Unique continuation from infinity for linear waves
aa r X i v : . [ m a t h . A P ] J a n UNIQUE CONTINUATION FROM INFINITYFOR LINEAR WAVES
SPYROS ALEXAKIS, VOLKER SCHLUE, AND ARICK SHAO
Abstract.
We prove various uniqueness results from null infinity, for linearwaves on asymptotically flat space-times. Assuming vanishing of the solutionto infinite order on suitable parts of future and past null infinities, we derivethat the solution must vanish in an open set in the interior. We find that theparts of infinity where we must impose a vanishing condition depend stronglyon the background geometry. In particular, for backgrounds with positive mass(such as Schwarzschild or Kerr), the required assumptions are much weakerthan the ones in the Minkowski space-time. The results are nearly optimal inmany respects. They can be considered analogues of uniqueness from infinityresults for second order elliptic operators. This work is partly motivated byquestions in general relativity.
Contents
1. Introduction 12. Results and Discussion 53. The Model Space-times 154. Carleman Estimates 205. Proof of Theorems 2.3-2.6. 31Appendix A. Kerr Metric in Comoving Coordinates 43References 451.
Introduction
We prove unique continuation results from infinity for wave equations overasymptotically flat backgrounds. In particular, consider solutions φ of a linearwave equation(1.1) L g φ := (cid:3) g φ + a α ∂ α φ + V φ = 0over a Lorentzian manifold (
M, g ), with (cid:3) g being the Laplace-Beltrami operatorfor g . We show that if the solution vanishes (in a suitable sense) at parts of futureand past null infinities I + , I − ⊂ ∂M , then the solution φ must vanish on an opendomain inside ( M, g ).One motivation for this paper comes from older and newer studies in general rel-ativity regarding the possibility of periodic-in-time solutions of the Einstein equa-tions. This has been considered both for vacuum space-times, [32, 33, 34], and forgravity coupled with matter fields, [7, 8]. In most settings, the problem reduces towhether solutions of the Einstein equations (vacuum or with matter) which emitno radiation towards the null infinities I + , I − must be stationary. In view of the techniques developed in [2], it seems that a positive answer to the uniqueness ques-tion for linear waves should be applicable towards the above problem. We intendto return to this in a subsequent paper; see also Section 1.2.2.The present paper, however, is primarily inspired by the challenge of derivinganalogues of unique continuation from infinity and from a point, which have beenstudied for second order elliptic operators, [4, 14, 19, 20, 26, 36, 30, 31], to thesetting of wave equations. We begin with a brief review of this subject; excellentbroader reviews can be found in [16, 38, 39] and references therein.1.1. Unique Continuation and Pseudoconvexity.
Unique continuation is es-sentially a question of uniqueness of solutions to PDEs. Consider a smooth function h defined over a domain D ⊂ R m , with d h = 0 everywhere in D . LetΣ := { h = 0 } , Σ + := { h ≥ } , Σ − = { h ≤ } .The uniqueness problem is the following: Question.
Assume that: • φ ∈ C ( D ) solves the second-order PDE p ( x, D ) φ = 0 in D , and • φ = 0 and d φ = 0 on Σ ∩ B ( P, δ ) , where P ∈ Σ , and where B ( P, δ ) denotesthe ball in R m about P of radius δ > .Is it true that φ = 0 in Σ + ∩ B ( P, δ ′ ) , for some (possibly smaller) δ ′ > ? Second-order Hyperbolic Equations.
When p ( x, D ) is a wave operator, withprincipal part of the form (cid:3) g , this question is of particular interest when Σ is non-characteristic and time-like (with respect to g , thought of as a Lorentzian metric).In that case, the Cauchy problem is ill-posed, [13], yet uniqueness may sometimeshold. The key condition that ensures uniqueness is H¨ormander’s strong pseudo-convexity condition, which in this setting takes the form [29]:(1.2) D h ( X, X ) <
0, if g ( X, X ) = g ( X, Dh ) = 0.In particular, the pseudo-convexity depends only on the principal symbol of p ( x, D )and its relation to the hypersurface Σ. In this setting, one has the following result: Theorem 1.1. [21]
Assume L g := (cid:3) g + a α ∂ α + V is a wave operator with smoothcoefficients, and suppose ψ ∈ C ( D ) solves L g φ = 0 in D . Assume furthermore that φ = 0 , d φ = 0 on Σ around P . Then, if Σ is strongly pseudo-convex at P in thesense of (1.2) , then φ vanishes in a (relatively open) domain in Σ + near P . The condition (1.2) can be interpreted geometrically as convexity of Σ + withrespect to null geodesics at P ∈ Σ. Specifically, (1.2) holds if and only if any nullgeodesic that is tangent to Σ at P locally lies in Σ − , with first order of contact at P . The necessity of the pseudo-convexity condition has been shown by Alinhac,[3]: He produced examples of wave operators L g for which unique continuation, asformulated in Theorem 1.1, fails across a non-pseudoconvex surface Σ.The proof of Theorem 1.1 relies on Carleman estimates for (cid:3) g . A key elementof such estimates is the construction of a weight function f whose level sets arepseudo-convex, and for which the level sets { f = C } ∩ Σ + are compact near P . For brevity, we present the discussion here for scalar equations.
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 3
Second-order Elliptic Equations.
Unique continuation holds always for sec-ond order elliptic operators across smooth hypersurfaces; see [10]. However, aninteresting modification is when one replaces the assumption of zero Cauchy dataon a hypersurface with data at a point . In that case, the necessary requirement,due to [4, 14], is vanishing to infinite order at that point: Definition 1.2.
We say that φ vanishes to infinite order at P ∈ R m if there exists δ > N ∈ N , Z B ( P,δ ) | φ | r − N < ∞ ,where r ( x ) = | x − P | . Theorem 1.3. [4, 14]
Assume that φ solves a second order elliptic equation (1.3) Hφ := − ∆ g φ + a α ∂ α φ + V φ = 0 , with smooth coefficients in a domain D ⊂ R m . Then, if φ vanishes to infinite orderat P ∈ D , it vanishes in a neighborhood of P . An analogue of this question is to assume infinite-order vanishing at infinity:
Question.
