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 ϕ(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (t,x)$$\end{document} of the scalar wave equation on a class of stationary asymptotically flat (3+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3+1)$$\end{document}-dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that ϕ(t,x)=ct-3+O(t-4+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$\end{document} for bounded |x|, where c∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in {\mathbb {C}}$$\end{document} 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-2l-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-2 l-3}$$\end{document} decay for waves with angular frequency at least l, and t-2l-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-2 l-4}$$\end{document} 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.

(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 1 − 2m r −1 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 .
On Cauchy surfaces which intersect the future event horizon transversally, Theorem 1.1 remains valid (upon replacing t by an appropriate time function) without the requirement that the initial data vanish near the horizon; see Sect. 1.1 below.
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. 1 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 Sect. 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; it has also featured in the physics literature, see e.g. [Lea86]. 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 [Vas20b,Vas20a] 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). We also mention the recent work by Bouclet-Burq [BB21] on sharp estimates for the spectral measure at low frequencies-which imply sharp decay results for Schrödinger and wave flows-for Laplace operators on very long range perturbations of Euclidean space (allowing for any positive fractional amount of symbolic decay) in any spatial dimension.

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 2 See Fig. 1. The family of subextremal Kerr metrics, described in Sect. 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.
Theorem 1.2 (Price's law with leading order term on subextremal Kerr spacetimes). Let g be a subextremal Kerr metric. Consider compactly supported initial data φ 0 , φ 1 ∈ C ∞ c (X • ). Then, for a constant c, explicitly computable in terms of φ 0 , φ 1 , the solution φ of the initial value problem g φ = 0, φ| t * =0 = φ 0 , ∂ t * φ| t * =0 = φ 1 , decays according to 3 φ − c t * + r t 2 * (t * + 2r ) 2 ≤ C t −4+ 494 P. Hintz Fig. 1. On the left: Penrose diagram of a Schwarzschild or subextremal Kerr spacetime, with level sets of t * (black), r (red), and t * /r (blue) 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.
Remark 1.4. In any region r m,a < r 0 < r < (1 − δ)t away from the event horizon and away from null infinity, we can replace t * in (1.6) by t − r up to t −4+ errors and thus obtain 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 in principle find a concrete value for N by carefully revisiting our arguments. 5 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 is nonzero for generic initial data (namely, it vanishes only on a codimension 1 subspace of C ∞ c (X • )⊕C ∞ c (X • )). 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 Sect. 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 Sect. 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 Kerrde 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 Sect. 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 Sect. 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 Figs. 1 and 3.
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 Sect. 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 [HHV21] 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.

Sharp asymptotics for wave equations with stationary potentials. On subextremal
Kerr spacetimes (or generalizations as in Sect. 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). Let V ∈ C ∞ (R 3 ) be a potential which in r > 1 is of the form Suppose that the resolvent 6 (−σ 2 + R 3 + V ) −1 : L 2 (R 3 ) → L 2 (R 3 ) extends analytically from Im σ 1 to Im σ > 0, and is continuous down as the (unique) solution of ( R 3 + V )u (0) = 0 such that u (0) → 1 as r → ∞, and denote by u * (0) = u (0) the corresponding solution of ( R 3 +V )u * (0) = 0. Given smooth, compactly supported initial data φ 0 , φ 1 ∈ C ∞ c (R 3 ), let φ(t, x) denote the solution of the initial value problem (1.8) Then, for x restricted to any fixed compact subset of R 3 , we have A simple class of examples for which the hypotheses of Theorem 1.9 are satisfied is given by nonnegative potentials V . See Sect. 3.3 for the general result, and the discussion following (3.42) 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.

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, 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 [HHV21, §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,Vas20a]. See 7 [HHV21, Theorem 4.3].
As in [HHV21], 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 [Vas20a,Vas20b]. 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 −iσ t and leading order spatial dependence r −1 e iσ r , thus an overall e −iσ t * 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 [Vas20b], 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 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 where we set 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 + σ u 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 8 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 Sect. 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 = σ r 1 and σ r 1); the resolved space X + res already played a prominent role in [Vas20b]. 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 (3.36) in the proof of Theorem 3.9).
The proof of the full Price law (1.5) in Sect. 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 [Mor20] 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); 9 (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; 10 (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 [HHV21]. Examples include Maxwell's equation or the equations of linearized gravity on Kerr spacetimes.

