#  Pointwise decay for semilinear wave equations on Kerr spacetimes

Mihai Tohaneanu  Department of Mathematics, University of Kentucky, Lexington,
KY 40506<EMAIL_ADDRESS>
###### Abstract.

In this article we prove pointwise bounds for solutions to the semilinear wave
equation with integer powers $p\geq 3$ on Kerr backgrounds with small angular
momentum and small initial data. We expect that the bounds proved in this
paper are optimal.

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of the result
    2. 1.2 History
    3. 1.3 Acknowledgments
  2. 2 The Kerr metric and local energy norms
    1. 2.1 The Kerr metric
    2. 2.2 Local energy norms
  3. 3 Local energy decay for the nonlinear problem
  4. 4 $r^{p}$ estimates
  5. 5 Setup for pointwise estimates
  6. 6 Estimates for the fundamental solution
  7. 7 An improved bound
  8. 8 Improved estimates in the interior and for derivatives
  9. 9 The bootstrap argument

##  1\. Introduction

###  1.1. Statement of the result

In this paper we establish pointwise bounds for solutions to the semilinear
wave equations on Kerr backgrounds. We consider the equation

(1.1)
$\Box_{K}\phi=\pm\phi^{p},\qquad\phi|_{{\tilde{t}}=0}=\phi_{0},\qquad\tilde{T}\phi|_{{\tilde{t}}=0}=\phi_{1}\
.$

Here the coordinate ${\tilde{t}}$ is chosen so that the slice ${\tilde{t}}=0$
is space-like and so that ${\tilde{t}}=t$ away from the black hole, and
$\tilde{T}$ is the future unit normal to ${\tilde{t}}=0$ (see Section 2 for
details). Moreover, $p\geq 3$ is any integer, $\Box_{K}$ denotes the
d’Alembertian in the Kerr metric, and the initial data are smooth and
supported in $\\{|x|\leq R_{1}\\}$ for some fixed (but arbitrary) $R_{1}$. We
also assume that, for some fixed $N\gg 1$, we have that

$\|\phi_{0}\|_{H^{N+1}}+\|\phi_{1}\|_{H^{N}}\leq\varepsilon.$

Let $\kappa=\min\\{2,p-2\\}$. Let ${\tilde{r}}$ equal to $r$ on a compact
region and to the Regge-Wheeler coordinate $r^{*}$ near infinity (see Section
2 for details). We will show that, for small enough $\varepsilon$,

(1.2)
$\phi\lesssim\frac{\varepsilon}{\langle{\tilde{t}}\rangle\langle{\tilde{t}}-{\tilde{r}}\rangle^{\kappa}},$

where $\langle x\rangle=\sqrt{2+|x|^{2}}$.

In future work, we will also show that this rate of decay is sharp.

###  1.2. History

The semilinear wave equation in ${\mathbb{R}}^{1+3}$

$\Box\phi=\pm\phi^{p},\qquad\phi|_{t=0}=\phi_{0},\qquad{\partial}_{t}\phi|_{t=0}=\phi_{1}$

has been studied extensively. There are many papers concerned with global
existence, uniqueness, and scattering, see for example [16], [33], [30], [31],
[5], [11]. In particular, it is well-known that for small initial data there
is a unique global solution if $p>1+\sqrt{2}$, see [15], [13], [37].

In terms of pointwise decay of solutions, there are a number of results, see
for example [29], [33], [40]. In the case of compactly supported smooth data,
the optimal decay rate is

$\phi\lesssim\frac{1}{\langle t\rangle\langle t-r\rangle^{p-2}}\ .$

This was shown for small initial data in [35] and for large data in the
defocusing case in [11] (when $p=5$) and [6] (when $3\leq p<5$).

There is also a vast literature concerned with establishing pointwise decay
estimates for solutions to the linear wave equation $\Box_{g}\phi=0$ for
various Lorentzian metrics $g$. In the case of the Schwarzschild metric, the
solution to the wave equation was conjectured to decay at the rate of
$\tilde{t}^{-3}$ on a compact region by Price [28], and this rate of decay was
shown to hold for a variety of spacetimes, including Schwarzschild and Kerr
with $|a|<M$, see [10], [38], [23], [21], [14], [3]. It is by now well
understood that once local energy estimates is established in a compact region
on an asymptotically flat region, one can obtain pointwise decay rates that
are related to how fast the metric coefficients decay to the Minkowski metric;
see, for example, [38], [23], [27], [26], [1], [2], [24], [25], [19].

Much less is known of the behavior of solutions of the semilinear wave
equations with power nonlinearity on Schwarzschild and Kerr backgrounds. For
small initial data, global existence and pointwise decay rates of $t^{-1}$
were shown for spherically symmetric solutions in Schwarzschild in the case
$p>4$ in [8]; a similar result was shown the case $p>3$ (without the
spherically symmetric assumption) in [7]. For Kerr with small angular momentum
and $p>1+\sqrt{2}$, global existence was shown in [17].

The goal of this paper is to establish sharp decay rates for solutions to
(1.1) with small initial data. For simplicity, we also pick the initial data
to be compactly supported, though one can do away with it assuming enough
decay in weighted Sobolev spaces. The ideas of the proof also apply to other
metrics and nonlinearities; in particular, see the upcoming result of Looi
[20], where a similar result for the quintic defocusing nonlinearity on
perturbations of Minkowski (and large initial data) is shown.

The paper is structured as follows. In Section 2 we introduce the Kerr metric,
our preferred coordinates, local energy estimates, and our main theorem.
Section 3 is dedicated to proving local energy estimates for our nonlinear
problem. These estimates imply an initial pointwise decay rate of
$\frac{(t-r)^{1/2}}{t}$, which is insufficient as a starting point for $p\leq
5$. Section 4 supplements this with $r^{p}$ estimates, which give better
weighted $L^{2}$ estimates. Section 5 rephrases the problem in a more
convenient way. Section 6 contains the proof of the main lemma used to improve
the pointwise decay. Section 7 uses the results of the previous sections to
yield an initial pointwise decay of $\frac{1}{r(t-r)^{\delta}}$. Section 8
obtains improved decay rates for the solution in the interior, and for
derivatives. Section 9 is the main bootstrap argument: starting from the decay
estimate of Section 7, we use the results of Sections 6 and Section 8 to
improve the decay rate to the optimal one.

###  1.3. Acknowledgments

The author would like to thank Hans Lindblad, Sung-Jin Oh and Shi-Zhuo Looi
for many useful conversations regarding the paper, and the Korea Institute for
Advanced Study for their hospitality during the spring of 2019. The author was
partly supported by the Simons collaboration Grant 586051.

##  2\. The Kerr metric and local energy norms

###  2.1. The Kerr metric

The Kerr geometry in Boyer-Lindquist coordinates is given by

$ds^{2}=g^{K}_{tt}dt^{2}+g_{t\phi}dtd\phi+g^{K}_{rr}dr^{2}+g^{K}_{\phi\phi}d\phi^{2},+g^{K}_{\theta\theta}d\theta^{2}$

where $t\in{\mathbb{R}}$, $r>0$, $(\phi,\theta)$ are the spherical coordinates
on ${\mathbb{S}}^{2}$ and

$g^{K}_{tt}=-\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}},\qquad
g^{K}_{t\phi}=-2a\frac{2Mr\sin^{2}\theta}{\rho^{2}},\qquad
g^{K}_{rr}=\frac{\rho^{2}}{\Delta},$
$g^{K}_{\phi\phi}=\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}}\sin^{2}\theta,\qquad
g^{K}_{\theta\theta}={\rho^{2}},$

with

$\Delta=r^{2}-2Mr+a^{2},\qquad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta.$

Here $M$ represents the mass of the black hole, and $aM$ its angular momentum.

A straightforward computation gives us the inverse of the metric:

$g_{K}^{tt}=-\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta}{\rho^{2}\Delta},\qquad
g_{K}^{t\phi}=-a\frac{2Mr}{\rho^{2}\Delta},\qquad
g_{K}^{rr}=\frac{\Delta}{\rho^{2}},$
$g_{K}^{\phi\phi}=\frac{\Delta-a^{2}\sin^{2}\theta}{\rho^{2}\Delta\sin^{2}\theta},\qquad
g_{K}^{\theta\theta}=\frac{1}{\rho^{2}}.$

The case $a=0$ corresponds to the Schwarzschild space-time. We shall
subsequently assume that $a$ is small $0<a\ll M$, so that the Kerr metric is a
small perturbation of the Schwarzschild metric. Note also that the
coefficients depend only $r$ and $\theta$ but are independent of $\phi$ and
$t$.

We denote the Kerr metric by $g_{K}$, and the Schwarzschild metric by $g_{S}$.
Let $\Box_{K}$ and $\Box_{S}$ be the d’Alembertian associated to the Kerr and
Schwarzschild metric, respectively. Similarly $dV_{K}$ and $dV_{S}$ are the
volume forms, and $d\Sigma_{K}$ and $d\Sigma_{S}$ are the restrictions of the
volume form to a hypersurface.

In the above coordinates the Kerr metric has singularities at $r=0$, on the
equator $\theta=\pi/2$, and at the roots of $\Delta$, namely
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$. To remove the singularities at $r=r_{\pm}$
we introduce functions $r_{K}^{*}=r_{K}^{*}(r)$, $v_{+}=t+r_{K}^{*}$ and
$\phi_{+}=\phi_{+}(\phi,r)$ so that (see [12])

$dr_{K}^{*}=(r^{2}+a^{2})\Delta^{-1}dr,\qquad dv_{+}=dt+dr_{K}^{*},\qquad
d\phi_{+}=d\phi+a\Delta^{-1}dr.$

Note that when $a=0$ the $r_{K}^{*}$ coordinate becomes the Schwarzschild
Regge-Wheeler coordinate

$r^{*}=r+2M\log(r-2M).$

The Kerr metric can be written in the new coordinates
$(v_{+},r,\phi_{+},\theta)$

$\begin{split}ds^{2}=&\
-(1-\frac{2Mr}{\rho^{2}})dv_{+}^{2}+2drdv_{+}-4a\rho^{-2}Mr\sin^{2}\theta
dv_{+}d\phi_{+}-2a\sin^{2}\theta drd\phi_{+}+\rho^{2}d\theta^{2}\\\ &\
+\rho^{-2}[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta]\sin^{2}\theta\,d\phi_{+}^{2},\end{split}$

which is smooth and nondegenerate across the event horizon up to but not
including $r=0$. We introduce the function

$\tilde{t}=v_{+}-\mu(r),$

where $\mu$ is a smooth function of $r$. In the
$(\tilde{t},r,\phi_{+},\theta)$ coordinates the metric has the form

$\begin{split}ds^{2}=&\
(1-\frac{2Mr}{\rho^{2}})d\tilde{t}^{2}+2\left(1-(1-\frac{2Mr}{\rho^{2}})\mu^{\prime}(r)\right)d\tilde{t}dr\\\
&\ -4a\rho^{-2}Mr\sin^{2}\theta
d\tilde{t}d\phi_{+}+\Bigl{(}2\mu^{\prime}(r)-(1-\frac{2Mr}{\rho^{2}})(\mu^{\prime}(r))^{2}\Bigr{)}dr^{2}\\\
&\ -2a(1+2\rho^{-2}Mr\mu^{\prime}(r))\sin^{2}\theta
drd\phi_{+}+\rho^{2}d\theta^{2}\\\ &\ +\rho^{-2}[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta]\sin^{2}\theta d\phi_{+}^{2}.\end{split}$

On the function $\mu$ we impose the following two conditions:

(i) $\mu(r)\geq r^{*}$ for $r>2M$, with equality for $r>{5M}/2$.

(ii) The surfaces $\tilde{t}=const$ are space-like, i.e.

$\mu^{\prime}(r)>0,\qquad 2-(1-\frac{2Mr}{\rho^{2}})\mu^{\prime}(r)>0.$

As long as $a$ is small, we can use the same function $\mu$ as in the case of
the Schwarzschild space-time in [22].