Given self-adjoint operators H of the form (1.3) and a solution φ to (1.4) Hφ = λφ with λ ≥ , in a neighborhood G of infinity in R n , then if φ satisfies (1.5) Z G φ r N < ∞ , for all N ∈ N ,and a α , V satisfy suitable decay conditions, does φ vanish in G ? This has been derived by many authors in various settings, for example [1, 19,20, 22, 26, 28, 30], to name a few. One of the motivations for this question isthat (in the case where g is the Euclidean metric or a small perturbation on R n ) anaffirmative answer implies the non-existence of L -solutions to (1.4) with λ >
0. Forif φ ∈ L is a solution to (1.4) with λ >
0, then the equation implies that φ vanishesto infinite order at infinity, and the question reduces to unique continuation frominfinity. (This is implicit in [19, Ch. XIV].) In the case where λ = 0, the infiniteorder vanishing cannot be derived and must be a priori assumed. Our results in thispaper can be thought of as direct analogues of the above problem for second-orderhyperbolic equations.1.2. Time-periodicity and applications.
We now turn to the connection withperiodic-in-time solutions.1.2.1.
Time-periodic solutions.
Let us note here that the absence of positive eigen-values λ > H is equivalent to the non-existence of periodic-in-time solutionsof the form(1.6) φ ( t, x ) := e i √ λt ψ ( x ) , to the corresponding wave equation(1.7) − ∂ tt φ ( t, x ) − Hφ ( t, x ) = 0 , The infinite-order vanishing is clearly necessary, as the example of homogenous harmonicpolynomials of any order shows.
SPYROS ALEXAKIS, VOLKER SCHLUE, AND ARICK SHAO over the ( n + 1)-dimensional Minkowski space-time R n +1 = { t ∈ R , x ∈ R n } ,with ψ ( x ) ∈ L ( R n ). Results on the absence of positive eigenvalues of Schr¨odingeroperators H := − ∆ + V , (with ∆ being the Euclidean (flat) Laplacian over R n and V a suitably decaying potential), have been derived by Kato, [26], and Agmon, [1],for potentials V obeying suitable pointwise decay conditions; see also [36]. Morerecently, results have been obtained for rough potentials V in suitable L p spaces,[22, 28]. One can also prove the absence of the zero eigenvalue of operators H ,which correspond to constant-in-time solutions of (1.7), but the decay assumptionson the solution must be strengthened to vanishing to infinite order at infinity, andthe potential V must decay faster; see [27]. Now a time-periodic solution of the form (1.6) to (1.7) would in fact vanish onthe entire I + , I − , in the sense that we will be considering here for solutions of(1.1), and would thus fall under the assumptions of our theorems below. In thissense, our work can be considered a generalization of the above results; however, wetreat time-dependent wave equations directly. Furthermore we will be consideringa more localized version of this problem: we show that vanishing of the solution of(1.1) on parts of I + , I − suffices to derive the vanishing of the solution in a part ofthe interior. Our results are robust in that they will hold for perturbations of theMinkowski metric. In fact a large part of this paper is concerned with understandinghow the (optimal) result depends on the geometry of the background metric. Theconditions we impose on the lower-order terms are similar to those one imposes forthe zero-eigenvalue problem for operators H = ∆ + a α ∂ α + V . Results on uniquecontinuation from spatial infinity for time-dependent Schr¨odinger equations havealso recently been obtained by Escauriaza, Kenig, Ponce and Vega; see [15] andreferences therein.1.2.2. Applications to general relativity.
A separate question to which our studyhere pertains directly, is that of inheritance of symmetry: whether matter fieldscoupled to the space-time via the Einstein equations must inherit the symmetries ofthe underlying space-time. In particular, Biˇc´ak, Scholtz, and Tod, [7, 8], considerEinstein-Maxwell and (various massless and massive) Einstein-scalar field space-times, for which the underlying space-time is stationary. Under the assumptions ofanalyticity at I − and asymptotic simplicity, they derive the inheritance of symme-try for some of the fields in question as follows: From the asymptotic simplicity ofthe space-time metric, the authors show that the T -derivative of the matter field˜ φ must vanish to infinite order on I − , where T is the stationary Killing field. Thereal-analyticity of the matter fields at I − then implies that the fields vanish off I − as well. Our Theorems 2.3 and 2.5 below, applied to T ˜ φ , allow us to remove theanalyticity assumption for those matter models for which the equations of motionreduce to a wave equation of the type considered there. On the other hand, as recalled in [7, 8], there are examples obtained by Bizon-Wasserman, [9], of massive Einstein-Klein-Gordon space-times in which the space-time is static but the field is in fact time-periodic, thus the underlying symmetry is not inherited. These examples do not contradict the theorems in [7, 8], since thesesolutions are manifestly non-analytic at I − , I + . Nonetheless, such examples raise In a separate direction, Finster, Kamran, Smoller, and Yau, [17], proved the non-existence ofperiodic-in-time solutions of the Dirac equation in the Kerr exterior. In view of [23], one can probably not hope that analyticity can in fact be derived from thenature of the problem.
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 5 the challenge of finding a suitable condition on the vanishing of the T -derivativesof the fields which would allow for the extension of symmetry (alternatively, thevanishing of T ˜ φ in the space-time) to hold. Theorems 2.4 and 2.6 below providesuch a condition, in the vanishing of the solution at a specific exponential rate; forgeneral wave equations, this is nearly optimal. Acknowledgments.
We are grateful to Sergiu Klainerman and Alex Ionescu formany helpful conversations. We thank them for generously sharing their resultsin [25]. The first author was partially supported by NSERC grants 488916 and489103, and a Sloan fellowship. The second author was supported by NSF grant0932078 000, while in residence at MSRI, Berkeley, CA, in the fall semester 2013.2.
Results and Discussion
Our main results deal with linear wave equations on various backgrounds. Theresults readily apply to semi-linear wave equations, under suitable pointwise decayassumptions on the solutions. For simplicity of the presentation, we consider thecase of one scalar equation, although the methods generalize readily to systems.2.1.