Outline of the paper.
• In Sect. 2, we describe the geometry (Sect. 2.1) and spectral theory (Sect. 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 (Sect. 2.2.2-2.2.3) as required for the precise analysis of the iteration (1.12). • In Sect. 3, we prove the main result giving the low energy resolvent expansion (Sect. 3.1) and use it to prove Price's law with leading order term (Sect. 3.2). The modifications required for stationary potentials and Theorem 1.9 are described in Sect. 3.3. • In Sect. 4, subextremal Kerr spacetimes are placed into our general framework.

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 . 502 P. Hintz 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 V = a∂ r + 3 j=1 b j ρ j , where a, b j ∈ C ∞ (X ), and 1 , 2 , 3 ∈ V(S 2 ) are the rotation vector fields. 11 In local coordinates (θ, ϕ) on S 2 , this means V = a∂ r + r −1b One can check that the coordinate vector fields Definition 2.3. We call a smooth Lorentzian 12 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 Sect. 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 Sect. 4.
Remark 2.5. It suffices to assume that g 00 and the ρ 2 C ∞ error terms of g 0X , g X X 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: of m-th order b-differential operators consists of all finite sums of up to m-fold products of b-vector fields. Finally, Lemma 2.7. The wave operator g of an admissible metric is given by where Q ∈ Diff 1 b (X ) and g (0) ∈ ρ 2 Diff 2 b (X ); near ∂ X , they are of the form 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 X X 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 13 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) (Mode stability.) The nullspace of (σ ) on A 1 (X ) is trivial for all σ ∈ C, Im σ ≥ 0.
(2) (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. 13 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 ).
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: . 15 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 −2iσ r 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 ). • tf: the front face; See Fig. 2. In ρ < 1, the functions are smooth defining functions of the respective boundary hypersurfaces. Away from zf, it is more convenient to work with the local defining functionsρ = ρ/σ = (σ r ) −1 and σ , and away from bf one can take ρ = r −1 andr = σ/ρ = σ r . Thus, tf captures the transition from the regime σ r 1 to σ r 1. (This is related to [ On the manifold with corners X + res , we consider conormal spaces with A 0,0,0 (X + res ) 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.
Definition 2.13. (1) Let E be an index set, α 0 := inf Re E, and α ∈ R; put β = min(α 0 , α). Then the space Then consists of all u ∈ A β bf −,β tf −,β zf − (X + res ) 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.
A typical index set is The regularity of the low energy resolvent is then as follows; this is similar to [HHV21, Propositions 12.4 and 12.12], though here we do not keep track of the number of derivatives used.
The claim (2.11) follows directly from (σ ) −1 f ∈ A β ([0, 1); A α− (X )); the latter is proved by a simple adaptation of (2.12). 16 The order at bf can be improved to 1− by taking < − 1 2 close to − 1 2 in the application of Theorem 2.11 in the proof below, though we do not need this precision here.

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 Sect. 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:

Lemma 2.19. We have
for states u (0) , u * (0) ∈ A ((0,0),1−) (X ) which are uniquely determined by their leading order behavior u (0) , u * (0) ∈ 1 + A 1− (X ). 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 Sects. 3.3 and 4. Correspondingly, the proof of this lemma will only use the structures which are present in these more general situations.
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 ρ 2 b ρ 2 tf Diff 2 b ; 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 ρ L 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.
Expand v and h into spherical harmonics Y lm , l = 0, 1, 2, . . . , |m| ≤ l, with / Y lm = l(l + 1), and let Y l := span{Y lm : m = −l, . . . , l}. Restricted to Y l , the inverse L 0 (ξ ) −1 | Y l = (−(iξ) 2 − iξ + l(l + 1)) −1 is meromorphic with simple poles precisely at iξ = −l − 1, l. Thus, writing h = h 0 + h with h orthogonal to Y 0 (i.e. with vanishing spherical average), then L 0 (ξ ) −1 h 0 (ξ ), resp. L 0 (ξ ) −1 h (ξ ), is meromorphic in Im ξ > −1 with (at most) a double, resp. simple pole at ξ = 0. In the inverse Mellin transform we can then shift the contour to Im ξ = −1 + ; the residue theorem gives the expansion The value of c can be determined from this, or directly by noting that L 0 (logr ) 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 leading order term at tf is equal toũ (2) , and the leading order term at zf is equal to −(log σ ρ )u (0) .
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 =r y.

