the introduction, our basic assumption will be that of a
(degenerate) spacetime-localised integrated local energy decay statement for
the inhomogeneous equation (1.2).

The statement is that a certain energy flux on $\Sigma(\tau)$ plus a
nonnegative bulk are controlled by an initial energy flux on
$\Sigma(\tau_{0})$ and a spacetime integral over the slab
$\mathcal{R}(\tau_{0},\tau)$ relating to the inhomogeneous term $F$. The
controlled bulk integral is allowed to degenerate where a certain degeneration
function $\chi=\chi(r)$ vanishes, but is assumed to control a zeroth order
term without degeneration (with decaying weight at infinity).

Let us thus define

$\chi:[r_{0},\infty)\to[0,1]$ (3.2)

to be a function such that $\chi=1$ for $r\geq R/4$ and $r\leq r_{2}$.

The assumed estimate is:

$\displaystyle\sup_{v:\tau\leq\tau(v)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{F}}(v,\tau_{0},\tau),$
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{E}}(\tau)+c\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{E}}_{\mathcal{S}}(\tau)+c\int_{\tau_{0}}^{\tau}{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}+c\int_{\tau_{0}}^{\tau}\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau^{\prime})d\tau^{\prime}$
(3.3)
$\displaystyle\leq\lambda\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{E}}(\tau_{0})+C\int_{\mathcal{R}(\tau_{0},\tau)}\left|(V_{0}^{\mu}\partial_{\mu}\psi+w_{0}\psi)F\right|+C\int_{\mathcal{R}(\tau_{0},\tau)}F^{2},$

for some constants $\lambda\geq 1$, $C\geq 1$, $0<c<1$, and where $V_{0}$ is a
fixed vector field and $w_{0}$ a fixed function satisfying

$|g_{0}(V_{0},L)|\lesssim 1,\qquad|g_{0}(V_{0},\underline{L})|\lesssim
1,\qquad\sum|g_{0}(V_{0},\Omega_{i})|^{2}\lesssim r^{2},\qquad|w_{0}|\lesssim
r^{-1}.$ (3.4)

Here, the unprimed energies are defined by

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{E}}(\tau)$
$\displaystyle:=\int_{\Sigma(\tau)}|L\psi|^{2}+|\not{\nabla}\psi|^{2}+\iota_{r\leq
R}|\underline{L}\psi|^{2}+r^{-2}\psi^{2},$ (3.5)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{E}}_{\mathcal{S}}(\tau)$
$\displaystyle:=\int_{\mathcal{S}(\tau_{0},\tau)}|L\psi|^{2}+|\not{\nabla}\psi|^{2}+|\underline{L}\psi|^{2}+\psi^{2},$
(3.6)
$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\mathcal{F}}(\tau_{0},\tau,v)$
$\displaystyle:=\int_{\underline{C}_{v}\cap\mathcal{R}(\tau_{0},\tau)}|\underline{L}\psi|^{2}+|\not{\nabla}\psi|^{2}+r^{-2}\psi^{2},$
(3.7)

while the primed energies are defined by

$\displaystyle{}^{\chi}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau)$
$\displaystyle:=\int_{\Sigma(\tau)}r^{-1-\delta}\chi(r)(|L\psi|^{2}+|\underline{L}\psi|^{2}+|\not{\nabla}\psi|^{2}),$
$\displaystyle\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}(\tau)$
$\displaystyle:=\int_{\Sigma(\tau)}r^{-3-\delta}\psi^{2}.$ (3.8)

(The ′ notation will be used in general to denote energies that naturally
appear in bulk terms.) The estimate (3.3) is to hold for all smooth $\psi$
such that the right hand side of (3.3) is finite. In the above, we already see
the $\delta$ fixed in (3.1). For future reference let us also define the
quantity

$\displaystyle\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}(\tau)$
$\displaystyle:=\int_{\Sigma(\tau)}r^{-1-\delta}(|L\psi|^{2}+|\underline{L}\psi|^{2}+|\not{\nabla}\psi|^{2})+r^{-3-\delta}\psi^{2}.$
(3.9)