Perturbations of Minkowski Space-time.
The first set of results dealswith ( n + 1)-dimensional Minkowski space-time R n +1 and a large class of its per-turbations. Recall that the Minkowski metric itself is given by(2.1) g M := − u d v + n − X A,B =1 r ˚ γ AB d y A d y B ,where u and v are the standard optical functions(2.2) u := 12 ( t − r ), v := 12 ( t + r ),and where r ( u, v ) = v − u . In addition, y A , y B are coordinates on the sphere S n − ,while ˚ γ AB is the round metric on S n − expressed in those coordinates. Recall alsothat the future and past null infinities I ± of R n +1 correspond to the boundaries I + := { v = + ∞ , u ∈ ( −∞ , + ∞ ) } , I − := { u = −∞ , v ∈ ( −∞ , + ∞ ) } .We make the convention, both here and below, that uppercase Roman indices A , B , etc. correspond to coordinates on the level spheres S u,v of ( u, v ), while Greekindices α , β , etc. correspond to all n + 1 coordinate functions u, v, y , . . . , y n − . Definition 2.1.
For any fixed ǫ > • Let f ǫ denote the function(2.3) f ǫ := ( v + ǫ ) − ( − u + ǫ ) − . • Moreover, for ω >
0, we define the corresponding domain(2.4) D ǫω := { < f ǫ < ω } . • Define also the following subsets of future and past null infinity:(2.5) I + ǫ := { v = + ∞ , u ≤ ǫ } ⊂ I + , I − ǫ := { u = −∞ , v ≥ − ǫ } ⊂ I − .The function f ǫ is a key quantity that will be used both to state and to proveour theorems. Thus, we collect some simple observations concerning f ǫ here: SPYROS ALEXAKIS, VOLKER SCHLUE, AND ARICK SHAO • The positive level sets of f ǫ turn out to be strongly pseudo-convex (seeSection 3). Moreover, these level sets all intersect I + and I − at the spheres { u = ǫ, v = + ∞} and { v = − ǫ, u = −∞} , respectively (see Figure 3). • Note that { f ǫ = 0 } corresponds to the segments I + ǫ ∪ I − ǫ of null infinity. • Note also that f ǫ ∼ r − near spatial infinity ι , while f ǫ ∼ r − at theinterior points of I + ǫ and I − ǫ .We will be considering perturbations of g M of the general form:(2.6) g = µ d u − K d u d v + ν d v + n − X A,B =1 r γ AB d y A d y B + n − X A =1 ( c Au d y A d u + c Av d y A d v ),where again r = v − u , and u, v are defined as in (2.2). In order to describe preciselythe asymptotic conditions required for our coefficients µ , ν , K , c Au , c Av , γ AB , wemake the following definitions: Definition 2.2.
Given a function G = G ( u, v ), we define the following: • A C -function ϕ belongs to O δ ( G ) iff | ϕ | ≤ δG . • A C -function ϕ belongs to O ( G ) iff ϕ ∈ O δ ( G ) for some δ > • If both G and ϕ are C , then we say that f ∈ O δ ( G ) iff ϕ ∈ O δ ( G ), | ∂ u ϕ | ≤ δ | ∂ u G | , | ∂ v ϕ | ≤ δ | ∂ v G | , | ∂ I ϕ | ≤ δG . • Likewise, we say that ϕ ∈ O ( G ) iff ϕ ∈ O δ ( G ) for some δ > g can now be stated as follows: • There exists δ > , sufficiently small with respect to ǫ , such that (2.7) K = 1 + O δ ( r − ), γ AB = ˚ γ AB + O δ ( r − ), c Au , c Av = O δ ( r − ), µ, ν = O δ ( r − ).We present two results for wave equations over such background metrics. Theresults depend on the asymptotic behaviour of the lower-order terms, and the as-sumptions vary accordingly. Roughly speaking, our first theorem applies to linearwave equations over ( R n +1 , g ) whose lower order terms decay sufficiently rapidly atinfinity. We will show that if solutions of such equations vanish to infinite order at I + ǫ and I − ǫ , then the solution must vanish in an open domain in R n +1 whichcontains I + ǫ ∪ I − ǫ on its boundary. Theorem 2.3.
Consider a perturbed Minkowski metric g over R n +1 of the form (2.6) , (2.7) . Consider also any wave operator L g := (cid:3) g + a α ∂ α + V , where (2.8) a u ∈ O (( v + ǫ ) − r − ) , a v ∈ O (( − u + ǫ ) − r − ) , a I ∈ O ( f r − ) , V ∈ O ( f η ) , The required fall-off matches the one needed to rule out the existence of the zero-eigenvaluefor the corresponding elliptic operator.
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 7 for some η > . Let ω > , and consider any C -solution φ on D ǫω of the equation L g φ = 0 , which in addition satisfies (2.9) Z D ǫω φ r N + Z D ǫω | ∂ v φ | r N + Z D ǫω | ∂ u φ | r N + n − X I =1 Z D ǫω | ∂ I φ | r N < ∞ ,for all N ∈ N . Then, there exists < ω ′ < ω so that φ ≡ in D ǫω ′ . We digress now to discuss the optimality of the above, as well as some alternateways of phrasing this result.
Remark . An alternate statement of Theorem 2.3 is in terms of a differential inequality . More specifically, one can rephrase Theorem 2.3 with φ satisfying | (cid:3) g φ | ≤ X α | a α || ∂ α φ | + | V || φ | ,rather than the linear equation L g φ ≡
0. Analogous variants also hold true withrespect to the remaining main theorems in this paper. Moreover, these alternativeformulations can be proved in exactly the same manner.