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:

X ). For positive frequencies, the resolvent acting on f is then of 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) (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].

First iteration
We begin by analyzing the first term in (3.3) in some detail: Proof. We have u := (0) −1 f ∈ A 1− (X ). But then, in the notation (2.3), (3.4) Let χ ∂ ∈ C ∞ (X ) denote a cutoff which is identically 1 near ∂ X and supported in a slightly larger neighborhood of ∂ X . Then, using Lemma 2.7, u ∂ := χ ∂ u ∈ A 1− satisfies an equation We use this equation to establish better decay of u ∂ using a b-normal operator argument similarly to the proof of Lemma 2.23. Passing to the Mellin transform in ρ, defined by u ∂ (ξ ) := ρ −iξ u ∂ (ρ) dρ ρ (dropping the dependence on the spherical variables from the notation), equation (3.5) becomes (The sign switch of ξ compared to the proof of Lemma 2.23 is due to the Mellin transform there being in the variabler ∼ ρ −1 .) We already know that u ∂ (ξ ) is holomorphic in Im ξ > −1 and satisfies estimates for all s, N ∈ R and > 0; the same estimate holds for f ∂ (ξ ) but in the larger region −2 < Im ξ < 0. Expanding u ∂ (ξ ) into spherical harmonics, so u ∂ (ξ ) = u ml (ξ )Y ml and f ∂ (ξ ) = f ml (ξ )Y ml , and noting that L 0 (ξ )| Y l = −(iξ) 2 + iξ + l(l + 1) (with Y l = span{Y lm }) is invertible for iξ = −l, l + 1, we conclude that u ml (ξ ) = ( L 0 (ξ )| Y l ) −1 f ml (ξ ) is holomorphic in Im ξ > −1 for all m, l except possibly for l = 0 where it has a simple pole. Therefore, in the inverse Mellin transform we can shift the contour of integration through the pole at ξ = −i to Im ξ = −2 + ; the residue theorem thus gives u ∂ (ρ) ∈ c (0) ρ + A 2− and therefore The constant c (0) can be evaluated as follows: 18 letting χ (ρ) = χ(ρ/ ) where χ ∈ C ∞ ([0, ∞)) vanishes near 0 and is identically 1 on [1, ∞), we have In the second summand, note that [ (0), χ ] ∈ ρ 2 Diff 1 b converges to 0 strongly as an operator g| by Lemma 2.7, hence convergence to 0 of the pointwise product of the two slots of the pairing in A 3+δ suffices for convergence of the inner product to 0.) Similarly, all subleading terms of (0) (i.e. terms in ρ 3 Diff 2 b ) do not contribute in the limit. Therefore, using the fact that the leading order term of u * (0) at ∂ X is 1, and using the explicit form of the term L 0 from (2.3), the second term in (3.7) is equal to upon substituting x = ρ/ . Plugging this into (3.7) gives c (0) = (4π) −1 f, u * (0) . We sharpen the asymptotics of u further by plugging the partial expansion (3.6) into equation (3.4): using L 0 ρ ≡ 0 and the explicit expression for L 1 in (2.3), we obtain with the a priori informationũ 0 ∈ A 2− . Localizing near ∂ X and using the Mellin transform as before, we now get a contribution toũ 0 from the pole of ( L 0 (ξ )| Y 1 ) −1 at ξ = −2i, and an additional contribution from the single pole of −2mc (0) ρ 2 χ ∂ (ξ ) at 18 This is an instance of the proof of the relative index formula in [Mel93,§6]. Roughly speaking, the lack of invertibility of L 0 (ξ ) on Y 0 for ξ = −i is due to its lack of injectivity (the kernel producing the c (0) ρ leading order term) or, equivalently, due to its lack of surjectivity which manifests itself in the existence of a cokernel, which here is the kernel of L * 0 on A 0 ; the constant c (0) then measures the failure of the right hand side f ∂ in (3.5) to be orthogonal to the cokernel. ξ = −2i (where L 0 (ξ ) acting on Y 0 does not have a pole). More directly, we can solve away −2mρ 2 c (0) by hand using L 0 (mc (0) ρ 2 ) = −2mc (0) ρ 2 , thus L 0ũ 0 ∈ A 2+α ,ũ 0 :=ũ 0 − mc (0) ρ 2 ; and thenũ 0 = ρ 2 Y (1) +ũ for some Y (1) ∈ Y 1 andũ ∈ A 2+α− . The proof is complete.
Denote the output of Lemma 3.2 by (3.8) By (2.4) and in the notation of Lemma 2.7, we have (3.9) and therefore, since Q 0 ρ = 0,
By a normal operator argument as in the proof of Lemma 3.2, we conclude that We plug this into the expansion (3.11). If we let (3.13) For the fourth term on the right, which is the main term at this step, we compute (3.14) using (3.9) again; note here that ρ Q 0ũ1 ∈ A 2+α− due to Q 0 ρ = 0. Therefore, we can apply Proposition 2.24 to f 2,0 = 4mc (0) ρ 2 to deduce with leading order term at zf equal to −(log σ ρ )4mc (0) u (0) , and leading order term at tf equal to 4mc (0)ũ (2) (in the notation of the proposition); these are our main terms. The remaining terms are error terms: Proposition 2.14 and Lemma 2.17 give, a fortiori, in the last term in (3.13) finally, σ 2 (σ ) −1 acts on an element of σ A 2− + σ 2 A 3− , hence lies in the space (3.17) as well.

