STRICHARTZ ESTIMATES ON SCHWARZSCHILD BLACK HOLE

STRICHARTZ ESTIMATES ON SCHWARZSCHILD BLACK HOLE BACKGROUNDS

arXiv:0802.3942v3 [math.AP] 28 Aug 2009

JEREMY MARZUOLA, JASON METCALFE, DANIEL TATARU, AND MIHAI TOHANEANU We study dispersive properties for the wave equation in the Schwarzschild space-time. The rst result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere. Abstract.

1.

Introduction

The aim of this article is to contribute to the understanding of the global-in-time dispersive properties of solutions to wave equations on Schwarzschild black hole backgrounds. Precisely, we consider two robust ways to measure dispersion, namely the local energy estimates and the Strichartz estimates. Let us begin with the local energy estimates. For solutions to the constant coecient wave equation in 3 + 1 dimensions, 2u = 0,

u(0) = u0 ,

we have the original estimates of Morawetz [33] Z tZ

(1.1)

0

R3

ut (0) = u1 ,

1

1 |6 ∇u|2 (t, x) dt dx . k∇u0 k2L2 + ku1 k2L2 |x|

where 6 ∇ denotes the angular derivative. To prove this one multiplies the wave equation by the multiplier (∂r + 1r )u and integrates by parts. Within dyadic spatial regions one can also control u, ∂t u and ∂r u. Precisely, we have the local energy estimates (1.2)

1

3

R− 2 k∇ukL2 (R×B(0,R)) + R− 2 kukL2 (R×B(0,R)) . k∇u0 kL2 + ku1 kL2 .

See for instance [20], [22], [40], [41], [42]. One can also consider the inhomogeneous problem, (1.3)

2u = f,

u(0) = u0 ,

ut (0) = u1 ,

The authors were supported in part by the NSF grants DMS0354539 and DMS0301122 . 1 There is another estimate commonly referred to as a Morawetz estimate. This corresponds to using the multiplier (t2 + r2 )∂t + 2tr∂r . We will reserve the term Morawetz estimate for (1.1) and shall call the latter estimate the Morawetz conformal estimate. 1

2

JEREMY MARZUOLA, JASON METCALFE, DANIEL TATARU, AND MIHAI TOHANEANU

In view of (1.2) we dene the local energy space LEM for the solution u by (1.4)

h j i 3j kukLEM = sup 2− 2 k∇ukL2 (Aj ) + 2− 2 kukL2 (Aj ) , j∈Z

where Aj = R × {2j ≤ |x| ≤ 2j+1 }..

For the inhomogeneous term f we introduce a dual type norm ∗ = kf kLEM

X

j

2 2 kf kL2 (Aj ) .

j∈Z

Then we have:

Theorem 1.1. The solution u to (1.3) satises the following estimate: (1.5)

∗ kukLEM . k∇u0 kL2 + ku1 kL2 + kf kLEM

One may ask whether similar bounds also hold for perturbations of the Minkowski space-time. Indeed, in the case of small long range perturbations the same bounds as above were established very recently by two of the authors, see [30, Proposition 2.2] or [28, (2.23)] (with no obstacle, Ω = ∅). See also [1], [27] for related local energy estimates for small perturbations of the d'Alembertian. For large perturbations one faces additional diculties, due on one hand to trapping for large frequencies and on the other hand to eigenvalues and resonances for low frequencies. The Schwarzschild space-time, considered in the present paper, is a very interesting example of a large perturbation of the Minkowski space-time, where trapping causes signicant diculties. The Schwarzschild space-time M is a spherically symmetric solution to Einstein's equations with an additional Killing vector eld K , which models the exterior of a massive spherically symmetric body. Factoring out the S2 component it can be represented via the Penrose diagram: The radius r of the S2 spheres is intrinsically determined and is a smooth function on M which has a single critical point at the center. The regions I and I 0 represent the exterior of the black hole, respectively its symmetric twin, and are characterized by the relation r > 2M . We can represent I as I = R × (2M, ∞) × S2

with a metric whose line element is (1.6)

2M 2 2M −1 2 ds2 = − 1 − dt + 1 − dr + r2 dω 2 r r

where dω 2 is the measure on the sphere S2 . The Killing vector eld K is given by K = ∂t , which is time-like within I . The dierential dt is intrinsic, but the function t is only dened up to translations on I . The regions II and II 0 represent the black hole, respectively its symmetric twin, the white hole, and are characterized by the relation r < 2M . The same metric as in (1.6) can be used. The Killing vector eld K is still given by K = ∂t , which is now space-like. Light rays can enter the black hole but not leave it. By symmetry light rays can leave the white hole but not enter it.


r=0

r=∞

3

r=∞ i

i+

+

r = 2M I0

I

II

i0

i0 I0

II 0

I

t=∞

t=0

i−

i− r=0

r=∞

r=∞

The Penrose diagram for the Kruskal extension of the Schwarzschild solution Figure 1.

The surface r = 2M is called the event horizon. While the singularity at r = 0 is a true metric singularity, we note that the apparent singularity at r = 2M is merely a coordinate singularity. Indeed, denote r∗ = r + 2M log(r − 2M ) − 3M − 2M log M,

so that

2M −1 dr∗ = 1 − dr, r∗ (3M ) = 0 r and set v = t + r∗ . Then in the (r, v, ω) coordinates the metric in region I is expressed

in the form

2M 2 ds2 = − 1 − dv + 2dvdr + r2 dω 2 , r which extends analytically into the black hole region I + II . In particular, given a choice of the function t in region I , this uniquely determines the function t in the region II via

the same change of coordinates. In a symmetric fashion we set w = t − r∗ . Then in the (r, w, ω) coordinates the metric is expressed in the form 2M 2 ds2 = − 1 − dw − 2dwdr + r2 dω 2 , r which extends analytically into the white hole region I + II 0 .

One can also introduce global nonsingular coordinates by rewriting the metric in the Kruskal-Szekeres coordinate system, v

v 0 = e 4M ,

w

w0 = −e− 4M .

4


However, this is of less interest for our purposes here. Further information on the Schwarzschild space can be found in a number of excellent texts. We refer the interested reader to, e.g., [18], [31], and [51]. As far as the results in this paper are concerned, for large r the Schwarzschild spacetime can be viewed as a small perturbation of the Minkowski space-time. The diculties in our analysis are caused by the dynamics for small r, where trapping occurs. The presence of trapped rays, i.e. rays which do not escape either to innity or to the singularity r = 0, are known to be a signicant obstacle to proving local energy, dispersive, and Strichartz estimates and, in some case, are known to necessitate a loss of regularity. See, e.g., [10] and [37]. There are two places where trapping occurs on the Schwarzschild manifold. The rst is the surface r = 3M which is called the photon sphere. Null geodesics which are initially tangent to the photon sphere will remain on the surface for all times. Microlocally the energy is preserved near such periodic orbits. However what allows for local energy estimates near the photon sphere is the fact that these periodic orbits are hyperbolic. The second is at the event horizon r = 2M , where the trapped geodesics are the vertical ones in the (r, v, ω) coordinates. However, this second family of trapped rays turns out to cause no diculty in the decay estimates since in the high frequency limit the energy decays exponentially along it as v → ∞. This is due to the fact that the frequency decays exponentially along the Hamilton ow, and in the physics literature it is well-known as the red shift eect. To describe the decay properties of solutions to the wave equation in the Schwarzschild space, it is convenient to use coordinates which make good use of the Killing vector eld and are nonsingular along the event horizon. The (r, v, ω) coordinates would satisfy these requirements. However the level sets of v are null surfaces, which would cause some minor diculties. This is why in I + II we introduce the function v˜ dened by v˜ = v − µ(r)

where µ is a smooth function of r. In the (˜v , r, ω) coordinates the metric has the form 2M 0 2M 2 d˜ v +2 1− 1− µ (r) d˜ v dr ds = − 1 − r r 2M 0 (µ (r))2 dr2 + r2 dω 2 . + 2µ0 (r) − 1 − r 2

On the function µ we impose the following two conditions: (i) µ(r) ≥ r∗ for r > 2M , with equality for r > 5M /2. (ii) The surfaces v˜ = const are space-like, i.e. 2M 0 2− 1− µ (r) > 0. r The rst condition (i) insures that the (r, v˜, ω) coordinates coincide with the (r, t, ω) coordinates in r > 5M /2. This is convenient but not required for any of our results. What is important is that in these coordinates the metric is asymptotically at asr → ∞. −1 0 In the proof of the Strichartz estimates, it is also required that µ (r) = 1 − 2M near r µ0 (r) > 0,


5

r = 3M , which in other words says that we can work in the (r, t) coordinates near the

photon sphere. However, this may be merely an artifact of our method. We introduce a symmetric function v˜1 in I 0 + II , as well as the the functions w˜ and w ˜1 in I + II 0 , respectively I 0 + II 0 . Given a parameter 0 < r0 < 2M we partition the Schwarzschild space into seven regions M = MR ∪ ML ∪ M0R ∪ M0L ∪ MT ∪ MC ∪ MB

as in Figure 2. The right/left top/bottom regions are

r = r0 < 2M

MT

r = 3M

ML

MC

MR

r = 3M

i0

i0

M0L

M0R MB r = r0 < 2M

The Schwarzschild space partition represented on the Penrose diagram Figure 2.