Remark . The assumption V ∈ O ( f η ), η >
0, implies that V vanishes ata rate r − − η at spatial infinity and at a rate of r − − η at (future and past) nullinfinities. In fact, straightforward examples for elliptic operators of the form ∆ + V reveal that if V is only assumed to decay at a rate r − η , the result would fail.Thus, our theorem is nearly optimal with regards to the decay conditions on thelower-order terms. Remark . Since this result only assumes smoothness of the solution in D ǫω , asopposed to all of R n +1 , the assumption of vanishing of infinite order, in the senseof (2.9), is necessary . Indeed, in Minkowski space-time, one can consider(2.10) φ N := ∂ i . . . ∂ i N r − n +2 ,where the ∂ i j ’s are spatial Cartesian coordinate derivatives on R n ; see also [18].This function solves ∆ R n φ N = 0 away from the origin, and it vanishes to order N + n − Remark . It is worth noting that in Minkowski space-time, our methods implythat a smooth solution φ vanishes in the whole domain f < ǫ − , which in particularincludes the entire double-null cone centered at the origin. Thus, standard energyestimates imply that in this case, φ vanishes in the entire space-time.Let us recall at this point the standard Penrose conformal compactification ofMinkowski space-time. Let(2.11) Ω := (1 + u ) − (1 + v ) − ,and consider the new metric ¯ g := Ω g M . Applying the change of coordinates U := tan − u and V := tan − v , we see that(2.12) ¯ g = − U d V + sin ( V − U )(d θ + sin θ d φ ). In fact, this decay condition can be relaxed a bit, using the generalization of the Carlemanestimates described in the remark at the end of Section 4. However, we prefer to give the cleanerstatement above, rather than burden the reader with the most general version of the result possible. This can be thought of as a Lorentzian analogue of the stereographic projection.
SPYROS ALEXAKIS, VOLKER SCHLUE, AND ARICK SHAO ι ι − ι + S R I + I − Figure 1.
Penrose conformal compactification of Minkowskispace-time.In particular, ¯ g extends smoothly to the boundaries V = + π and U = − π , whichcorrespond to I + and I − , respectively. In fact, the compactified manifold ( ¯ R n +1 , ¯ g )is isometric to a relatively compact domain in the Einstein cylinder S n × R (withthe natural product metric); cf. Figure 1. Remark . The methods of [3] strongly suggest that in the Minkowski setting,the assumption of (infinite-order) vanishing on at least I +0 ∪ I − = { v = ∞ , u ≤ } ∪ { u = −∞ , v ≥ } is also necessary. In particular, [3] showed that unique continuation across a(smooth) hypersurface H requires pseudo-convexity of H , in general. Well-knownexamples of hypersurfaces that are not pseudo-convex in Minkowski space-timeare the one-sheeted hyperboloids H C = {− t + r = C } . The failure of pseudo-convexity is captured by the fact that any null geodesic tangent to such a hyper-boloid H C will remain on H C .Recalling the Penrose compactification above, and using the conformal Laplacianand its conformal covariance (see Section 5.2), we observe that ¯ φ := Ω − ( n − / φ solves an analogous wave equation with respect to the compactified metric ¯ g . Now,the hyperboloids H C are mapped to smooth surfaces ¯ H C in the Einstein cylinderthat converge to I +0 ∪ I − (embedded in the Einstein cylinder) as C ր ∞ . Further-more, by the conformal invariance of null geodesics, the ¯ H C ’s continue to be ruledby null geodesics. In particular, the existence of these non-pseudoconvex surfaces seem to suggestthat solutions to the equation in the compactified picture which vanish to infiniteorder on less than I +0 ∪ I − will not necessarily vanish in the interior near thoseboundaries. It would follow that φ also does not vanish. This property can also be seen by examining the null geodesics in the compactified space-time¯ R n +1 ⊂ S n × R . The geodesic motion can be decomposed into a constant motion in the R -directionand a geodesic motion on S n . Then, the focusing of the ¯ H C ’s (and of the null geodesics on them)is just a feature of the positive curvature of S n . NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 9
Remark . In certain cases, one would like to have assumptions in terms of C ∞ -norms instead of the weighted Sobolev norms in (2.9). Moreover, based on a formalanalysis of the wave equation on the characteristic surfaces I + ∪ I − , one wouldexpect that, under sufficient regularity assumptions on the solution on I + ∪ I − ,one should be able to only assume vanishing of the radiation field of the solutionto derive the above uniqueness.This is indeed true in Minkowski space-time. Transforming the wave equationto the Penrose compactified picture, as before, we can argue that if the solution ¯ φ in the Penrose setting were C ∞ up to the entire boundary I + ∪ I − , then only theassumption that ¯ φ = 0 on I + ∪ I − suffices to derive that the solution vanishes inthe interior. To see this, one uses the vanishing of d ¯ φ at ι , along with the waveequation for ¯ φ and its derivatives (as propagation equations along null geodesics on I ± ), to show that all derivatives of ¯ φ vanish on I + ∪ I − . This implies infinite-ordervanishing in the physical picture, and the result will follow from Theorem 2.3.However, the assumption that a function should be C ∞ in the compactifiedsetting is quite strong. For example, all the functions φ N defined in (2.10) yieldcorresponding functions ¯ φ N in the compactified picture which fail to be C ∞ preciselyat ι . Furthermore, it should be noted that generic perturbed space-times of thetype (2.6) yield non-smooth metrics in the compactified picture [12], and hence theabove formal analysis is not possible.A question of separate interest is wave equations on perturbed Minkowski space-times with potentials of order O (1). This in particular includes massive Klein-Gordon equations, where, as observed by examples in [8, 9], the previous resultfails. It turns out that one must assume faster than polynomial vanishing of thesolution on I + ǫ ∪ I − ǫ to derive uniqueness in this setting. In particular, we requirethe solution to vanish faster than the exponential rate exp( r − / ). Theorem 2.4.
Consider a perturbed Minkowski metric g over R n +1 in the form (2.6) and (2.7) , and consider a wave operator L g := (cid:3) g + a α ∂ α + V , where (2.13) a u ∈ O ( v − f − r − ) , a v ∈ O (( − u ) − f − r − ) , a I ∈ O ( f r − ) , V ∈ O (1) .Let ω > , and consider any C -solution φ on D ǫω of the equation L g φ = 0 , whichin addition satisfies (2.14) Z D ǫω φ e Nr + Z D ǫω | ∂ v φ | e Nr + Z D ǫω | ∂ u φ | e Nr + n − X I =1 Z D ǫω | ∂ I φ | e Nr < ∞ ,for all N ∈ N . Then, there exists < ω ′ < ω so that φ ≡ in D ǫω ′ . In view of the results of [30, 31], the required rate of vanishing for the solution isnearly optimal, given the assumptions on the bounds of the lower-order terms. Inparticular, it was shown that when V is allowed to be complex-valued, there existsolutions of (∆ + V ) φ = 0 in R \ B obeying the bound | φ | ≤ C exp( − cr / ) whichdo not vanish. In the sense that the function admits a C ∞ -extension to the entire ambient manifold S n × R . Schwarzschild and positive mass space-times.