Asymptotic behavior of waves.
For simplicity, we first restrict our attention to the long time asymptotics in compact spatial sets in Sect. 3.2.1 before describing the global asymptotics in Sect. 3.2.2.

Asymptotics in spatially compact sets
Theorem 3.4. Let α ∈ (0, 1), 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 iσ t * 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 (1) in Definition 2.9.
Let χ ∈ C ∞ c ((−1, 1)) be identically 1 near 0; we then split φ = φ 0 + φ 1 + φ 2 , (3.20) 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 (3.13) as well as of the regular piece (3.16) 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 (3.23) The term φ 1 (σ ) has an extra factor of σ , hence lies in the remainder space in (3.23). Now, the smooth first term in (3.23) 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.
Remark 3.5. The constant arising in (3.22) 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 + I I , where We estimate |I | . In I I , we fix k ∈ N 0 , k > β + 1, and write Expanding the derivative and substitutingσ = σ t * , we can estimate |I I | 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
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 in σ ; here, H (x) = x 0 + 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. Fig. 4. Writingr = σ/ρ, and factoring σ 2 = ρ 2r 2 , this is equal to (3.36) here we integrated by parts twice using iv −1 ∂r e −ir v = e −ir v . The integral can be evaluated explicitly. 20 Indeed, since ∂ 2 rr 2 = (r ∂r + 2)∂rr , the arguments in Remark 2.25 show that We then regularize the integral in (3.36) 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 (3.36) 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 (3.29), (3.30), (3.31), and (3.35), 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 + .
That is, we have an equality of leading order terms at We finally note that we can iterate the normal operator argument at I + based on (3.38), until the v 3+α decay of f prevents getting further terms in the expansion; the result is that φ ∈ A ((1,0),3+α),((3,0),3+α) (M ). The proof is complete.
Initial value problems can be reduced to forcing problems: . 21 Then the solution φ of the initial value problem has the asymptotic behavior stated in Theorem 3.9 for any α < 1, with the constant c M ( f ) replaced by Proof. Fix R > 1 so that supp φ 0 and supp φ 1 are contained in the ball B(0, R) ⊂ X • of radius R. Then there exists > 0 such that φ(t * , x) = 0 for |x| ≥ 2R and t * ∈ [0, 3 ]. Fix a cutoff χ(t * ) which is identically 0 for t * ≤ 0 and equal to 1 for t * ≥ 2 . Then φ + = χφ vanishes in t * ≤ and satisfies (3.39) 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 − 2ρ Qφ 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 satisfies the asymptotics stated in Theorems 3.4 and 3.9, with u (0) and u * (0) given by (3.41). (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 (3.40) 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 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; Theorem 3.12 thus remains valid upon changing the mass m in the definition of the constant c M ( f ) in (3.18) by the effective mass 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 (3.8), and the remainder of the calculation is unaffected. As a concrete example, one can take g to be the Minkowski metric g = −dt 2 + dr 2 + r 2 / g, put t * = t − r (thus, g satisfies Definition 2.3 with m = 0), and take V to be a smooth potential of the form * leading order term can be computed as in Corollary 3.11 with m(V ) = 1 2V 0 in place of m. In particular, modifying t * to be equal to t near supp φ 0 ∪ supp φ 1 , this gives c M (φ 0 , φ 1 ) = −V 0 π φ 1 , u * (0) and thus proves Theorem 1.9. 22 Remark 3.13. Similarly to Remark 2.5, one can relax (3.42) to ρ 3 C ∞ (X ) + A 4 (X ) without causing any changes in the argument, and to ρ 3 C ∞ (X ) + A 3+β (X ), β > 0, with extra work.