Let $\tilde{r}$ denote a smooth strictly increasing function (of $r$) that
equals $r$ for $r\leq R$ and $r^{*}$ for $r\geq 2R$ for some large $R$. We
will use the coordinates $(\tilde{t},x^{i})$, where $x^{i}={\tilde{r}}\omega$.
We use Latin indices $i,j=1,2,3$ for spatial summation and Greek indices
$\alpha,\beta=0,1,2,3$ for space-time summation. By $\langle r\rangle$ we
denote a smooth radial function which agrees with $r$ for large $r$ and
satisfies $\langle r\rangle\geq 1$. Note that, since $r\approx{\tilde{r}}$, we
can use $r^{k}$ and ${\tilde{r}}^{k}$ interchangeably when defining our spaces
of functions.

We fix $r_{e}$ satisfying $r_{-}<r_{e}<r_{+}$. The choice of $r_{e}$ is
unimportant, and for convenience we may simply use $r_{e}=M$ for all Kerr
metrics with $a/M\ll 1$. Let $\mathcal{M}=\\{\tilde{t}\geq 0,\ r\geq
r_{e}\\}$, $\Sigma(T)=\mathcal{M}\cap\\{\tilde{t}=T\\}$, and
$\Sigma^{-}=\Sigma(0)$.

A major difficulty in understanding dispersion properties for solutions to the
linear wave equation on $\mathcal{M}$ is the presence of trapped null
geodesics (i.e. null geodesics which do not escape either to infinity or to
the singularity). One family of such geodesics occurs at the event horizon;
however, due to the red-shift effect, the energy decays exponentially (in the
high-frequency approximation) along such rays. A second family of such
geodesics occurs in the compact region $|r-3M|\lesssim|a|$. Here the energy
does not decay exponentially, but due to the hyperbolic nature of the
trapping, it disperses after a time proportional to the logarithm of the
frequency, and thus space-time estimates can still be recovered, albeit with a
loss of derivatives.

Our favorite sets of vector fields will be

$\partial=\\{\partial_{\tilde{t}},\partial_{i}\\},\qquad\Omega=\\{x^{i}\partial_{j}-x^{j}\partial_{i}\\},\qquad
S=\tilde{t}\partial_{\tilde{t}}+\tilde{r}\partial_{\tilde{r}},$

namely the generators of translations, rotations and scaling. We set
$Z=\\{\partial,\Omega,S\\}$.

For a triplet $\alpha=(i,j,k)$ we define $|\alpha|=i+3j+9k$ and

$u_{\alpha}={\partial}^{i}\Omega^{j}S^{k}.$

The numerology is borrowed from [23], and takes into account the loss of
derivatives that occurs when applying weak local energy estimates to vector
fields.

We define the classes $S^{Z}(r^{k})$ of functions in
${\mathbb{R}}^{+}\times{\mathbb{R}}^{3}$ by

$f\in S^{Z}(r^{k})\Longleftrightarrow|Z^{j}f(t,x)|\leq c_{j}\langle
r\rangle^{k},\quad j\geq 0.$

By $S^{Z}_{rad}(r^{k})$ we denote spherically symmetric functions in
$S^{Z}(r^{k})$, and by $S^{Z}_{der}(r^{k})$ the space of functions so that

$f\in S^{Z}(r^{k}),\quad{\partial}f\in S^{Z}(r^{k-1}).$

Given a vector $g$, we will also use the notation

$f\in S^{Z}(r^{k})g$

to mean that

$f=\sum h_{i}g_{i},\quad h_{i}\in S^{Z}(r^{k}),$

and similarly for $S^{Z}_{rad}(r^{k})$, $S^{Z}_{der}(r^{k})$.

In particular, a quick computation yields that

(2.1) $g_{K}^{\alpha\beta}-g_{S}^{\alpha\beta}\in S^{Z}_{der}(r^{-2}).$

Note that due to (2.1) we also have

(2.2) $\Box_{g_{K}}u-\Box_{g_{S}}u\lesssim
r^{-2}(|{\partial}^{2}u|+|{\partial}u|).$

The main theorem of the paper is the following:

######  Theorem 2.1.

Let $p\geq 3$ be an integer. Assume that $\phi$ solves the wave equation

(2.3)
$\Box_{K}\phi=\pm\phi^{p},\qquad\phi|_{\Sigma^{-}}=\phi_{0},\qquad\tilde{T}\phi|_{\Sigma^{-}}=\phi_{1}\
.$

Let $\kappa=\min\\{2,p-2\\}$. Fix $m\in{\mathbb{N}}$ and $R_{1}>r_{e}$. Then
there are $N\gg m$ and $\varepsilon>0$ so that, for any initial data
$(\phi_{0},\phi_{1})$ supported in $\\{r_{e}\leq r\leq R_{1}\\}$ with

$\|\phi_{0}\|_{H^{N+1}}+\|\phi_{1}\|_{H^{N}}\leq\varepsilon,$

then $\phi$ exists globally in $\mathcal{M}$ and satisfies the pointwise
bounds

(2.4) $\phi_{\leq
m}\lesssim\frac{\varepsilon}{\langle{\tilde{t}}\rangle\langle{\tilde{t}}-{\tilde{r}}\rangle^{\kappa}}\
.$

###  2.2. Local energy norms

We consider a partition of ${\mathbb{R}}^{3}$ into the dyadic sets
$A_{R}=\\{R\leq\langle r\rangle\leq 2R\\}$ for $R\geq 1$. We will use the
notation $A\lesssim B$ to mean that there is a constant $C$ independent of $u$
and $\varepsilon$ so that $|A|\leq CB$; the value of $C$ might change from
line to line. Similarly, $A\ll B$ means that $|A|\leq cB$ for a small enough
constant $c$.

We now introduce the local energy norm $LE$

(2.5) $\begin{split}\|u\|_{LE}&=\sup_{R}\|\langle
r\rangle^{-\frac{1}{2}}u\|_{L^{2}(\mathcal{M}\cap{\mathbb{R}}\times A_{R})}\\\
\|u\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]}&=\sup_{R}\|\langle
r\rangle^{-\frac{1}{2}}u\|_{L^{2}(\mathcal{M}\cap[\tilde{t}_{0},\tilde{t}_{1}]\times
A_{R})},\end{split}$

its $H^{1}$ counterpart

(2.6) $\begin{split}\|u\|_{LE^{1}}&=\|\nabla u\|_{LE}+\|\langle
r\rangle^{-1}u\|_{LE}\\\ \|u\|_{LE^{1}[\tilde{t}_{0},\tilde{t}_{1}]}&=\|\nabla
u\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]}+\|\langle
r\rangle^{-1}u\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]},\end{split}$

as well as the dual norm

(2.7) $\begin{split}\|f\|_{LE^{*}}&=\sum_{R}\|\langle
r\rangle^{\frac{1}{2}}f\|_{L^{2}(\mathcal{M}\cap{\mathbb{R}}\times A_{R})}\\\
\|f\|_{LE^{*}[\tilde{t}_{0},\tilde{t}_{1}]}&=\sum_{R}\|\langle
r\rangle^{\frac{1}{2}}f\|_{L^{2}(\mathcal{M}\cap[\tilde{t}_{0},\tilde{t}_{1}]\times
A_{R})}.\end{split}$

We also define similar norms for higher Sobolev regularity

$\begin{split}\|u_{\leq m}\|_{LE^{1}}&=\sum_{|\alpha|\leq
m}\|u_{\alpha}\|_{LE^{1}}\\\ \|u_{\leq
m}\|_{LE^{1}[\tilde{t}_{0},\tilde{t}_{1}]}&=\sum_{|\alpha|\leq
m}\|u_{\alpha}\|_{LE^{1}[\tilde{t}_{0},\tilde{t}_{1}]}\\\ \|u_{\leq
m}\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]}&=\sum_{|\alpha|\leq
m}\|u_{\alpha}\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]},\end{split}$

respectively

$\begin{split}\|f\|_{LE^{*,k}}&=\sum_{|\alpha|\leq
k}\|\partial^{\alpha}f\|_{LE^{*}}\\\
\|f\|_{LE^{*,k}[\tilde{t}_{0},\tilde{t}_{1}]}&=\sum_{|\alpha|\leq
k}\|\partial^{\alpha}f\|_{LE^{*}[\tilde{t}_{0},\tilde{t}_{1}]}.\end{split}$

Finally, we introduce a weaker version of the local energy decay norm 111In
Kerr one can actually control a stronger norm, where the $r$-derivative does
not degenerate at the trapped set. However, we do not need the stronger norm
in this paper.

$\begin{split}\|u\|_{LE^{1}_{w}}&=\|(1-\chi_{ps})\nabla u\|_{LE}+\|\langle
r\rangle^{-1}u\|_{LE}\\\
\|u\|_{LE^{1}_{w}[\tilde{t}_{0},\tilde{t}_{1}]}&=\|(1-\chi_{ps})\nabla
u\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]}+\|\langle
r\rangle^{-1}u\|_{LE[\tilde{t}_{0},\tilde{t}_{1}]},\end{split}$

as well as the dual norms

$\begin{split}\|f\|_{LE^{*}_{w}}&=\|\chi_{ps}\nabla
f\|_{L^{2}L^{2}}+\|f\|_{LE^{*}}\\\
\|f\|_{LE^{*}_{w}[\tilde{t}_{0},\tilde{t}_{1}]}&=\|\chi_{ps}\nabla
f\|_{L^{2}[\tilde{t}_{0},\tilde{t}_{1}]L^{2}}+\|f\|_{LE^{*}[\tilde{t}_{0},\tilde{t}_{1}]}.\end{split}$

Here $\chi_{ps}$ is a smooth, compactly supported spatial cutoff function that
equals $1$ in a neighborhood of the trapped set. We also define the higher
order weak norms as above.

We define the (nondegenerate) energy

$E[u](\tilde{t})=\int_{\Sigma(\tilde{t})}|{\partial}u|^{2}d\Sigma_{K}(\tilde{t}).$

We will need the following local energy estimates, which were proved in [39]
(for derivatives) and [23] (for vector fields):

(2.8) $\|u_{\leq
m}\|_{LE_{w}^{1}[\tilde{t}_{0},\tilde{t}_{1}]}+\|{\partial}u_{\leq
m}\|_{L^{\infty}L^{2}}\lesssim E[u_{\leq
m}](\tilde{t}_{0})+\|(\Box_{K}u)_{\leq
m}\|_{L^{1}L^{2}+LE^{*}_{w}[\tilde{t}_{0},\tilde{t}_{1}]}.$

We will also use a version that replaces the loss near the trapped set with a
derivative loss:

(2.9) $\|u_{\leq
m}\|_{LE^{1}[\tilde{t}_{0},\tilde{t}_{1}]}+\|{\partial}u_{\leq
m}\|_{L^{\infty}L^{2}}\lesssim E[u_{\leq
m+1}](\tilde{t}_{0})+\|(\Box_{K}u)_{\leq
m+1}\|_{L^{1}L^{2}+LE^{*}[\tilde{t}_{0},\tilde{t}_{1}]}.$

##  3\. Local energy decay for the nonlinear problem

In order to prove the main theorem from this section, we will use two results
from [17]. We need the following weighted Sobolev estimates, see Lemma 3.1
from [17]:

For $R\geq 10$, $2\leq q<\infty$, and any $b\in{\mathbb{R}}$, we have

(3.1) $\|r^{b}v\|_{L_{r}^{\frac{2q(q-1)}{q-2}}L_{\omega}^{\infty}(r\geq
R+1)}\lesssim\|r^{b-\frac{1}{q-1}}v_{\leq 6}\|_{L_{r}^{q}L_{\omega}^{2}(r\geq
R)},$ (3.2) $\|r^{b}v\|_{L^{\infty}_{x}(|x|\geq
R+1)}\lesssim\|r^{b-\frac{2}{q}}v_{\leq 6}\|_{L^{q}_{r}L^{2}_{\omega}(r\geq
R)}.$

Moreover, we will use Theorem 3.2 from [17]:

######  Theorem 3.1.

(Theorem 3.2, [17])

Let $p\in[2,\infty)$. Suppose $w$ solves

$\Box_{K}w=G_{1}+G_{2},\qquad w(0,\cdot)={\partial}_{t}w(0,\cdot)=0,$

Additionally, suppose that $w$ vanishes in $[0,\infty]\times\\{|x|\leq c\\}$
for some $c>0$ .