Note that if $\chi=1$ identically, then
$\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}+\underaccent{\scalebox{0.8}{\mbox{\tiny$-\mkern-3.0mu1$}}}{\mathcal{E}}^{\prime}=\accentset{\scalebox{0.6}{\mbox{\tiny$(-1\mkern-6.0mu-\mkern-6.0mu\delta)$}}}{\mathcal{E}}^{\prime}$.
We note that all expressions defined above are $T$-invariant.

We distinguish the constants $C$ and $\lambda$ in (3.3) to highlight the
significance of the case $\lambda=1$, when (3.3) is derived from a suitable
energy identity and the energies are replaced by exact fluxes. See already
Section 3.4.1.

Again, the assumptions are motivated by our model cases of Minkowski space,
Schwarzschild and sub-extremal Kerr black hole exteriors. Note that (3.3)
indeed holds in the case of Kerr in the full subextremal range $|a|<M$
[DRSR16]. In fact, in all of the above cases more precise estimates than (3.3)
are available with respect to what can actually be controlled by the right
hand side; we shall not need to make use of this here.

######  Remark 3.2.1.

We in fact allow more generally on the right hand side of (3.3) a finite sum
of terms of the form
$\left|(V_{0}^{\mu}\partial_{\mu}\psi+w_{0}\psi)F\right|$, each with distinct
$V_{0}$ and $w_{0}$, but where all $V_{0}$ and $w_{0}$ satisfy (3.4). For
notational convenience, we will always write a unique $V_{0}$, $w_{0}$,
however. The inclusion of the term $\int F^{2}$ on the right hand side of
(3.3) is due to the fact that the estimate, as in [DRSR16], may be proven
applying cutoff functions to $\psi$, which result in such a term. In cases (i)
and (ii) we shall see that this (3.3) holds without having to include such a
term. In case (iii), in practice, we can often restrict this term to be
supported only in $r\leq R$.

###  3.3 Physical space identities on a general Lorentzian metric

As explained in the introduction, for nonlinear applications to (1.1), it is
most convenient when (3.3) is the result of a physical space energy identity
(1.4) for solutions $\psi$ of (1.2). Thus, we will recall some general
properties of such identities here.

####  3.3.1 Definition of energy currents

Physical space energy identities can be associated to a quadruple
$(V,w,q,\varpi)$ where $V^{\mu}$ is a vector field on $\mathcal{M}$, $w$ is a
scalar function, $q_{\mu}$ is a one-form and $\varpi_{\mu\nu}$ is a two-form.
Given such a quadruple, a general Lorentzian metric $g$ and a suitably regular
function $\psi$, we define

$\displaystyle J^{V,w,q,\varpi}_{\mu}[g,\psi]$
$\displaystyle:=T_{\mu\nu}[g,\psi]V^{\nu}+w\psi\partial_{\mu}\psi+q_{\mu}\psi^{2}+*d\big{(}\psi^{2}\varpi\big{)}_{\mu}$
(3.10) $\displaystyle K^{V,w,q}[g,\psi]$
$\displaystyle:=\pi^{V}_{\mu\nu}[g]T^{\mu\nu}[g,\psi]+\nabla^{\mu}w\psi\partial_{\mu}\psi+w\nabla^{\mu}\psi\partial_{\mu}\psi+\nabla^{\mu}q_{\mu}\psi^{2}+2\psi
q_{\mu}g^{\mu\nu}\partial_{\nu}\psi$ (3.11) $\displaystyle H^{V,w}[\psi]$
$\displaystyle:=V^{\mu}\partial_{\mu}\psi+w\psi,$ (3.12)

where

$T_{\mu\nu}[g,\psi]=\partial_{\mu}\psi\partial_{\nu}\psi-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi,\qquad\pi^{X}_{\mu\nu}[g]=\frac{1}{2}(\mathcal{L}_{X}g)_{\mu\nu}=\frac{1}{2}(\nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu}),$

and where $*\colon\Lambda^{3}\mathcal{M}\to\Lambda^{1}\mathcal{M}$ denotes the
Hodge star operator.

