A sharp version of Price's law for wave decay on asymptotically flat spacetimes

We prove Price's law with an explicit leading order term for solutions $\phi(t,x)$ of the scalar wave equation on a class of stationary asymptotically flat $(3+1)$-dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $\phi(t,x)=c t^{-3}+\mathcal O(t^{-4+})$ for bounded $|x|$, where $c\in\mathbb C$ is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk-Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $t^{-2 l-3}$ decay for waves with angular frequency at least $l$, and $t^{-2 l-4}$ decay for waves which are in addition initially static. This definitively settles Price's law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.


Introduction
The Schwarzschild spacetime [Sch16] with mass m > 0 is a spherically symmetric solution of the Einstein vacuum equation given by on R t × (2m, ∞) r × S 2 , where / g is the standard metric on S 2 . To describe our main result in a simple setting, we consider the initial value problem (1.2) with compactly supported and smooth initial data φ 0 , φ 1 ∈ C ∞ c ((2m, ∞) × S 2 ). (1) There exists a constant c ∈ C so that φ(t, x) decays according to for any > 0. Derivatives of φ − ct −3 along any finite number of the vector fields t∂ t and ∂ x satisfy the same estimate (with different C ). Explicitly, c is given by φ 1 (r, θ, ϕ) r 2 sin θ dr dθ dϕ. (1.4) (2) If l ∈ N 0 and φ 0 , φ 1 are supported in angular frequencies ≥ l (meaning that for all r, φ j (r, −) ∈ C ∞ (S 2 ) is orthogonal to the eigenspaces of ∆ / g with eigenvalues k(k + 1) for k = 0, . . . , l − 1), then (1.5) and the same decay holds for derivatives of φ along t∂ t and ∂ x . This decay rate is generically sharp.
(3) In both cases, if φ is initially static, i.e. φ 1 ≡ 0, then the decay rate of φ is faster by one power of t −1 .
We describe a more general result momentarily. Price [Pri72a,Pri72b], as clarified by Price and Burko [PB04], conjectured these decay rates in the 1970s. Pointwise t −3 decay was proved by Donninger-Schlag-Soffer [DSS12] for Schwarzschild spacetimes and by Tataru [Tat13] on a general class of stationary asymptotically flat spacetimes which includes Schwarzschild and subextremal Kerr spacetimes [Ker63]. Parts (2)-(3) (see Corollary 5.4) constitute the definitive resolution of Price's conjecture for linear scalar waves; they improve on the pointwise t −2l−2 decay (t −2l−3 for initially static perturbations) established by Donninger-Schlag-Soffer [DSS11] by one power of t −1 ; in fact, we control the infinite sum over all spherical harmonic modes with frequency at least l, rather than merely individual modes. (See also [Lea86] for a heuristic description of the full time evolution.) Angelopoulos-Aretakis-Gajic [AAG18a] gave the first rigorous derivation of the leading order term in (1.4) on a class of spherically symmetric, stationary, and asymptotically flat spacetimes including Schwarzschild and subextremal Reissner-Nordström spacetimes. Theorem 1.1(1) is a consequence of a partial expansion of the resolvent g (σ) −1 at σ = 0. Using a novel systematic and, to a large degree, algorithmic method, we show, roughly speaking, that the strongest singularity of g (σ) −1 , acting on inputs with compact support (or more generally satisfying almost-sharp decay assumptions), is σ 2 log(σ + i0), and we compute its coefficient explicitly; see §1.3 and Theorem 3.1 for further details. The study of the low energy resolvent has a long history, starting with the work by Jensen-Kato [JK79] on Euclidean space. Recent works describe qualitative [BH10,VW13] and quantitative bounds [RT15] as well as Hahn-meromorphic properties [MS14] of the resolvent, and the connection between the low energy resolvent behavior and the cohomology of the spatial manifold [SW19]. Here, we adopt Vasy's perspective [Vas19b,Vas19a] and obtain the resolvent expansion in a direct manner, rather than by adapting the Schwartz kernel constructions of Guillarmou-Hassell and Sikora [GH08, GH09,GHS13] (discussed further below).
1.1. Sharp asymptotics on subextremal Kerr spacetimes. In order to describe radiation falling into the black hole or escaping to infinity, it is convenient to introduce a new time coordinate t * whose level sets are transversal to the future event horizon and to future null infinity. Indeed, one can choose t * to be roughly equal to t + r * near the event horizon r = 2m and t − r * for large r, where r * = r + 2m log(r − 2m) is the Regge-Wheeler tortoise coordinate. The Schwarzschild metric g, expressed using t * instead of t, can then be extended smoothly across the event horizon, and is a stationary Lorentzian metric on 1 See Figure 1.1. The family of subextremal Kerr metrics, described in §4, generalizes the Schwarzschild metric and describes rotating black holes with angular momentum a ∈ (−m, m) and event horizon at r = r m,a := m + √ m 2 − a 2 > m.
See Theorem 4.5. Figure 1.1 shows the Penrose diagram and a resolution (blow-up) well-adapted to the description of global asymptotics (described in detail in Definition 3.7). indicated. Also shown are I + (future null infinity), H + (the event horizon), and i + (future timelike infinity). A hypersurface t * /r = v ∈ (0, ∞) is a timelike cone asymptotic to the cone r = 1 1+v t. On the right: Resolution of the Penrose diagram obtained by first blowing up i + (obtaining K + ) and then the lift of the future boundary of I + (obtaining I + ). The asymptotics ct −3 * govern decay at K + , while the profile (1.6) gives asymptotics at I + (which match ct −3 * at K + ∩ I + ).
Remark 1.3. For initial data supported away from the event horizon, a simple explicit expression for the constant c using Boyer-Lindquist coordinates is given in Corollary 4.7.
We note that t/(t 2 − r 2 ) 2 is an exact solution of the wave equation of Minkowski space. For more details, see Remark 4.6.
Remark 1.5. The constant C in (1.6) is bounded by a universal constant times φ 0 H N + φ 1 H N −1 for some large N as long as φ 0 , φ 1 have support in a fixed compact subset of X • . In order to simplify the exposition, we do not keep track of the number of derivatives or the precise decay assumptions (except for forcing problems, see Theorems 3.4, 3.9, 4.5). The interested reader can find a concrete value for N by carefully revisiting our arguments.
Remark 1.6. In the context of part (2) of Theorem 1.1, we prove t −l−3 * decay of φ in future timelike cones, and t −l−2 * decay of the radiation field of φ; for initially static perturbations, the decay rates are improved by 1. See Theorem 5.1. Generalizations of such l-dependent decay rates to Kerr spacetimes have been discussed in the physics literature [GPP08,BK14].
The constant c in Theorem 1.2 only vanishes on a codimension 1 subspace of initial data. Thus, the restriction of |∂ t * φ| 2 to the event horizon of the black hole generically obeys a pointwise lower (and upper) bound of t −6 * . This proves Conjecture 1.9 in the paper [LS16] by Luk-Sbierski and thus implies that generic smooth and compactly supported Cauchy data on subextremal Kerr spacetimes give rise to solutions of the scalar wave equation for which the nondegenerate energy on any spacelike hypersurface transversal to the Cauchy horizon is infinite (see [LS16,Conjecture 1.7]). Indeed, the upper and lower bounds in assumptions (i) and (ii) of their main theorem, [LS16, Theorem 3.2], hold for q = 5, δ = 0.
The asymptotic behavior (1.6) holds more generally on any stationary and asymptotically flat (with mass m ∈ R) spacetime, a notion we introduce in §2. Roughly speaking, these are spacetimes whose metrics have a 2m/r long range term just like the Schwarzschild metric g m , plus lower order perturbations which decay at a rate of at least r −2 . We need to assume the absence of zero energy bound states or resonances (smooth stationary solutions of g φ = 0 with |φ| r −1 for large r), the absence of nontrivial mode solutions which are purely oscillatory or exponentially growing as t * → ∞, and high energy estimates for the resolvent; in concrete situations, the latter can typically be proved easily using microlocal methods. See Definitions 2.3 and 2.9 and the results in §3.2.