We present our next the-orem for Schwarzschild space-times and general perturbations, which include allKerr metrics. Surprisingly, the theorem here requires a weaker assumption at in-finity than in Minkowski space-time; in particular only vanishing on arbitrarilysmall portions of I + , I − near ι is assumed. Morally, this is due to the stronger pseudo-convexity arising from the positive mass, rather than from where the chosenhyperboloids are anchored, as in the Minkowski case.Recall the form of the Schwarzschild metric in the exterior region (2.15) g S := − (cid:18) − m S r (cid:19) d t + (cid:18) − m S r (cid:19) − d r + r X A,B =1 ˚ γ AB d y A d y B ,where the mass m S is a positive constant, and where r > m S . To express themetric in null coordinates, we recall the definition of the Regge-Wheeler coordinate:(2.16) r ∗ ( r ) := Z rr (cid:18) − m S s (cid:19) − d s , r > m S .For a fixed constant r > m S , we denote the corresponding optical functions by u = u r := t − r ∗ v = v r = t + r ∗ g S = − (cid:18) − m S r (cid:19) d u d v + r X A,B =1 ˚ γ AB d y A d y B .While future and past null infinity are identified (for any choice of r ) with I + := { v = + ∞ , − ∞ < u < ∞} , I − = { u = −∞ , − ∞ < v < ∞} ,respectively, we note that { u = 0 } and { v = 0 } intersect on { t = 0 } at the sphereof radius r = r ; c.f. Fig. 2. Thus, the region D r = { v > u < } correspondsto a subdomain of the entire Schwarzschild exterior. More precisely, by choosing r as large as we wish, D r corresponds to the exterior region of a bifurcate nullsurface emanating from a sphere of arbitrarily large radius r , or equivalently, anarbitrarily small neighborhood of spacelike infinity, i = { u = −∞ , v = ∞} .Our next result is that for certain perturbations of the Schwarzschild metric,analogues of Theorems 2.3 and 2.4 hold, with the assumption of (infinite-order)vanishing on I + ǫ ∪ I − ǫ replaced by vanishing just on the regions I + r := { v = + ∞ , u < } , I − r := { u = −∞ , v > } , for any chosen r > m S , and the domain replaced by(2.18) D r ω := { < f r < ω } where f r := 1( − u ) v .Since this feature solely relies on the positivity of the mass m S > non-constant Bondi energy and angularmomentum at portions of I + and I − which “join up” at ι . For the sake of clarity, we stress that we are referring to the Schwarzschild metric in n +1 = 4dimensions only. We note that the higher-dimensional Schwarzschild metrics actually fall underthe assumptions of Theorems 2.3, 2.4 above, not the theorems below. NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 11 t = 0 u = 0 v = 0 r = r ι I + r I − r Figure 2.
Any small neighborhood of i can be chosen to be theregion u < v > D = ( −∞ , × (0 , ∞ ) × S n − ,and we let u and v denote the projections from D to its first and second components.On D , we consider metrics of the form:(2.20) g = µ d u − K d u d v + ν d v + n − X A,B =1 r γ AB d y A d y B + n − X A =1 ( c Au d y A d u + c Av d y A d v ) , where r and m are now smooth positive functions on D which satisfy certain boundsthat we impose below, and y A : 1 , . . . , n − S u,v of ( u, v ).We impose the following assumptions: • The components of g satisfy the following bounds:(2.21) K = 1 − mr , γ AB = ˚ γ AB + O (cid:18) v − u (cid:19) , c Au , c Av = O (cid:18) v − u (cid:19) , µ, ν = O (cid:18) v − u ) (cid:19) , where ˚ γ is the round metric. • m satisfies the uniform lower bound(2.22) m ≥ m min > and d m satisfies the following uniform bounds, for some η > (2.23) | ∂ I m | = O (( v − u )( − uv ) − η ), | ∂ u m | , | ∂ v m | = O (cid:0) ( v − u ) − (cid:1) . • r is bounded on a level set of v − u , i.e., there exist C, M > ≪ | r ( p ) | ≤ C , whenever v ( p ) − u ( p ) = M .Moreover, r satisfies the differential inequality (cid:20) O (cid:18) v − u ) (cid:19)(cid:21) d v − (cid:20) O (cid:18) v − u ) (cid:19)(cid:21) d u =(2.25) = (cid:18) mr (cid:19) d r + n − X I =1 O (cid:18) v − u (cid:19) d y I . • The following estimate holds for some η > (cid:12)(cid:12)(cid:12) (cid:3) g (cid:16) mr (cid:17)(cid:12)(cid:12)(cid:12) = O (cid:16) ( − uv ) − − η (cid:17) .Note that the conditions imposed on the metrics above allow non-constant massand angular momentum at I + , I − and are consistent with (and weaker than) theones imposed by Sachs, [37].We define(2.27) f := 1( − u ) v , D ω := { < f < ω } . Theorem 2.5.
Consider a metric g on D of the form (2.20) , satisfying the condi-tions (2.21) - (2.26) , and consider a wave operator L g := (cid:3) g + a α ∂ α + V , with (2.28) a u ∈ O ( v − r − ) , a v ∈ O (( − u ) − r − ) , a I ∈ O ( f r − ) , V ∈ O ( f η ) ,for some η > . Let ω > , and consider any C -solution φ on D ω of the equation L g φ = 0 , which in addition satisfies (2.29) Z D ω φ r N + Z D ω | ∂ v φ | r N + Z D ω | ∂ u φ | r N + n − X I =1 Z D ω | ∂ I φ | r N < ∞ ,for all N ∈ N . Then, there exists < ω ′ < ω so that φ ≡ in D ω ′ . Finally, we present an analogue of Theorem 2.4 for this class of metrics.