Then for any $\delta_{1}>0$ and $1/2-1/q<s<1/2$ we have

(3.3)
$\|r^{\frac{3}{2}-\frac{4}{q}-s}w\|_{L^{q}L^{q}L^{2}}\lesssim\|r^{-\frac{1}{2}-s}G_{1}\|_{L^{1}L^{1}L^{2}}+\|r^{\frac{3}{2}-s+\delta_{1}}G_{2}\|_{L^{2}L^{2}}.$

The goal of this section is to prove the following:

######  Theorem 3.2.

Let $m\geq 6$ be a positive integer. Suppose that the initial data
$(\phi_{0},\phi_{1})$ satisfies

(3.4) $\|(\phi_{0})_{\leq m+6}\|_{H^{1}}^{2}+\|(\phi_{1})_{\leq
m+6}\|_{L^{2}}^{2}\leq\varepsilon^{2},$

where $\varepsilon$ is small enough. Then the equation (2.3) has a global
solution that satisfies

(3.5) $\sup_{\tilde{t}}E[\phi_{\leq m}](\tilde{t})+\|\phi_{\leq
m}\|_{LE^{1}}\leq c_{m}\varepsilon$ (3.6) $\|\phi_{\leq m}\|_{L^{p}L^{2p}}\leq
c_{m}\varepsilon.$

###### Proof.

Let $\chi\in C^{\infty}({\mathbb{R}})$ satisfy $0\leq\chi(r)\leq 1$,
$\chi(r)\equiv 0$ for $r\leq R_{1}$, and $\chi(r)\equiv 1$ for $r>R_{1}+1$.
Pick any $1+\sqrt{2}<q<3$. For
$\gamma=\frac{4}{q}-\frac{2}{q-1}=\frac{2(q-2)}{q(q-1)}$, we define

(3.7) $\|\phi\|_{X^{m}}=\|r^{-\gamma}\chi\phi_{\leq
m}\|_{L^{q}L^{q}L^{2}}+\|{\partial}\phi_{\leq
m}\|_{L^{\infty}L^{2}}+\|\phi_{\leq m-1}\|_{LE^{1}}$ (3.8)
$\|g\|_{N^{m}}=\|r^{-\gamma q}\chi^{q}g_{\leq m}\|_{L^{1}L^{1}L^{2}}+\|g_{\leq
m}\|_{L^{1}L^{2}}.$

We first prove the following linear estimate:

(3.9) $\|\phi\|_{X^{m}}\lesssim\|(\phi_{0})_{\leq
m+6}\|_{H^{1}}+\|(\phi_{1})_{\leq m+6}\|_{L^{2}}+\|\Box_{K}\phi\|_{N^{m}}\ .$

Indeed, the last two terms in (3.7) can be estimated by using (2.9).

In order to estimate the first term, we apply (3.3) (with
$s=\frac{3}{2}-\frac{2}{q-1}$ and small $\delta_{1}$).

Pick any $|\alpha|\leq m$. Note first that, due to the support properties of
$\chi$ and the initial data, we have

$\chi\phi_{\alpha}(0,\cdot)={\partial}_{t}\left(\chi\phi_{\alpha}(0,\cdot)\right)=0.$

We have

$\Box_{K}(\chi\phi_{\alpha})=\chi(\Box_{K}\phi)_{\alpha}+[\Box_{K},\chi]\phi_{\alpha}+\chi[\Box_{K},Z^{\alpha}]\phi$

An easy computation gives

$[\Box_{K},{\partial}]\phi\in S^{Z}(r^{-2}){\partial}{\partial}^{\leq
1}\phi,\quad[\Box_{K},\Omega]\phi\in S^{Z}(r^{-2}){\partial}{\partial}^{\leq
1}\phi,$ $[\Box_{K},S]\phi\in
S^{Z}(1)\Box_{K}\phi+S^{Z}(r^{-2+}){\partial}\phi+S^{Z}(r^{-2+}){\partial}\Omega\phi+S^{Z}(r^{-2}){\partial}{\partial}^{\leq
1}\phi,$

and thus by induction we obtain that

(3.10) $[\Box_{K},Z^{\alpha}]\phi=F_{1}+F_{2},\quad F_{1}\in
S^{Z}(1)(\Box_{K}\phi)_{\leq|\alpha|},\quad F_{2}\in
S^{Z}(r^{-2+}){\partial}\phi_{\leq|\alpha|}.$

We now pick $G_{1}=\chi(\Box_{K}\phi)_{\alpha}+\chi F_{1}$ and
$G_{2}=[\Box_{K},\chi]\phi_{\alpha}+\chi F_{2}$.

Since $\chi-\chi^{q}$ is supported in $[R,R+1]$, and $\frac{1}{2}+s=\gamma q$,
we see that

$\|r^{-\frac{1}{2}-s}G_{1}\|_{L^{1}L^{1}L^{2}}\lesssim\|r^{-\gamma
q}\chi(\Box_{K}\phi)_{\leq|\alpha|}\|_{L^{1}L^{1}L^{2}}\lesssim\|r^{-\gamma
q}\chi^{q}(\Box_{K}\phi)_{\leq|\alpha|}\|_{L^{1}L^{1}L^{2}}+\|(\Box_{K}\phi)_{\leq|\alpha|}\|_{L^{1}L^{2}}.$

On the other hand, we see that for small enough $\delta_{1}$:

$\|r^{\frac{3}{2}-s+\delta_{1}}\chi F_{2}\|_{L^{2}L^{2}}\lesssim\|\chi
r^{-\frac{1}{2}-s+\delta_{1}+}{\partial}\phi_{\leq|\alpha|}\|_{L^{2}L^{2}}\lesssim\|\phi_{\leq|\alpha|}\|_{LE^{1}}.$

We also have that

$\|r^{\frac{3}{2}-s+\delta_{1}}[\Box_{K},\chi]\phi_{\alpha}\|_{L^{2}L^{2}}\lesssim\|{\partial}^{\leq
1}\phi_{\alpha}\|_{L^{2}L^{2}(R\leq|x|\leq
R+1)}\lesssim\|\phi_{\leq|\alpha|}\|_{LE^{1}}.$

Theorem 3.1 now implies (3.9).

We now finish the proof of Theorem 3.2. We want to show that

(3.11) $\|\phi\|_{X^{m}}\leq C_{m}\varepsilon.$

It is enough to show that, for any $\psi$,

(3.12) $\|\psi^{p}\|_{N^{m}}\lesssim\|\psi\|_{X^{m}}^{p}.$

Indeed, let $\psi_{0}\equiv 0$ and recursively define $\psi_{k+1}$ to be the
solution to the linear equation

(3.13)
$\Box_{K}\psi_{k+1}=\pm\psi_{k}^{p},\qquad\psi_{k+1}|_{\Sigma^{-}}=\phi_{0},\qquad\tilde{T}\psi_{k+1}|_{\Sigma^{-}}=\phi_{1}.$

(3.9) and (3.12) imply

$\|\psi_{k+1}\|_{X^{m}}\lesssim C(\epsilon+\|\psi_{k}\|_{X^{m}}^{p}),$

and a continuity argument implies that for small enough $\epsilon$ the
sequence $\psi_{k}$ is Cauchy in $X^{m}$ and thus converges to a solution
$\phi$ that satisfies (3.11).

Let us now prove (3.12). We have for any $|\alpha|\leq m$:

(3.14)
$Z^{\alpha}(\psi^{p})\lesssim\sum_{\begin{subarray}{c}|\alpha_{1}|\leq\dots\leq|\alpha_{p}|\leq|\alpha|\\\
\alpha_{1}+\dots\alpha_{p}=\alpha\end{subarray}}|Z^{\alpha_{1}}\psi|\dots|Z^{\alpha_{p}}\psi|.$

In a compact region, we have by Hölder and Sobolev embeddings

$\begin{split}&\Bigl{\|}Z^{\alpha_{1}}\psi\dots
Z^{\alpha_{p}}\psi\Bigr{\|}_{L^{1}L^{2}(r\leq
R+2)}\leq\|Z^{\alpha_{p-1}}\psi\|_{L^{2}L^{\infty}(r\leq
R+2)}\|Z^{\alpha_{p}}\psi\|_{L^{2}L^{2}(r\leq
R+2)}\prod_{i=1}^{p-2}\|Z^{\alpha_{i}}\psi\|_{L^{\infty}L^{\infty}(r\leq
R+2)}\\\ &\lesssim\|\psi_{\leq|\alpha_{p-1}|+2}\|_{L^{2}L^{2}(r\leq
R+3)}\|\psi_{\leq|\alpha_{p}|}\|_{L^{2}L^{2}(r\leq
R+2)}\|{\partial}\psi_{\leq|\alpha_{p-2}|+1}\|^{p-2}_{L^{\infty}L^{2}}\\\
&\lesssim\|\psi\|_{X^{\left\lfloor
m/2\right\rfloor+2}}^{p-1}\|\psi\|_{X^{m}}\leq\|\psi\|^{p}_{X^{m}}.\end{split}$

On the other hand, for $r\geq R+2$ we have, using (3.1) (with
$b=\frac{\gamma}{q-1}$), (3.2) (with $b=\gamma$) and the fact that
$\frac{\gamma-1}{q-1}\leq-\gamma$:

$\begin{split}&\Bigl{\|}Z^{\alpha_{1}}u\dots
Z^{\alpha_{p}}\psi\Bigr{\|}_{L^{1}L^{2}(r\geq R+2)}\lesssim\|r^{-\gamma}\chi
Z^{\alpha_{p}}\psi\|_{L^{q}L^{q}L^{2}}\|r^{\gamma}Z^{\alpha_{1}}\psi\dots
Z^{\alpha_{p-1}}\psi\|_{L^{\frac{q}{q-1}}L^{\frac{2q}{q-2}}L^{\infty}(r\geq
R+2)}\\\ &\lesssim\|r^{-\gamma}\chi
Z^{\alpha_{p}}\psi\|_{L^{q}L^{q}L^{2}}\|\psi_{\leq|\alpha_{p-1}|}\|^{p-q}_{L^{\infty}L^{\infty}(r\geq
R+2)}\|r^{\gamma}(\psi_{\leq|\alpha_{p-1}|})^{q-1}\|_{L^{\frac{q}{q-1}}L^{\frac{2q}{q-2}}L^{\infty}(r\geq
R+2)}\\\ &\lesssim\|\psi\|_{X^{m}}\|\psi\|^{p-q}_{X^{\left\lfloor
m/2\right\rfloor+6}}\|r^{\frac{\gamma}{q-1}}\psi_{\leq|\alpha_{p-1}|}\|^{q-1}_{L^{q}L^{\frac{2q(q-1)}{q-2}}L^{\infty}(r\geq
R+2)}\lesssim\|\psi\|^{1+p-q}_{X^{m}}\|r^{\frac{\gamma-1}{q-1}}\psi_{\leq|\alpha_{p-1}|}\|^{q-1}_{L^{q}L^{q}L^{\infty}(r\geq
R+1)}\\\
&\lesssim\|\psi\|^{1+p-q}_{X^{m}}\|r^{-\gamma}\chi\psi_{\leq|\alpha_{p-1}|+6}\|^{q-1}_{L^{q}L^{q}L^{2}}\lesssim\|\psi\|^{1+p-q}_{X^{m}}\|\psi\|^{q-1}_{X^{\left\lfloor
m/2\right\rfloor+2}}\lesssim\|\psi\|^{p}_{X^{m}}.\end{split}$

Finally,

$\begin{split}&\Bigl{\|}r^{-\gamma q}\chi^{q}Z^{\alpha_{1}}\psi\dots
Z^{\alpha_{p}}\psi\Bigr{\|}_{L^{1}L^{1}L^{2}}\lesssim\|r^{-\gamma}\chi
Z^{\alpha_{p}}\psi\|_{L^{q}L^{q}L^{2}}\|r^{-\gamma(q-1)}\chi^{q-1}Z^{\alpha_{1}}\psi\dots
Z^{\alpha_{p-1}}\psi\|_{L^{\frac{q}{q-1}}L^{\frac{q}{q-1}}L^{\infty}}\\\
&\lesssim\|r^{-\gamma}\chi
Z^{\alpha_{p}}\psi\|_{L^{q}L^{q}L^{2}}\|\psi_{\leq|\alpha_{p-1}|}\|^{p-q}_{L^{\infty}L^{\infty}}\|r^{-\gamma(q-1)}\chi^{q-1}(\psi_{\leq|\alpha_{p-1}|})^{q-1}\|_{L^{\frac{q}{q-1}}L^{\frac{q}{q-1}}L^{\infty}}\\\
&\lesssim\|\psi\|_{X^{m}}\|\psi\|^{p-q}_{X^{\left\lfloor
m/2\right\rfloor+2}}\|r^{-\gamma}\chi\psi_{\leq|\alpha_{p-1}|}\|^{q-1}_{L^{q}L^{q}L^{\infty}}\lesssim\|\psi\|^{p}_{X^{m}}.\end{split}$

The proof of (3.12) is now complete.

