Localized energy estimates for wave equations on high dimensional Schwarzschild space-times

The localized energy estimate for the wave equation is known to be a fairly robust measure of dispersion. Recent analogs on the $(1+3)$-dimensional Schwarzschild space-time have played a key role in a number of subsequent results, including a proof of Price's law. In this article, we explore similar localized energy estimates for wave equations on $(1+n)$-dimensional hyperspherical Schwarzschild space-times.


Introduction
One of the more robust measures of dispersion for the wave equation is the so called localized energy estimates. These estimates have played a role in understanding scattering theory, as means of summing local in time Strichartz estimates to obtain global in time Strichartz estimates on a compact set, as an essential tool for proving long time existence of quasilinear wave equations in exterior domains, and as means to handle errors in certain parametrix constructions.
Estimates of this form originated in [28] where it was shown that solutions to the constant coefficient, homogeneous wave equation (∂ 2 t − ∆)u = 0, u(0) = u 0 , ∂ t u(0) = u 1 on R + × R n , n ≥ 3 satisfy where ∇ denotes the angular derivatives. Though not the original method of proof, one can obtain this estimate by multiplying 2u by ∂ r u + n−1 2 u r , integrating over a space-time slab, and integrating by parts. One can also obtain control on ∂ r and ∂ t in a fixed dyadic annulus. Attempting to sum over these dyadic annuli comes at the cost of a logarithmic blow up in time. One may otherwise introduce an additional component of the weight that permits summability. In this case, it can be shown that for u as above and n ≥ 4 t,x ([0,T ]×R n ) ∇u 0 2 + u 1 2 .
A particular case of interest which does not fall into these latter categories is the wave equation on (1+3)-dimensional Schwarzschild space-times. While asymptotically flat and time independent, this metric is not, however, nontrapping. There is trapping which occurs on the so called photon sphere which necessitates a loss in the estimates as compared to those for the Minkowski wave equation. Despite this complication, a number of proofs of estimates akin to (1.1) exist for the wave equation on Schwarzschild space-times. See [4]- [9], [14,15], [22]. In [35], these estimates have been extended to Kerr space-times with small angular momentums. Here an additional difficulty is encountered as [2] shows that the estimates will not follow from an analog of the argument above with any first order differential multiplier.
The goal of this article is to extend the known localized energy estimates on (1 + 3)dimensional Schwarzschild space-times to (1 + n)-dimensional Schwarzschild space-times for n ≥ 3. There are multiple notions of the Schwarzschild space-time for n ≥ 4, and we restrict our attention to the hyperspherical case. For a derivation and discussion of such black hole space times, see e.g. [33], [27]. An alternative notion of higher dimensional Schwarzschild space-times is discussed e.g. in [17]. We shall not explore these hypercylindrical Schwarzschild manifolds or other notions of higher dimensional black holes here.
The exterior of a (1 + n)-dimensional hyperspherical Schwarzschild black hole (M, g) is described by the manifold M = R × (r s , ∞) × S d+2 and the line element Here d = n − 3, r s denotes the Schwarzschild radius (i.e. r = r s is the event horizon), and dω is the surface measure on the sphere S d+2 = S n−1 . Here, the d'Alembertian is given by The Killing vector field ∂ t yields the conserved energy That is, when We now define our localized energy norm. To this end, we set Here r ps = d+3 We note that by modifying the lower order portion of the multiplier which appears in the next section that it is straightforward to obtain an estimate on ∂ t φ as well. Moreover, one can obtain an analogous estimate for the inhomogeneous wave equation by placing the forcing term in an appropriate dual norm. The necessary decay of the coefficient c r at ∞ can also be significantly improved. It is relatively simple to carry out these modifications, but for the sake of clarity, we omit the details.