MR = {˜ v ≥ 0, r ≥ r0 } ⊂ I + II,

ML = {˜ v1 ≥ 0, r ≥ r0 } ⊂ I 0 + II,

M0R = {w ˜ ≤ 0, r ≥ r0 } ⊂ I + II 0 ,

M0L = {w ˜1 ≤ 0, r ≥ r0 } ⊂ I 0 + II 0 ,

the top and bottom regions are

MT = {r < r0 } ∩ II,

MB = {r < r0 } ∩ II 0 ,

and the central region MC is the remainder of M. Moreover, dene Σ− v = 0} R = MR ∩ {˜ Σ+ R = MR ∩ {r = r0 }.

and similarly for the other regions. In what follows we consider the Cauchy problem ˜ |Σ = φ1 (1.7) 2g φ = f, φ|Σ0 = φ0 , Kφ 0 where for convenience we choose the initial surface Σ0 to be the horizontal surface of symmetry Σ0 = {t = 0} ∩ (I + I 0 )

6


˜ is smooth, everywhere timelike and equals K on Σ0 outside MC . Observe that and K we cannot use K on all of Σ0 since it is degenerate at the center (i.e. on the bifurcate sphere). The equation (1.7) can be solved as follows: (i) Solve the equation in MC with Cauchy data on Σ0 . Since MC is compact and has forward and backward space-like boundaries, this is a purely local problem. (ii) Solve the equation in MR with Cauchy data on Σ− R . The forward boundary of MR is Σ+ , which is space-like. This is the most interesting part, where we are interested R in the decay properties as v˜ → ∞. In a similar manner solve the equation in ML , M0R and M0L . (iii) Solve the equation in MT with initial data on the space-like surface ΣT = {r = r0 } ∩ II . Here one can track the solution up to the singularity and encounter a mix of local and global features. This part of the analysis in not pursued in the present article. A signicant role in our analysis is played by the Killing vector eld K , which in the (r, v˜) coordinates equals ∂v˜ . This is time-like outside the black hole but space-like inside it. Furthermore, it is degenerate at the center. Using the Killing vector eld outside the black hole we obtain a conserved energy E0 [φ] for solutions φ to the homogeneous equation 2g φ = 0. On surfaces t = const in the (r, t) coordinates the energy E0 [φ](t) has the form

(1.8)

Z

∞ h

Z

1−

E0 [φ] = S2

2M

i 2M 2M −1 (∂t φ)2 + 1 − (∂r φ)2 + |6 ∇φ|2 r2 drdω. r r

Since the vector eld K is degenerate at the center, so is the corresponding energy E0 at r = 2M . Hence it would be natural to replace it with a nondegenerate energy, which on the initial surface Σ0 can be expressed as (1.9) E[φ](Σ0 ) =

Z

Z

S2

∞ h

2M

3

1

i 2M − 2 2M 2 1− (∂t φ)2 + 1 − (∂r φ)2 + |6 ∇φ|2 r2 drdω. r r

Unfortunately this is no longer conserved, and this is one of the diculties which we face in our analysis. We remark that a related form of a nondegenerate energy expression was introduced in [14] and proved to be bounded in the exterior region on surfaces t = const. Part of the novelty of our approach is to prove bounds not only in the exterior region, but also inside the event horizon. This is natural if one considers the fact that the singularity at r = 2M is merely a removable coordinate singularity. In order to do this, it is no longer suitable to measure the evolution of the energy on the surfaces t = const (see below). Thus the above energy E[φ](Σ0 ) is relegated to a secondary role here and is used only to measure the size of the initial data. A priori the energy E[φ](t) of φ only determines its Cauchy data at time t modulo constants. However, in what follows we implicitly assume that φ decays at ∞, in which case φ can be also estimated via a Hardy-type inequality, (1.10)

Z

1

1−

2M − 2 −2 2 2 r φ r drdω . r

Z

1

1−

2M 2 (∂r φ)2 r2 drdω. r

This is proved in a standard manner; the details are left to the reader.


7

We shall now further describe our main estimates in the region MR : the local energy decay, the WKB analysis which yields a local energy decay with only a logarithmic loss, and nally the Strichartz estimates. For the initial energy on Σ− R we use E[φ](Σ− R) =

Z Σ− R

|∂r φ|2 + |∂v˜ φ|2 + |6 ∇φ|2 r2 drdω.

For the nal energy on Σ+ R we set E[φ](Σ+ R) =

Z Σ+ R

|∂r φ|2 + |∂v˜ φ|2 + |6 ∇φ|2 r02 d˜ v dω.

We also track the energy on the space-like slices v˜ = const, Z

|∂r φ|2 + |∂v˜ φ|2 + |6 ∇φ|2 r2 drdω.

E[φ](˜ v0 ) = MR ∩{˜ v =˜ v0 }

Thus = E[φ](0). For the local energy estimates one may rst consider a direct analogue of the Minkowski bound (1.5). Unfortunately such a bound is hopeless due to the trapping which occurs at r = 3M . Instead, for our rst result we dene a weaker preliminary local energy space LE0 with norm (1.11) ! E[φ](Σ− R)

kφk2LE0 =

Z

MR

2 3M 1 1 1 1 2 2 2 |∂ φ| + 1 − |∂ φ| + |6 ∇ φ| + 4 φ2 r v ˜ r2 r r2 r r

r2 drd˜ v dω.

Compared to the LEM norm we note the power loss in the angular and v˜ derivatives at r = 3M . The LE0 norm is also weaker than LEM as r → ∞, but this is merely for convenience. At the same time we would like to also consider the inhomogeneous problem 2g φ = f . To measure the inhomogeneous term f , we introduce the norm LE0∗ , which is stronger ∗ : than LEM (1.12)

kf k2LE0∗

−2 3M = 1− r2 f 2 r2 drd˜ v dω. r MR Z

Again the important dierence is at r = 3M . Our rst local energy estimate is the following:

Theorem 1.2. Let φ solve the inhomogeneous wave equation 2g φ = f on the Schwarzschild manifold. Then we have (1.13)

2 E[φ](Σ+ v ) + kφk2LE0 . E[φ](Σ− R ) + sup E[φ](˜ R ) + kf kLE0∗ . v ˜≥0

Here we made no eort to optimize the weights at r = 3M and r = ∞. This is done later in the paper. On the other hand the above estimate follows from a relatively simple application of the classical positive commutator method. The advantage of having even such a weaker estimate is that it is sucient in order to allow localization near the interesting regions r = 3M and r = ∞, which can then be studied in greater detail using specic tools.

8


The rst related results regarding the solution of the wave equation on Schwarzschild backgrounds were obtained in [50] and [24] which proved uniform boundedness in region I (including the event horizon). The rst pointwise decay result (without, however, a rate of decay) was obtained in [49]. Heuristics from [36] suggest that solutions to the wave equation in the Schwarzschild case should locally decay like v −3 . For spherically symmetric data a v −3+ decay rate was obtained in [16], and under the additional assumption of the initial data vanishing near the event horizon, the v −3 decay rate was proved in [23]. In general the best known decay rate, proved in [14], is v −1 (see also [7]). We also refer the reader to [38], where optimal pointwise decay rates for each spherical harmonic are established for a closely related problem. Estimates related to (1.13) were rst proved in [25] for radially symmetric Schrödinger equations on Schwarzschild backgrounds. In [2, 3, 4], those estimates are extended to allow for general data for the wave equation. The same authors, in [5, 6], have provided studies that give improved estimates near the photon sphere r = 3M . Moreover, we note that variants of these bounds have played an important role in the works [7] and [14] which prove analogues of the Morawetz conformal estimates on Schwarzschild backgrounds. This allows one to deduce a uniform decay rate for the local energy away from the event horizon, though there is necessarily a loss of regularity due to the trapping that occurs at the photon sphere. Instead in this paper we restrict ourselves to time translation invariant estimates, and we aim to clarify/streamline these as much as possible. All of the above articles use the conserved (degenerate) energy E0 [φ] on time slices, obtained using the Killing vector eld ∂t . As such, their estimates are degenerate near the event horizon. Further progress was made in [14], where an additional vector eld was introduced near the event horizon, in connection to the red shift eect. This enabled them to obtain bounds in the exterior region involving a nondegenerate form of the energy related to (1.9). The approach of [25], [2], [7] and [14] is to write the equation using the Regge-Wheeler tortoise coordinate and to expand in spherical harmonics. For the equation corresponding to each spherical harmonic, one uses a multiplier which changes sign at the critical point of the eective potential. Here we work in the coordinates (r, v˜, ω), though this is not of particular signicance, and we do not expand into spherical harmonics. We prove (1.13) using a positive commutator argument which requires a single dierential multiplier. We hope that this makes the methods more robust for other potential applications. During nal preparations of this article, localized energy estimates proved without using the spherical harmonic decomposition also appeared in [15]. The methods contained therein are somewhat dierent from ours. ∗ Compared to the stronger norms LEM , LEM the weights in (1.13) have a polynomial singularity at r = 3M , which corresponds to the family of trapped geodesics on the photon sphere. As a consequence of the results we prove later, see Theorem 3.2, the latter fact can be remedied to produce a stronger estimate.

Theorem 1.3. Let φ solve the inhomogeneous wave equation 2g φ = f on the Schwarzschild manifold. Then (1.13) still holds if the coecient (1 − 3M/r)2 in the LE0 and the


LE0∗

norms is replaced by

9

−2 3M 1 − ln 1 − . r

Now we have only a logarithmic singularity at r = 3M . The result above is only stated in this form for the reader's convenience. The full result in Theorem 3.2 is stronger but also more complicated to state since it provides a more precise microlocal local energy estimate. The logarithmic loss is not surprising, since it is characteristic of geometries with trapped hyperbolic orbits (see for instance [9], [12], [34]). Indeed, a similar estimate in the semiclassical setting is obtained in [13] using entirely dierent techniques. Note, however, that the aforementioned estimate only involves logarithmic loss of the frequency; our result is stronger since it also implies bounds for k(ln |r∗ |)−1 ukL2 , which are necessary in order to prove Strichartz estimates. There are two regions on which the analysis is distinct. The metric is asymptotically at, and thus, near innity, one can retrieve the classical Morawetz type estimate. On the other hand, around the photon sphere r = 3M we take an expansion into spherical harmonics as well as a time Fourier transform. Then it remains to study an ordinary dierential equation which is essentially similar to (∂x2 − λ2 (x2 + ))u = f,

|| 1,

|x| . 1.

For this we use a rough WKB approximation in the hyperbolic region combined with energy estimates in the elliptic region. Airy type dynamics occur near the zeroes of the potential. Even though it is weaker, the initial bound in Theorem 1.2 plays a key role in the analysis. Precisely, it allows us to glue together the estimates in the two regions described above. We next consider the Strichartz estimates. For solutions to the constant coecient wave equation on R × R3 , the well-known Strichartz estimates state that (1.14) k|Dx |−ρ1 ∇ukLpt 1 Lqx1 . k∇u(0)kL2 + k|Dx |ρ2 f k p02 q20 . Lt Lx

Here the exponents (ρi , pi , qi ) are subject to the scaling relation (1.15)

3 1 3 + = −ρ p q 2

and the dispersion relation (1.16)

1 1 1 + ≤ , p q 2

2 < p ≤ ∞.