Let us note already that it is often more natural to parametrise choices of
currents in a slightly different way, as twisted currents [HW14], for
instance, given a scalar function, one may define a twisted energy momentum
tensor $\tilde{T}_{\mu\nu}[g,\psi]$ according to (A.8) and then consider for
instance currents of the form
$\tilde{J}^{V}_{\mu}[g,\psi]:=\tilde{T}_{\mu\nu}[g,\psi]V^{\nu}$, etc. Such a
current can always be rewritten as (3.10) for some $w$, $q$, $\varpi$.

####  3.3.2 The divergence identity

With the above definitions, one can compute the following identity for a
general function $\psi$:

$\nabla_{g}^{\mu}J^{V,w,q,\varpi}_{\mu}[g,\psi]=K^{V,w,q}[g,\psi]+H^{V,w}[\psi]\Box_{g}\psi.$
(3.13)

In particular, for solutions of the covariant wave equation $\Box_{g}\psi=0$,
one obtains a divergence relation between the currents $J$ and $K$ which both
depend only on the $1$-jet of $\psi$. See [Chr00] for a discussion of the
classification of currents with this property. Note that $\varpi$ does not
contribute to the bulk current $K$, and neither $\varpi$ nor $q$ contribute to
$H$, which moreover is independent of the metric $g$.

The significance of identity (3.13) is that it can be integrated in a
spacetime region bounded by homologous hypersurfaces to obtain a relation
between boundary fluxes of $J$ and a bulk integral of $K$. We give the form of
this relation below in the special case of the region
$\mathcal{R}(\tau_{0},\tau_{1},v)\subset\mathcal{M}$.

####  3.3.3 The integrated identity on $\mathcal{R}(\tau_{0},\tau_{1},v)$

For a solution of (1.2), identity (3.13) upon integration in
$\mathcal{R}(\tau_{0},\tau_{1},v)$ yields

$\displaystyle\int_{\Sigma(\tau_{1},v)}J[\psi]\cdot
n+\int_{\mathcal{S}(\tau_{0},\tau_{1})}J[\psi]\cdot
n+\int_{\underline{C}_{v}(\tau_{0},\tau_{1})}J[\psi]\cdot
n+\int_{\mathcal{R}(\tau_{0},\tau_{1},v)}K[\psi]$
$\displaystyle=\int_{\Sigma(\tau_{0},v)}J[\psi]\cdot
n-\int_{\mathcal{R}(\tau_{0},\tau_{1},v)}H[\psi]F,$ (3.14)

where the normals and volume forms are with respect to the metric $g_{0}$
according to the convention of Section 2.5.4.

Identity (3.14) will be useful when it satisfies suitable positivity
properties for its bulk and boundary terms.

###  3.4 Assumed unweighted first order physical space identities: cases
(i)–(iii)

As discussed already in the introduction, we shall not in general require that
(3.3) is the result of an integrated divergence identity (3.14) associated to
currents with a pointwise coercivity property. We shall, however, require, _in
addition_ to assuming (3.3), the existence of currents generating an identity
(3.14) with much weaker non-negativity properties, properties which in
particular are insensitive to the presence and structure of possible trapping.

It is indeed useful to see first, however, how the existence of a physical
space proof of (3.3) via an identity of type (3.14) simplifies the
considerations. Thus, we shall distinguish two simpler cases, to be called
case (i) and (ii), to be discussed in Sections 3.4.1 and 3.4.2 below, where
indeed (3.3) is proven via an identity (3.14), (with case (i) corresponding to
the even simpler setting where there is no degeneration at all in the
coercivity).