Remark 1.7. Waves on dynamical spacetimes which merely settle down to a stationary spacetime can not be described in one fell swoop using spectral methods. Rather, as demonstrated on asymptotically de Sitter [HV15,Hin16] and Kerr-de Sitter spacetimes [HV16], in particular in the proof of the nonlinear stability of slowly rotating Kerr-de Sitter black holes [HV18], the analysis of the stationary wave equation (in these settings based on [SBZ97,BH10,MSBV14,Dya11,Vas13,Hin17]) is one step in a two-step analysis. Namely, microlocal methods allow the control of high frequencies of waves on dynamical spacetimes, while their decay is controlled using precise decay results on the stationary model spacetime (typically with a loss of regularity); together, this controls waves up to compact error terms (on a scale of weighted Sobolev spaces) which can then be dropped in perturbative regimes. Details of this approach on asymptotically flat spacetimes will be given in future work.
In order to put Theorems 1.1 and 1.2 into context, we recall that Angelopoulos, Aretakis, and Gajic are pursuing a program aimed at a detailed asymptotic description of waves on spherically symmetric spacetimes, including both subextremal and extremal black hole spacetimes. In particular, in [AAG18b], they prove almost sharp inverse polynomial decay on spherically symmetric, stationary, asymptotically flat spacetimes using energy estimate and vector field methods. In the aforementioned companion paper [AAG18a], they give the first rigorous proof of a t −3 * leading order term in compact spatial regions, a t −2 * leading order term for the radiation field (confirming predictions of Gundlach-Price-Pullin [GPP94]), and the asymptotic profile in the full forward light cone; key to their arguments are certain conservation laws at null infinity. Our results on Kerr (or more general) spacetimes remove the assumption that the underlying spacetime be spherically symmetric; as we shall discuss in §1.2 below, we also allow the coupling of scalar waves to stationary potentials. On the other hand, unlike [AAG18a], we do not keep careful track of the number of derivatives used. The subsequent work [AAG19] goes one step further and computes the first subleading t −3 * log t * term of the radiation field for spherically symmetric waves, confirming heuristic arguments by Gómez-Winicour-Schmidt [GWS94]. These leading and subleading terms are the first two terms of a (conjectural) full polyhomogeneous expansion of linear scalar waves φ on Kerr (or more general) spacetimes.
Remark 1.8. In §3.2.2, we define a compactification of [0, ∞) t * × X • to a manifold with corners on which we conjecture φ to be polyhomogeneous; see Figures 1.1 and 3.1.
On asymptotically Minkowski spacetimes, Baskin-Vasy-Wunsch [BVW15,BVW18] show the polyhomogeneity of scalar waves on a resolution of the radial compactification of R 4 at the boundaries at infinity of the future and past light cones. The spacetimes under consideration are required to be well-behaved with respect to the dilation action (t, x) → (λt, λx); in particular, stationary perturbations are not permitted. Baskin-Marzuola [BM19b] (see also [BM19a]) extended these results to allow for conic singularities of the metric on a cross section of the dilation action. This is directly related to the profile appearing in (1.6): in the terminology of [BVW15,BVW18,BM19b], this profile is, under suitable identifications, a resonant state of exact hyperbolic space with a conic singularity at r = 0; and indeed I + is equal to the blow-up of the 'north cap', denoted C + in the references, at the 'north pole'.
Guillarmou-Hassell and Sikora [GH08, GH09,GHS13] give a complete description of the Schwartz kernel of the low energy resolvent (−σ 2 + ∆ g + V ) −1 for potential scattering on asymptotically conic (or flat) spaces as a polyhomogeneous distribution on a suitable resolved space (which includes (0, 1) σ × X • × X • as an open dense submanifold). Via the inverse Fourier transform, this (together with bounds for bounded and high energies) gives full polyhomogeneous expansions of linear waves on a compactification of the spacetime. Their setup does not directly apply to Schwarzschild or Kerr spacetimes but, in concert with [HV01] does cover wave equations on Riemannian manifolds whose metrics, in dimension 3, have a long range mass term 2m/r of the same type as the Schwarzschild metric.
The proofs in [DSS11] of the first l-dependent pointwise decay rates t −2l−2 in the context of Theorem 1.1, as well as the subsequent [DSS12], control the spectral measure for low frequencies using separation of variables techniques. (See Finster-Kamran-Smoller-Yau [FKSY06] for a similar approach on Kerr spacetimes.) Tataru [Tat13] proves t −3 decay in large generality on a class of asymptotically flat and stationary spacetimes under the assumption that local energy decay estimates hold; these estimates hold on subextremal Kerr spacetimes, as discussed below (see also [MST20]). The metric asymptotics assumed in [Tat13] are quite weak: Tataru allows even the long range perturbations to be merely conormal, in contrast to our 2m/r leading order term which, however, is key for getting the leading order term rather than merely a O(t −3 * ) upper bound. (Our assumptions on short range perturbations in Definition 2.3 can easily be relaxed to conormality, see Remark 2.5.) His method allows for the coupling to stationary potentials with r −3 decay; these fit into our framework as well, as discussed in §1.2 below. Metcalfe-Tataru-Tohaneanu [MTT12] subsequently established Price's law on nonstationary spacetimes with suitable decay towards stationarity. Unlike [Tat13] and the present paper, the proof in [MTT12] does not make use of the Fourier transform in time, but rather combines local energy decay with the explicit solution of the constant coefficient d'Alembertian. The same authors also prove t −4 decay for the Maxwell equation on Schwarzschild spacetimes [MTT17]. In this case, there is a zero resonance, which gives rise to the stationary Coulomb solution (and is dealt with in an ad hoc manner). On the spectral side, this corresponds to a first order pole of the resolvent, the sharp analysis of which is beyond the scope of the present paper; see [HHV19] for weaker results in a related context.
There is a large amount of literature on wave decay on perturbations of Minkowski space; besides the above references, we mention in particular the work by Christodoulou and Klainerman [Kla80,Chr86,CK93], Lindblad-Rodnianski [LR10], and references therein.
Local energy decay estimates and pointwise decay of linear scalar waves on Kerr spacetimes were first obtained for very small angular momenta by Andersson-Blue [AB15a], Dafermos-Rodnianski [DR10], and Tataru-Tohaneanu [TT11], and established in the full subextremal range by Dafermos-Rodnianski-Shlapentokh-Rothman [DRSR16]; the spectral theoretic input is the mode stability proved by Whiting [Whi89] and Shlapentokh-Rothman [SR15]. Strichartz estimates were proved in [MMTT10,Toh12]. See Aretakis [Are12] for the extremal case. Results for semilinear and quasilinear equations were proved by Luk [Luk13], Lindblad-Tohaneanu [LT18,LT20] 1.2. Sharp asymptotics for wave equations with stationary potentials. On subextremal Kerr spacetimes (or generalizations as in §2), we can couple scalar waves to stationary complex-valued potentials V with r −3 decay at infinity under spectral conditions on g + V as before (absence of bound states and nontrivial nondecaying mode solutions; high energy estimates). The asymptotics (1.6) continue to hold in every cone δt * < r < (1 − δ)t * , δ > 0; however, the asymptotic behavior in compact spatial sets is modified: one has where u (0) is an 'extended bound state': u (0) is the unique stationary solution of ( g + V )u (0) = 0 which for large r is equal to a constant c plus O(r −1+ ) corrections. Here, c is equal to the L 2 inner product of a linear combination of the initial data with an 'extended dual bound state' u * (0) which solves ( g + V ) * u * (0) = 0. We illustrate this on Minkowski space R t × R 3 x with metric −dt 2 + dx 2 ; even in this setting, the result appears to be new: Theorem 1.9 (Sharp asymptotics for wave equations with stationary potentials in a simple special case).