Theorem 2.6.
Consider a metric g on D of the form (2.20) , satisfying the condi-tions (2.21) - (2.26) , and consider a wave operator L g := (cid:3) g + a α ∂ α + V , with (2.30) a u ∈ O ( v − f − r − ) , a v ∈ O (( − u ) − f − r − ) , a I ∈ O ( f r − ) , V ∈ O (1) .Let ω > , and consider any C -solution φ on D ω of the equation L g φ = 0 , whichin addition satisfies (2.31) Z D ω φ e Nr + Z D ω | ∂ v φ | e Nr + Z D ω | ∂ u φ | e Nr + n − X I =1 Z D ω | ∂ I φ | e Nr < ∞ ,for all N ∈ N . Then, there exists < ω ′ < ω so that φ ≡ in D ω ′ . Note that (2.22) and (2.23) imply m is uniformly bounded from above and has limits at I ± . NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 13
Remark . It turns out that the above class of perturbations (2.20), (2.21) alsoincludes all Kerr metrics (both sub- and super-extremal). While this is not apparentin the Boyer-Lindquist coordinates, there is a special coordinate transformation,discussed in Appendix A, which brings the Kerr metrics in the above form. Roughly,the transformation is designed to undo the 2 π -periodic rotation of the componentsin Boyer-Lindquist coordinates. The Kerr metric in these co-moving coordinates isthen one order closer to the Schwarzschild metric in terms of powers of r − andhence satisfies (2.21). Remark . As we will explain in more detail in the next section, the moralreason behind this weaker vanishing assumption in Theorem 2.5 is precisely the extra convexity (towards null infinity) of certain null geodesics in Schwarzschildcompared to Minkowski. The pseudo-convexity of the above function f r directlyimplies that any rotational null geodesic in Schwarzschild which is δ -close to ι (inthe inverted picture, with respect to the inverted null coordinates U , V ) will infact intersect both I + , I − √ δ -close to ι . This is a manifestation of the blow-up ofthe ρ -component of the Weyl curvature in the inverted metric. Remark . We note that an analogue of our theorems for free wave equations onstatic warped product backgrounds (which fall under the zero-mass class consideredin Thm. 2.3) has been derived in [5] using the Radon transform; see earlier work[18] for free waves on the Minkowski background. Analogous results were recentlyobtained in [6] on the Schwarzschild space-time for spherically symmetric waveswhich are trivial near i .2.3. Discussion of the Ideas.
The proof of all the above theorems will be basedon new Carleman estimates. Such estimates are a common tool in unique continua-tion problems. In fact, we derive our estimates in a uniform way, essentially provingall theorems above together.The approach we use in deriving our Carleman estimates follows the methodadopted in [24], based on energy currents associated to the energy-momentum ten-sor of the wave equations under consideration. There are several challenges thatwe must overcome in deriving Carleman estimates in our setting, which generallyarise from the geometry of infinity in asymptotically Minkowskian spaces. We high-light some of these in the next section which deals only with the Minkowski andSchwarzschild space-times as model cases. It is useful to synopsize all of them here,however: • Degenerating pseudo-convexity:
The first step in deriving our estimates isto construct a function f for each background whose level sets are pseudo-convex. These will be the functions f ǫ , f r defined above (we collectively de-note these by f ). A first difficulty arises here, in that the pseudo-convexityof the level sets of f degenerates towards infinity. • A conformal inversion:
Partly forced by the methods for deriving Carlemanestimates, we consider a conformal transformation of the domains D where Indeed, this cannot be achieved by defining u, v purely in terms of the usual coordinates t, r .The obstruction is the coefficient of d r in these coordinates. See the next section. As discussed in Remark 2.3.5, in the absence of pseudo-convexity of some sort, the presenceof time-dependent lower-order terms in the equation would not allow for the generalization ofthese results, even on the Minkowski background. we seek to derive the vanishing of the solutions of (1.1), so as to convert thenull infinities I + ǫ , I − ǫ , etc. above into boundaries at finite (affine) distance.It is natural to expect one has the freedom to perform such conformaltransformations in view of the conformal invariance of null geodesics, whosegeometry is the key to the pseudo-convexity requirement.The standard conformal transformation in our setting would be the Pen-rose conformal compactification. However, it turns out that we can not derive our Carleman estimates with that tool. To overcome this, we con-sider the equation (1.1) above with respect to the new metric ¯ g := K − f g .This is actually a very natural transformation, and should be thought of asa warped version of the conformal isometry of the Minkowski space-time, g M → u − v − g M . In this setting, the null infinities I + ǫ , I − ǫ are transformedto a complete double null cone with vertex at ι . One has a certain ex-tra convexity near these cones in our picture (see the next section), whichmakes our result possible. • Reparametrizations:
After this inversion, certain aspects of the analysisresemble the strong unique continuation discussed above. In particular, itturns out it is necessary to work not with the function f , but a new function F ( f ) with the same level sets that “accelerates” away from I ± ǫ faster. Itis this choice of function F (given the foliation by the level sets of f ) thatdifferentiates between Theorems 2.3, and 2.5 from Theorems 2.4 and 2.6. • Absorption of Error Terms:
To derive our weighted L -estimates, variouserror terms that arise in the analysis must be absorbed into the main terms.This can be done readily in the setting of Theorem 1.1, where the initialsurface Σ is smooth and strongly pseudo-convex. However in our settingthe degeneration of the pseudo-convexity makes this very delicate, due tothe presence of weights in the Carleman estimates that vanish/blow-up atdifferent rates towards the boundary. Our choice of the conformal inversionis essential here in ensuring that error terms can be absorbed.2.4. Outline of the Paper.