∎

##  4\. $r^{p}$ estimates

The local energy spaces from the previous section are enough to obtain a weak
decay estimate, see Theorem 6.1 from [18].

######  Theorem 4.1.

(Theorem 6.1, [18])

Let $T$ be a fixed time. We then have for $T\leq\tilde{t}\leq 2T$:

(4.1) $|\phi_{\leq_{|\alpha|}}|\leq
C_{m}\langle\tilde{t}\rangle^{-1}\langle\tilde{t}-{\tilde{r}}\rangle^{1/2}\|\phi_{\leq|\alpha|+13}\|_{LE^{1}[T,2T]}.$

In particular, this implies

(4.2)
$|\phi_{\leq|\alpha|}|\lesssim\varepsilon\langle\tilde{t}\rangle^{-1}\langle\tilde{t}-{\tilde{r}}\rangle^{1/2}$

for all $|\alpha|\leq N-19$.

One idea to continue here is to rewrite the equation as

$\Box\phi=(\Box_{\phi}-\Box_{K})\phi\pm\phi^{p},$

and use the fundamental solution for the Minkowski, combined with (4.2), to
improve the rate of decay. This works for $p>5$, but not for smaller $p$.
Instead we first use $r^{p}$ estimates inspired by the work of Dafermos-
Rodnianski [9] to improve the pointwise decay.

Let ${\partial}_{v}={\partial}_{t}+{\partial}_{\tilde{r}}$, and
${\not\negmedspace\partial}$ denote angular derivatives. We introduce the
weighted local energy norm $LE_{\gamma}$ for $\gamma>0$:

(4.3)
$\|\phi\|_{LE_{\gamma}}=\|r^{\frac{\gamma-1}{2}}\phi\|_{L^{2}(\mathcal{M})},$

and its $H^{1}$ counterpart

(4.4)
$\|\phi\|_{LE_{\gamma}^{1}}=\|{\partial}_{v}\phi\|_{LE_{\gamma}}+\|{\not\negmedspace\partial}\phi\|_{LE_{\gamma}}+\|r^{-1}\phi\|_{LE_{\gamma}}.$

We also define the degenerate norm

$\|\phi\|_{LE^{1}_{w,\gamma}}=\|(1-\chi_{ps}){\partial}_{v}\phi\|_{LE_{\gamma}}+\|(1-\chi_{ps}){\not\negmedspace\partial}\phi\|_{LE_{\gamma}}+\|r^{-1}\phi\|_{LE_{\gamma}},$

and the weighted energy

$E_{\gamma}[\phi](\tilde{t})=\int_{\Sigma(\tilde{t})}r^{\gamma}\left(|{\partial}_{v}\phi|^{2}+|{\not\negmedspace\partial}\phi|^{2}+r^{-2}\phi^{2}\right)d\Sigma_{K}(\tilde{t}).$

For the dual norm, we define

(4.5) $\|f\|_{LE_{\gamma}^{*}}=\|r^{\frac{\gamma}{2}}f\|_{LE^{*}}.$

We will prove the following linear estimate:

######  Theorem 4.2.

Assume that $\phi$ solves

$\Box_{K}\phi=F,\qquad\phi|_{{\tilde{t}}=0}=\phi_{0},\qquad\tilde{T}\phi|_{{\tilde{t}}=0}=\phi_{1}\
.$

Then for any $0<\gamma<2$ and compactly supported $(\phi_{0},\phi_{1})$ we
have

(4.6)
$\sup_{\tilde{t}}E_{\gamma}[\phi](\tilde{t})+\|\phi\|_{LE^{1}_{w,\gamma}}^{2}\lesssim
E[\phi](0)+E[\phi](T)+\|\phi\|^{2}_{LE^{1}_{w}[0,T]}+\|F\|^{2}_{LE_{\gamma}^{*}}.$

A similar result appears in a paper by Stogin [32].

###### Proof.

Recall that the energy-momentum tensor is given by

$Q_{\alpha\beta}[g]=\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}g_{\alpha\beta}\partial^{\mu}\phi\partial_{\mu}\phi.$

Its contraction with respect to a vector field $X$ is denoted by

$P_{\alpha}[g,X]=Q_{\alpha\beta}[g]X^{\beta}.$

Its divergence is given by

$\nabla^{\alpha}P_{\alpha}[g,X]=\Box_{g}\phi\cdot
X\phi+\frac{1}{2}Q[g,X],\qquad
Q[g,X]=Q_{\alpha\beta}[g]\pi_{X}^{\alpha\beta},$

where $\pi_{X}^{\alpha\beta}$ is the deformation tensor of $X$, which is given
in terms of the Lie derivative by

$\pi_{\alpha\beta}^{X}=\nabla_{\alpha}X_{\beta}+\nabla_{\beta}X_{\alpha}=({\mathcal{L}}_{X}g)_{\alpha\beta}.$

In coordinates, one can write

(4.7)
$Q[g,X]=-\frac{1}{\sqrt{|g|}}(X(\sqrt{|g|}g^{\alpha\beta}))\partial_{\alpha}\phi\,\partial_{\beta}\phi\\\
+(g^{\alpha\gamma}\partial_{\gamma}X^{\beta}+g^{\beta\gamma}\partial_{\gamma}X^{\alpha})\partial_{\alpha}\phi\partial_{\beta}\phi-\partial_{\gamma}X^{\gamma}\,g^{\alpha\beta}\partial_{\alpha}\phi\,\partial_{\beta}\phi.$

More generally, for a vector field $X$, a scalar function $q$ and a one-form
$m$, we define

$P_{\alpha}[g,X,q,m]=P_{\alpha}[g,X]+q\phi\partial_{\alpha}\phi-\frac{1}{2}(\partial_{\alpha}q)\phi^{2}+\frac{1}{2}m_{\alpha}\phi^{2}.$

The divergence of $P$ is

(4.8)
$\nabla^{\alpha}P_{\alpha}[g,X,q,m]=\Box_{g}\phi\Bigl{(}X\phi+qu\Bigr{)}+Q[g,X,q,m],$

where

$Q[g,X,q,m]=\frac{1}{2}Q[g,X]+q\partial^{\alpha}\phi\,\partial_{\alpha}\phi+m_{\alpha}\phi\,\partial^{\alpha}\phi+(\nabla^{\alpha}m_{\alpha}-\frac{1}{2}\nabla^{\alpha}\partial_{\alpha}q)\,\phi^{2}.$

Let $\mathcal{M}_{[0,T]}=\mathcal{M}\cap[0,T]\times{\mathbb{R}}^{3}$. The
divergence theorem yields, assuming that $X$, $q$ and $m$ are supported in
$\\{r>4M\\}$, that

(4.9)
$\int_{\mathcal{M}_{[0,T]}}Q[g,X,q,m]dV_{g}=-\int_{\mathcal{M}_{[0,T]}}\Box_{K}\phi\left(X\phi+q\phi\right)dV_{g}+BDR^{g},$

where $BDR^{g}$ denotes the boundary terms

$BDR^{g}=\int_{\Sigma(\tilde{t})}\langle d\tilde{t},P[g,X,q,m]\rangle
d\Sigma\Big{|}_{0}^{T}.$

Even though one can do the following computations in Kerr, it is easier to
first do the Schwarzschild case, and treat Kerr perturbatively.

Fix $\gamma<2$ and pick $0<\delta$ small so that

(4.10) $(1-\delta)^{2}-2(1-\gamma\delta)<0.$

Let

$X=r^{\gamma}{\partial}_{v},\qquad
q(r)=r^{\gamma-1}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}},\qquad
m=\gamma(1-\delta)r^{\gamma-2}dv.$

We compute, using the fact that
${\partial}_{{\tilde{r}}}={\Bigl{(}1-\frac{2M}{r}\Bigr{)}}{\partial}_{r}$

$\begin{split}&P_{0}[g_{S},X,q,m]=\frac{r^{\gamma}}{2}\left(|{\partial}_{t}\phi|^{2}+|{\partial}_{\tilde{r}}\phi|^{2}+{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}|{\not\negmedspace\partial}\phi|^{2}+2{\partial}_{t}\phi{\partial}_{\tilde{r}}\phi\right)+r^{\gamma-1}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}\phi{\partial}_{t}\phi+\frac{\gamma(1-\delta)r^{\gamma-2}}{2}\phi^{2}\\\
&=\frac{r^{\gamma}}{2}\left(|{\partial}_{v}\phi|^{2}+\frac{1}{r}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}\phi\right)^{2}+\frac{r^{\gamma}}{2}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}|{\not\negmedspace\partial}\phi|^{2}-r^{\gamma-1}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}^{2}\left(\phi{\partial}_{r}\phi+\frac{1}{2r}\phi^{2}\right)+\frac{\gamma(1-\delta)r^{\gamma-2}}{2}\phi^{2}.\end{split}$

Recall that for the Schwarzschild metric $dV_{S}=r^{2}d\tilde{t}drd\omega$,
and $d\Sigma_{S}=r^{2}drd\omega$. Let $R_{2}$ be large enough, and
$\chi_{R_{2}}(r)$ be a smooth cutoff equal to $1$ when $r\geq R_{2}$ and
supported in $\\{r\geq\frac{R_{2}}{2}\\}$. After integrating by parts we
obtain

$\begin{split}&-\int_{\Sigma(\tilde{t})}\langle
d\tilde{t},P[g_{S},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\rangle
d\Sigma_{S}=\int_{\Sigma(\tilde{t})}\chi_{R_{2}}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}^{-1}\left[\frac{r^{\gamma}}{2}\left({\partial}_{v}\phi+\frac{1}{r}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}\phi\right)^{2}\right.\\\
&+\frac{r^{\gamma}}{2}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}|{\not\negmedspace\partial}\phi|^{2}-r^{\gamma-1}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}^{2}\left(\phi{\partial}_{r}\phi+\frac{1}{2r}\phi^{2}\right)+\frac{\gamma(1-\delta)r^{\gamma-2}}{2}\phi^{2}\Bigl{.}\Bigr{]}d\Sigma_{S}=\\\
&\int_{\Sigma(\tilde{t})}\chi_{R_{2}}\frac{r^{\gamma}}{2}\left[{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}^{-1}\left({\partial}_{t}\phi+{\partial}_{\tilde{r}}\phi+\frac{1}{r}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}\phi\right)^{2}+|{\not\negmedspace\partial}\phi|^{2}+\left(\gamma(2-\delta)-\frac{2M\gamma}{r}\right)\frac{\phi^{2}}{r^{2}}\right]\\\
&+\chi_{R_{2}}^{\prime}\frac{r^{\gamma-1}-2Mr^{\gamma-2}}{2}\phi^{2}d\Sigma_{S}.\end{split}$

Since the support of $\chi_{R_{2}}^{\prime}$ is contained in
$\\{\frac{R_{2}}{2}\leq r\leq R_{2}\\}$, we have by Hardy’s inequality that

$\int_{\Sigma(\tilde{t})}\chi_{R_{2}}^{\prime}\frac{r^{\gamma-1}-2Mr^{\gamma-2}}{2}\phi^{2}d\Sigma_{S}\lesssim
E[\phi](\tilde{t}).$

Moreover, due to (2.1) we see that

$Q_{\alpha\beta}[g_{K}]-Q_{\alpha\beta}[g_{S}]\lesssim\frac{1}{r^{2}}|{\partial}\phi|^{2},$

which immediately implies, since $\gamma<2$, that

$\langle
d\tilde{t},P[g_{S},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\rangle-\langle
d\tilde{t},P[g_{K},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\rangle\lesssim
r^{\gamma-2}|{\partial}\phi|^{2}\lesssim|{\partial}\phi|^{2}.$

Finally, we have that $\sqrt{|g_{K}|}\approx\sqrt{|g_{S}|}$.