Then, for x restricted to any fixed compact subset of R 3 , we have See §3.3 for the general result, and the discussion following (3.40) for the proof of Theorem 1.9; see Remark 4.8 for the extension to Kerr spacetimes, and Remark 3.13 regarding relaxed regularity requirements. The existence of a leading order term, and also its explicit form, can in principle also be obtained using the methods of [GHS13] upon supplementing the reference with high energy resolvent estimates. The asymptotic behavior of solutions of (1.8) for compactly supported V is drastically different (resonance expansions, exponential decay); for a detailed discussion, we refer to [DZ19, Chapter 3] and references therein.
1.3. Method of proof; outlook. We work almost entirely on the spectral side and solve forward problems for forced waves, by means of the Fourier transform, given byφ(σ, x) = e iσt * φ(t * , x) dt * . Thus, where g (σ) is defined in terms of g by replacing all ∂ t * derivatives by multiplication by −iσ. The integral is well-defined and produces the forward solution of (1.9); we refer the reader to the discussion in [HHV19, §1.1] for details, and only briefly recall the main aspects here. Roughly speaking, if one integrates over the contour Im σ = C 1, one does obtain the forward solution by the Paley-Wiener theorem. One can then shift the contour down to the real axis using the mode stability assumption and the absence of zero energy resonances; high energy estimates (polynomial bounds on g (σ) −1 acting on suitable Sobolev spaces) justify the contour shifting. As mentioned before, mode stability is known on subextremal Kerr spacetimes [Whi89,SR15]; high energy estimates on the other hand are known to hold using semiclassical microlocal techniques, combining radial point estimates at the event horizons [Vas13] (see also [Zwo16] and [DZ19, Appendix E] for streamlined presentations), estimates at the normally hyperbolic trapped set [WZ11,Dya16,Dya15], and radial point (microlocal Mourre) estimates at spatial infinity [Mel94,Vas19a]. See 5 [HHV19, Theorem 4.3].
As in [HHV19], the main task is thus to control the regularity of g (σ) −1 at σ = 0; higher regularity means faster temporal decay of φ. A key aspect of our analysis is that we use the Fourier transform in the coordinate t * (whose level sets are transversal to null infinity), rather than the 'usual' time coordinate t (which is not as well-suited to scattering theoretic considerations in the context of wave equations) as for example in [BH10,VW13,Vas18]. The advantage of this point of view was pointed out by Vasy [Vas19a,Vas19b]. Namely, the limiting resolvent g (σ) −1 for nonzero real σ produces outgoing solutions; working on the Minkowski spacetime and setting t * = t − r for concreteness, this corresponds to solutions with time dependence e −iσt and leading order spatial dependence r −1 e iσr , thus an overall e −iσt * r −1 ; therefore, the 'outgoing' condition for the output of g (σ) −1 in (1.10) means nonoscillatory, σ-independent r −1 behavior at infinity. In fact, the output is conormal, i.e. has r −1 decay upon repeated application of r∂ r and rotation vector fields.
As shown in [Vas19b], one can then analyze the low energy resolvent g (σ) −1 uniformly as σ → 0 on spaces A α of conormal functions with suitable decay rates α at infinity, corresponding to r −α decay. Besides needing to allow outgoing r −1 asymptotics in the target space, the decay rate α of the target space need to be chosen to ensure the invertibility also of the zero energy operator. The model is the Euclidean Laplacian ∆ R 3 , which is invertible with domain given by functions decaying faster than r 0 (avoiding the nullspace given by constants) and less than r −1 (since ∆ −1 R 3 with Schwartz kernel (4π|x − y|) −1 typically does produce r −1 asymptotics). Overall, one expects invertibility (1.11) and thus uniform bounds for g (σ) −1 : A 2+α → A α for small σ.
We can now illustrate the basic mechanism underlying the proof of our main theorems. On the Schwarzschild spacetime, g (σ) is, to the relevant precision, equal to L(σ) = −2iσr −1 (r∂ r + 1) + ∆ R 3 + 2mr −3 (r∂ r ) 2 for large r. In terms of weights, the first term typically only gains one order of decay (mapping r α to r α−1 ), the second term (which is L(0)) two. Let us now formally expand the resolvent near zero frequency by writing The assumption of very small angular momenta, which is used in the precise low energy resolvent analysis of the reference, is not used in the proof of high energy estimates.
The first term, u 0 , is σ-independent. In the second term, we gain a factor of σ (thus suggesting this is a more regular term); however, L(0) −1 loses two orders of decay, while σ −1 (L(σ) − L(0)) only gains back one order, thus f 1 has one order of decay less than f . That is, the σ-gain comes at the cost of losing one order of decay in the argument of L(σ) −1 .
One would like to iterate as often as possible while remaining in the invertible range (1.11); the resolvent expansion is then u 0 + σu 1 + · · · + σ k u k + · · · . However, one cannot continue the iteration once f k only has r −2 decay or less: the correction term in the expansion is then typically no longer uniformly bounded as σ → 0. In fact, we show that if f k has borderline r −2 decay, then L(σ) −1 r −2 has a logarithmic singularity at σ = 0 with explicit coefficient, hence (1.13) is σ k log(σ + i0). Upon taking the inverse Fourier transform in (1.10), this singular term gives rise to a t −k−1 * leading order term 6 of the solution of the wave equation g φ = f . On the (3 + 1)-dimensional stationary and asymptotically flat spacetimes under consideration here, we obtain a borderline term (1.13) for k = 2, giving the desired t −3 * decay; see the beginning of §3.1 for a brief sketch of why it is indeed f 2 that has borderline r −2 decay, and how the mass term m enters.
The precise analysis of a borderline term (1.13) is accomplished by geometric microlocal means: one constructs an approximate solution of L(σ)ũ = f k on a resolution X + res of the total space [0, 1) σ × ([0, 1) ρ × S 2 ), ρ = r −1 , obtained by blowing up σ = ρ = 0 (thus separating the regimes σ/ρ = σr 1 and σr 1); the resolved space X + res already played a prominent role in [Vas19b]. The model problem on the front face is the spectral family at frequency 1 of a rescaled exact Minkowski space and can be analyzed in detail; the desired approximate solution is shown to have a log(σ/ρ) singularity (see Lemma 2.23). To find the true solutionũ, one merely needs to apply L(σ) −1 to the remaining error which has a logarithmic singularity in σ but better than r −2 spatial decay. (Overall, the coefficient of the logarithmic term ofũ is an element in ker L(0) of size 1 for large r, thus equal to 1 for wave equations, and equal to u (0) as in Theorem 1.9 in the presence of potentials; see Proposition 2.24.) The leading order term in the full forward light cone asserted in (1.6) drops out of this construction as well, via the inverse Fourier transform (in σ/ρ) of the solution of the model problem (see equation (3.34) in the proof of Theorem 3.9).
The proof of the full Price law (1.5) in §5 proceeds along the same lines; the higher regularity is mainly due to the fact that the invertibility (1.11) holds for the larger range α ∈ (−l, l + 1) of weights when restricting to angular frequency l ∈ N, which allows for more iterations (1.12).
Remark 1.10. The regularity and pointwise estimates for φ in Theorems 1.1 and 1.2 are consequences of the conormal regularity at σ = 0 of the resolvent (i.e. regularity of L(σ) −1 f with respect to repeated application of σ∂ σ ) with values in appropriate conormal spatial function spaces.
Potential future extensions and applications of the methods developed here include: (1) a new proof of Morgan's results [Mor] on decay for stationary spacetimes asymptotic to Minkowski space at an inverse polynomial rate using the same approach (albeit possibly requiring more derivatives on the metric and the initial data); (2) a proof of the polyhomogeneity of φ on a compactification of the spacetime mentioned in Remark 1.8. We expect that this can be done using same iteration (1.12) upon keeping track of polyhomogeneous expansions of all the u k , f k , and extending the analysis of borderline (or worse) terms (1.13) so as to keep track of the polyhomogeneous expansion of the solution of the model problem, as well as of the remaining error term. The main ingredient is the analysis of L(0) −1 on inputs which are polyhomogeneous on the resolved space X + res ; (3) an analysis of the effect of the angular momentum parameter a of Kerr spacetimes.