All pairs (ρ, p, q) satisfying (1.15) and (1.16) are called Strichartz pairs. Those for which the equality holds in (1.16) are called sharp Strichartz pairs. Such estimates rst appeared in the seminal works [8], [43, 44] and as stated include contributions from, e.g., [17], [35], [19], [26], and [21]. If one allows variable coecients, such estimates are well-understood locally-in-time. For smooth coecients, this was rst shown in [32] and later for C 2 coecients by [39] and [45, 46, 47].

10


Globally-in-time, the problem is more delicate. Even a small, smooth, compactly supported perturbation of the at metric may refocus a group of rays and produce caustics. Thus, constructing a parametrix for incoming rays proves to be quite dicult. At the same time, one needs to contend with the possibility of trapped rays at high frequencies and with eigenfunctions/resonances at low frequencies. Global-in-time estimates were shown for small, long range perturbations of the metric in [29] using an outgoing parametrix. In order to keep the parametrix outgoing one must allow evolution both forward and backward in time. This construction is based on an earlier argument in [48] for the Schrödinger equation. The smallness assumption, however, precludes trapping and does not permit a direct application to the current setup. On the other hand, a second result of [29] asserts that even for large, long range perturbations of the metric one can still establish global-in-time Strichartz estimates provided that a strong form of the local energy estimates holds. This switches the burden to the question of proving local energy estimates. The result in [29] cannot be applied directly to the present problem due to the logarithmic losses in the local energy estimates near the trapped rays. However, it can be applied for the near innity part of the solution. In a bounded spatial region, on the other hand, we take advantage of the local energy estimates to localize the problem to bounded sets, in which estimates are shown using the local-in-time Strichartz estimates of [39], [45]. Thus we obtain

Theorem 1.4. If φ solves 2g φ = (ρ1 , p1 , q1 ) and (ρ2 , p2 , q2 ) we have (1.17)

f

in MR then for all nonsharp Strichartz pairs 2

E[φ](Σ+ v ) + k∇φkLp1 H˙ x−ρ1 ,q1 . E[φ](Σ− R ) + sup E[φ](˜ R ) + kf k v ˜

v ˜

2

p0

0 ρ ,q2

˙ x2 Lv˜2 H

.

Here the Sobolev-type spaces H˙ s,p coincide with the usual H˙ s,p homogeneous spaces in R3 expressed in polar coordinates (r, ω). As a corollary of this result one can consider the global solvability question for the energy critical semilinear wave equation in the Schwarzschild space,  in M  2g φ = ±φ5 (1.18)  ˜ = φ1 in Σ0 . φ = φ0 , Kφ

Theorem 1.5. Let r0 > 0. Then there exists > 0 so that for each initial data (φ0 , φ1 ) which satises E[φ](Σ0 ) ≤

the equation (1.18) admits an unique solution φ in the region {r > r0 } which satises the bound E[φ](Σr0 ) + kφkH˙ s,p ({r>r0 }) . E[φ](Σ0 )

for all indices s, p satisfying

1 4 =s+ , p 2

0≤s
r0 }) appearing above, in view of the ambiguity due to the choice of coordinates. In a compact neighbourhood of the center region MC this is nothing but the classical H s,p norm. By compactness, dierent choices of coordinates lead to equivalent norms. Consider now the upper exterior region MR (as well as its three other mirror images). Using the coordinates (˜v , x) with x = ωr, we dene H˙ s,p (MR ) as the restrictions to R+ × {|x| > r0 } of functions in the homogeneous Sobolev space H˙ s,p (R × R3 ). Acknowledgements: The authors are grateful to M. Dafermos and I. Rodnianski for pointing out their novel way of taking advantage of the red shift eect in [14], and to N. Burq and M. Zworski for useful conversations concerning the analysis near trapped null geodesics. 2.

The Morawetz-type estimate

In this section, we shall prove Theorem 1.2. We note that the estimate (1.13) is trivial over a nite v˜ interval by energy estimates for the wave equation; the diculty consists in proving a global bound in v˜. By the same token, once we prove (1.13) for some choice of r0 < 2M , we can trivially make the transition to any r0 < 2M due to the local theory. Thus in the arguments which follow we reserve the right to take r0 suciently close to 2M . We consider solutions to the inhomogeneous wave equation on the Schwarzschild manifold in MR , which is given by 2g φ = ∇α ∂α φ = f.

Here ∇ represents the metric connection. Associated to this equation is an energymomentum tensor given by 1 Qαβ [φ] = ∂α φ∂β φ − gαβ ∂ γ φ∂γ φ. 2 A simple calculation yields the most important property of Qαβ , namely that if φ solves the homogeneous wave equation then Qαβ [φ] is divergence-free: ∇α Qαβ [φ] = 0,

More generally, we have

if ∇α ∂α φ = 0.

∇α Qαβ [φ] = ∂β φ 2g φ.

In order to prove Theorem 1.2, we shall contract Qαβ with a vector eld X to form the momentum density Pα [φ, X] = Qαβ [φ]X β .

Computing the divergence of this vector eld, we have ∇α Pα [φ, X] = 2g φXφ + Qαβ [φ]π αβ ,

where παβ =

is the deformation tensor of X .

1 (∇α Xβ + ∇β Xα ) 2

12


If X is the Killing vector eld K then the above divergence vanishes, (2.1)

∇α Pα [φ, K] = 0 if 2g φ = 0.

This gives rise to the E0 [φ] conservation law outside the black hole. Naively, one may seek vector elds X so that the quadratic form Qαβ [φ]π αβ is positive denite. However, this may not always be possible to achieve. Instead we note that it may be just as good to have the symbol of this quadratic form positive on the characteristic set of 2g . Then it would be possible to make the above quadratic form positive after adding a Lagrangian correction term of the form q∂ γ φ∂γ φ. Such a term can be conveniently expressed in divergence form modulo lower order terms. Precisely, for a vector eld X , a scalar function q and a 1-form m we dene 1 1 Pα [φ, X, q, m] = Pα [φ, X] + qφ∂α φ − ∂α qφ2 + mα φ2 2 2

where m allows us to modify the lower order terms in the divergence formula. Then we obtain the modied divergence relation (2.2)

∇α Pα [φ, X, q, m] = 2g φ Xφ + qφ + Q[φ, X, q, m], 1 Q[φ, X, q, m] = Qαβ [φ]π αβ + q∂ α φ ∂α φ + mα φ ∂ α φ + (∇α mα − ∇α ∂α q) φ2 . 2

Theorem 1.2 is proved by making appropriate choices for X , q and m so that the quadratic form Q[φ, X, q, m] dened by the divergence relation is positive denite. In what follows we assume that X , q and m are all spherically symmetric and invariant with respect to the Killing vector eld K .

Lemma 2.1. There exist smooth, spherically symmetric, K -invariant X , q, and m in satisfying the following properties: (i) X is bounded , |q(r)| . r−1 , |q0 (r)| . r−2 and m has compact support in r . (ii) The quadratic form Q[φ, X, q, m] is positive denite,

r ≥ 2M

2

Q[φ, X, q, m] & r

−2

3M |∂r φ| + 1 − r 2

2

(r−2 |∂v˜ φ|2 + r−1 |6 ∇φ|2 ) + r−4 φ2 .

(iii) X(2M ) points toward the black hole, X(dr)(2M ) < 0, and hm, dri(2M ) > 0. We postpone the proof of the lemma and use it to conclude the proof of Theorem 1.2. Let X , q and m be as in the lemma. We extend them smoothly beyond the event horizon preserving the spherical symmetry and the K -invariance. By (2.1) we can modify the vector eld X without changing the quadratic form Q in (2.2), ∇α Pα [φ, X + CK, q, m] = 2g φ (X + CK)φ + qφ + Q[φ, X, q, m].

Here C is a large constant. We integrate this relation in the region D = {0 < v˜ < v˜0 , r > r0 } In the (r, v˜) coordinates

2


13

using the (r, v˜, ω) coordinates. This yields Z

(2g φ ((X + CK)φ + qφ) + Q[φ, X, q, m]) r2 drd˜ v dω

D

Z =

v˜=˜v0 Z hd˜ v , P [φ, X + CK, q, m]ir drdω − 2

v ˜=0

hdr, P [φ, X + CK, q, m]ir02 d˜ v dω.

r=r0

We claim that if C is large enough and r0 suciently close to 2M then the integrals on the right have the correct sign, (2.3)

Z

hd˜ v , P [φ, X + CK, q, m]ir2 drdω . CE[φ](˜ v1 ),

E[φ](˜ v1 ) . −

v1 ∈ R

v ˜=˜ v1

(2.4)

hdr, P [φ, X + CK, q, m]i & |∂r φ|2 + |∂v˜ φ|2 + |∂ω φ|2 + φ2 ,

r = r0 .

If these bounds hold then the conclusion of the theorem follows by (ii) and CauchySchwarz. Indeed, a direct computation yields 1 hd˜ v , P [φ, ∂v˜ ]i = − 2

2M 02 2M 2 2 −2 2 2µ − 1 − µ |∂v˜ φ| + 1 − |∂r φ| + r |∂ω φ| , r r 0

respectively 2M (∂r − µ0 ∂v˜ )φ∂v˜ φ. hdr, P [φ, ∂v˜ ]i = |∂v˜ φ|2 + 1 − r

On the other hand hd˜ v , P [φ, ∂r ]i =

2M 0 2M 02 1− 1− µ |∂r φ|2 − 2µ0 − 1 − µ ∂v˜ φ∂r φ, r r

while hdr, P [φ, ∂r ]i = −

2M 02 2M 1 − 2µ0 − 1 − µ |∂v˜ φ|2 − 1 − |∂r φ|2 + r−2 |∂ω φ|2 . 2 r r

We compute hd˜ v , P [φ, X + CK]i = (X(d˜ v ) + C)hd˜ v , P [φ, ∂v˜ ]i + X(dr)hd˜ v , P [φ, ∂r ]i.