As in the (1+3)-dimensional case, the higher dimensional hyperspherical Schwarzschild space-times have a photon sphere where trapping occurs. Rays initially located on this sphere and moving initially tangent to it will stay on the surface for all times. Such trapping is an obstacle to many types of dispersive estimates, and the vanishing in the coefficient c ω at the trapped set is a loss, when compared to the nontrapping Minkowski setting, which results from this trapping.
The proof follows by constructing an appropriate differential multiplier. This is carried out in the next section, and the construction most closely resembles that which appears in [22]. Some significant technicalities needed to be resolved, however, in order to find a construction which works in all dimensions d ≥ 1.

Construction of the multiplier
Associated to 2 g is the energy-momentum tensor whose most important property is the following divergence condition Contracting Q αβ with the radial vector field X = f (r) 1 − rs r d+1 ∂ r , we form the Calculating the divergence of this quantity we have The last term involving the Lagrangian is not signed. In order to eliminate it, we utilize a lower order term in the multiplier. To this end, we modify the momentum densitỹ and recompute the divergence Ideally one would choose f (r) to be smooth, bounded, and so that the coefficients in the last three terms on the right are all nonnegative. Indeed, this is precisely what is done in, e.g., [31] and [24] when r s = 0, i.e. in Minkowski space-time. The localized energy estimate then follows from an application of the divergence theorem as well as an application of a Hardy inequality and the energy estimate in order to handle the time boundary terms. Unfortunately, in the current setting it does not appear possible to construct such an f . We shall instead construct f so that the last term in the right can be bounded below by a positive quantity minus a fractional multiple of the second term in the right.
We let g(r) = . The multiplier will be defined piecewise, and it will be convenient to parametrize in terms of the values of h(r). To this end, we fix the notation r θ to denote the value of r so that h(r θ ) = θ. Note, e.g., that r −∞ = r s and r ps = r 0 . More explicitly, r d+1 θ = r d+1 s d+1 2 e θ + 1 . An approximate multiplier is r ps r d+1 s r d+2 h(r). We must, however, smooth out the logarithmic blow up at r = r s . We must also smooth out h(r) near ∞ in order to prevent a term in f ′ (r) which has an unfavorable sign.
In order to accomplish this, set Here α = 5 − δ 0 for some 0 < δ 0 ≪ 1. Then, set r ps r d+1 s r d+2 a(h(r)). We notice that f is C 2 with the exception of a jump in the second derivative at r −1/ε .
Using that 2 g φ = 0, integrating (2.1) over [0, T ] × (r s , ∞) × S d+2 , and applying the divergence theorem, we have We first make a note about the boundary terms at r −1/ε . Indeed, an elementary calculation shows Of particular interest is that the coefficient of the resulting boundary term at r −1/ε is O(εδ).
We now proceed to showing that the sum of the first three terms in the right of (2.2) produce a positive contribution. To do so, we will examine f on a case by case basis and show • f ′ (r) > 0 for all r > r s • f (r) < 0 for r < r ps and f (r) > 0 for r > r ps .
• l(f )φ 2 r d+2 dr dω dt is bounded below by a positive term minus a fractional multiple of the (∂ r φ) 2 term and a r −1/ε boundary term.
By absorbing these latter pieces into those previously shown to positively contribute, the estimate shall nearly be in hand. It will only remain to examine the time boundary terms, and in particular, establish the Hardy inequality which will permit a direct application of conservation of energy.
The multiplier in this region is constructed to smooth out the logarithmic blow up at the event horizon. This is the only case in which we shall not be able to just show that Noting that a(x) ≤ −1/ε for x ≤ −1/ε, we immediately see that f (r) < 0 on this range. We also compute Each of these summands is nonnegative on the given range, yielding the desired sign for f ′ (r).