Here, a enters at one order lower (in terms of r-decay) than the mass parameter m, and destroys spherical symmetry, thus leading to the coupling of spherical harmonics when computing further terms (i.e. beyond what we do in the present paper) in the resolvent expansion. It would be interesting to see how the presence of nonzero angular momentum affects the full asymptotic expansion, in particular, whether there are extra logarithmic terms which are not present for a = 0; (4) sharp asymptotics for equations with zero energy resonances or bound states. This requires significantly more work, as the resolvent now has strong singularities at σ = 0; see [HHV19]. Examples include Maxwell's equation or the equations of linearized gravity on Kerr spacetimes.
1.4. Outline of the paper.
• In §2, we describe the geometry ( §2.1) and spectral theory ( §2.2) of the class of stationary and asymptotically flat spacetime under investigation. We give a detailed account of the regularity and mapping properties of the low energy resolvent ( § §2.2.2-2.2.3) as required for the precise analysis of the iteration (1.12). • In §3, we prove the main result giving the low energy resolvent expansion ( §3.1) and use it to prove Price's law with leading order term ( §3.2). The modifications required for stationary potentials and Theorem 1.9 are described in §3.3. • In §4, subextremal Kerr spacetimes are placed into our general framework.

Asymptotically flat spacetimes
2.1. Metrics and wave operators. The model for the large scale behavior of the spacetimes we have in mind here is the Schwarzschild spacetime: given the mass m > 0, it has the metric where / g = dθ 2 + sin 2 θ dϕ 2 is the standard metric on S 2 . Denote by r * = r + 2m log(r − 2m) the tortoise coordinate, and put t * = t − r * ; then The spacetimes we consider here are 'short range' perturbations of this. To capture the asymptotics in a compact fashion, we define: Definition 2.1. The compactified spatial manifold X is the radial compactification Thus, smooth functions on X are precisely those smooth functions on R 3 which in r > 1 are smooth functions of r −1 and the spherical variables. Near ∂X = ρ −1 (0), we shall work in the collar neighborhood [0, ) ρ × S 2 .
Definition 2.2. The scattering tangent bundle sc T X → X is the unique vector bundle for which the space of smooth sections consists of all smooth vector fields V on X • which for r > 1 are of the form One can check that the coordinate vector fields ∂ x 1 , ∂ x 2 , ∂ x 3 on R 3 form a basis of sc T X down to ∂X. For example, the restriction of g −1 m on S 2 T * X, (1 − 2mρ)∂ 2 r + r −2 (∂ 2 θ + sin −2 θ ∂ 2 ϕ ), lies in C ∞ (X; S 2 sc T X) upon cutting it off to a neighborhood of ρ = 0.
Definition 2.3. We call a smooth Lorentzian 8 metric g on M • = R t * × X • stationary and asymptotically flat (with mass m ∈ R) if ∂ t * is a Killing vector field, and if moreover (1) dt * is everywhere future timelike, i.e. g 00 < 0; (2) the coefficients of the dual metric Remark 2.4. The function t * defined previously on the Schwarzschild spacetime does not satisfy (1), but a small modification does; see §4. Even then, this definition excludes Schwarzschild and Kerr metrics due to the existence of horizons. Since the low energy behavior of the resolvent is only sensitive to large end of the spacetime however, the adaptations to deal with Kerr are small and will be discussed in §4.
Remark 2.5. It suffices to assume that g 00 and the ρ 2 C ∞ error terms of g 0X , g XX are merely conormal, i.e. of class A 2 (X) in the notation of Definition 2.8, in order for all arguments in this paper to apply unchanged. One can further relax their decay to A 1+β (X) for β > 0, though this does affect the spatial decay rate and the σ-regularity of various terms in the resolvent expansion.
The wave operator g = −|g| −1/2 ∂ µ (|g| 1/2 g µν ∂ ν ) ∈ Diff 2 (M • ) of such a metric g is invariant under time translations. We compute its form in the following terms: The wave operator g of an admissible metric is given by where the dilation-invariant (in ρ) operators Q 0 , L 0 , L 1 are given by Proof. The coefficients of second order derivatives are of course equal to (minus) the coefficients of the dual metric function; noting that ∂ r = −ρ 2 ∂ ρ , this verifies the first order term of Q 0 and the second order terms of L 0 , L 1 . To compute the lower order terms, note that in polar coordinates (θ, ϕ) on S 2 and up to an overall sign, we have Thus, up to ∂ t * • ρ 3 Diff 1 b error terms (captured by ρQ), the t * -X-cross terms are given by For the zero energy operator, the ∂ 2 r term of the dual metric gives, modulo ρ 2 Diff 2 b and using that which gives the terms in L j involving ∂ ρ . The ∂ 2 θ term gives −ρ 2 (sin θ) −1 ∂ θ sin θ ∂ θ ); and the ∂ 2 ϕ * coefficient finally gives −(sin θ) −2 ρ 2 ∂ 2 ϕ * . The ρ 2 C ∞ error terms in g XX contribute to the ρ 2L ∈ ρ 2 · ρ 2 Diff 2 b error terms of g (0).

Spectral theory.
We fix a stationary and asymptotically flat metric g with mass m. We denote the spectral family of g by We equip X with the volume density |dg X | (2.5) defined via |dg| = |dt * ||dg X |, where |dg| is the volume density of g. (Thus, in local coordinates on X • , |dg X | = | det g(t * , x)| 1/2 |dx|, with the determinant independent of t * .) We write L 2 (X) := L 2 (X; |dg X |); formal L 2 adjoints of differential operators on X shall always be with respect to this L 2 space.
Definition 2.8. On X as in Definition 2.1, we define the following function spaces.
(1) For s ∈ N 0 and ∈ R, we define the where the inner sum is over a finite set of operators A jk which span Diff j b (X) over C ∞ (X). 9 The spaces H s b (X) for s ∈ R are defined by duality and interpolation.
as a vector space, but with norm given by That is, each b-derivative comes with an extra factor of h.
Definition 2.9. The metric g is spectrally admissible if the following conditions are satisfied: (1) (Absence of bound states.) The nullspace of (0) on A 1 (X) is trivial. 9 In the concrete setting at hand, we can take the operators A jk to be all up to j-fold compositions of the b-vector fields ∂ x i and x i ∂ x j for i, j = 1, 2, 3, since these vector fields span V b (X) over C ∞ (X).
(3) (High energy estimates.) There exists δ ∈ R such that for s ∈ R, < − 1 2 , s+ > − 1 2 , there exists C > 0 such that the estimate holds for all u for which the norms on both sides are finite.
Typically, the estimate (2.7) follows from assumptions on the dynamics of the nullgeodesic flow on (M • , g): if there is no trapping, one can take δ = 0; if there is normally hyperbolic trapping, one needs to take δ > 0 though it can be arbitrarily small.
10 By this, we mean that ∂ m The conditions arise from (1) allowing the outgoing behavior of u, which means r −1 e iσ(r−t) behavior on spacetime for r 1, and thus r −1 on the spectral side, with r −1 ∈ H ∞, b precisely for < − 1 2 ; (2) enforcing the absence of ingoing spherical waves r −1 e iσ(−r−t) on spacetime, thus r −1 e −2iσr on the spectral side, which is accomplished by requiring a decay order s + > − 1 2 for nonzero oscillations at r = ∞; (3) working in a range of weights on which (0) is invertible, which gives the range of weights (− 3 2 , − 1 2 ) relative to L 2 (X).