For large enough C we have X(d˜v ) + C & C . Therefore the rst term on the right is negative denite for r > 2M . More precisely, it is only the coecient of the |∂r φ|2 term which degenerates at r = 2M . However, due to condition (iii) in the lemma we have X(dr)(2M ) < 0; therefore we pick up a negative |∂r φ|2 coecient at r = 2M . Thus we obtain 2M 2 2 −2 2 −hd˜ v , P [φ, X +CK]i ≈ C |∂v˜ φ| + 1 − |∂r φ| + r |∂ω φ| +|∂r φ|2 , r

r > 2M.

Since all the coecients in the quadratic form on the left are continuous, it follows that the above relation extends to r > r0 for some r0 < 2M depending on C , namely (2.5)

0 < 2M − r0 C −1 .

14


In order to prove (2.3) it remains to estimate the lower order terms P [φ, 0, q, m] in terms of the positive contribution above. Since |q| . r−1 and m has compact support in r, we can bound 1 |hd˜ v , P [φ, 0, q, m]i| . r−1 |φ| |∂v˜ φ|2 + |∂r φ|2 2 + r−2 |φ|2 .

Then by Cauchy-Schwarz it suces to estimate Z

∞

1

r−2 |φ|2 r2 dr . C − 2

r0

Z

∞

r0

2M + 1 |∂r φ|2 r2 dr C 1− r

which is a routine Hardy-type inequality. We next turn our attention to (2.4) and begin with the principal part hdr, P [φ, X + CK]i = (X(d˜ v ) + C)hdr, P [φ, ∂v˜ ]i + X(dr)hdr, P [φ, ∂r ]i.

Examining the expressions for the two terms above, we see that for r0 subject to (2.5) we have 2M hdr, P [φ, X + CK]i & C|∂v˜ φ|2 + |∂ω φ|2 − 1 − |∂r φ|2 , r = r0 . r0 Next we consider the lower order terms. The contribution of m is 1 hm, driφ2 & φ2 2 due to condition (iii) in the Lemma. The contribution of q is 1 qφhdr, dφi − φ2 hdr, dqi. 2 2M The coecient of the second term is 1 − r q 0 , which is negligible for r0 close to 2M .

In the rst term we have

2M 2M 0 hdr, dφi = 1 − ∂r φ + 1 − 1 − µ ∂v˜ φ. r r are negligible, and since q is bounded we get All terms involving 1 − 2M r qφ∂v˜ φ C|∂v˜ φ|2 + φ2

for large enough C .

Proof of Lemma 2.1. It is convenient to look for X in the (r, t) coordinates, where we choose the vector eld X of the form X = X1 + δX2 ,

δ1

where 2M 2M −1 2M ∂r , X2 = b(r) 1 − ∂r − 1 − ∂t X1 = a(r) 1 − r r r and a and b 1 − 2M will be chosen to be smooth. Note that X is a smooth vector eld r

in the nonsingular coordinates (r, v), since in these coordinates we have X1 = a(r)

1−

2M ∂r + ∂v , r

2M X2 = b(r) 1 − ∂r . r


15

We remark that the vector eld X2 is closely related to the vector eld Y introduced earlier in [14] in order to take advantage of the red shift eect. However, in their construction Y is in a form which is nonsmooth near the event horizon and which is restricted to the exterior region. The primary role played by X2 here is to ensure that X + CK is time-like near the event horizon. The red-shift eect largely takes care of the rest. For convenience, we set 2M 1 2 ∂r r a(r) . t1 (r) = 1 − 2 r r

A direct computation yields (2.6)

2M 2 0 r − 3M ∇α Pα [φ, X1 ] = 1 − a (r)(∂r φ)2 + a(r) |6 ∇φ|2 r r2 1 − t1 (r)∂ γ φ ∂γ φ + X1 φ2g φ, 2

respectively 2 1 0 2M ∇ Pα [φ, X2 ] = b (r) 1 − ∂r φ − ∂t φ 2 r r − 3M 1 2M 0 + b(r) − 1− b (r) |6 ∇φ|2 r2 2 r 2M 1 1− b(r)∂ γ φ ∂γ φ + X2 φ2g φ − r r α

(2.7)

where ∂ γ φ ∂γ φ = −

1−

2M −1 2M (∂t φ)2 − 1 − (∂r φ)2 − |6 ∇φ|2 . r r

We choose a so that the rst line of the right side of (2.6) is positive. This requires that (2.8)

a0 (r) & r−2 ,

a(3M ) = 0.

We choose b so that the rst line of the right hand side of (2.7) is positive. Precisely, we take b supported in r ≤ 3M with b=−

b0 (r) , 1 − 2M r

r ∈ [2M, 3M ].

with b0 smooth, decreasing in [2M, 3M ) and supported in {r ≤ 3M }. In particular that guarantees that b0 (2M ) > 0, which is later used to verify the condition (iii) in the Lemma. The exact choice of b0 is not important, and in eect b only plays a role very close to the event horizon r = 2M . Even though b is singular at 2M , the second term of the coecient of |6 ∇φ|2 in the second line of (2.7) is nonsingular. Hence if δ is suciently small this term is controlled by the rst line in (2.6).

16


Taking the above choices into account, we have 2 1 0 2M 2 0 2M 2 a (r)(∂r φ) + δ b (r) 1 − ∂r φ − ∂t φ Q[φ, X, 0, 0] = 1 − r 2 r (r − 3M )2 +O |6 ∇φ|2 − q0 ∂ γ φ ∂γ φ r3

(2.9) where

q0 (r) =

1 1 2M . t1 (r) + δ b(r) 1 − 2 r r

The last term in (2.9) is a Lagrangian expression and is accounted for via the q term. The rst three terms give a nonnegative quadratic form in ∇φ. This form is in eect positive denite for r < 3M , where b0 > 0. However for larger r it controls ∂r φ and 6 ∇φ but not ∂t φ. This can be easily remedied with the Lagrangian term. Precisely, we choose q of the form q = q0 + δ1 q1 ,

q1 (r) = χ{r>5M/2}

(r − 3M )2 . r4

where χ{r>5M/2} is a smooth nonnegative cuto which is supported in {r > 5M/2} and equals 1 for r > 3M . The positive parameter δ1 is chosen so that δ1 δ . Then the only nonnegligible contribution of δ1 q1 is the one involving ∂t φ. We obtain (2.10)

2 2M 2 2M 1 Q[φ, X, q, 0] = 1 − O(r−2 )(∂r φ)2 + δ b0 (r) 1 − ∂r φ − ∂t φ r 2 r −1 2 (r − 3M ) 1 2M +O |∂t φ|2 − ∇α ∂α q φ2 . |6 ∇φ|2 + δ1 q1 1 − r3 r 2

The contribution of q1 can be made arbitrarily small by taking δ1 small. Hence it will be neglected in the sequel. At this stage it would be convenient to be able to choose a so that ∇α ∂α t1 (r) < 0. A direct computation yields ∇α ∂α t1 (r) = −La

with

oi 1 h 2M 2 n 2M 1 2 ∂ ∂ 1 − r ∂ r a(r) . 1 − r r r r2 r r 2r2 Unfortunately it turns out that the condition La > 0 and (2.8) are incompatible, in the sense that there is no smooth a which satises both. However, one can nd a with a logarithmic blow-up at 2M which satises both requirements. Such an example is r − 2M −2 2 . a(r) = r (r − 3M )(r + 2M ) + 6M log M La(r) = −

This is in no way unique, it is merely the simplest we were able to produce. One veries directly that a0 (r) & r−2 ,

La(r) & r−4 .

To eliminate the singularity of a above we replace it by a (r) =

1 f (R) r2


17

where is a small parameter, R = (r − 3M )(r + 2M ) + 6M 2 log

r − 2M M

,

and f (R) = −1 f (R)

where f is a smooth nondecreasing function such that f (R) = R on [−1, ∞] and f = −2 on (−∞, −3]. The condition (2.8) is satised uniformly with respect to small ; therefore the choice of δ is independent of the choice of . With this modication of a we recompute 2M −1 La = f 0 (R)La + O()f 00 (R) + O 2 1 − f 000 (R). r

This is still positive except for the region {R < −1}. To control it we introduce an m term in the divergence relation as follows. Let γ(r) be a function to be chosen later. We set 2M 2 γ, mt = δb0 (r) 1 − r

Then

2M mr = δb0 (r) 1 − γ, r

mω = 0.

2M 2M mα ∂ α φ = δb0 (r)γ(r) 1 − 1− ∂r φ − ∂t φ , r r

while ∇α mα = δr−2 ∂r

1−

2M 2 2 0 r b (r)γ(r) . r

Hence, completing the square we obtain (r − 3M )2 2M 2 −2 2 O(r )(∂r φ) + O |6 ∇φ|2 Q[φ, X, q, m] = 1 − r r3 2M −1 + δ1 q1 1 − |∂t φ|2 + nφ2 r 2 2M 1 0 2M + δ b (r) 1 − ∂r φ − ∂t φ + 1 − γφ 2 r r

where the coecient n is given by 2M 2M b0 (r) 2M 2 1 −2 2 −1 ∂r r b(r) 1 − −δ 1− γ(r)2 n = La − δr ∂r r 1 − 2 r r 2 r 1 2M 2 0 1 2M 2 0 + δr−2 γ∂r r2 1 − b (r) + δγ 0 1 − b (r). 2 r 2 r

We assume that γ is supported in {r < 3M } and satises 0 ≤ γ ≤ 1,

γ 0 > −1.

Then for r > 3M we have n = La & r−4 ,

18


while for r ≤ 3M we can write 2M 2 0 n = La + δγ 0 (r) 1 − b (r) + O(δ). r If R > −1 then, using the bound from below on γ 0 , we further have n ≥ La + O(δ)

which is positive provided that δ is suciently small. On the other hand in the region {R ≤ −1}, we have n≥

2M −1 1 2M 2 0 b (r)γ 0 (r) + O(δ) + O()f 00 (R) + O 2 1 − δ 1− f 000 (R). 2 r r

The γ 0 term can be taken positive, while all the other terms may be negative so they must be controlled by it. The restriction we face in the choice of γ 0 comes from the fact that 0 ≤ γ ≤ 1. Hence we need to verify that 2M −1 000 δ + |f 00 (R)| + 2 1 − |f (R)| δ. r R≤−1

Z I=

Indeed, the interval of integration has size ≤ e−c ; therefore the above integral can be bounded by −1

I . e−c

−1

+

which suces provided that is small enough. Finally, note that

2M 2M a(r) 1 − + δb(r) 1 − (2M ) < 0, r r 2M 2 hm, dri(2M ) = δb0 (r) 1 − γ (2M ) > 0. r

X(dr)(2M ) =

So (iii) is also satised.