It remains to show an appropriate lower bound for the l(f )φ 2 term of (2.2). Calculating l(f ) we find As a ′ (h(r)) = (δεh(r) + δ − 1) −2 and r < r ps in this regime, the second term in the right has the desired sign. The third term in the right of (2.4) also has the desired sign as The key step is to control the contribution of the last term in the right of (2.4). We shall abbreviate R(r) = δεh(r) + δ − 1. By the Fundamental Theorem of Calculus, we observe Evaluating the derivative in the left side yields To the second term in the right, we apply the Schwarz inequality to obtain This shows that We now use (2.8) and the fact that 13 12 ≤ d+5 4 for d ≥ 1 to account for the last term in (2.4). We then see that Here we have also used (2.3). As l(g) remains bounded in the relevant region and as the coefficient in the integrand of the second term in the right of (2.9) is ε 2 e 1/ε ≫ 1 on the said region, the first term in the right can be controlled by a fraction of the second provided ε is sufficiently small. This finally yields The second term on the right can be bootstapped into the positive contribution provided by the first term in the right of (2.2). The remaining boundary term at r −1/ε will be controlled at the end of this section using pieces from the subsequent case.
For r in this range, we simply have s which is negative, as is desired in order to guarantee a positive contribution from the | ∇φ| 2 term of (2.2). Moreover, we have s whose every term is positive for r s < r ≤ r ps .
It only remains to examine l(f (r)) for this region. Here, we first note that Since (2.12) is nonnegative for r ≤ r ps , we have for r ≤ r ps . Summing this with (2.11), we have which is clearly nonnegative for r ≥ r s .
This region corresponds precisely to h(r) ∈ [0, α]. We, thus, see that As a(h(r)) takes on a maximum value of 8α 15 and as (d+2 8α 15 > 0 for d ≥ 1 and α < 5, the sum of the first two terms is positive. And as the last term is clearly positive, we see that f ′ (r) > 0 as desired.
It remains to verify that l(f ) is positive. We begin by calculating Setting it remains to show that p(r) + n 1 (r) + n 2 (r) + n 3 (r) > 0.
For the case d = 1, which is an everywhere positive quadratic.
Proof of (2.14): Here, again, we use that we are studying the regime that r ≥ r ps and that h 2 − α 2 is minimized when h(r) = 0. It is thus obtained that (2.16) Here, we make the change of variables x = h(r). Thus, r d+1 − r d+1 s = d+1 2 r d+1 s e x . The right side of (2.16) can be rewritten as Setting and noticing that q(0) > 0, it will suffice to show that q ′ (x) ≥ 0 for x between 0 and α. We compute As d+5 d+3 ≤ 3 2 for d ≥ 1 and as 1 + x ≤ e x , it follows that The latter inequality follows from the fact that d 2 ≥ 6 25 (d + 1) provided d ≥ 1. Since the above inequality holds for α = 5, by continuity, we have that it also holds for α = 5 − δ 0 for some δ 0 > 0, which completes the proof.
Proof of (2.15): Using that we are examining the region r ≥ r ps and rewriting l(g) as in the previous case, we have Using that 24r d+1 when r ≥ r ps , we obtain Proceeding as above with the change of variables x = h(r), this is we first note that s(0) > 0. For x ≤ 5, we furthermore have For α = 5, this is 1 which is easily seen to be positive for d ≥ 1. By continuity, positivity also follows for α = 5 − δ 0 provided δ 0 is sufficiently small.
In this regime, r ps r d+1 s r d+2 , which is clearly positive. Moreover, which is also positive since α < 5 ≤ 15 16 (d+3) 2 d+2 for d ≥ 1. Finally, we notice that l(f (r)) = l(g(r)) when in this case. Thus, as in the proof of (2.14), we have for r ≥ r ps , we see that l(f (r)) > 0 as desired.
Boundary term at r −1/ε : In order to finish showing that the right side of (2.2) is nonnegative, it remains to examine the r −1/ε boundary term in (2.2) as well as the subsequent contribution from (2.10). Here, we simply utilize the Fundamental Theorem of Calculus to control these terms via the positive contributions of the first and third term in the right of (2.2) in the range [r −1/ε , r ps ]. The scaling parameter δ insures the necessary smallness.