As a consequence of the Sobolev embedding (2.6), we have In order to capture the output of the resolvent (σ) −1 precisely near ρ = σ = 0, we work on a resolved space: Definition 2.12. The resolved space (for positive frequencies) X + res is the blow-up The boundary hypersurfaces are denoted as follows: • tf: the front face; On the manifold with corners X + res , we consider conormal spaces consisting of all locally bounded functions that remain such upon application of any finite number of vector fields tangent to all boundary hypersurfaces of X + res . We also need more precise function spaces capturing partial polyhomogeneous expansions. Recall that an index set is a subset E ⊂ C × N 0 such that the number of (z, k) ∈ E with Re z < C is finite for any fixed C ∈ R, and so that (z, k) ∈ E implies (z + 1, k) ∈ E and, if k ≥ 1, (z, k − 1) ∈ E. We let inf Re E denote the smallest value of Re z among all (z, k) ∈ E.
(1) Let E be an index set, α 0 := inf Re E, and α ∈ R; put β = min(α 0 , α). Then the space A (E,α) (X) ⊂ A β− (X) consists of all u which are smooth in X • and which near ∂X have a partial expansion which have partial expansions with conormal remainders at all boundary hypersurfaces. That is, in a collar neighborhood where the exponents on the right are, in this order, the weights at [0, ) × ∂zf and {0} × zf. Likewise, u has partial expansions at the remaining two boundary hypersurfaces tf, bf.
(3) Partially polyhomogeneous spaces such as A α bf ,α tf ,(E zf ,α zf ) (X + res ) have partial expansions only at the boundary hypersurfaces at which an index set is given.
The regularity of the low energy resolvent is then as follows; this is similar to [HHV19, Propositions 12.4 and 12.12], though here we do not keep track of the number of derivatives used.
Remark 2.15. Upon taking the inverse Fourier transform in σ, this already suffices to show For inputs living on the resolved space, we record: (2.14) , proving (2.14). These arguments apply verbatim also to (0) −1 f in view of (2.9).
2.2.3. Action of the resolvent on large inputs. We first record a simple estimate for less decaying inputs which will be used to estimate error terms in resolvent expansions later on: The same conclusion holds if, more generally, f ∈ A 0 ([0, 1), A 2−α (X)).
Remark 2.18. Following the general strategy outlined in §1.3, this lemma can also be proved more systematically by solving a model problem involving (1) and applying the standard resolvent to the remaining error term which has better decay.
The key technical result for obtaining the precise nature of the first singular term of the resolvent concerns the regularity of (σ) −1 acting on borderline ρ 2 C ∞ (X) input, see Proposition 2.24 below. To set this up, we first show: In the present setting we simply have However, we keep the notation more general in order for our derivation of Price's law to apply unchanged to more general situations such as Kerr or wave equations with potential, see § §3.3 and 4. Correspondingly, the proof of this lemma will only use the structures which are present in these more general situations.
The arguments for u * (0) are completely analogous.
The analysis of (σ) −1 f , f ∈ ρ 2 C ∞ (X), proceeds by constructing an approximate solution of (σ)u = f near ρ = σ = 0 explicitly, and then correcting it to a true solution using (σ) −1 acting on a function space with more decay. The relevant model problem already prominently featured in [Vas19b,§5] in the context of the proof of Theorem 2.11.
Thus, (1) ∈ ρ bf ρ −2 zf Diff 2 b (tf). The importance of (1) in relation to (σ) stems from the following calculation: Lemma 2.21. The operator (σ), as a second order differential operator on X + res , is a b-differential operator of class (σ) ∈ ρ b ρ 2 tf Diff 2 b (X + res ). Its b-normal operators are: • 2iσ 2ρ (ρ∂ρ − 1) at bf, i.e. (σ) differs from this by an element of Proof. Working near bf ⊂ X + res with coordinates σ ≥ 0,ρ ∈ [0, ∞), and a factor of ∂X, consider the form (2.4) of (σ) in the notation of Lemma 2.7: the terms Q 0 and L 0 give rise to (1). On the other hand, elements of σ l ρ k Diff m b (X) lift along the stretched projection X + res → X to elements of ρ k bf ρ l+k tf ρ l zf Diff m b (X + res ); henceQ and ρL 1 +L (as well as g 00 σ 2 ) lift to b-differential operators on X + res which vanish cubically at tf and quadratically at bf. Near zf ⊂ X + res on the other hand, all terms with a factor of σ vanish at zf, hence the b-normal operator at zf is (0) as claimed.
These types of arguments are frequently formulated in the opposite order: one first explicitly solves away the leading term (here the spherically symmetric part off ) using the logr term, and then solves away the remaining error term, acting on which L −1 0 does not produce any logarithmic singularities anymore (i.e. poles on the Mellin transform side).
The imaginary part of the constant term ζ ofũ (2) is equal to real part of the O(r) term of I(r) := ∞ 0 e −2t log(r + it) dt. The proof of Lemma 2.23 gives the structure of the expansion and the coefficient of the logarithmic term in I(r) = −c + ir logr + ζr + A 2− . Thus, Re ζ = ∂r Re I(r)|r =0 = limr →0 ∞ 0 e −2tr r 2 +t 2 dt = π 2 upon substituting t =ry.

Price's law with a leading order term
We fix a stationary and asymptotically flat (with mass m) metric g (see Definition 2.3), which we moreover assume is spectrally admissible (see Definition 2.9). We abbreviate where the density |dg X | is defined after (2.5).

Resolvent expansion.
The key result of the paper is: . For positive frequencies, the resolvent acting on f is then the form , where the leading terms of σ −2 u sing (σ) at zf and tf are, respectively, −(log σ ρ )c X (f )u (0) and c X (f )ũ (2) withũ (2) = (1) −1 (ρ 2 ) (see Proposition 2.24); here, The subscript 'sing' refers to the fact that u sing captures the most singular (σ 2 log σ) behavior of the resolvent at zf (i.e. as σ → 0 in X • ), while u reg collects those terms in the resolvent expansion which are smooth down to σ = 0 or at least more regular than u sing .
As already used in the proof of Proposition 2.24, the strategy is to write In the second term, we gain a power of σ due to (σ) − (0) ∈ σρDiff 1 b (X); however, (0) −1 typically loses (at least) two orders of decay, while (σ) − (0) typically only gains back one order, thus the σ-gain comes at the cost of reducing the decay of the argument of (σ) −1 . One can iterate the rewriting (3.1) while keeping track of the precise decay of the terms on which (σ) −1 on the right in (3.1) acts. The outline of the proof of Theorem 3.1 is then: (3) (σ) −1 f 2 is logarithmically divergent as σ → 0 since f 2 (barely) fails to have sufficient decay, cf. Proposition 2.24. This produces u sing (σ).
The total resolvent expansion being step (3) above provides the main contribution to the singular term u sing (σ). We remark that the importance of the O(ρ 2 ) subleading term of u 0 is also explained in the discussion of [Tat13, Proposition 6.14].
Proof. By a normal operator argument as in the proof of Lemma 3.2, we conclude that u 1 ∈ ρY (1) + A 1+α− for some Y (1) ∈ Y 1 .
3.2. Asymptotic behavior of waves. For simplicity, we first restrict our attention to the long time asymptotics in compact spatial sets in §3.2.1 before describing the global asymptotics in §3.2.2.

Asymptotics in spatially compact sets.
Theorem 3.4. Let α ∈ (0, 1), and let f = f (t * , x) ∈ C ∞ c (R t * ; A 4+α (X)). Then the unique forward solution φ of φ = f satisfies for x ∈ X lying in a fixed compact set K X • . Derivatives of φ decay accordingly to Proof. Using the Fourier transformφ(σ, x) = R e iσt * φ(t * , x) dt * , we express φ as The fact that this gives the (unique) forward solution follows from the Paley-Wiener theorem upon deforming the integration contour to iC + R and letting C → ∞; this uses the mode stability assumption (2) in Definition 2.9.