3.

Log-loss local energy estimates

The aim of this section is to prove a local energy estimate for solutions to the wave equation on the Schwarzschild space which is stronger than the one in Theorem 1.2. Consequently, we strengthen the norm LE0 to a norm LE and we relax the norm LE0∗ to a norm LE ∗ which satisfy the following natural bounds: (3.11) kφk2LE0 . kφk2LE . kφk2LEM , respectively 2 2 (3.12) kf k2LEM ∗ . kf kLE ∗ . kf kLE ∗ . 0 We note that these bounds uniquely determine the topology of the LE and LE ∗ spaces away from the photon sphere and from innity. This is due to the fact that the local energy estimates in Theorem 1.2 have no loss in any bounded region away from the photon sphere. To dene the LE , respectively LE ∗ , norms we consider a smooth partition of unity 1 = χeh (r) + χps (r) + χ∞ (r)


19

where χeh is supported in {r < 11M/4}, χps is supported in {5M/2 < r < 5M } and χ∞ is supported in {r > 4M }. Then we set (3.13)

kφk2LE = kχeh φk2LEM + kχps φk2LEps + kχ∞ φk2LEM ,

respectively (3.14)

2 2 kφk2LE ∗ = kχeh φk2LEM ∗ + kχps φkLE ∗ + kχ∞ φkLE ∗ . ps M

∗ The norms LEps and LEps near the photon sphere are dened in Section 3.1 below, see ∗ (3.20), respectively (3.21); their topologies coincide with LEM , respectively LEM , away from the photon sphere. With these notations, the main result of this section can be phrased in a manner similar to Theorem 1.2:

Theorem 3.2. For all functions φ which solve 2g φ = f in MR we have (3.15)

− 2 2 sup E[φ](˜ v ) + E[φ](Σ+ R ) + kφkLE . E[φ](ΣR ) + kf kLE ∗ . v ˜>0

We continue with the setup and estimates near the photon sphere in Section 3.1, the setup and estimates near innity in Section 3.2 and nally the proof of the theorem in Section 3.3. 3.1. The analysis near the photon sphere. Here it is convenient to work in the Regge-Wheeler coordinates given by r∗ = r + 2M log(r − 2M ) − 3M − 2M log M.

Then r = 3M corresponds to r∗ = 0, and a neighbourhood of r = 3M away from innity and the event horizon corresponds to a compact set in r∗ . In these coordinates the operator 2g has the form

2M (3.16) r 1 − r

2g r−1 = LRW = ∂t2 − ∂r2∗ −

r − 2M ∂ω + V (r), r3

V (r) = r−1 ∂r2∗ r.

For r∗ in a compact set the energy has the form Z E[φ] ≈

(∂t φ)2 + (∂r∗ φ)2 + (∂ω φ)2 drdω,

and the initial local smoothing norms are expressed as kφk2LE0 ≈

Z

(∂r∗ φ)2 + r∗ 2 ((∂ω φ)2 + (∂t φ)2 ) + φ2 drdωdt,

respectively kf k2LE0∗ ≈

Z

r∗ −2 f 2 drdωdt.

On the other hand kφk2LEM ≈

Z

(∂r∗ φ)2 + (∂ω φ)2 + (∂t φ)2 + φ2 drdωdt, kf k2LEM ∗

Z ≈

f 2 drdωdt.

20


In the sequel we work with spatial spherically symmetric pseudodierential operators in the (r∗ , ω) coordinates where ω ∈ S2 . We denote by ξ the dual variable to r∗ , and by 1 λ the spectral parameter for (−∂ω ) 2 . Thus the role of the Fourier variable is played by the pair (ξ, λ), and all our symbols are of the form a(r∗ , ξ, λ)

To such a symbol we associate the corresponding Weyl operator Aw . Since there is no symbol dependence on ω , one can view this operator as a combination of a one dimensional 1 Weyl operator and the spectral projectors Πλ associated to the operator (−∆S2 ) 2 , namely Aw =

X

aw (λ)Πλ

λ

All of our L2 estimates admit orthogonal decompositions with respect to spherical harmonics, therefore in order to prove them it suces to work with the xed λ operators aw (λ), and treat λ as a parameter. However, in the proof of the Strichartz estimates later on we need kernel bounds for operators of the form Aw , which is why we think of λ as a second Fourier variable and track the symbol regularity with respect to λ as well. Of course, this is meaningless for λ in a compact set; only the asymptotic behavior as λ → ∞ is relevant. Let γ0 : R → R+ be a smooth increasing function so that γ0 (y) =

1 y

y < 1, y ≥ 2.

Let γ1 : R+ → R+ be a smooth increasing function so that γ1 (y) =

1

y2 1

y < 1/2, y ≥ 1.

Let γ : R2 → R+ be a smooth function with the following properties:  

1 pz < C, γ (y) y < γ(y, z) = 0 pz/2, z ≥ C,  1 2 2 z γ1 (y /z) y ≥ z/2, z ≥ C,

where C is a large constant. In the sequel z is a discrete parameter, so the lack of smoothness at z = C is of no consequence. Consider the symbol aps (r∗ , ξ, λ) = γ(− ln(r∗ 2 + λ−2 ξ 2 ), ln λ),

and its inverse ∗ a−1 ps (r , ξ, λ) =

γ(− ln(r∗ 2

1 . + λ−2 ξ 2 ), ln λ)

We note that if λ is small then they both equal 1, while if λ is large then they satisfy the bounds 1

(3.17)

1 ≤ aps (r∗ , ξ, λ) ≤ aps (r∗ , 0, λ) ≤ (ln λ) 2 , 1

∗ −1 ∗ (ln λ)− 2 ≤ a−1 ps (r , 0, λ) ≤ aps (r , ξ, λ) ≤ 1.


We also observe that the region where y 2 > z corresponds to r∗ 2 + λ−2 ξ 2 < e− Thus dierentiating the two symbols we obtain the following bounds (3.18)

√

|∂rα∗ ∂ξβ ∂λν aps (r∗ , ξ, λ)| ≤ cα,β,ν λ−β−ν (r∗ 2 + λ−2 ξ 2 + e−

ln λ − α+β 2

)

21

√

ln λ

.

,

respectively √

∗ −2 ∗ −β−ν ∗ 2 (3.19) |∂rα∗ ∂ξβ ∂λν a−1 (r + λ−2 ξ 2 + e− ps (r , ξ, λ)| ≤ cα,β,ν aps (r , ξ, λ)λ

ln λ − α+β 2

)

,

where α + β + ν > 0. These show that we have a good operator calculus for the corresponding pseudodierential operators. In particular in terms of the classical symbol classes we have δ aps , a−1 ps ∈ S1,0 ,

δ > 0.

Then we introduce the Weyl operators Aps =

X

aw ps (λ)Πλ ,

λ

respectively A−1 ps =

X

w (a−1 ps ) (λ)Πλ .

λ

By (3.18) and (3.19) one easily sees that these operators are approximate inverses. More precisely for small λ, ln λ < C , they are both the identity, while for large λ −1 w −1 kaw e ps (λ)(aps ) (λ) − IkL2 →L2 . λ

√

ln λ

,

ln λ ≥ C.

Choosing C large enough we insure that the bound above is always much smaller than 1. We use these two operators in order to dene the improved local smoothing norms (3.20)

1 kφkLEps = kA−1 ≈ kA−1 ps φkHt,x ps ∇t,x φkL2 ,

(3.21)

∗ = kAps f kL2 . kf kLEps

Due to the inequalities (3.17) we have a bound from above for awps (λ), ∗ ∗ kaw ps (λ)f kL2 . kaps (r , 0, λ)f kL2 . k| ln |r ||f kL2 , w respectively a bound from below for (a−1 ps ) (λ), w −1 ∗ ∗ −1 k(a−1 f kL2 ps ) (λ)f kL2 & kaps (r , 0, λ)f kL2 & k| ln |r ||

for f supported near r∗ = 0. In particular this shows that for f supported near the photon sphere we have (3.22)

kφkLEps & k| ln |r∗ ||−1 ∇φkL2 ,

∗ ∗ . k| ln |r ||f kL2 kf kLEps

which makes Theorem 1.3 a direct consequence of Theorem 3.2. Our main estimate near the photon sphere is

Proposition 3.3. a) Let φ be a function supported in {5M/2 < r < 5M } which solves Then

2g φ = f .

(3.23)

kφk2LEps . kf k2LEps ∗ .

22


∗ b) Let f ∈ LEps be supported in {11M/4 < r < 4M }. Then there is a function φ supported in {5M/2 < r < 5M } so that (3.24) sup E[φ] + kφk2LEps + k2g φ − f k2LE0∗ . kf k2LEps ∗ . t

Proof. Due to (3.16) we canrecast the problem in the Regge-Wheeler coordinates. Denoting u = rφ, g = 1 − 2M rf we have LRW u = g . Also it is easy to verify that for φ r and f supported in a xed compact set in r∗ we have kφkLEps ≈ kukLEps ,

∗ ≈ kgkLE ∗ . kf kLEps ps

Hence in the proposition we can replace φ and f by u and g , and 2g by LRW . To prove part (a) we expand in spherical harmonics with respect to the angular variable and take a time Fourier transform. We are left with the ordinary dierential equation (3.25) (∂r2∗ + Vλ,τ (r∗ ))u = g, where Vλ,τ (r∗ ) = τ 2 −

r − 2M 2 λ + V. r3

Depending on the relative sizes of λ and τ we consider several cases. In the easier cases it suces to replace the bound (3.23) with a simpler bound (3.26) k∂r∗ ukL2 + (|τ | + |λ|)kukL2 . kgkL2 .