Fix a smooth cutoff β which is identity for, say, r ≤ r −1 and which vanishes for r ≥ r ps . Then, for r ≤ r −1 , we have φ(r) = − rps r ∂ s (βφ) ds.

Using the Schwarz inequality, this yields
Applying this at r −1/ε yields Multiplying both sides by δ and integrating over [0, T ]×S d+2 , we see that these boundary terms can be bootstrapped into the contributions of Case 2.

A Hardy inequality and the time boundary terms
In the previous section, we constructed a multiplier so that the right side of (2.2) provides a positive contribution. By inspection, the coefficients are easily seen to correspond to those in (1.2). What remains is to control the left side of (2.2) in terms of the initial energy. For the first term, this is straightforward. For the second term in the left side of (2.2), a Hardy-type inequality is employed, which shall be proved below.
For the first term in (2.2), we need only apply the Schwarz inequality to see that And thus, by conservation of energy, these terms are controlled by E[φ](0) as desired.
For the second term in (2.2), we again apply the Schwarz inequality. It remains to show that as a subsequent application of conservation of energy will complete the proof.
In order to show (3.1), we shall prove a Hardy-type inequality which is in the spirit of that which appears in [14]. Indeed, we notice that the coefficient in the integrand in the left side of (3.1) is O((log(r − r s )) −2 (r − r s ) −1 ) as r → r s and is O(r −2 ) as r → ∞. Thus, it will suffice to show that Notice that ρ(r) ∼ r d+1 as r → ∞ and ρ(r) ∼ 1 − log r−rs r −1 as r → r s .

Introduction
One of the more robust measures of dispersion for the wave equation is the so called localized energy estimates. These estimates have played a role in understanding scattering theory, as means of summing local in time Strichartz estimates to obtain global in time Strichartz estimates on a compact set, as an essential tool for proving long time existence of quasilinear wave equations in exterior domains, and as means to handle errors in certain parametrix constructions.
Estimates of this form originated in [30] where it was shown that solutions to the constant coefficient, homogeneous wave equation where ∇ denotes the angular derivatives. Though not the original method of proof, one can obtain this estimate by multiplying 2u by ∂ r u + n−1 2 u r , integrating over a space-time slab, and integrating by parts. One can also obtain control on ∂ r and ∂ t in a fixed dyadic annulus. Attempting to sum over these dyadic annuli comes at the cost of a logarithmic blow up in time. One may otherwise introduce an additional component of the weight that permits summability. In this case, it can be shown that for u as above and n ≥ 4 Here x = 1 + |x| 2 denotes the Japanese bracket and u ′ = (∂ t u, ∇ x u) represents the space-time gradient. An estimate akin to (1.1) also holds for n = 3 but in this case the weight x −3/2 in the second term in the left side must be replaced by x −3/2− . To prove (1.1), one multiplies the equation instead by f (r)∂ r u+ n−1 The second author was supported in part by the NSF through grant DMS0800678 . 1 and integrates by parts. For fixed j, this yields an estimate for the left side over |x| ≈ 2 j with weight x −1/2 in the first term. Introducing the additional weight and summing over j produces (1.1). See, e.g., [20], [21], [22], [25], [32], [34], [35]. Analogous estimates have been shown for small, asymptotically flat, possibly time-dependent perturbations of the d'Alembertian in [1], [26], [27,28] as well as for time-independent, asymptotically flat, nontrapping perturbations in e.g. [12], [13], [33].
A particular case of interest which does not fall into these latter categories is the wave equation on (1+3)-dimensional Schwarzschild space-times. While asymptotically flat and time independent, this metric is not, however, nontrapping. There is trapping which occurs on the so called photon sphere which necessitates a loss in the estimates as compared to those for the Minkowski wave equation. Despite this complication, a number of proofs of estimates akin to (1.1) exist for the wave equation on Schwarzschild spacetimes. See [5]- [10], [15,16], [24]. In [38], [17], and [3], these estimates have been extended to Kerr space-times with small angular momentums. Here an additional difficulty is encountered as [2] shows that the estimates will not follow from an analog of the argument above with any first order differential multiplier.