Let χ ∈ C ∞ c ((−1, 1)) be identically 1 near 0; we then split (3.18) The high energy piece φ 2 has strong decay: since we are not counting derivatives, we simply observe that by Lemma 2.10, φ 2 (σ) is Schwartz in σ with values in A 1− (X). Therefore, Consider next the low energy piece φ 0 . Since we are restricting to ρ > 0, Theorem 3.1 shows that φ 0 is of class for some real-valued u ∈ C ∞ (X • ) (from the O(r 0 ) term of Proposition 2.24); the constant iπ 2 comes from Remark 2.25 (see also Remark 3.5 below). An inspection of the explicit expansion (3.11) as well as of the regular piece (3.14) show that the smooth pieces for σ > 0 and for σ < 0 fit together in a smooth fashion at σ = 0. (See §3.2.2 for a 'resolved space' picture of this.) For real σ, we have (σ) −1 f = (−σ) −1f ; but note that for σ < 0, and with the branch cut of log along −i[0, ∞), , hence the logarithmic terms from ±σ > 0 combine to a σ 2 (log(σ +i0)− iπ 2 ) term. Absorbing the constant − iπ 2 into the smooth parts, we thus obtain (3.21) The term φ 1 (σ) has an extra factor of σ, hence lies in the remainder space in (3.21). Now, the smooth first term in (3.21) has rapidly vanishing (as t * → ∞, in compact subsets of X • ) inverse Fourier transform. Either of the conormal terms has Fourier transform which are bounded by t −3−α+ * together with all their derivatives along t * ∂ t * and ∂ x ; see Lemma 3.6 below. The main contribution to φ 0 and thus to φ comes from the logarithmic singularity.
Recall then that the inverse Fourier transform of (σ + i0) z (with the sign convention used to pass to the spectral family) is To evaluate this, we note that in t * > 0, In summary, F −1 (σ 2 log(σ + i0)) = 2t −3 * , t * > 0, (3.23) and this vanishes in t * < 0. The logarithmic term in (3.21) thus gives the t −3 * leading order term with the stated constant. This proves (3.17).
Remark 3.5. The constant arising in (3.20) is necessarily iπ 2 by causality considerations. Indeed, in the derivation of the asymptotics above, only the imaginary component of this constant matters; let us write it as i( π 2 + c) for some c ∈ R. The asymptotics of φ would then have an additional term c ; this would be the only contribution to φ which is not rapidly decaying as t * → −∞. But since φ vanishes for large negative t * , this is impossible unless c = 0.
In the proof, we used the following standard result on inverse Fourier transforms of conormal distributions, whose proof we include for completeness: Proof. We work in t * > 0. Let χ ∈ C ∞ c ([0, 1)) be 1 near 0. Then 2πφ(t * ) = I + II, where We estimate |I| . In II, we fix k ∈ N 0 , k > β + 1, and write Expanding the derivative and substitutingσ = σt * , we can estimate |II| by the sum of two types of terms: the first arises from having all σ-derivatives fall onφ, giving in the second type of terms, j = 1, . . . , k derivatives fall on χ, similarly giving

Global asymptotics.
To cleanly describe the asymptotic behavior of φ in Theorem 3.4 in the full forward light cone, we pass to a compactification of the spacetime manifold in t * > 0. (1) the Cauchy surface Σ = t −1 * (0) ∼ = X; (2) null infinity I + : the lift of [0, ∞] t * × ∂X; (3) Minkowski future timelike infinity I + : the front face; (4) spatially compact future infinity K + : the lift of τ −1 (0). See Figure 3.1. Here, Σ is an interior hypersurface, whereas the remaining three are boundary hypersurfaces 'at infinity'. Correspondingly, the correct notion of regularity at Σ is smoothness in the usual sense: Definition 3.8. Denoting by ρ I + , ρ I + , ρ K + ∈ C ∞ (M + ) boundary defining functions of the respective boundary hypersurfaces, we define the space of conormal functions A α,β,γ (M + ) := ρ α I + ρ β I + ρ γ K + A 0,0,0 (M + ) which are smooth at Σ. That is, φ ∈ A 0,0,0 (M + ) if and only if φ remains bounded upon application of any finite number of smooth vector fields on M + which are tangent to I + , I + , and K + (but not necessarily at Σ). Function spaces with partial polyhomogeneous expansions at some of the boundary hypersurfaces are denoted A (E,α),β,γ (M + ) etc. as in Definition 2.13. are defining functions of I + and I + , respectively; away from I + on the other hand, are defining functions of K + and I + , respectively. In particular, with I + ∩ ρ −1 I + ,1 (0) being the future boundary of I + . Theorem 3.9. Let α ∈ (0, 1), f ∈ C ∞ c ((0, ∞) t * ; A 4+α (X)), and denote by φ the unique forward solution of φ = f . Then Setting c M (f ) = − 2m π f, u * (0) , the leading order terms of φ are: (3) at I + : The leading order terms match up, as they should: the leading order term at I + in part (2) has asymptotics at I + and K + matching (1) and (3).
Remark 3.10. As long as u (0) is constant (which is the case here, but not in the more general context of §3.3 below), we can capture the leading order behavior of φ in a more condensed form by writing v = t * /r and noting that t * (t * + r) −1 is a global defining function for I + ; thus, In terms of t := t * + r, the leading order term is c M (f )t/(t 2 − r 2 ) 2 .
For the proof, it is convenient to glue together the resolved spaces for positive and negative frequencies, thus forming which contains X ± res := ±[0, 1) σ × X; {0} × ∂X as submanifolds with corners; see Figure 3.2. The point is that X res allows us to track smoothness across σ = 0 (see the discussion leading up to (3.21)) while at the same time resolving the delicate zero energy behavior. Proof of Theorem 3.9. At first, we shall only keep very rough track of the decay at I + ; we will recover sharp asymptotics there at the end of the proof. We revisit the proof of Theorem 3.4 and use the splitting φ = φ 0 + φ 1 + φ 2 as in (3.18). By (3.19), we have • Low energy contribution: regular part. Consider next φ 0 , starting with the contribution from the regular part u reg (σ) in the notation of Theorem 3.1. The contribution φ 0,reg,1 from the smooth first piece of u reg (σ) is rapidly vanishing, that is, φ 0,reg,1 ∈ A 1−,∞,∞ (M ). (3.28) The second piece of u reg (σ), σ > 0, glues together with its negative frequency analogue to give an element u reg,2 = u reg,2,0 + ± u reg,2,± , u reg,2,0 ∈ A α−,2+α− (X res ), u reg,2,± ∈ A α−,2+α−,2+α− (X ± res ).
The conclusion is that the ρ 3 I + leading order term of φ at I + is equal to that of the inverse Fourier transform of the leading order term of u sing at tf, extended by (degree −2) homogeneity along (ρ, σ) → (λρ, λσ), λ > 0, i.e. to the inverse Fourier transform of + is the Heaviside function. Thus, the two leading order terms at tf from X ± res get glued together at the front face of X res , with a resulting mild logarithmic singularity at the 'seam' σ = 0 of the formr 2 log(r + i0); cf. Figure 3.2. Writingr = σ/ρ, and factoring σ 2 = ρ 2r2 , this is equal to (3.34) here we integrated by parts twice using iv −1 ∂re −irv = e −irv . The integral can be evaluated explicitly. 16 Indeed, since ∂ 2 rr 2 = (r∂r + 2)∂rr, the arguments in Remark 2.25 show that We then regularize the integral in (3.34) by inserting a factor e − r and letting 0, giving This integral is now easily evaluated, and its real part is 2π(v+1) (v+2) 2 . Plugging this into (3.34) and using that c X (f (0)) = − 1 2 c M (f ), this gives the desired leading order term at I + . Putting this together with (3.27), (3.28), (3.29), and (3.33), and noting that the piece φ 1 has an extra order of t * -decay relative to φ 0 , we have now shown with the claimed leading order terms at I + and K + .