Case 1: λ, τ . 1. Then we solve (3.25) as a Cauchy problem with data on one side and obtain a pointwise bound , |u| + |ur∗ | . kgkL2

which easily implies (3.26). Case 2: λ τ . Then Vλ,τ (r∗ ) ≈ τ 2 for r∗ in a compact set; therefore (3.25) is hyperbolic in nature. Hence we can solve (3.25) as a Cauchy problem with data on one side and obtain τ |u| + |ur∗ | . kgkL2 ,

which implies (3.26). Case 3: λ τ . Then Vλ,τ (r∗ ) ≈ −λ2 for r∗ in a compact set; therefore (3.25) is elliptic. Then we solve (3.25) as an elliptic problem with Dirichlet boundary conditions on a compact interval and obtain 3

1

λ 2 |u| + λ 2 |ur∗ | . kgkL2 ,

which again gives (3.26). Case 4: λ ≈ τ 1. In this case (3.26) is no longer true, and we need to prove (3.23) which in this case can be written in the form w w (3.27) k∂r∗ ukL2 + λk(a−1 ps ) (λ)ukL2 . kaps (λ)gkL2 , where u, g are subject to (3.25). The ∂r∗ u term above is present in order to estimate the high frequencies |ξ| λ. For lower frequencies it is controlled by the second term on the left of (3.27).


23

The potential V in (3.25) can be treated perturbatively in (3.23) and is negligible. The remaining part of Vλ,τ (r∗ ) has a nondegenerate minimum at r = 3M which corresponds to r∗ = 0. Hence we express it in the form Vλ,τ (r∗ ) = λ2 (W (r∗ ) + ),

where W is smooth and has a nondegenerate zero minimum at r∗ = 0 and || . 1. We now prove the following:

Proposition 3.4. Let W be a smooth function satisfying W (0) = W 0 (0) = 0, W 00 (0) > 0, and || . 1. Let w be a solution of the ordinary dierential equation ∂r2∗ + λ2 (W (r∗ ) + ))w(r∗ ) = g

supported near r∗ = 0. Then (3.27) holds. It would be convenient to replace the norm on the right in (3.27) by kaps (r∗ , 0, λ)gkL2 . This is not entirely possible since this is a stronger norm. However, we can split g into a component g1 with aps (r∗ , 0, λ)g1 ∈ L2 plus a high frequency part:

Lemma 3.5. Each function in the form

g ∈ L2

supported near the photon sphere can be expressed

g = g1 + λ−2 ∂r2∗ g2

with g1 and g2 supported near the photon sphere so that (3.28)

kaps (r∗ , 0, λ)g1 kL2 + k|r∗ 2 + e−

√ ln λ

1

| 8 g2 kL2 + λ−2 k∂r∗ g2 kL2 . kaw ps (λ)gkL2 .

Proof. The symbols aps (r∗ , 0, λ) and aps (r∗ , ξ, λ) are comparable provided that ln(r∗ 2 + e−

√

ln λ

) ≈ ln(r∗ 2 + e−

√

ln λ

+ λ−2 ξ 2 ).

This includes a region of the form D=

ln(λ−2 ξ 2 )
0.

To obtain the decomposition of g we set g2 = q w g,

where the symbol of q is q(r∗ , ξ, λ) = λ2 ξ −2 (1 − χD ).

24


w w Since (a−1 ps ) is an approximate inverse for aps , the estimate for g2 in the lemma can be written in the form √ 1 −2 w w k∂r∗ q w (a−1 (3.29) k(r∗ 2 + e− ln λ ) 8 q w (a−1 ps ) (λ)f kL2 + λ ps ) (λ)f kL2 . kf kL2 . In the rst term it suces to look at the principal symbol of the operator product since −1+δ the remainder belongs to OP S1,δ for all δ > 0. To verify that the product of the symbols is bounded we note that a−1 ps is bounded. For the other two factors we consider two cases. If |ξ| & λ then both factors are bounded. On the other hand if |ξ| . λ then in the support of q we have

λ−2 ξ 2 & (r∗ 2 + e−

which gives q . (r∗ 2 + e−

√

√ ln λ

1

)8 ,

ln λ − 81

)

.

The estimate for the second term in (3.29) is similar but simpler. It remains to consider the bound for g1 , which is given by g1 = (1 + λ−2 Dr2∗ q w )g,

Dr ∗ =

As above, the bound for g1 can be written in the form

1 ∂r∗ i

w kaps (r∗ , 0, λ)(1 + λ−2 Dr2∗ q w )(a−1 ps ) (λ)f kL2 . kf kL2 . δ 0 δ The three operators above belong respectively to S1,δ , S1,δ , and S1,δ for all δ > 0. Hence δ the product belongs to S1,δ , and it suces to show that its principal symbol is bounded. But the principal symbol of the product is given by ∗ aps (r∗ , 0, λ)χD a−1 ps (r , ξ, λ),

which is bounded due to the choice of D. Finally we remark that as constructed the functions g1 and g2 are not necessarily supported near the photon sphere. This is easily rectied by replacing them with truncated versions, g1 := χ1 (r∗ )g1 ,

g2 := χ(r∗ )g2

where χ1 is a smooth compactly supported cuto which equals 1 in the support of g . It is clear that the bound (3.28) is still valid after truncation.

Using the above decomposition of g we write u in the form u = λ−2 g2 + u ˜.

For the rst term we use the above lemma to estimate λkλ−2 g2 kL2 + kλ−2 ∂r∗ g2 kL2 . kaw ps (λ)gkL2 ,

which is stronger than what we need. For u˜ we write the equation (3.30) (∂r2∗ + λ2 (W + ))˜ u = g˜, g˜ = g1 − (W + )g2 . 1 For g˜ we only use a weighted L bound, 1

k(λ−1 + |W + |)− 4 g˜kL1 . kaw ps (λ)gkL2 ,


25

which is obtained from the weighted L2 bounds on g1 and g2 by Cauchy-Schwarz. For u˜ on the other hand, it suces to obtain a pointwise bound:

Lemma 3.6. For each λ−1 < σ < 1 and each function u˜ with compact support, we have 1

1

w ˜kL∞ . λk(a−1 ukL2 . k(σ + |W + |)− 4 ∂r∗ u ˜kL∞ + λk(σ + |W + |) 4 u ps ) (λ)˜

Proof. Since W has a nondegenerate zero minimum at 0, if > −σ then σ + |W + | ≈ σ+||+W . Hence without any restriction in generality we can replace (, σ) by (0, σ+||). Thus in the sequel we can assume that either = 0 or < −σ . We consider three cases: √ Case 1: ||, σ < e− ln λ . We consider an almost orthogonal partition of u˜ in dyadic regions with respect to r∗ : X

u ˜=u ˜R φk2LEM . E[χ>R φ](0) + k2g (χ>R φ)k2LEM ∗ . E[φ](0) + k[2g , χ>R ]φk2L2 . E[φ](0) + kφk2LE0 ,

where in the last two steps we have used the compact support of 2g (χ>R φ) = [2g , χ>R ]φ. b) Let R be large enough, as in part (a). For |x| > R the Schwarzchild metric g is a small long range perturbation of the Minkowski metric, according to the denition in [29]. We consider a second metric g˜ in R3+1 which coincides with g in {|x| > R} but which is globally a small long range perturbation of the Minkowski metric. Let ψ be the forward solution to 2g˜ ψ = f . Then we set φ = χ>R ψ.

The estimate (1.5) holds for the metric g˜, therefore we obtain sup E[ψ](t) + kψkLEM . kf k2LEM ∗ t

Then the same bound holds as well for φ. Furthermore, we can compute the error 2g φ − f = (χ>R − 1)f + [2g , χ>R ]ψ

This has compact spatial support, and can be easily estimated in L2 as in part (a).

3.3. Proof of Theorem 3.2. Given f ∈ LE ∗ we split it into f = χeh f + χps f + χ∞ f.

For the last two terms we use part (b) of the Propositions 3.3,3.10 to produce approximate solutions φps and φ∞ near the photon sphere, respectively near innity. Adding them up we obtain an approximate solution φ0 = φps + φ∞

for the equation 2g φ = f . Due to (3.24) and (3.45) we obtain for φ0 the bound (3.47) sup E[φ0 ](˜ v ) + kφ0 k2LE . kf k2LE ∗ , v ˜

while the error

f1 = 2g (φps + φ∞ ) − f is supported away from r = 3M and r = ∞ and satises kf1 kLE0∗ ≈ kf1 kL2 . kf kLE ∗ .

Then we nd φ = φ0 + φ1 by solving 2g φ1 = f1 ∈ LE0∗ ,

φ1 [0] = φ[0] − φ0 [0],

By Theorem 1.2 we obtain the LE0 bound for φ1 . It remains to improve this to an LE bound for φ1 . By part (a) of Proposition 3.10 we can estimate kχ∞ φkLEM . Near the photon sphere we would like to apply part (a) of Proposition 3.3 to χps φ. However we cannot proceed in an identical manner because part (a) of Proposition 3.3


33

does not involve the Cauchy data of φ at t = 0, and instead applies to functions φ dened on the full real axis in t. To address this issue we extend φ1 backward in t to the set M0R , by solving the homogeneous problem 2g φ1 = 0 in M0R , with matching Cauchy data on the common boundary of MR and M0R . The extended function φ1 belongs to both LE(MR ) and LE(MR0 ), and now we can estimate χps φ1 via part (a) of Proposition 3.3. 4.

Strichartz estimates

In this section we prove Theorem 1.4. The theorem follows from the following two propositions. The rst gives the result for the right hand side, f , in the dual local energy space:

Proposition 4.11. Let (ρ, p, q) be a nonsharp Strichartz pair. Then for each with 2g φ ∈ LE ∗ + L1v˜ L2 we have (4.48)

φ ∈ LE

k∇φk2Lp H˙ −ρ,q . E[φ](0) + kφk2LE + k2g φk2LE ∗ +L1 L2 . v ˜

v ˜

The second one allows us to use Lp2 Lq2 in the right hand side of the wave equation. 0

0

Proposition 4.12. There is a parametrix K for 2g so that for all nonsharp Strichartz pairs (ρ1 , p1 , q1 ) and (ρ2 , p2 , q2 ) we have 2 2 2 (4.49) sup E[Kf ](˜v ) + E[Kf ](Σ+ ˙ −ρ1 ,q1 . kf k p02 ˙ ρ2 ,q0 R ) + kKf kLE + k∇Kf kLp1 H 2 Lv˜ H

v ˜

v ˜

and the error estimate (4.50) k2g Kf − f kLE ∗ +L1v˜ L2 . kf k p02 ˙ ρ2 ,q0 . 2 L H v ˜

We rst show how to use the propositions in order to prove the Theorem.