The goal of this article is to extend the known localized energy estimates on (1 + 3)dimensional Schwarzschild space-times to (1 + n)-dimensional Schwarzschild space-times for n ≥ 3. There are multiple notions of the Schwarzschild space-time for n ≥ 4, and we restrict our attention to the hyperspherical case. For a derivation and discussion of such black hole space times, see e.g. [36], [29]. An alternative notion of higher dimensional Schwarzschild space-times is discussed e.g. in [19]. We shall not explore these hypercylindrical Schwarzschild manifolds or other notions of higher dimensional black holes here.
The exterior of a (1 + n)-dimensional hyperspherical Schwarzschild black hole (M, g) is described by the manifold M = R × (r s , ∞) × S d+2 and the line element Here d = n − 3, r s denotes the Schwarzschild radius (i.e. r = r s is the event horizon), and dω is the surface measure on the sphere S d+2 = S n−1 . Here, the d'Alembertian is given by The Killing vector field ∂ t yields the conserved energy That is, when We now define our localized energy norm. To this end, we set Here r ps = d+3 We note that by modifying the lower order portion of the multiplier which appears in the next section that it is straightforward to obtain an estimate on ∂ t φ as well. Moreover, one can obtain an analogous estimate for the inhomogeneous wave equation by placing the forcing term in an appropriate dual norm. The necessary decay of the coefficient c r at ∞ can also be significantly improved. It is relatively simple to carry out these modifications, but for the sake of clarity, we omit the details.
As in the (1+3)-dimensional case, the higher dimensional hyperspherical Schwarzschild space-times have a photon sphere where trapping occurs. Rays initially located on this sphere and moving initially tangent to it will stay on the surface for all times. Such trapping is an obstacle to many types of dispersive estimates, and the vanishing in the coefficient c ω at the trapped set is a loss, when compared to the nontrapping Minkowski setting, which results from this trapping.
The proof follows by constructing an appropriate differential multiplier. This is carried out in the next section, and the construction most closely resembles that which appears in [24]. Some significant technicalities needed to be resolved, however, in order to find a construction which works in all dimensions d ≥ 1.
1 Upon completion of this work, the authors learned that a version of Theorem 1.1 was also obtained independently in the forthcoming [31].

Construction of the multiplier
Associated to 2 g is the energy-momentum tensor whose most important property is the following divergence condition Contracting Q αβ with the radial vector field X = f (r) 1 − rs r d+1 ∂ r , we form the Calculating the divergence of this quantity we have The last term involving the Lagrangian is not signed. In order to eliminate it, we utilize a lower order term in the multiplier. To this end, we modify the momentum densitỹ and recompute the divergence Ideally one would choose f (r) to be smooth, bounded, and so that the coefficients in the last three terms on the right are all nonnegative. Indeed, this is precisely what is done in, e.g., [34] and [26] when r s = 0, i.e. in Minkowski space-time. The localized energy estimate then follows from an application of the divergence theorem as well as an application of a Hardy inequality and the energy estimate in order to handle the time boundary terms. Unfortunately, in the current setting it does not appear possible to construct such an f . We shall instead construct f so that the last term in the right can be bounded below by a positive quantity minus a fractional multiple of the second term in the right.
We let g(r) = . The multiplier will be defined piecewise, and it will be convenient to parametrize in terms of the values of h(r). To this end, we fix the notation r θ to denote the value of r so that h(r θ ) = θ. Note, e.g., that r −∞ = r s and r ps = r 0 . More explicitly, r d+1 θ = r d+1 s d+1 2 e θ + 1 . An approximate multiplier is r ps r d+1 s r d+2 h(r). We must, however, smooth out the logarithmic blow up at r = r s . We must also smooth out h(r) near ∞ in order to prevent a term in f ′ (r) which has an unfavorable sign.