We thus have

(4.11) $E_{\gamma}[\phi](\tilde{t})\lesssim-\int_{\Sigma(\tilde{t})}\langle
d\tilde{t},P[g_{K},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\rangle
d\Sigma_{K}+E[\phi](\tilde{t}).$

Moreover, since the initial data has compact support, we clearly have that
when $\tilde{t}=0$:

(4.12) $-\int_{\Sigma(0)}\langle
d\tilde{t},P[g_{K},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\rangle
d\Sigma_{K}\approx E[\phi](0).$

We now compute the spacetime term. Using (4.7) we have for any $f(r)$:

$Q[g_{S},f(r){\partial}_{t}]=2f^{\prime}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}{\partial}_{t}\phi{\partial}_{r}\phi$
$\begin{split}Q\left[g_{S},{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}f(r){\partial}_{r}\right]=&f^{\prime}({\partial}_{t}\phi)^{2}+f^{\prime}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}^{2}({\partial}_{r}\phi)^{2}+\left[2f\left(1-\frac{3M}{r}\right)-{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}f^{\prime}\right]|{\not\negmedspace\partial}\phi|^{2}\\\
&-\frac{1}{2r}{\Bigl{(}1-\frac{2M}{r}\Bigr{)}}f{\partial}^{\gamma}\phi{\partial}_{\gamma}\phi\end{split}$

and thus

$Q[g_{S},X,q,0]=\frac{\gamma
r^{\gamma-1}}{2}({\partial}_{v}\phi)^{2}+r^{\gamma-1}\left(\frac{2-\gamma}{2}+\frac{(\gamma-3)M}{r}\right)|{\not\negmedspace\partial}\phi|^{2}-\frac{1}{2}\Box_{g_{S}}q.$

We also compute

$-\frac{1}{2}\Box_{g_{S}}q=\gamma(1-\gamma)r^{\gamma-3}\left(1+O(\frac{1}{r})\right),$

which unfortunately has the wrong sign when $\gamma>1$. On the other hand, we
have

$\nabla^{\alpha}m_{\alpha}=(1-\delta)\gamma^{2}r^{\gamma-3}\left(1+O(\frac{1}{r})\right)$

and thus

$\nabla^{\alpha}m_{\alpha}-\frac{1}{2}\Box_{g_{S}}q=\gamma(1-\delta\gamma)r^{\gamma-3}\left(1+O(\frac{1}{r})\right).$

Due to (4.10) we see that

$\frac{\gamma
r^{\gamma-1}}{2}({\partial}_{v}\phi)^{2}+\gamma(1-\delta)r^{\gamma-2}\phi{\partial}_{v}\phi+\gamma(1-\delta\gamma)r^{\gamma-3}\phi^{2}\gtrsim
r^{\gamma-1}({\partial}_{v}\phi)^{2}+r^{\gamma-3}\phi^{2}$

and thus for $R_{2}$ large enough and $r\geq R_{2}$ we have

$Q[g_{S},X,q,m]\gtrsim
r^{\gamma-1}\left(|{\partial}_{v}\phi|^{2}+|{\not\negmedspace\partial}\phi|^{2}\right)+r^{\gamma-3}\phi^{2}.$

Due to the support properties of $\chi_{R_{2}}$ we get

(4.13)
$\int_{\mathcal{M}_{[0,T]}}Q[g_{S},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]dV_{K}\gtrsim\|\phi\|^{2}_{LE^{1}_{w,\gamma}[0,T]}-\|\phi\|^{2}_{LE^{1}_{w}[0,T]}.$

Using (2.1) we see that

(4.14)
$Q[g_{S},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]-Q[g_{K},\chi_{R_{2}}X,\chi_{R_{2}}q,\chi_{R_{2}}m]\lesssim\chi_{R_{2}/2}\left(r^{\gamma-3}|{\partial}\phi|^{2}+r^{\gamma-4}|\phi|^{2}\right).$

We thus obtain from (4.9), (4.11), (4.12), (4.13) and (4.14) that

$E_{\gamma}[\phi](T)+\|\phi\|^{2}_{LE^{1}_{w,\gamma}[0,T]}\lesssim\int_{\mathcal{M}_{[0,T]}}\left|(\Box_{K}\phi)\chi_{R_{2}}\left(X\phi+q\phi\right)\right|dV_{K}+E[\phi](0)+E[\phi](T)+\|\phi\|^{2}_{LE^{1}_{w}[0,T]}.$

The result (4.6) now follows by Cauchy Schwarz.

∎

##  5\. Setup for pointwise estimates

Note first that we can consider, instead of $\Box_{K}$, an operator that looks
like $\Box$ (with respect to ${\tilde{r}}$) plus a long range perturbation;
this will allow us to directly apply the results in [23]. Indeed, let

$P=|g_{K}|^{1/4}(-g_{K}^{\tilde{t}\tilde{t}})^{-1/2}\Box_{K}(-g_{K}^{\tilde{t}\tilde{t}})^{-1/2}|g_{K}|^{-1/4}.$

$P$ is self-adjoint with respect to $d\tilde{t}dx$. More importantly, a quick
computation yields that

$P={\partial}_{\alpha}\left(g_{K}^{\alpha\beta}(-g_{K}^{\tilde{t}\tilde{t}})\partial_{\beta}\right)+V,\quad
V=|g_{K}|^{1/4}(-g_{K}^{\tilde{t}\tilde{t}})^{-1/2}\Box_{K}\left((-g_{K}^{\tilde{t}\tilde{t}})^{-1/2}|g_{K}|^{-1/4}\right).$

It is easy to see that $V\in S^{Z}(r^{-3})$. Moreover, in Schwarzschild we
have that for large $r$, $-g_{S}^{\tilde{t}\tilde{t}}=g_{S}^{r^{*}r^{*}}$ and
$g_{S}^{\tilde{t}r^{*}}=0$. We thus have

$P=\Box+P_{lr},$

where the long range spherically symmetric part $P_{lr}$ has the form

(5.1) $P_{lr}=g_{lr}(r)\Delta_{\omega}+V,\qquad g_{lr}\in
S_{rad}^{Z}(r^{-3}),\qquad V\in S^{Z}(r^{-3}).$

For the Kerr metric, using (2.1) yields

(5.2) $P=\Box+P_{lr}+P_{sr},$

where the short-range part $P_{sr}$ has the form

(5.3) $P_{sr}=\partial_{\alpha}g_{sr}^{\alpha\beta}\partial_{\beta},\quad
g_{sr}^{\alpha\beta}\in S^{Z}_{der}(r^{-2}).$

We now see that $\phi$ satisfies

$P\phi=(-g_{K}^{\tilde{t}\tilde{t}})\phi^{5}+h_{1}\phi+h_{2}{\partial}\phi,\quad
h_{1}\in S^{Z}(r^{-3}),\quad h_{2}\in S^{Z}_{der}(r^{-2}).$

Now pick any $|\alpha|\leq N$. After commuting with vector fields, using
(5.2), (5.1), and (5.3), we obtain

(5.4)
$P\phi_{\alpha}=F_{\alpha}+G_{\alpha},\qquad\phi_{\alpha}|_{{\tilde{t}}=0}=\phi_{0}^{\alpha},\qquad\tilde{T}\phi_{\alpha}|_{{\tilde{t}}=0}=\phi_{1}^{\alpha},$

with

(5.5)
$F_{\alpha}=\left((-g_{K}^{\tilde{t}\tilde{t}})\phi^{5}\right)_{\alpha}\in
S^{Z}(1)\sum_{\begin{subarray}{c}|\alpha_{1}|\leq\dots\leq|\alpha_{p}|\leq|\alpha|\\\
\alpha_{1}+\dots\alpha_{p}=\alpha\end{subarray}}\prod_{j=1}^{p}\phi_{\alpha_{j}},$
(5.6) $G_{\alpha}\in
S^{Z}(r^{-3})\phi_{\leq|\alpha|+6}+S^{Z}_{der}(r^{-2}){\partial}\phi_{\leq|\alpha|+5},$

and

(5.7)
$\|\phi_{0}^{\alpha}\|_{H^{1}}+\|\phi_{1}^{\alpha}\|_{L^{2}}\lesssim\varepsilon.$

We will use (5.4) to control the solution when ${\tilde{r}}$ is small.

On the other hand, for large ${\tilde{r}}$ it is more convenient to work with
$\Box$ and treat the rest perturbatively. Let $\chi_{out}$ be a cutoff equal
to $1$ for large $r$, and $\psi_{\alpha}=\chi_{out}\phi_{\alpha}$. Then
$\psi_{\alpha}$ satisfies

(5.8)
$\Box\psi_{\alpha}=F_{\alpha}+G_{\alpha},\qquad\psi_{\alpha}|_{{\tilde{t}}=0}=\tilde{T}\psi_{\alpha}|_{{\tilde{t}}=0}=0,$

where $F_{\alpha}$ and $G_{\alpha}$ are supported away from $0$ and satisfy
(5.5), (5.6).

We will decompose $\psi_{\alpha}$ as

$\psi_{\alpha}=\psi_{1}+\psi_{2},$

where

(5.9)
$\Box\psi_{1}=G_{\alpha},\qquad\psi_{1}|_{{\tilde{t}}=0}=0,\qquad{\partial}_{t}\psi_{1}|_{{\tilde{t}}=0}=0,$

(5.10)
$\Box\psi_{2}=F_{\alpha},\qquad\psi_{1}|_{{\tilde{t}}=0}=0,\qquad{\partial}_{t}\psi_{1}|_{{\tilde{t}}=0}=0.$

By finite speed of propagation, $F_{\alpha}$ and $G_{\alpha}$ are supported in
the forward light cone $\\{|x|<\tilde{t}+CR_{1}\\}$. By a time translation, we
may assume that $F_{\alpha}$ and $G_{\alpha}$ are supported in the forward
light cone $\\{|x|<\tilde{t}\\}$.

Finally, in the next sections $n$ will represent a large constant, which does
not depend on $\alpha$, but may increase from one estimate to the next.

##  6\. Estimates for the fundamental solution

The goal of this section is to prove pointwise estimates for solutions to the
inhomogeneous wave equation on Minkowski backgrounds.

We will first prove the following lemma, which gives pointwise bounds for the
solution assuming the inhomogeneity lies in certain weighted $L^{\infty}$
spaces.

For any $\beta,\gamma,\eta\in{\mathbb{R}}$, we define the weighted
$L^{\infty}$ norms

(6.1) $\|G\|_{L_{\beta,\gamma,\eta}^{\infty}}=\|\langle
r\rangle^{\beta}\langle t\rangle^{\gamma}\langle
t-r\rangle^{\eta}H(t,r)\|_{L_{t,r}^{\infty}},\quad
H(t,r)=\sum_{0}^{2}\|\Omega^{i}G(t,r\omega)\|_{L^{2}({\mathbb{S}}^{2})}.$

######  Lemma 6.1.

i) Let $\psi$ solve

(6.2) $\Box\psi=G,\qquad\psi(0)=0,\quad{\partial}_{t}\psi(0)=0,$

where $G$ is supported in $\\{|x|\leq t\\}$. For any $1<\beta\leq 3$ and
$\eta\neq 1$ we have:

(6.3) $\psi(t,x)\lesssim\frac{1}{\langle r\rangle\langle
t-r\rangle^{\beta+\tilde{\eta}}}\|G\|_{L_{\beta,1,\eta}^{\infty}},$

where we define, for any arbitrary $\delta>0$,

$\tilde{\eta}=\left\\{\begin{array}[]{cc}\eta-\delta-2&\eta<1,\cr-1&\eta>1\end{array}\right..$

ii) Let $\psi$ solve

(6.4)
$\Box\psi={\partial}_{t}{\widetilde{G}},\qquad\psi(0)=0,\quad{\partial}_{t}\psi(0)=0,$

where ${\widetilde{G}}$ is supported in $\\{t/2\leq|x|\leq t\\}$. For any
$2<\beta\leq 3$ and $\eta\neq 1$ we have:

(6.5) $\psi(t,x)\lesssim\frac{1}{\langle r\rangle\langle
t-r\rangle^{1+\tilde{\eta}}}\left(\|{\widetilde{G}}\|_{L_{\beta-1,1,\eta}^{\infty}}+\|S{\widetilde{G}}\|_{L_{\beta-1,1,\eta}^{\infty}}+\|\Omega{\widetilde{G}}\|_{L_{\beta-1,1,\eta}^{\infty}}+\|\langle
t-r\rangle{\partial}{\widetilde{G}}|\|_{L_{\beta-1,1,\eta}^{\infty}}\right).$

We remark here that (6.3) is similar to previous classical results, see for
instance [15], [4], [34], [36].