• Decay at null infinity. The asymptotic behavior can be determined by integration along approximate characteristics as in [HV20,§5]. Let us use more compact notation, v := ρ I + ,1 = ρt * , τ = ρ I + ,1 = t −1 * , near I + ⊂ M + , so that ∂ t * = τ (−τ ∂ τ + v∂ v ) and ρ∂ ρ = v∂ v ; we work in the neighborhood M := [0, 1) τ × [0, 1) v × ∂X of I + ∩ I + ⊂ M + . Then Lemma 2.7 implies 16 This is not surprising: in the context of works by Baskin-Vasy-Wunsch [BVW15,BVW18] and, more directly, Baskin-Marzuola [BM19b], the leading order behavior at I + is directly related to resonances of exact hyperbolic space with a cone point at the origin (i.e. forgetting that hyperbolic space is smooth across the origin), this being the model of Minkowski space with a line of conic points along r = 0.
its asymptotic behavior as t * → ∞ can thus be computed using Theorem 3.9. The constant c M ([ , χ]φ) must be independent of χ. One can thus evaluate it by taking χ to be the Heaviside function H(t * ), which by Lemma 2.7 gives which pairs against u * (0) to − 2ρQφ 0 + g 00 φ 1 , u * (0) L 2 (X) , as claimed.
3.3. Wave equations with stationary potentials. As a simple generalization, we briefly consider wave equations with a stationary potential V decaying like r −4 , or more precisely we can allow V to be complex-valued. With = g the wave operator of a stationary and asymptotically flat (with mass m) metric g, we consider the operator We assume that P V spectrally admissible, that is, P V has no nontrivial zero energy bound states, no nontrivial mode solutions with frequency 0 = σ ∈ C, Im σ ≥ 0, and that high energy estimates hold-these assumptions are precisely those in Definition 2.9 but for P V in place of . Following the proof of Lemma 2.19, there exist extended zero energy (dual) states where u (0) and u * (0) are uniquely determined by u (0) , u * (0) ∈ 1 + A 1− (X). (The two are related by u * (0) = u (0) , and thus equal for real-valued potentials.) Theorem 3.12. For P V as above, the unique forward solution of c (R t * ; A 4+α (X)) satisfies the asymptotics stated in Theorems 3.4 and 3.9, with u (0) and u * (0) given by (3.39). (In particular, the shape u + of the leading order term at I + does not depend on V .) Proof. The analysis of the resolvent P V (σ) −1 acting on A 4+α (X) in Theorem 3.1 goes through verbatim. Indeed, the decay assumption (3.38) ensures that in the zero energy operator ρ −2 P V (0) = ρ −2 (0) + ρ −2 V , the potential enters at the same level as the error termL in (2.2); this term did not play any role in the arguments above.
Long range potentials V ∈ ρ 3 C ∞ (X) (3.40) can be handled as well. Denote by V 0 ∈ C ∞ (∂X) the leading order term, i.e. V − V 0 ρ 3 ∈ ρ 4 C ∞ (X), V 0 ∈ C ∞ (∂X). (We do not require V 0 to be spherically symmetric, which removes a requirement made in [Tat13].) Then the spherically symmetric partV 0 of V 0 enters at the same level as the mass parameter m; Indeed, ρ −2 P V (0)(ρ + m(V )ρ 2 ) ∈ ρ 3 C ∞ (X); thus, for f ∈ A 4+α (X), we have P V (0) −1 f ≡ c (0) (ρ + m(V )ρ 2 ) up to terms in ρ 2 C ∞ (X) with vanishing spherical average (just like Y (1) in Lemma 3.2-the vanishing of the spherical average of Y (1) , i.e. the orthogonality to spherically symmetric functions, is all that was used in subsequent arguments). This gives the claimed correction to u 0 in (3.6), and the remainder of the calculation is unaffected.

Asymptotics on subextremal Kerr spacetimes
We recall the definition of subextremal Kerr spacetimes in §4.1 and describe the (minimal) adaptations of the setup of §2 and the main theorems on wave decay in § §4.2-4.3.
Proof. The choice of F ensures g −1 m,a (dt * , dt * ) ∈ r −2 a C ∞ (X) ⊂ ρ 2 C ∞ (X); the asymptotics of the remaining metric coefficients can be readily verified by inspection of (4.3).
See Figure 1.1. We note that in r > r m,a , the zero energy operator can be computed directly in Boyer-Lindquist coordinates; changing to the coordinate ϕ 0 := ϕ + r 3m a ∆r dr (which is (4.2) without the cutoff), which is regular across r = r m,a , we obtain For the rest of this section, we fix a subextremal Kerr metric g m,a , ≡ gm,a , −, − ≡ −, − L 2 (X;|dg X |) , |dg X | = r 2 a sin θ dr dθ dϕ * ; (4.7) as in (2.5), the density on X is defined so that |dt * ||dg X | = |dg m,a |. (1) For s, ∈ R, the spaceH s, consisting of all those elements which are supported in X.
(3) For α ∈ R, we defineĀ α (X) = ρ αĀ0 (X) as the space of restrictions of elements of Thus,H s, b (X) is a Hilbert space as the quotient of H s, b (R 3 ) by the subspace of elements with support in the closure of R 3 \ X. Elements ofĀ α (X) are smooth down to Σ X,f , and the Sobolev embedding (2.6) holds for the extendible spaces.
Proof. For Schwarzschild metrics g m,0 , mode stability and the absence of bound states are proved using simple integration by parts arguments; see [HHV19, Theorem 6.1] for detailed arguments in the function spaces used in the present paper. In the reference, it is also shown that mode stability and absence of bound states are open conditions in a, thus hold for slowly rotating Kerr black holes as well. In the general subextremal Kerr case, mode stability for nonzero σ was proved by Whiting [Whi89] and Shlapentokh-Rothman [SR15]; see also [AMPW17] for generalizations; the absence of zero energy bound states was argued for in [PT73,Teu72].
The estimates for bounded nonzero energies as well as the high energy estimates are proved in [HHV19, Theorem 4.3] (note that the compact error terms in the estimate for bounded energies can be dropped due to mode stability). We recall that for bounded frequencies, these estimates rely on (1) radial point estimates at the event horizon [Vas13,Mel94] (requiring the regularity to be above the threshold 1 2 in order to exclude singular behavior there, cf. u * (0) ∈ H 1/2− loc (X • ) in Lemma 4.4 below, and see [HHV19, Footnote 3]) combined with real principal type propagation [DH72]; (2) scattering theory near ∂X in second microlocal function spaces [Vas19a]; (3) standard elliptic estimates in m r < ∞. For high energy estimates, one uses the semiclassical versions of these estimates in [Vas13,Vas19a], see also [VZ00], as well as estimates at the normally hyperbolically trapped set of Kerr spacetimes; this structural nature of the trapped set was first noted for small a by Wunsch-Zworski [WZ11], together with the requisite high energy estimates, and proved in the full subextremal range by Dyatlov [Dya15]. The high energy estimates lose only a logarithmic power of h, whereas we simply allow ourselves a loss of a full power (see also [Dya16]).
Remark 4.6. In r > r 0 > 2r m,a , we can use Boyer-Lindquist coordinates; since t * = t − r + O(log r), we can replace t * + r in (4.8) by t upon committing a lower order (in terms of powers of t −1 ) error as long as in addition we stay in any fixed forward timelike cone r < (1 − δ)t. This gives the asymptotics This implies a corresponding result for the initial value problem precisely as in Corollary 3.11. For initial data supported away from the horizon, the constant in the leading order term takes a particularly simple form: Corollary 4.7. Suppose φ 0 , φ 1 ∈ C ∞ c ((r m,a , ∞) × S 2 ) are initial data with compact support disjoint from the event horizon. Then the asymptotic behavior of the solution φ of the initial value problem in Boyer-Lindquist coordinates has the asymptotic behavior given in Theorem 4.5 (with α < 1 arbitrary) upon replacing the constant c(f ) there by the constant (r 2 + a 2 ) 2 ∆ r − a 2 sin 2 θ φ 1 (r, θ, ϕ) + 4mar ∆ r φ 0 (r, θ, ϕ) sin 2 θ dr dθ dϕ. (4.10) For initial data with support intersecting the event horizon, the constant is given by Corollary 3.11.