For the reader’s convenience, we start with a sep-arate Section 3, where the pseudo-convexity of the functions f ǫ , f r is derived inthe model Minkowski and Schwarzschild space-times. In particular, the stronger pseudo-convexity that the Schwarzschild space-times exhibit becomes apparent. Wealso briefly present the Carleman estimates we will be deriving in those space-times.In Section 4, we derive the Carleman estimates needed for our theorems. Theestimates we derive here are adapted to the conformally inverted metric, and allowfor the degenerating pseudo-convexity.In Section 5, we prove our results in a uniform way for all Theorems 2.3, 2.4,2.5, and 2.6 together. We first transform the operators under consideration to newoperators in the conformally inverted picture. We then show that the assumptionsof the Carleman estimates, Propositions 4.1 and 4.2, are satisfied in this invertedsetting. We end the section with the (standard) proof that a Carleman estimateimplies the desired vanishing.Finally, in Appendix A, we discuss co-moving coordinates for the Kerr exteriors,which show that they fall under the assumptions of Theorem 2.5. In choosing F ( f ) correctly (in the conformally inverted picture) for Theorem 2.3, we wereguided by [25]. NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 15 ι ε I + I − Figure 3.
Pseudoconvexity in Minkowski space: The nullgeodesics (black) tangential to the level sets of f (red) remain inthe outer component.3. The Model Space-times
Since the proof of our theorems (presented in Sections 4 and 5) somewhat con-ceals the role of the underlying geometry in our results, we present here somekey constructions in the two basic space-times to which our theorems apply: theMinkowski and Schwarzschild space-times.3.1.
Pseudo-convexity in Minkowski Space-time.
In double null coordinates,the Minkowski metric takes the form(3.1) g = − u d v + r ˚ γ AB d y A d y B , r ( u, v ) = v − u , where the outgoing and ingoing null hypersurfaces are precisely the level sets of theoptical functions u , v , respectively, and the hyperboloids H C are expressed as(3.2) − uv = C > , intersecting the null infinities at the endpoints of the asymptotes u = 0 and v = 0;cf. Figure 3. We recall the function f = f ǫ = 1( − u + ǫ )( v + ǫ ) ,whose level sets can be thought of as perturbations of the hyperboloid (3.2) whichasymptote to u = ǫ and v = − ǫ .We shall show that there exists a function h such that(3.3) π = h g − ∇ f restricted to the tangent space of the level sets of f is strictly positive-definite.This positivity encodes the pseudo-convexity and plays a key role in our Carlemanestimates. We check the positivity of π in a suitable frame. Orthonormal frame.
We introduce an orthonormal frame (
N, T, E , . . . , E n − ) ad-apted to the level sets of f . Let ( E , . . . , E n − ) be an orthonormal frame on thesphere:(3.4) r ˚ γ ( E A , E B ) = δ AB . Clearly, the (timelike future-directed) unit vector tangent to the level sets of f ǫ andorthogonal to E , . . . , E n − is then(3.5) T = 12 1 √ f v + ε ∂∂u + 12 1 √ f − u + ε ∂∂v , and the (spacelike) unit normal N to our level sets is given by(3.6) N = 12 1 √ f v + ε ∂∂u −
12 1 √ f − u + ε ∂∂v . Pseudo-convexity.
Since (cid:0) ∇ f (cid:1) uu = 2 f ( − u + ε ) , (cid:0) ∇ f (cid:1) uv = − f , (cid:0) ∇ f (cid:1) vv = 2 f ( v + ε ) , (3.7a) (cid:0) ∇ f (cid:1) uA = 0 , (cid:0) ∇ f (cid:1) AB = − f r (cid:0) r + 2 ε (cid:1) ˚ γ AB , (3.7b)we find that( ∇ f )( T, T ) = 12 f , ( ∇ f )( T, N ) = 0 , ( ∇ f )( N, N ) = 32 f , (3.8) ( ∇ f )( E A , E B ) = − f δ AB − εr f δ AB , ( ∇ f )( E A , T ) = 0 . (3.9)Thus, choosing(3.10) h = − f − εr f , we ensure that the tensor π is diagonal with respect to ( N, T, E , . . . , E n − ), andpositive on the level sets of f : π ( T, T ) = 12 εr f > π ( E A , E B ) = 12 εr f δ AB > . (3.12)These formulas make precise the qualitative picture of Fig. 3, namely, that thepseudo-convexity is stronger the larger ǫ is, but also degenerates as the level setsapproach the null hypersurfaces at infinity. Since the pseudo-convexity as defined in (1.2) is a conformally invariant property,inspired by the conformal inversion of Minkowski space-time, we define:(3.13) ¯ g = f g. In [24], the above foliation by level sets of f ǫ with ǫ < f is then pseudo-convex in the opposite direction.) Their data is prescribed on abifurcate null hypersurface in Minkowski space, emanating from a sphere of radius r > u + v = 0. The surfaces ( − u + ǫ )( v + ǫ ) = c for any − r / < ǫ < c > ( r / ǫ )( r / ǫ ). Note that as thebifurcation sphere is shrunk to a point, r →
0, the pseudoconvexity of the foliation degenerates,as ǫ → ǫ = 0, which they generously shared with us, [25]. NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 17 ι f = c − U V = c I + I − ¯ I Figure 4.
Depiction of the warped inversion of Minkowski space.Upon introducing the new coordinates U = − ( − u + ǫ ) − , V = ( v + ǫ ) − , we find(3.14) ¯ g = − U d V + f r ˚ γ , and see that the conformal transformation (3.13) in fact represents an inversionthat maps the level sets of f to hyperbolas in the ( U, V )-plane, cf. Figure 4:(3.15) f = − U V .
Pseudo-convexity in the Inverted Picture.