Proof of Theorem 1.4. Suppose that 2g φ = f with f ∈ Lp2 H˙ ρ2 ,q2 . We write φ as 0

0

φ = φ1 + Kf

with K as in Proposition 4.12. By (4.49) the Kf term satises all the required estimates; therefore it remains to consider φ1 . Using also (4.50) we obtain k2g φ1 k2LE ∗ +L1 L2 + E[φ1 ](0) . E[φ](0) + kf k2 p02 L

v ˜

˙ ρ2 ,q20 . H

Then Theorem 3.2 combined with Duhamel's formula yields kφ1 k2LE + k2g φ1 k2LE ∗ +L1 L2 + sup E[φ1 ](˜ v ) . E[φ](0) + kf k2 p02 v ˜

v ˜

Finally the Lp1 H˙ −ρ1 ,q1 bound for ∇φ1 follows by Proposition 4.11.

L

˙ ρ2 ,q20 . H

We continue with the proofs of the two propositions.

Proof of Proposition 4.11. By Duhamel's formula and Theorem 3.2 we can neglect the

L1 L2 part of 2g φ. Hence in the sequel we assume that 2g φ ∈ LE ∗ .

We use cutos to split the space into three regions, namely near the event horizon, near the photon sphere and near innity, φ = χeh φ + χps φ + χ∞ φ.

34


Due to the denition of the LE and LE ∗ norms we have E[φ](0) + kφk2LE + k2g φk2LE ∗ & E[χeh φ](0) + kχeh φk2H 1 + k2g (χeh φ)k2L2 + E[χps φ](0) + kχps φk2LEps + k2g (χps φ)k2LEps ∗ + E[χ∞ φ](0) + kχ∞ φk2LEM + k2g (χ∞ φ)k2LEM ∗ .

Proving this requires commuting 2g with the cutos. However this is straightforward since the LE and LE ∗ norms are equivalent to the H 1 , respectively L2 , norm in the support of ∇χeh , ∇χps and ∇χ∞ . It remains to prove the Lpv˜ H˙ −ρ,q bound for each of the three terms in ∇φ. We consider the three cases separately: I. The estimate near the event horizon. This is the easiest case. Given φ supported in {r < 11M/4}, we partition it on the unit scale with respect to v˜, φ=

X

χ(˜ v − j)φ,

j∈Z

where χ is a suitable smooth compactly supported bump function. Commuting the cutos with 2g one easily obtains the square summability relation X

kχ(˜ v − j)φk2H 1 + k2g (χ(˜ v − j)φ)k2L2 + E[χ(˜ v − j)φ](0) . kφk2H 1 + k2g φk2L2 + E[φ](0),

j∈N

where the energy term on the left is nonzero only for nitely many j . Since each of the functions χ(˜v − j)φ have compact support, they satisfy the Strichartz estimates due to the local theory; see [32], [39], [47]. The above square summability with respect to j guarantees that the local estimates can be added up. II. The estimate near the photon sphere. For φ supported in {5M/2 < r < 5M } we need to show that k∇φk2Lp H −ρ,q . E[φ](0) + kφk2LEps + k2g φk2LEps ∗ . v ˜

We use again the Regge-Wheeler coordinates. Then the operator 2g is replaced by LRW . The potential V can be neglected due to the straightforward bound ∗ . kφkLE kV φkLEps . ps

Indeed, for φ at spherical frequency λ we have 1

1

− ∗ . | ln(2 + λ)| 2 kφkL2 . λ| ln(2 + λ)| 2 kφkL2 . kφkLE . kV φkLEps ps

We introduce the auxiliary function ψ = A−1 ps φ.

By the denition of the LEps norm we have (4.51) kψkH 1 . kφkLEps . We also claim that (4.52) kLRW ψkL2 . kφkLEps + kLRW φkL2 . −1 2 Since Aps is L bounded, this is a consequence of the commutator bound 2 [A−1 ps , LRW ] : LEps → L ,


35

or equivalently 1 2 (4.53) [A−1 ps , LRW ]Aps : H → L . It suces to consider the rst term in the symbol calculus, as the remainder belongs to δ OP S1,δ , mapping H δ to L2 for all δ > 0. The symbol of the rst term is 2 −3 q(ξ, r∗ , λ) = {a−1 (r − 2M )λ2 }aps (λ) ps (λ), ξ + r 1+δ and a-priori we have q ∈ S1,δ . For a better estimate we compute the Poisson bracket

q(ξ, r∗ , λ) = a−1 ps (λ)γy (y, ln λ)

4ξr∗ − 2ξ∂r∗ (r−3 (r − 2M )) r∗ 2 + λ−2 ξ 2

where y = r∗ 2 + λ−2 ξ 2 . The rst two factors on the right are bounded. The third is bounded by λ since ∂r∗ (r−3 (r − 2M )) vanishes at r∗ = 0. In addition, q is supported in 0 |ξ| . λ. Hence we obtain q ∈ λS1−δ,δ . Then the commutator bound (4.53) follows. Given (4.51) and (4.52), we argue as in the rst case, namely we localize ψ to time intervals of unit length and then apply the local Strichartz estimates. By summing over these strips we obtain k∇ψkLp H −ρ,q . kφkLEps + kLRW φkL2

for all sharp Strichartz pairs (ρ, p, q). To return to φ we invert A−1 ps , φ = Aps ψ + (1 − Aps A−1 ps )φ.

The second term is much more regular, k∇(1 − Aps A−1 ps )φkL2 H 1−δ . kφkLEps ,

δ > 0;

therefore it satises all the Strichartz estimates simply by Sobolev embeddings. For the main term Aps ψ we take advantage of the fact that we only seek to prove the nonsharp Strichartz estimates for φ. The nonsharp Strichartz estimates for ψ are obtained from the sharp ones via Sobolev embeddings, 3 3 + ρ2 = + ρ1 , q2 q1

k∇ψkH −ρ2 ,q2 . k∇ψkH −ρ1 ,q1 ,

ρ1 < ρ2 .

To obtain the nonsharp estimates for φ instead, we need a slightly stronger form of the above bound, namely

Lemma 4.13. Assume that 1 < q1 < q2 < ∞. Then (4.54)

3 3 + ρ2 = + ρ1 . q2 q1

kAps ukH −ρ2 ,q2 . kukH −ρ1 ,q1 ,

Proof. We need to prove that the operator B = Opw (ξ 2 + λ2 + 1)−

ρ2 2

Aps Opw (ξ 2 + λ2 + 1)

maps Lq1 into Lq2 . The principal symbol of B is b0 (r∗ , ξ, λ) = (ξ 2 + λ2 + 1)

ρ1 −ρ2 2

aps (r∗ , ξ, λ),

and by the pdo calculus the remainder is easy to estimate, ρ1 −ρ2 −1+δ B − bw , 0 ∈ OP S1,0

δ > 0.

ρ1 2

36


The conclusion of the lemma will follow from the Hardy-Littlewood-Sobolev inequality if we prove a suitable pointwise bound on the kernel K of bw0 , namely 1

1

(4.55) |K(r1∗ , ω1 , r2∗ , ω2 )| . (|r1∗ − r2∗ ||ω1 − ω2 |2 )−1+ q1 − q2 . For xed r∗ we consider a smooth dyadic partition of unity in frequency as follows: 1 = χ{|ξ|>λ} +

X

χ{λ≈µ}

χ{|ξ|.ν0 } +

µ X

! χ{|ξ|≈ν}

,

ν=ν0

µ dyadic

where ν0 = ν0 (λ, r∗ ) is given by

√ ln ν0 (λ, r∗ ) = ln λ + max{ln r∗ , − ln λ}.

This leads to a similar decomposition for b0 , namely b0 = b00 +

X

bµ,R φkLE ∗ M

+ E[χR which is supported in r > R and equals 1 in the support of χ>R . Then we dene >R K∞ f =χ ˜>R ψ∞ . >R It remains to show that K∞ satises the appropriate bounds, >R >R >R f k2Lp1 H˙ −ρ1 ,q1 . kχ>R f k2 p02 sup E[K∞ f ](t) + kK∞ f k2LEM + k∇K∞ L

t

˙ ρ2 ,q20 , H

respectively the error estimate >R ∗ . kχ>R f k p0 k2g K∞ f − χ>R f kLEM ˙ ρ2 ,q20 . L 2H

These are easily obtained by applying the following lemma to ψ∞ :

Lemma 4.16. Let f bound (4.65)

∈ Lp2 H˙ ρ2 ,q2 . 0

0

Then the forward solution ψ to 2g˜ψ = f satises the

sup E[ψ](t) + kψk2LEM + k∇ψk2Lp1 H˙ −ρ1 ,q1 . kf k2 p02 L

t

˙ ρ2 ,q20 . H

It remains to prove the lemma. This largely follows from [29, Theorem 6], but there is an interesting technical issue that needs clarication. Precisely, [29, Theorem 6] shows that we have the bound (4.66)

sup E[ψ](t) + k∇ψk2X 0 + k∇ψk2Lp1 H˙ −ρ1 ,q1 . kf k2 p02 L

t

˙ ρ2 ,q20 . H

By Lemma 4.14, we are left with proving that (4.67)

3j

sup 2− 2 kψkL2 (Aj ) . kf kLp02 H˙ ρ2 ,q20 . j∈Z

We note that this does not follows from Lemma 4.14; this is a forbidden endpoint of the Hardy inequality in [29, Lemma 1(b)]. However, the bound (4.67) can still be obtained, although in a roundabout way. Precisely, from (4.66) we have (4.68)

sup E[ψ](t) . kf k2 p02 t

L

˙ ρ2 ,q20 H

for the forward in time evolution, and similarly for the backward in time problem. On the other hand, a straightforward modication of the classical Morawetz estimates (see e.g. [27]) for the wave equation shows that the solutions to the homogeneous wave equation 2g˜ ψ = 0 satisfy (4.69)

sup 2−3j kψk2L2 (Aj ) . E[ψ](0). j∈Z

Denote by 1t>s H(t, s) the forward fundamental solution for 2g˜ and by H(t, s) its backward extension to a solution to the homogeneous equation, 2g˜ H(t, s) = 0. Combining the bounds (4.68) and (4.69) shows that −3j

sup 2 j

Z

2

H(t, s)f (s)ds

2 R

L (Aj )

. kf k2 p02 L

˙ ρ2 ,q20 . H


41

Since p02 < 2, by the Christ-Kiselev lemma [11], it follows that −3j

sup 2 j

Z

∞

t

2

H(t, s)f (s)ds

L2 (Aj )

. kf k2 p02 L

˙ ρ2 ,q20 , H

which is exactly (4.67).