In order to accomplish this, set We first make a note about the boundary terms at r −1/ε . Indeed, an elementary calculation shows Of particular interest is that the coefficient of the resulting boundary term at r −1/ε is O(εδ).
We now proceed to showing that the sum of the first three terms in the right of (2.2) produce a positive contribution. To do so, we will examine f on a case by case basis and show • f ′ (r) > 0 for all r > r s • f (r) < 0 for r < r ps and f (r) > 0 for r > r ps .
• l(f )φ 2 r d+2 dr dω dt is bounded below by a positive term minus a fractional multiple of the (∂ r φ) 2 term and a r −1/ε boundary term.
By absorbing these latter pieces into those previously shown to positively contribute, the estimate shall nearly be in hand. It will only remain to examine the time boundary terms, and in particular, establish the Hardy inequality which will permit a direct application of conservation of energy.
The multiplier in this region is constructed to smooth out the logarithmic blow up at the event horizon. This is the only case in which we shall not be able to just show that l(f ) ≥ 0.
Noting that a(x) ≤ −1/ε for x ≤ −1/ε, we immediately see that f (r) < 0 on this range. We also compute Each of these summands is nonnegative on the given range, yielding the desired sign for f ′ (r).
The key step is to control the contribution of the last term in the right of (2.4). We shall abbreviate R(r) = δεh(r) + δ − 1. By the Fundamental Theorem of Calculus, we observe The dominant term is p(r), and we shall show 1 3 p(r) + n 1 (r) > 0, (2.13) 1 2 p(r) + n 2 (r) ≥ 0, (2.14) Proof of (2.13): Using that we are in the regime r ≥ r ps and that (h(r) 2 − α 2 ) 2 is maximized when h(r) = 0, we have The last quantity is clearly positive, as desired, for d > 1.
For the case d = 1, which is an everywhere positive quadratic.
Boundary term at r −1/ε : In order to finish showing that the right side of (2.2) is nonnegative, it remains to examine the r −1/ε boundary term in (2.2) as well as the subsequent contribution from (2.10). Here, we simply utilize the Fundamental Theorem of Calculus to control these terms via the positive contributions of the first and third term in the right of (2.2) in the range [r −1/ε , r ps ]. The scaling parameter δ insures the necessary smallness.
Fix a smooth cutoff β which is identity for, say, r ≤ r −1 and which vanishes for r ≥ r ps . Then, for r ≤ r −1 , we have φ(r) = − rps r ∂ s (βφ) ds.
Multiplying both sides by δ and integrating over [0, T ]×S d+2 , we see that these boundary terms can be bootstrapped into the contributions of Case 2.

A Hardy inequality and the time boundary terms
In the previous section, we constructed a multiplier so that the right side of (2.2) provides a positive contribution. By inspection, the coefficients are easily seen to correspond to those in (1.2). What remains is to control the left side of (2.2) in terms of the initial energy. For the first term, this is straightforward. For the second term in the left side of (2.2), a Hardy-type inequality is employed, which shall be proved below.
And thus, by conservation of energy, these terms are controlled by E[φ](0) as desired.
For the second term in (2.2), we again apply the Schwarz inequality. It remains to show that as a subsequent application of conservation of energy will complete the proof.
In order to show (3.1), we shall prove a Hardy-type inequality which is in the spirit of that which appears in [15]. Indeed, we notice that the coefficient in the integrand in the left side of (3.1) is O((log(r − r s )) −2 (r − r s ) −1 ) as r → r s and is O(r −2 ) as r → ∞. Thus, it will suffice to show that Notice that ρ(r) ∼ r d+1 as r → ∞ and ρ(r) ∼ 1 − log r−rs r −1 as r → r s .