Proof. The expression (4.10) comes directly from Corollary 3.11 upon (1) choosing the time function t * in such a way that it is equal to the Boyer-Lindquist time coordinate t near the support of the initial data, (2) using the form (4.1) of the dual metric g −1 m,a , and (3) recalling the spatial volume density from (4.7).
Remark 4.8. Under a spectral admissibility condition as discussed in §3.3, one can couple scalar waves on subextremal Kerr spacetimes to a potential V ∈ ρ 4 C ∞ (X), obtaining the same decay rates, though the coefficient of the t −3 * leading order term in compact spatial sets is modified by the potential precisely as in Theorem 3.12. Long range potentials V ∈ ρ 3 C ∞ (X) are covered as well, with additional modifications as discussed after (3.40).

The full Price law on Schwarzschild spacetimes
As another application of our methods, we prove the full Price law on the Schwarzschild spacetime which predicts pointwise t −2l−3 decay of linear scalar waves with fixed 'angular momentum' l ∈ N 0 . Concretely, recall the space Y l := span{Y lm : m = −l, . . . , l} ⊂ C ∞ (S 2 ) of degree l spherical harmonics; / ∆Y = l(l + 1)Y for Y ∈ Y l . Recalling the Schwarzschild metric for m > 0, given by g m = g m,0 , from (2.1), we can introduce a time coordinate t * as in Lemma 4.1. We use the notation X from (4.4) for the compactification of the spatial manifold X • = [m, ∞) r × S 2 ω , andĀ α (X) for functions conormal at ∂X = ρ −1 (0), ρ = r −1 , with weight ρ α , and smooth across the artificial interior hypersurface r = m. We say that a function f on M • = R t * × X • is supported in angular frequency l ∈ N 0 if / ∆f = l(l + 1)f , or equivalently f (t * , r, −) ∈ Y l for all t * , r; and we say that f is supported in angular frequencies ≥ l if the L 2 (S 2 )-orthogonal projection of f (t * , r, −) to Y 0 ⊕ · · · ⊕ Y l−1 vanishes for all t * , r.
Theorem 5.1. Let α ∈ (0, 1) and l ∈ N 0 , and write ≡ gm , m > 0. Let f ∈ C ∞ c (R t * ;Ā 4+l+α (X)) and suppose f is supported in angular frequencies ≥ l. Then the unique forward solution φ of φ = f obeys the pointwise decay bounds |φ(t * , r, ω)| ≤ Ct −2l−3 * , together with all derivatives along t * ∂ t * , ∂ r , and spherical vector fields, for x = (r, ω) restricted to any fixed compact subset K X • . Moreover, this decay rate is generically sharp when K has nonempty interior: it can be improved if and only if f satisfies 2l + 1 linearly independent constraints.
In the full future causal cone t * ≥ 0, the pointwise decay rate of φ (and of its derivatives along any product of powers of t * ∂ t * , r∂ r , Ω) is t −l−3 * , and the radiation field has t −l−2 * decay (as do its derivatives along any product of powers of t * ∂ t * , Ω).
Remark 5.2. One can explicitly compute the leading order coefficient ) in compact subsets of X • ; it is an element in the (2l + 1)dimensional space of solutions u (l) of (0)u (l) = 0 with leading order asymptotic behavior u (l) −r l Y ∈Ā −l+1− (X), Y ∈ Y l . For example, for l = 1, this space is spanned by (r−m)Y 1m , m = −1, 0, 1 (see [HHV19, Proposition 6.2]). Moreover, one can, in principle, prove global asymptotics similar to those in Theorem 3.9, with explicit leading order term. However, we shall not pursue this here beyond the rough description given in Theorem 5.1.
Remark 5.3. The proof of Theorem 5.1 below works verbatim for all spherically symmetric, stationary and asymptotically flat metrics with mass m ∈ R (see Definition 2.3) which are spectrally admissible (see Definition 2.9).
Proof of Theorem 5.1. We shall assume l ≥ 1, the case l = 0 being a special case of Theorem 4.5. In order to unburden the notation, we write A α forĀ α throughout. The strategy of the proof is the same as in §3. The restriction to angular frequencies ≥ l increases the range of weights for which the zero energy operator is invertible, and the range of weights on which the low energy resolvent remains bounded. For clarity, we discuss this for fixed angular frequencies, as well as the inversion of the resolvent on borderline inputs, in the first part of the proof. In the second part, we explain how the iteration based on (3.1) works, and how the resolvent acting on A 4+l+α (X) inputs has a σ 2l+2 log(σ + i0) singularity; this will imply the theorem, as shown in the third step. In the final step of the proof, we explain the minimal changes required for bounding the 'infinite sum over angular frequencies ≥ l'.
The term l (σ) −1 (f (σ) −f (0)) has an additional order of vanishing in σ, hence can be absorbed into the error term in (5.7). Upon taking the inverse Fourier transform, the resolvent expansion (5.7) thus implies the desired t −2l−3 decay of the forward solution φ of φ = f ; and the explicit description (5.8) justifies Remark 5.2.
We argue that this decay rate is generically sharp. It follows from the argument that all constants c k and d k are nonzero for k ≥ 1, hence so is u (l) in view of (5.8), provided only that Y (0) = 0. But a pairing argument completely analogous to (3.5) (with the role of u * (0) now played by u * (l) (Y ) ∈ ker l (0), Y ∈ Y l , with leading order behavior ρ −l Y ) produces a formula for Y (0) : identifying Y l ∼ = C 2l+1 via the basis Y lm , it reads Y (0) = c f, u * (l) (Y lm ) , where c = 0 is an explicit constant. Therefore, φ has a nontrivial t −2l−3 leading order term plus a O(t −2l−3−α+ ) remainder, unless f satisfies the 2l +1 linearly independent constraints f, u * (l) (Y lm ) = 0. Since the leading order term u (l) has radial dependence given by the solution of an ODE, it cannot vanish on a nonempty open subset of X • unless it vanishes identically.
This in particular implies t −l−3 * decay of φ as t * → ∞ for ρt * restricted to compact subsets of (0, ∞); the claimed t −l−2 * decay of the radiation field, or equivalently the t −l−3 * decay of lim r→∞ (ρt * ) −1 φ, follows from the t −l−3 * decay of φ towards (I + ) • as in the last step of the proof of Theorem 3.9.
• Modifications for forcing supported in angular frequencies ≥ l. One can analyze (σ) directly on the subspace of C ∞ c (X • ) consisting of functions supported in angular frequencies ≥ l and follow the above arguments. The only modification is that u 0 in (5.5) attains an additional contribution ρ l+2 Y (0) with Y (0) ∈ Y l+1 ; one can keep track of the effect of this term simply by using the above arguments for angular frequency equal to l+1. In particular, it does not contribute to the most singular σ 2l+2 log σ term of the resolvent.
Proof. We only need to prove (5.10). We can choose R 0 < R 1 so that R 0 < r < R 1 on supp φ j , j = 0, 1, and then define t * to be equal to the static time coordinate t for r ∈ [R 0 , R 1 ]; thus, Theorem 5.1 is applicable. The reduction of (5.9) to a forcing problem in static coordinates reads (5.11) there are no spatial derivatives falling on φ 0 , φ 1 since mixed time-space derivatives are absent in the Schwarzschild wave operator in static coordinates. For initially static perturbations, the right hand side is (1 − 2m r ) −1 φ 0 (x)δ (t), and the point is that its Fourier transform vanishes simply at σ = 0. The extra factor of σ of the forcing on the spectral side renders the main singularity of the resolvent a multiple of σ · σ 2l+2 log(σ + i0), thus giving an extra order of time decay upon taking the inverse Fourier transform.