5.

The critical NLW

In this section we prove Theorem 1.5. We rst consider (1.18) in the compact region

MC . We denote by ψ the solution to the homogeneous equation ˜ |Σ = φ1 2g ψ = 0, ψ|Σ = φ0 , Kψ 0

0

and by T f the solution to the inhomogeneous problem 2g (T f ) = f,

˜ f|Σ = 0. T F|Σ0 = 0, KT 0

Then we can rewrite the nonlinear equation (1.18) in the form (5.70) φ = ψ ± T (φ5 ). We dene Sobolev spaces in MC by restricting to MC functions in the same Sobolev space which are compactly supported in a larger open set. By the local Strichartz estimates we have kψk

and kT f k

1

H 2 ,4 (MC )

1

H 2 ,4 (MC )

. E[φ](Σ0 )

. kf k

1 4

H 2 , 3 (MC )

.

At the same time we have the multiplicative estimate kφ5 k

1 4

H 2 , 3 (MC )

. kφk5

1

H 2 ,4 (MC )

.

Then for small initial data we can use the contraction principle to solve (5.70) and obtain 1 a solution φ ∈ H 2 ,4 (MC ). In addition, still by local Strichartz estimates, the solution φ will have nite energy on any space-like surface, in particular on the forward and backward space-like boundary of MC . Thus we obtain E[φ](Σ− R ) . E[φ](Σ0 ).

It remains to solve (1.18) in MR (and its other three symmetrical copies). Using the (˜ v , r, ω) coordinates in MR we dene ψ and T as above, but with Cauchy data on Σ− R. By the global Strichartz estimates in Theorem 1.4, for (s, p) as in the theorem we have kψkLp H˙ s,p (MR ) . E[φ](Σ− R)

and kT f kLp H˙ s,p (MR ) . kf kL1 L2 .

In particular we can take p = 5 which corresponds to s = we have kφkL5 L10 . kφk ˙

3

H 10 ,5

;

3 10

. By Sobolev embeddings

42


therefore kφ5 kL1 L2 . kφk5˙

3

H 10 ,5

.

Hence we can solve (5.70) using the contraction principle and obtain a solution φ ∈ H˙ 10 ,5 . This implies that φ5 ∈ L1 L2 , which yields all of the other Strichartz estimates, as well as the energy bound on the forward boundary Σ+ R of MR . This concludes the proof of the theorem. 3

References

[1] S. Alinhac: On the MorawetzKeel-Smith-Sogge inequality for the wave equation on a curved background. Publ. Res. Inst. Math. Sci. 42 (2006), 705720. [2] P. Blue and A. Soer: Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates. Adv. Dierential Equations 8 (2003), 595614. [3] P. Blue and A. Soer: The wave equation on the Schwarzschild metric II: Local decay for the spin-2 Regge-Wheeler equation. J. Math. Phys. 46 (2005), 9pp. [4] P. Blue and A. Soer: Errata for Global existence and scatttering for the nonlinear Schrödinger equation on Schwarzschild manifolds, Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates, and The wave equation on the Schwarzschild metric II: Local decay for

the spin 2 Regge Wheeler equation , preprint. [5] P. Blue and A. Soer: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256 (2009), no. 1, 190 . [6] P. Blue and A. Soer: Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole,http://xxx.lanl.gov, math/0612168. [7] P. Blue and J. Sterbenz: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Comm. Math. Phys. 268 (2006), 481504. [8] P. Brenner: On Lp − Lp0 estimates for the wave-equation. Math. Z. 145 (1975), 251254. [9] N. Burq: Smoothing eect for Schrödinger boundary value problems. Duke Math. J. 123 (2004), 403427. [10] N. Burq, P. Gérard, and N. Tzvetkov: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126 (2004), 569605. [11] M. Christ and A. Kiselev:Maximal functions associated to ltrations, J. Funct. Anal. 179(2001), 409425. [12] H. Christianson: Dispersive estimates for manifolds with one trapped orbit, Comm. Partial Dierential Equations 33 (2008), p. 11471174. [13] Y. Colin de Verdière and B. Parisse: Equilibre Instable en Regime Semi-classique: I - Concentration Microlocale. Commun. PDE. 19, (1994), p. 1535-1563 [14] M. Dafermos and I. Rodnianski: The red-shift eect and radiation decay on black hole spacetimes, arXiv:gr-qc/0512119v1. [15] M. Dafermos and I. Rodnianski: A note on energy currents and decay for the wave equation on a Schwarzschild background, arXiv:0710.0171v1. [16] M. Dafermos, I. Rodnianski: A proof of Price's law for the collapse of a self-gravitating scalar eld, Invent. Math. 162 (2005), 381- 457. [17] J. Genibre and G. Velo: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995), 5068. [18] S. W. Hawking and G. F. R. Ellis: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. London, New York: Cambridge University Press 1973. [19] L. Kapitanski: Some generalizations of the Strichartz-Brenner inequality. Leningrad Math. J. 1 (1990), 693726. [20] M. Keel, H. Smith, and C. D. Sogge: Almost global existence for some semilinear wave equations. J. Anal. Math. 87 (2002), 265279. [21] M. Keel and T. Tao: Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955980. [22] C. E. Kenig, G. Ponce and L. Vega: On the Zakharov and Zakharov-Schulman systems. J. Funct. Anal. 127 (1995), 204234.


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[23] J. Kronthaler Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry. arXiv:0709.3703 [24] B. Kay and R. Wald: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere Classical Quantum Gravity4 (1987), 893-898 [25] I. Laba and A. Soer: Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds. Helv. Phys. Acta 72 (1999), 274294. [26] H. Lindblad and C. D. Sogge: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130 (1995), 357426. [27] J. Metcalfe and C. D. Sogge: Long-time existence of quasilinear wave equations exterior to starshaped obstacles via energy methods. SIAM J. Math. Anal. 38 (2006), 391420. [28] J. Metcalfe and C. D. Sogge: Global existence of null-form wave equations in exterior domains. Math. Z. 256 (2007), 521549. [29] J. Metcalfe and D. Tataru: Global parametrices and dispersive estimates for variable coecient wave equations, arXiv:0707.1191 [30] J. Metcalfe and D. Tataru: Global parametrices and dispersive estimates for variable coecient wave equations in exterior domains, arXiv:0806.3409v1. [31] C. W. Misner, K. S. Thorne, and J. A. Wheeler: Gravitation. San Francisco, CA: W. H. Freeman and Co., 1973. [32] G. Mockenhaupt, A. Seeger, and C. D. Sogge: Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. J. Amer. Math. Soc. 6 (1993), 65130. [33] C. Morawetz: Time decay for the nonlinear Klein-Gordon equations. Proc. Roy. Soc. Ser. A. 306 (1968), 291296. [34] S. Nonnenmacher and M. Zworski: Quantum decay rates in chaotic scattering, preprint (2007). [35] H. Pecher: Nonlinear small data scattering for the wave and Klein-Gordon equations. Math. Z. 185 (1984), 261270. [36] R. Price: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations Phys. Rev. D (3) 5 (1972), 2419-2438 [37] J. V. Ralston: Solutions of the wave equation with localized energy. Comm. Pure Appl. Math. 22 (1969), 807823. [38] W. Schlag, A. Soer, W. Staubach: Decay for the wave and Schroedinger evolutions on manifolds with conical ends, Part II. arXiv:0801.2001 [39] H. F. Smith: A parametrix construction for wave equations with C 1,1 coecients. Ann. Inst. Fourier (Grenoble) 48 (1998), 797835. [40] H. F. Smith and C. D. Sogge: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Dierential Equations 25 (2000), 21712183. [41] J. Sterbenz: Angular regularity and Strichartz estimates for the wave equation. With an appendix by I. Rodnianski. Int. Math. Res. Not. 2005, 187231. [42] W. A. Strauss: Dispersal of waves vanishing on the boundary of an exterior domain. Comm. Pure Appl. Math. 28 (1975), 265278. [43] R. S. Strichartz: A priori estimates for the wave equation and some applications. J. Funct. Anal. 5 (1970), 218235. [44] R. S. Strichartz: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), 705714. [45] D. Tataru: Strichartz estimates for operators with nonsmooth coecients and the nonlinear wave equation. Amer. J. Math. 122 (2000), 349376. [46] D. Tataru: Strichartz estimates for second order hyperbolic operators with nonsmooth coecients II. Amer. J. Math. 123 (2001), 385423. [47] D. Tataru: Strichartz estimates for second order hyperbolic operators with nonsmooth coecients III. J. Amer. Math. Soc. 15 (2002), 419442. [48] D. Tataru: Parametrices and dispersive estimates for Schroedinger operators with variable coecients, to appear, Amer. J. Math. [49] F. Twainy: The Time Decay of Solutions to the Scalar Wave Equation in Schwarzschild Background Thesis, University of California San Diego 1989 [50] R. M. Wald: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20 (1979), 1056?1058 [51] R. M. Wald: General relativity. University of Chicago Press, Chicago, IL, 1984.

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Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027

Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250

Department of Mathematics, University of California, Berkeley, CA 94720-3840

Department of Mathematics, University of California, Berkeley, CA 94720-3840