###### Proof.

We use the ideas from [23]. Let us first prove (6.3). Define

(6.6)
$H(t,r)=\sum_{0}^{2}\|\Omega^{i}G(t,r\omega)\|_{L^{2}({\mathbb{S}}^{2})}.$

By Sobolev embeddings on the sphere, we have $|G|\lesssim H$. Let $v$ be the
radial solution to

(6.7) $\Box v=H,\qquad v[0]=0.$

By the positivity of the fundamental solution, we have that
$|\psi|\lesssim|v|$. On the other hand, we can write $v$ explicitly:

(6.8) $rv(t,r)=\frac{1}{2}\int_{D_{tr}}\rho H(s,\rho)dsd\rho,$

where $D_{tr}$ is the rectangle

$D_{tr}=\\{0\leq s-\rho\leq t-r,\quad t-r\leq s+\rho\leq t+r\\}.$

We partition the set $D_{tr}$ into a double dyadic manner as

$D_{tr}=\bigcup_{R\leq t}D_{tr}^{R},\quad D_{tr}^{R}=D_{tr}\cap\\{R<r<2R\\}$

and estimate the corresponding parts of the above integral.

We clearly have

$\int_{D_{tr}^{R}}\rho
Hdsd\rho\lesssim\|G\|_{L_{\beta,1,\eta}^{\infty}}\int_{D_{tr}^{R}}\rho^{1-\beta}\langle
s\rangle^{-1}\langle s-\rho\rangle^{-\eta}d\rho ds.$

We now consider two cases:

(i) $R<(t-r)/8$. Here we have $\rho\sim R$ and $s\approx s-\rho\approx\langle
t-r\rangle$; therefore we obtain

$\int_{D_{tr}^{R}}\rho^{1-\beta}\langle s\rangle^{-1}\langle
s-\rho\rangle^{-\eta}d\rho ds\lesssim R^{3-\beta}\langle
t-r\rangle^{-1-\eta},$

and after summation, using that $\beta\leq 3$, we obtain

(6.9) $\sum_{R<(t-r)/8}\int_{D_{tr}^{R}}\rho Hdsd\rho\lesssim\frac{\ln\langle
t-r\rangle\langle t-r\rangle^{3-\beta}}{\langle
t-r\rangle^{1+\eta}}\lesssim\frac{1}{\langle
t-r\rangle^{\beta+\tilde{\eta}}}.$

(ii) $(t-r)/8<R<t$. Here we have $\rho\sim R$ and $s\gtrsim R$. Denote
$u=s-\rho$; then

$\int_{D_{tr}^{R}}\rho^{1-\beta}\langle s\rangle^{-1}\langle
s-\rho\rangle^{-\eta}d\rho ds\lesssim R^{1-\beta}\int_{0}^{t-r}\langle
u\rangle^{-\eta}du\lesssim R^{1-\beta}\langle t-r\rangle^{\mu(\eta)},$

where

$\mu(\eta)=\left\\{\begin{array}[]{cc}1-\eta&\eta<1,\cr
0&\eta>1\end{array}\right..$

Since $\beta>1$, we obtain after summation

(6.10) $\sum_{R>(t-r)/8}\int_{D_{tr}^{R}}\rho Hdsd\rho\lesssim\langle
t-r\rangle^{1-\beta+\mu(\eta)}.$

The conclusion (6.3) follows from (6.9) and (6.10).

We now prove (6.5). Let ${\widetilde{\psi}}$ be the solution to

(6.11) $\Box{\widetilde{\psi}}={\widetilde{G}},\qquad{\widetilde{\psi}}[0]=0.$

Clearly $\psi={\partial}_{t}{\widetilde{\psi}}$. We also note that in the
support of ${\widetilde{G}}$ we have

$(t\partial_{i}+x_{i}\partial_{t}){\widetilde{G}}\lesssim|S{\widetilde{G}}|+|\Omega{\widetilde{G}}|+\langle
t-r\rangle|{\partial}_{r}{\widetilde{G}}|.$

By (6.3), applied to ${\widetilde{\psi}}$, $\nabla{\widetilde{\psi}}$,
$\Omega{\widetilde{\psi}}$, $S{\widetilde{\psi}}$ and
$(t\partial_{i}+x_{i}\partial_{t}){\widetilde{\psi}}$ we obtain

$|{\widetilde{\psi}}|+|\nabla{\widetilde{\psi}}|+|S{\widetilde{\psi}}|+|\Omega{\widetilde{\psi}}|+\sum_{i}|(t\partial_{i}+x_{i}\partial_{t}){\widetilde{\psi}}|\lesssim\frac{1}{\langle
r\rangle\langle
t-r\rangle^{\beta+\tilde{\eta}-3}}\|{\widetilde{G}}\|_{L_{\beta-1,1,\eta}^{\infty}}.$

The above left hand side dominates $\langle
t-r\rangle\partial_{t}{\widetilde{\psi}}$; therefore the proof of the lemma is
complete.

∎

We will also use the following lemma, which gives pointwise control of
$\psi_{2}$ by the $r^{p}$ norms of Section 4.

######  Lemma 6.2.

For some $\alpha$, let $\psi_{2}$ solve (5.10), and assume also that (4.2)
holds. We then have for any $\gamma<2$, that

(6.12) $\psi_{2}(\tilde{t},x)\lesssim\frac{1}{\langle
r\rangle(t-{\tilde{r}})^{\gamma-\frac{3}{2}}}\left\|r^{-1}\phi_{\leq|\alpha|+6}\right\|_{LE_{\gamma}}^{2}.$

###### Proof.

The proof is similar to that of Lemma 6.1. Define

(6.13)
$H(\tilde{t},{\tilde{r}})=\|\phi_{\leq|\alpha|}\|^{p-2}_{L^{\infty}({\mathbb{S}}^{2})}\|\phi_{\leq|\alpha|+6}\|_{L^{2}({\mathbb{S}}^{2})}^{2}.$

By Sobolev embeddings on the sphere, we have $|F_{\alpha}|\lesssim H$. Let $v$
be the radial solution to

$\Box v=H,\qquad v[0]=0.$

By the positivity of the fundamental solution, we have that
$|\phi_{2}|\lesssim|v|$. On the other hand, we can write $v$ explicitly as in
(6.8):

$rv(t,r)=\frac{1}{2}\int_{D_{tr}}\rho H(s,\rho)dsd\rho$

where $D_{tr}$ is the rectangle

$D_{tr}=\\{0\leq
s-\rho\leq\tilde{t}-{\tilde{r}},\quad\tilde{t}-{\tilde{r}}\leq
s+\rho\leq\tilde{t}+{\tilde{r}}\\}.$

Moreover, due to (4.2), and the fact that $p\geq 3$, we see that

(6.14)
$H(\tilde{t},{\tilde{r}})\lesssim\frac{\langle\tilde{t}-{\tilde{r}}\rangle^{1/2}}{\langle\tilde{t}\rangle}\|\phi_{\leq|\alpha|+6}\|_{L^{2}({\mathbb{S}}^{2})}^{2}.$

We again consider two cases:

(i) $R<(\tilde{t}-{\tilde{r}})/8$. Since $\rho,\langle
s-\rho\rangle\lesssim\langle s\rangle$ and $s\gtrsim\tilde{t}-{\tilde{r}}$, we
have

$\begin{split}\int_{D_{tr}^{R}}\rho
Hdsd\rho&\lesssim\frac{1}{(\tilde{t}-{\tilde{r}})^{1/2}}\int_{D_{tr}^{R}}\rho\|\phi_{\leq|\alpha|+6}\|_{L^{2}({\mathbb{S}}^{2})}^{2}d\rho
ds\lesssim\frac{R^{2-\gamma}}{(\tilde{t}-{\tilde{r}})^{1/2}}\int_{D_{tr}^{R}}\rho^{\gamma-3}\|\phi_{\leq|\alpha|+6}\|_{L^{2}({\mathbb{S}}^{2})}^{2}\rho^{2}d\rho
ds\\\
&\lesssim\frac{R^{2-\gamma}}{(\tilde{t}-{\tilde{r}})^{1/2}}\left\|r^{-1}\phi_{\leq|\alpha|+6}\right\|_{LE_{\gamma}}^{2}.\end{split}$

(ii) $(\tilde{t}-{\tilde{r}})/8<R<\tilde{t}$. In this case, since
$\tilde{t}-{\tilde{r}}\lesssim\rho\leq s$, we have for small $0<\delta$ that

$\frac{\rho^{2-\gamma}\langle
s-\rho\rangle^{1/2}}{s}\lesssim\frac{R^{-\delta}}{(\tilde{t}-{\tilde{r}})^{\gamma-3/2-\delta}},$

and thus

$\int_{D_{tr}^{R}}\rho
Hdsd\rho\lesssim\int_{D_{tr}^{R}}\frac{\rho^{2-\gamma}\langle
s-\rho\rangle^{1/2}}{s}\rho^{\gamma-3}\|\phi_{\leq|\alpha|+6}\|_{L^{2}({\mathbb{S}}^{2})}^{2}\rho^{2}d\rho
ds\lesssim\frac{R^{-\delta}}{(\tilde{t}-{\tilde{r}})^{\gamma-3/2-\delta}}\left\|r^{-1}\phi_{\leq|\alpha|+6}\right\|_{LE_{\gamma}}^{2}.$

The conclusion now follows after summing over $R$.

Finally, in the next section we will use the following result, see Lemma 3.10
from [23], to improve the bounds for $\psi_{1}$:

######  Lemma 6.3.

Assume that $\psi$ solves (6.2). The following estimate holds for large enough
$n$:

(6.15) $\psi(t,x)\lesssim\frac{\log\langle t-r\rangle}{\langle r\rangle\langle
t-r\rangle^{\frac{1}{2}}}\|rG_{\leq n}\|_{LE^{*}}.$

∎

##  7\. An improved bound

We now pick $n$ suitably large, and assume that $m+n\ll N$. The main goal of
the section is to prove that, for any $0<q<\frac{1}{2}$, we have

(7.1) $\phi_{\leq m+n}\lesssim\varepsilon\langle
r\rangle^{-1}\langle\tilde{t}-{\tilde{r}}\rangle^{-q}.$

Clearly this holds for $r\lesssim 1$ by (4.2). For $r$ large, this will follow
by a continuity argument. We will assume that the following a-priori bounds
hold for some large constant $\tilde{C}$ independent of $\varepsilon$ and
$\tilde{t}$:

(7.2) $|\phi_{\leq m+n}|\leq\tilde{C}\varepsilon\langle
r\rangle^{-1}\langle\tilde{t}-{\tilde{r}}\rangle^{-q}.$

Clearly such a bound holds for small $\tilde{t}$ by Sobolev embeddings and the
compact support of the initial data. We now assume that (7.2) holds for all
$0\leq\tilde{t}\leq T$, and we will improve the constant on the right hand
side by a factor of $\frac{1}{2}$. For the rest of the section, the implicit
constants will not depend on $\tilde{C}$.