The most general case, case (iii), which represents the main goal of this
paper, will be discussed in Section 3.4.3. We will introduce some helpful
common notation in Section 3.4.4.

Finally, in Section 3.4.5, for the case $\mathcal{S}\neq\emptyset$, we will
provide a family of currents with enhanced red-shift control at
$\mathcal{H}^{+}$ and in the black hole interior, parametrised by a parameter
$\varsigma$, which will be useful for obtaining higher order estimates in
Section 3.6.

####  3.4.1 Case (i)

The simplest case to consider is when (3.3) indeed follows from a physical
space energy identity (3.14), and when moreover, there is in fact no
degeneration in the estimate (3.3), i.e. the function $\chi$ of (3.2)
satisfies $\chi=1$ identically.

That is to say, we assume that there exists a $T$-invariant quadruple
$(V,w,q,\varpi)$, where $V$ is a vector field, $w$ is a scalar function, $q$
is a one form and $\varpi$ is a two form, satisfying

$\displaystyle|g_{0}(V,L)|\lesssim 1,\qquad|g_{0}(V,\underline{L})|\lesssim
1,\qquad\sum|g_{0}(V,\Omega_{i})|^{2}\lesssim 1,$ (3.15)
$\displaystyle|w|\lesssim r^{-1},\qquad|q_{\mu}L^{\mu}|\lesssim
r^{-2},\qquad|q_{\mu}\underline{L}^{\mu}|\lesssim r^{-2},$
$\displaystyle|(*\,d\varpi)_{\mu}L^{\mu}|\lesssim
r^{-2},\qquad|(*\,d\varpi)_{\mu}\underline{L}^{\mu}|\lesssim r^{-2},$
$\displaystyle|*(\varrho^{L}\wedge\varpi)_{\mu}L^{\mu}|\lesssim
r^{-1},\quad|*(\varrho^{L}\wedge\varpi)_{\mu}\underline{L}^{\mu}|\lesssim
r^{-1},$
$\displaystyle|*(\varrho^{\underline{L}}\wedge\varpi)_{\mu}L^{\mu}|\lesssim
r^{-1},\quad|*(\varrho^{\underline{L}}\wedge\varpi)_{\mu}\underline{L}^{\mu}|\lesssim
r^{-1},$ $\displaystyle|*(\varrho^{i}\wedge\varpi)_{\mu}L^{\mu}|\lesssim
r^{-1},\quad|*(\varrho^{i}\wedge\varpi)_{\mu}\underline{L}^{\mu}|\lesssim
r^{-1},\qquad i=1,2,3,$

such that, defining $J^{V,w,q,\varpi}[\psi]$, $K^{V,w,q}$ by (3.10)–(3.11),
the energy identity (3.14) corresponding to these currents has the following
properties: (a) the boundary terms of (3.14) on $\Sigma(\tau)$ are coercive,
(b) the remaining boundary terms nonnegative, and (c) the bulk term $K$ is
nonnegative and coercive, with no degeneration, except “at infinity”, where
standard derivatives are in general only controlled with weight
$r^{-1-\delta}$.

More precisely, we assume the following pointwise bulk coercivity relation

$\displaystyle K^{V,w,q}[\psi]$ $\displaystyle\gtrsim
r^{-1-\delta}\left((L\psi)^{2}+(\underline{L}\psi)^{2}+|\not{\nabla}\psi|^{2}\right)+r^{-3-\delta}\psi^{2},$
(3.16)

and pointwise boundary coercivity relations:

$\displaystyle\qquad J^{V,w,q,\varpi}_{\mu}[\psi]n_{\Sigma(\tau)}^{\mu}$
$\displaystyle\gtrsim(L\psi)^{2}+|\not{\nabla}\psi|^{2}+\iota_{r\leq
R}(\underline{L}\psi)^{2}+r^{-2}\psi^{2},$ (3.17) $\displaystyle
J^{V,w,q,\varpi}_{\mu}[\psi]n_{\underline{C}_{v}}^{\mu}$