Asymptotics on Subextremal Kerr Spacetimes
We recall the definition of subextremal Kerr (B(0, R 0 )), and then couple this with a radial potential V = V (r ) by solving the forced equation Approximating V (s)s 2 4π s ≈V 0 4π s 2 ≈V 0 π T 2 and similarly replacing 1 T −s by 2 T , one obtains r := r 2 − 2mr + a 2 , r 2 a := r 2 + a 2 cos 2 θ, in Boyer-Lindquist coordinates t ∈ R, r ∈ (r m,a , ∞), θ ∈ (0, π), ϕ ∈ (0, 2π), where r m,a := m + m 2 − a 2 ; the dual metric g −1 m,a is r 2 a g −1 m,a = − −1 r (r 2 + a 2 )∂ t + a∂ ϕ 2 + r ∂ 2 r + ∂ 2 θ + sin −2 θ ∂ ϕ + a sin 2 θ ∂ t 2 .(4.1) In order to extend g m,a across r = r m,a on the one hand, and place it into the setting of Definition 2.3 for large r on the other hand, we define new coordinates , and F(r ) ∈ C ∞ (R) will be chosen below. The dual metric then is (4.3) For F ≡ 0, this can be extended analytically from r m,a < r < 3m to r > r − = m − √ m 2 − a 2 . In r ≥ 6m, we want the inner product r 2 a g −1 m,a (dt * , dt * ) = a 2 sin 2 θ − 2(r 2 + a 2 )F + (F ) 2 r to be negative; this holds for example for F = m 2 r 2 +a 2 , for which r 2 a g −1 m,a (dt * , dt * ) = −2m 2 + a 2 sin 2 θ + O(r −2 ) is indeed negative for r ≥ r 0 6m. Choosing F in r ≤ r 0 suitably, we can then ensure that dt * is future timelike everywhere in r ≥ m. Define the spatial manifold θϕ * ; we then state the asymptotic behavior of the metric on the closure of X • inside the radial compactification R 3 of R 3 (see Definition 2.1); by a slight abuse of notation, we write for the boundary of X at infinity, resp. the artificial boundary inside the black hole. Let ρ = r −1 denote a defining function of ∂ X . Then: Lemma 4.1. For a suitable choice of F ∈ C ∞ (R r ), with F (r ) = m 2 r 2 +a 2 for r 1, the dual metric g −1 m,a in the coordinates t * , r, θ, ϕ * defined in (4.2), is a smooth, nondegenerate Lorentzian dual metric on M • = R t * × X • so that dt * is everywhere future timelike, and so that g −1 m,a satisfies the assumptions in part (2) of Definition 2.3 near ∂ X.
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).
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 X,f , and the Sobolev embedding (2.6) holds for the extendible spaces. Lemma 4.3. Subextremal Kerr metrics g m,a are spectrally admissible in the sense that they satisfy condition (1) in Definition 2.9 on the corresponding extendible function spacesĀ 1 (X ). Moreover, they satisfy the high energy estimates (2.7) in condition (2) onH s, b,|σ | −1 (X ) for δ = 1, under the additional assumption that s > 1 2 . Moreover, the estimate (2.8) for bounded nonzero energies holds on extendible function spaces as well for s > 1 2 .