We will now show, that under assumption (7.2), we have that the solution to
(2.3) satisfies, for small enough $\varepsilon$,

(7.3) $\sup_{0\leq\tilde{t}\leq T}E_{\gamma}[\phi_{\leq
m+n+6}](\tilde{t})+\|\phi_{\leq
m+n+6}\|^{2}_{LE_{\gamma}^{1}[0,T]}\lesssim\varepsilon^{2}.$

Indeed, pick any $|\alpha|\leq m+n+6$ and apply Theorem 4.2 with
$\gamma=\frac{3}{2}+q$ to $\phi_{\alpha}$. We obtain, using also (3.10)

(7.4) $\begin{split}&\sup_{0\leq\tilde{t}\leq
T}E_{\gamma}[\phi_{\alpha}](\tilde{t})+\|\phi_{\alpha}\|^{2}_{LE_{\gamma}^{1}[0,T]}\lesssim
E_{[}\phi_{\alpha}](0)+E[\phi_{\alpha}](T)+\|\phi_{\alpha}\|^{2}_{LE^{1}_{w}[0,T]}\\\
&+\|(\phi^{p})_{\alpha}\|_{LE_{\gamma}^{*}[0,T]}^{2}+\|r^{-2+}{\partial}\phi_{\leq|\alpha|+1}\|^{2}_{LE_{\gamma}^{*}[0,T]}.\end{split}$

By (3.5) we have that

$E[\phi_{\alpha}](T)+\|\phi_{\alpha}\|^{2}_{LE^{1}_{w}[0,T]}\lesssim\varepsilon^{2}.$

Since $\gamma<2$, we have

(7.5)
$\|r^{-2+}{\partial}\phi_{\leq|\alpha|+1}\|^{2}_{LE_{\gamma}^{*}[0,T]}\lesssim\|\phi_{\leq
m+n+9}\|^{2}_{LE^{1}}\lesssim\varepsilon^{2}.$

Moreover, (3.14) and (7.2) yield

$\begin{split}\|(\phi^{p})_{\alpha}\|^{2}_{LE_{\gamma}^{*}[0,T]}&\lesssim(\tilde{C}\varepsilon)^{2(p-1)}\|\left(\langle
r\rangle^{-1}\langle\tilde{t}-{\tilde{r}}\rangle^{-\gamma}\right)^{p-1}\phi_{\leq|\alpha|}\|^{2}_{LE_{\gamma}^{*}[0,T]}\leq(\tilde{C}\varepsilon)^{2(p-1)}\|\langle
r\rangle^{-2}\phi_{\leq|\alpha|}\|^{2}_{LE_{\gamma}^{*}[0,T]}\\\
&\leq(\tilde{C}\varepsilon)^{2(p-1)}\|\phi_{\leq|\alpha|}\|^{2}_{LE_{\gamma}^{1}[0,T]}.\end{split}$

and this term can be absorbed in the LHS of (7.4) for small enough
$\varepsilon$. The conclusion (7.3) now follows.

In particular, we have that

(7.6) $\|\langle r\rangle^{-1}\phi_{\leq
m+n+6}\|_{LE_{\gamma}}\lesssim\varepsilon.$

We now finish the argument. By Lemma 6.3, (5.6) and (3.5), we obtain

(7.7) $\psi_{1}\lesssim\frac{\log\langle t-r\rangle}{\langle r\rangle\langle
t-r\rangle^{\frac{1}{2}}}\|r(G_{\alpha|})_{\leq
n}\|_{LE^{*}}\lesssim\frac{1}{\langle r\rangle\langle
t-r\rangle^{q}}\|\phi_{m+n}\|_{LE^{1}}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle t-r\rangle^{q}}.$

Moreover, (7.6) and (6.12) yield

(7.8) $\psi_{2}\lesssim\frac{1}{\langle
r\rangle(t-{\tilde{r}})^{q}}\left\|r^{-1}\phi_{\leq|\alpha|+6}\right\|_{LE_{\gamma}}^{2}\lesssim\frac{\varepsilon^{2}}{\langle
r\rangle(t-{\tilde{r}})^{q}}.$

Then (7.7) and (7.8) imply that

$\psi\lesssim\frac{\varepsilon}{\langle r\rangle\langle t-r\rangle^{q}},$

and the conclusion follows if $\tilde{C}$ is large enough.

##  8\. Improved estimates in the interior and for derivatives

For the forward cone $C=\\{r\leq t\\}\cap\mathcal{M}$ we consider a dyadic
decomposition in time into sets

$C_{T}=\\{T\leq t\leq 2T,\ \ r\leq t\\}\cap\mathcal{M}.$

For each $C_{T}$ we need a further double dyadic decomposition of it with
respect to either the size of $t-r$ or the size of $r$, depending on whether
we are close or far from the cone,

$C_{T}=\bigcup_{1\leq R\leq T/4}C_{T}^{R}\cup\bigcup_{1\leq U<T/4}C_{T}^{U},$

where for $R>4M$ , $U>1$ we set

$C_{T}^{R}=C_{T}\cap\\{R<r<2R\\},\qquad C_{T}^{U}=C_{T}\cap\\{U<t-r<2U\\}$

while for $R=4M$ and $U=1$ we have

$C_{T}^{R=4M}=C_{T}\cap\\{r_{e}<r<4M\\},\qquad
C_{T}^{U=1}=C_{T}\cap\\{0<t-r<2\\}.$

By $\tilde{C}_{T}^{R}$ and $\tilde{C}_{T}^{U}$ we denote enlargements of these
sets in both space and time on their respective scales. We also define

$C_{T}^{<T/2}=\bigcup_{R<T/4}C_{T}^{R}.$

and $\tilde{C}_{T}^{<T/2}$ an enlargement in space-time on its scale.

We will use Propositions 3.15 and 3.16 from [23], stated below. The role of
Proposition 3.15 is twofold: it will allow us to obtain a better bound for the
derivative in the region $r<t/2$, and improve the decay estimate of the
function in the interior region. The role of Proposition 3.16 is to obtain a
better bound for the derivative in the region $r>t/2$.

######  Proposition 8.1.

Assume that $Pu=f$. We have for any $m$ and large enough (but $m$-independent)
$n$ that

(8.1) $\begin{split}\|u_{\leq m}\|_{L^{\infty}(C_{T}^{<T/2})}&\ +\|\langle
r\rangle\nabla u_{\leq m}\|_{L^{\infty}(C_{T}^{<T/2})}\lesssim
T^{-\frac{3}{2}}\|u_{\leq m+n}\|_{LE(\tilde{C}_{T}^{<T/2})}\\\ &\
+T^{-\frac{1}{2}}\Bigl{(}\|f_{\leq
m+n}\|_{LE^{*}(\tilde{C}_{T}^{<T/2})}+\|\langle r\rangle^{2}\nabla f_{\leq
m+n}\|_{LE(\tilde{C}_{T}^{<T/2})}\Bigr{)}.\end{split}$

######  Proposition 8.2.

We have

(8.2) $\begin{split}U\|\nabla u_{\leq m}\|_{L^{\infty}(C_{T}^{U})}\lesssim&\
\|u_{\leq
m+n}\|_{L^{\infty}(\tilde{C}_{T}^{U})}+T^{-\frac{1}{2}}U^{\frac{1}{2}}\|f_{\leq
m+n}\|_{L^{2}(\tilde{C}_{T}^{U})}\\\ &\
+T^{-\frac{1}{2}}U^{\frac{3}{2}}\|\nabla f_{\leq
m+n}\|_{L^{2}(\tilde{C}_{T}^{U})}.\end{split}$

As a quick corollary of (8.1), assume that $\phi$ is the solution to (2.3),
and moreover satisfies

$|\phi_{\leq m+n+1}|\lesssim\frac{1}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{\gamma}},\quad\gamma>0.$

Let $\gamma_{1}=\min\\{\gamma,p\gamma-1\\}$. We then we obtain the improved
pointwise bound in $C_{T}^{<T/2}$

(8.3) $|\phi_{\leq m}|+|\langle r\rangle\nabla\phi_{\leq
m}|\lesssim\frac{1}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{\gamma_{1}}},$

and the improved bound in $C_{T}^{U}$

(8.4) $|\nabla\phi_{\leq
m}|\lesssim\frac{1}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+\gamma_{1}}}.$

Indeed, it is easy to check that

$\|r^{-1/2}\phi_{\leq m+n}\|_{L^{2}(\tilde{C}_{T}^{R})}\lesssim
T^{1/2-\gamma}.$

Moreover, using the fact that in $\tilde{C}_{T}^{<T/2}$ we have

$\phi^{p}_{\leq m+n+1}\lesssim\frac{1}{T^{p\gamma}\langle r\rangle^{p}}$

and furthermore using that $p\geq 3$:

$\|(\phi^{p})_{\leq m+n}\|_{LE^{*}(\tilde{C}_{T}^{<T/2})}+\|\langle
r\rangle^{2}\nabla((\phi^{p})_{\leq m+n})\|_{LE(\tilde{C}_{T}^{<T/2})}\lesssim
T^{\frac{1}{2}-p\gamma}.$

This finishes the proof of (8.3).

In order to prove (8.4), we see that, since in $\tilde{C}_{T}^{U}$ we have

$\phi^{p}_{\leq m+n+1}\lesssim\frac{1}{U^{p\gamma}T^{p}}.$

Since $U\lesssim T$ and $p\geq 3$, we obtain:

$T^{-\frac{1}{2}}U^{\frac{1}{2}}\|(\phi^{p})_{\leq
m+n}\|_{L^{2}(\tilde{C}_{T}^{U})}+T^{-\frac{1}{2}}U^{\frac{3}{2}}\|\nabla(\phi^{p})_{\leq
m+n}\|_{L^{2}(\tilde{C}_{T}^{U})}\lesssim T^{1-p}U^{2-p\gamma}\lesssim
T^{-1}U^{4-p-p\gamma}\lesssim T^{-1}U^{1-p\gamma}.$

##  9\. The bootstrap argument

We will now finish the proof. We assume that the bound (7.1) holds for some
$q$ close to $\frac{1}{2}$, and improve it to (2.4).

Note first that (7.1) already gives the desired rate near the cone, when
$\tilde{t}-{\tilde{r}}\approx 1$, and thus we can assume from now on that
$\tilde{t}-{\tilde{r}}>1$.

Let $\chi_{0}$ be a smooth cutoff supported in the region
$\frac{\tilde{t}}{2}\leq\tilde{t}-{\tilde{r}}\leq 2\tilde{t}$ and identically
one when $\frac{3\tilde{t}}{4}\leq\tilde{t}-{\tilde{r}}\leq\tilde{t}$ . We now
write $G_{\alpha}$ in the form

$G_{\alpha}=g_{1}\phi_{\leq|\alpha|+6}+\partial_{t}(g_{2}\phi_{<|\alpha|+6}),\qquad
g_{1}\in S^{Z}(r^{-3}),\ \ g_{2}\in S^{Z}_{der}(r^{-2}).$

Here we can confine ourselves to $\partial_{t}$ derivatives in the last term
because for any $S$ and $\Omega$ component we gain a factor of $r^{-1}$ and
include it in the first term. We now split $G_{\alpha}$ into two parts, so
that later on we can apply Lemma 6.1:

$G_{\alpha}=G_{\alpha}^{1}+\partial_{t}G_{\alpha}^{2},$

with

$G_{\alpha}^{1}=g_{1}\phi_{\leq|\alpha|+6}+\partial_{t}\left((1-\chi_{0})g_{2}\phi_{<|\alpha|+6}\right),\quad
G_{\alpha}^{2}=\chi_{0}g_{2}\phi_{<|\alpha|+6}.$

Let $q_{1}=\min\\{q,pq-1\\}$; more precisely, $q_{1}=q$ when $p\geq 4$, and
$q_{1}=3q-1$ when $p=3$. Using (8.3) and (8.4), we see that

(9.1) $|\phi_{\leq m+n}|+|\langle r\rangle\nabla\phi_{\leq
m+n}|\lesssim\frac{1}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{q_{1}}},$
$|\nabla\phi_{\leq
m+n}|\lesssim\frac{1}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+q_{1}}}.$

In particular, we have, for $|\alpha|\leq m+n-12$

(9.2)
$\begin{split}&\|G_{\alpha}^{1}\|_{L_{3,1,q_{1}}^{\infty}}\lesssim\|\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{q_{1}}\phi_{\leq|\alpha|+12}\|_{L^{\infty}L^{\infty}}\lesssim\varepsilon\\\
&\|G_{\alpha}^{2}\|_{L_{2,1,q_{1}}^{\infty}}+\|SG_{\alpha}^{2}\|_{L_{2,1,q_{1}}^{\infty}}+\|\Omega
G_{\alpha}^{2}\|_{L_{2,1,q_{1}}^{\infty}}+\|\langle\tilde{t}-{\tilde{r}}\rangle{\partial}G_{\alpha}^{2}\|_{L_{2,1,q_{1}}^{\infty}}\lesssim\|\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{q_{1}}\phi_{\leq|\alpha|+12}\|_{L^{\infty}L^{\infty}}\lesssim\varepsilon.\end{split}$

Consider first the case $p=3$. We note that, due to (9.1)

$F_{\alpha}\lesssim\frac{\varepsilon}{\langle\tilde{t}\rangle^{3}\langle\tilde{t}-{\tilde{r}}\rangle^{3q_{1}}}\lesssim\frac{\varepsilon}{\langle
r\rangle^{2}\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{3q_{1}}}.$

Lemma 6.1 (with $\beta=2$, $\gamma=1$ and $\eta=3q_{1}$) yields, provided that
we pick $q_{1}>1/3$ (which requires $q>4/9$):

(9.3) $\psi_{2}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle}.$

On the other hand, due to (9.2) and (9.1), we can apply Lemma 6.1 with
$\beta=3$, $\gamma=1$ and $\eta=q_{1}$. We obtain

(9.4) $\psi_{1}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+q_{1}-\delta}}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle}.$

We get by (9.1), (9.3) and (9.4) that

$\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle}.$

Using (8.3) we can turn the $r$ into a $t$ for $|\alpha|\leq m$:

(9.5)
$\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle}.$

which finishes the proof when $p=3$. We remark that it is the nonlinearity
that dictates the rate of the decay in this case.