Proof. For Schwarzschild metrics g m,0 , mode stability and the absence of bound states are proved using simple integration by parts arguments; see [HHV21, 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 with Im σ ≥ 0 was proved by Whiting [Whi89] and Shlapentokh-Rothman [SR15] (see also [AMPW17] for generalizations). We argue that these results prove mode stability in the sense of Definition 2.9(1). Following [SR15, Definition 1.1], a mode solution in the black hole exterior is of the form ψ(t, r, θ, ϕ) = e −iσ t e imϕ S σ ml (θ )R(r ) where e imϕ Y σ ml (σ ) is a smooth function on S 2 , is smooth at r = r m,a , and finally where < r −1 m,a , and r * = r + 2m log(r − 2m) is the tortoise coordinate as in §2.1. As remarked at the end of [SR15, §1.2], the condition on R near r = r m,a is equivalent to the smoothness of ψ across the future event horizon; thus, writing ψ = e −iσ t * u(r, θ, ϕ * ), the function u (which in r > r m,a is a smooth multiple of e imϕ S σ ml (θ )R(r ) and vice versa) is smooth down to r = r m,a . Moreover, since t * is equal to t − r * plus a smooth function of r −1 , the required behavior of R(r ) as r → ∞ implies that u is a smooth function of r −1 vanishing at r −1 = 0; this implies u ∈Ā 1 (X ∩ {r ≥ r m,a }). Application of [SR15, Theorems 1.5, 1.6] implies that u = 0. Returning to Definition 2.9(1), we have shown that any mode solution u ∈Ā 1 (X ) must vanish in r ≥ r m,a . A simple unique continuation result (for the radial ODE) for separated mode solutions in r < r m,a then upgrades the infinite order vanishing of u at r = r m,a to the vanishing of u in r < r m,a , as desired. 23 The absence of zero energy bound states can be proved via a study of the expressions for zero energy mode solutions in terms of hypergeometric functions [PT73,Teu72].
The estimates for bounded nonzero energies as well as the high energy estimates are proved in [HHV21,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 [HHV21, Footnote 3]) combined with real principal type propagation [DH72]; (2) scattering theory near ∂ X in second microlocal function spaces [Vas20a]; (3) standard elliptic estimates in m r < ∞. For high energy estimates, one uses the semiclassical versions of these estimates in [Vas13,Vas20a], 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]).
The extended zero energy states from Lemma 2.19 are as follows: 23 Alternatively, one can use analytic continuation and the recently proved fact that for analytic t * , mode solutions are analytic [PV21]. Lemma 4.4. For s < 1 2 and ∈ (− 5 2 , − 3 2 ), we have there. Atr = 0, uniqueness in the stated space is again due to the enlarged indicial gap arising already in (5.1).) The leading order logarithmic term ofũ atr = 0 is −(2l + 1) −1r l (logr )Y l , and thereforeũ has the claimed logarithmic singularity at zf =r −1 (0), with leading order term −(2l + 1) −1 ρ −l (logr )Y l . (5. 3) The conclusion is that which is the analogue of Proposition 2.24 but with spatial decay orders decreased by l.
− Step 1. Terms in the expansion with spatial decay. For k = 1, . . . , l−1, suppose that for some c k , c k , c k ∈ C and Y (1),k ∈ Y l ; we shall see that the main term we are interested in is the logarithmic one. Then where l (0) −1 : A −l+3 (X ) → A −l+1− (X ) is the inverse (5.1). Therefore, u (l) is the