Assume now that $p\geq 4$. In this case, (9.1) and (9.2) imply

(9.6)
$\begin{split}&F_{\alpha}\lesssim\frac{\varepsilon}{\langle\tilde{t}\rangle^{p}\langle\tilde{t}-{\tilde{r}}\rangle^{pq}}\lesssim\frac{\varepsilon}{\langle
r\rangle^{3}\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{q}},\qquad\|G_{\alpha}^{1}\|_{L_{3,1,q}^{\infty}}\lesssim\varepsilon\\\
&\|G_{\alpha}^{2}\|_{L_{2,1,q}^{\infty}}+\|SG_{\alpha}^{2}\|_{L_{2,1,q}^{\infty}}+\|\Omega
G_{\alpha}^{2}\|_{L_{2,1,q}^{\infty}}+\|\langle\tilde{t}-{\tilde{r}}\rangle{\partial}G_{\alpha}^{2}|\|_{L_{2,1,q}^{\infty}}\lesssim\varepsilon.\end{split}$

Lemma 6.1 (with $\beta=3$ and $\eta=q$) yields

(9.7) $\psi_{2},\psi_{3}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+q}},\qquad q<1/2.$

We thus obtain by (9.1) and (9.7)

$\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+q}},\qquad q<1/2,$

and by (8.3) we can turn the $r$ into a $t$:

(9.8) $\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle
t\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{1+q}},\qquad q<1/2.$

We now iterate one last time. Due to (9.1) and (9.2), we obtain

$\begin{split}&F_{\alpha}\lesssim\frac{\varepsilon}{\langle\tilde{t}\rangle^{p}\langle\tilde{t}-{\tilde{r}}\rangle^{pq+p}}\lesssim\frac{\varepsilon}{\langle
r\rangle^{3}\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{q+1}},\qquad\|G_{\alpha}^{1}\|_{L_{3,1,q+1}^{\infty}}\lesssim\varepsilon\\\
&\|G_{\alpha}^{2}\|_{L_{2,1,q+1}^{\infty}}+\|SG_{\alpha}^{2}\|_{L_{2,1,q+1}^{\infty}}+\|\Omega
G_{\alpha}^{2}\|_{L_{2,1,q+1}^{\infty}}+\|\langle\tilde{t}-{\tilde{r}}\rangle{\partial}G_{\alpha}^{2}|\|_{L_{2,1,q+1}^{\infty}}\lesssim\varepsilon.\end{split}$

Lemma 6.1 (with $\beta=3$ and $\eta=q+1$) yields

(9.9) $\phi_{2},\phi_{3}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{2}}.$

We thus obtain by (9.8) and (9.9)

$\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle
r\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{2}},$

and by (8.3) we can turn the $r$ into a $t$:

(9.10)
$\phi_{\alpha}\lesssim\frac{\varepsilon}{\langle\tilde{t}\rangle\langle\tilde{t}-{\tilde{r}}\rangle^{2}},$

which finishes the proof.

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# The High-Density Ionized Gas in the Central Parsecs of the Galaxy

Jun-Hui Zhao, Ray Blundell, James M. Moran  D. Downes, Karl F. Schuster  Dan
Marrone

###### Abstract

We report the results from observations of H30$\alpha$ line emission in Sgr A
West with the Submillimeter Array at a resolution of 2′′ and a field of view
of about 40′′. The H30$\alpha$ line is sensitive to the high-density ionized
gas in the minispiral structure. We compare the velocity field obtained from
H30$\alpha$ line emission to a Keplerian model, and our results suggest that
the supermassive black hole at Sgr A* dominates the dynamics of the ionized
gas. However, we also detect significant deviations from the Keplerian motion,
which show that the impact of strong stellar winds from the massive stars
along the ionized flows and the interaction between Northern and Eastern arms
play significant roles in the local gas dynamics.

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS 78,
Cambridge, MA 02138

Institut de Radio Astronomie Millimétrique, 38406 Saint Martin d’Hères, France

Department of Astronomy, University of Chicago, Chicago, IL 60637

##  1\. Introduction

The Galactic Center harbors a supermassive black hole (SMBH) with a mass of
$4.2\times 10^{6}$ M⊙ at the position of Sgr A* (Ghez et al. 2008; Gillessen
et al. 2009). Previous radio recombination line studies (Schwarz, Bregman, &
van Gorkom 1989; Roberts & Goss 1993; Roberts, Yusef-Zadeh, & Goss 1996)
revealed complex structures of ionized gas around Sgr A*, including the
“minispiral” and “Bar.” Parts of these structures have been modeled as gas
orbiting Sgr A* (Schwarz, Bregman, & van Gorkom 1989; Sanders 1998; Paumard,
Maillard, & Morris 2004; Vollmer & Duschl 2000; Liszt 2003; Zhao et al. 2009)
while other parts of the structures may be gravitationally unbound (Yusef-
Zadeh et al. 1998). In particular, the strong stellar winds from the clusters
of massive stars in the vicinity of Sgr A*, with the help of gravitational
focusing, create large-scale, unbound features (Lutz, Krabbe, & Genzel 1993).
There are a number of difficulties in existing line observations of these
structures. In the cm–radio maps from the Very Large Array (VLA), the
recombination line strength is low, the line-to-continuum ratio is low, the
bandwidth coverage is limited or irregular, and it is especially difficult to
detect high-velocity line wings. In the IR, the picture given by the emission
lines from the ionized gas is distorted, because of high obscuration by dust.
In this paper, we summarize the main results from our observations of the
H30$\alpha$ line at 1.3 mm with the SMA from three array configurations, the
combination of which provides an angular resolution of 2′′.

##  2\. Distribution of the High-Density Ionized Gas

Figure 1 presents the image integrated H30$\alpha$ line emission observed in
the central 1.5 parsecs, showing the Northern and Eastern arms of the
minispiral. The Western Arc does not shown well in our image in the
H30$\alpha$ line since most of the line emission is below the 4$\sigma$
sensitivity cutoff. The H30$\alpha$ line spectra are shown toward selected
regions with IR sources. A detailed comparison of the H30$\alpha$ line spectra
and the H92$\alpha$ line spectra observed with the VLA yields line ratios of
H30$\alpha$/H92$\alpha$ in most of the regions that are close to the frequency
ratio $\nu_{H30\alpha}/\nu_{H92\alpha}\sim 28$, suggesting that the ionized
gas in the central parsecs is optically thin and under LTE conditions. In some
regions, however (e.g., IRS 1W and 10W), the line ratios are significantly
smaller than the LTE value.

Figure 1.: The image of the H30$\alpha$ line emission around Sgr A* (plus-dot)
in the central 1.5-parsec region observed with the SMA with a resolution of
2′′. The H30$\alpha$ line spectra of the HII sources within the minispiral are
plotted and labelled with the names of the corresponding IRS sources.

To assess the physical conditions of the ionized gas, we have also fit a model
to the observed lines based on the assumption that the HII sources are
isothermal and homogeneous. The derived typical electron densities in the HII
clumps are 5–10$\times 10^{4}$ cm-3, consistent with the values ($10^{4}$
cm-3) derived from Pa$\alpha$ line emission at 1.87 $\mu$m (Scoville et al.
2003) for a given uncertainty in extinction corrections in the near IR. The
typical electron temperature in the minispiral arms is 5000–7000 K, in good
agreement with the previous determination from H92$\alpha$ (Roberts & Goss
1993). The highest density of $2\times 10^{5}$ cm-3 occurs in the IRS 13E
region while the highest temperatures of up to 13000 K were found in the Bar.
Both the H30$\alpha$ and H92$\alpha$ line profiles are broadened due to the
large velocity gradients along the line of sight and across the 2′′ beam
produced by the large dynamical motions near Sgr A*. For the H92$\alpha$ line,
the line width due to pressure broadening appears to be comparable to that of
thermal Doppler broadening while for the H30$\alpha$ lines, the pressure
broadening is negligible.

##  3\. Kinematics and Dynamics

Figure 2a shows the image of radial velocities determined from fitting the
peak velocities of the H30$\alpha$ line in the central 40′′, showing large
velocity gradients along the Northern and Eastern arms. The observed radial
velocity field is, in general, consistent with that predicted from a Keplerian
model (Zhao et al. 2009), suggesting that the SMBH at Sgr A* governs the
dynamics of the ionized gas. However, in several regions, the observed
kinematics deviate significantly from those predicted from the Keplerian
model, as described below.

Figure 2.: The observed radial velocity field (a) around Sgr A* (plus-dot)
from the SMA observations of the H30$\alpha$ line is compared with that
computed from the Keplerian model (b). The velocity wedge is in units of km
s-1. The inset (c) is the H30$\alpha$ spectrum toward the IRS 34 region
outlined with the black circle NW of Sgr A* in which five massive stars 7SW
(WN8), 34W (Ofpe/WN9), 34E (O9-9.5), 34NW (WN7), and 3E (WC5/6) are located
(shown with “+” symbols, Paumard et al. 2006). The arrow indicates the
direction of the averaged transverse velocity of the four stars that have
proper-motion information. The red circle indicates the region with high
electron temperature where the interaction between the Northern and Eastern
arms occurs. The arrow SE of Sgr A* indicates the transverse velocity of the
IRS 9W star (WN8).

IRS 34: In the northwest end of the Eastern Arm, a kinematic feature with
radial velocity of up to $-200$ km s-1 shows a large deviation from the value
of $-100$ km s-1 predicted from the Keplerian model (see Figure 2b). Near this
region, there is a group of Wolf-Rayet (WR) stars, evolved from early-type O
stars with a mass loss rate of 10-5 M⊙ yr-1. The suspected strong stellar
winds from these massive stars may play a considerable role in the local
kinematics.

IRS 9W: In the “tip” region (Paumard, Maillard, & Morris 2004), a peculiar
radial velocity of +300 km s-1 was detected from our H30$\alpha$ observations,
showing a large deviation from the radial velocity of +150 km s-1 predicted
from the Keplerian model. The impact of the strong stellar wind from the IRS
9W star (WN8) on the orbiting ionized flow in the Eastern Arm is likely
responsible for the high velocity of the tip.

The Bar: Deviations from Keplerian motion are also observed in the Bar region.
These non-Keplerian kinematics could be attributed to the interaction between
the ionized flows in both Northern and Eastern arms and the interaction of
ionized flows with the strong winds from the massive star clusters, e.g., IRS
16 and 13.

Finally, in the locations with large offsets from Sgr A*, we noticed that the
deviations in radial velocity from the Keplerian motions become significant
for the Northern Arm ($>$ 50 km s-1) while for the Eastern Arm the deviations
are less significant. The difference in the deviation of the observed radial
velocities from the Keplerian velocities between the Northern and Eastern arms
might suggest that the distribution of mass has a preference for the plane of
the circumnuclear disk, which is coplanar with the orbit of the Northern Arm
flow.

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