We recall the transformation law of theHessian of a function under conformal rescalings. In general, if g = Ω g and ∇ , ∇ are the Levi-Civita connections corresponding to g, g , respectively, then:(3.16) ∇ µν f = (cid:0) ∇ f (cid:1) µν + Ω ∂ µ (cid:0) Ω − (cid:1) ∂ ν f + Ω ∂ ν (cid:0) Ω − (cid:1) ∂ µ f − Ω g αβ ∂ α (cid:0) Ω − (cid:1) ( ∂ β f ) g µν . Thus, evaluating this formula on tangential null vectors X , X ( f ) = 0, we see thatthe level sets of f are still pseudoconvex with respect to g . Furthermore, given theconformal inversion (3.13), we can simply define(3.17) h := hf − + 1to ensure that π = h g − ∇ f is positive when restricted to the level sets of f . Infact, with respect to the orthonormal frame for g ,(3.18) ¯ N := f − N , ¯ T := f − T , ¯ E A := f − E A we then immediately obtain:(3.19) π ( T , T ) = 12 εr , π ( E A , E B ) = 12 εr δ AB . Moreover all off-diagonal terms of π vanish in this frame. Pseudo-convexity in the Schwarzschild Exterior.
We now discuss thepseudo-convexity properties of the function f r defined in (2.18) in the Schwarzschildspace-times. It turns out that the behavior of null geodesics is here substantially dif-ferent, yielding stronger (pseudo-)convexity for the level sets of f r , and ultimatelya stronger Carleman estimate. Inversion.
Recall the metric (2.15) for the Schwarzschild exterior in double nullcoordinates. Here, u = 0 and v = 0 are fixed by a choice of r > m S , and(3.20) f = f r = 1( − u ) v . We now consider the inverted metric(3.21) g = 11 − mr f g , on the domain D = { ( u, v ) : u < , v > } .Introducing coordinates U = u − , V = v − , the metric ¯ g takes the form:(3.22) g = − U d V + f r − mr · ˚ γ , and f takes the same form as in the Minkowski setting:(3.23) f = − U V .
We now proceed to show in the inverted space-time that the level sets of f areindeed pseudo-convex. Hessian of f . We calculate: ∇ UU f = 0 , ∇ UV f = − , ∇ V V f = 0 , ∇ UA f = 0 , ∇ V A f = 0 , (3.24)while the non-trivial contribution is now contained in: ∇ AB f = 12 ( U − V ) f r ˚ γ AB + f r − mr ˚ γ AB + 14 f r − mr mr (cid:16) V − U (cid:17) = − r ∗ r g AB + g AB + 34 2 mr r ∗ r g AB , r ∗ = v − u . (3.25) Orthonormal Frame.
We construct a frame ( ¯
N , ¯ T , ¯ E , . . . , ¯ E n − ) which is analogousto the one in the (inverted) Minkowski space-time:(3.26) ¯ N = 12 1 √ f h U ∂ U + V ∂ V i ¯ T = 12 1 √ f h − U ∂ U + V ∂ V i and ¯ E i tangent to the spheres S U,V satisfying(3.27) g ( ¯ E A , ¯ E B ) = R ˚ γ ( ¯ E A , ¯ E B ) = δ AB . Then we obtain: ∇ T ¯ T f = − , ∇ N ¯ N f = 12 , ∇ T ¯ N f = 0 , (3.28a) ∇ E A ¯ E B f = − r ∗ r δ AB + δ AB + 34 2 mr r ∗ r δ AB , ∇ E A ¯ T = 0 . (3.28b) NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 19
Pseudoconvexity.
Let then¯ h = 12 − (cid:16) r ∗ r − (cid:17) + 38 2 mr r ∗ r , (3.29) ¯ π = ¯ hg − ∇ f . (3.30)As previously discussed, the positivity of ¯ π restricted to the orthogonal complementof ¯ N signals the pseudo-convexity of the level sets of f . We find π ¯ T ¯ T = 14 2 mr log | r − m | −
38 1 r (cid:16) mr r ∗ + 23 r ∗ (cid:17) (3.31) π AB = 14 2 mr log | r − m | δ AB −
38 1 r (cid:16) mr r ∗ + 23 r ∗ (cid:17) δ AB . (3.32)In particular, irrespective of how large r ∗ is chosen, all tangential components arepositive for r large enough (depending on the choice of r ∗ ). This is what allowsunique continuation to hold in an arbitrarily small neighborhood of spatial infinityfor Schwarzschild space-times of positive mass m S > Reparametrizations and Carleman Estimates.
For both of our modelspacetimes, after the conformal inversion, we have a metric of the common form(3.33) ¯ g = − U d V + R ˚ γ AB d y A d y B ,while the foliating functions f ǫ and f r are now given by f = − U V .Consider a smooth function φ on the (inverted) space-time. The process forderiving the Carleman estimates for φ roughly follows the geometric method in[24]: we can compare this to an energy estimate for ψ = e − λF ( f ) φ , where F ( f )denotes an appropriate reparametrization of the level sets of f . More precisely, oneintegrates the divergence of a modified energy current,(3.34) J wβ [ ψ ] = Q αβ [ ψ ] ¯ ∇ α f + 12 ( ∂ β w ) · ψ − w · ∂ β ( ψ ) + P ♭β ,where Q [ ψ ] denotes the standard energy-momentum tensor for the wave equation(see (4.44)). In particular, one utilizes the gradient of f as a multiplier vector fieldin (3.34). In contrast to the usual energy estimates, here we wish for the bulk termsof the integrals to be positive and for the boundary terms to vanish .By choosing w in (3.34) appropriately, depending on h in the preceding discus-sions (see (4.5)) and on f , the divergence of (3.34) will produce precisely the tensors¯ π from the previous discussions, capturing the pseudo-convexity of the level sets of f . This pseudo-convexity produces positive bulk terms that are quadratic in ¯ ∇ ¯ T ψ and ¯ ∇ ¯ E a ψ , i.e., the derivatives of ψ in directions tangent to the level sets of f .On the other hand, to obtain positivity for the normal derivative ¯ ∇ ¯ N ψ and for ψ itself, one relies on the choice of reparametrization F ( f ) of f . From computations,one can see that at least a logarithmic blowup of F ( f ) as f ց F ( f ) := log f (which would correspond to the decayingpotential cases of Theorems 2.3 and 2.5) does not suffice to produce sufficientlypositive weights; to obtain the desired results, one adds a small extra accelerationto the above F ( f ); see (4.20). We also note that the bounded potential cases ofTheorems 2.4 and 2.6 correspond to the reparametrization F ( f ) = − f − / .The above ultimately results in an inequality of the form Z D ω ′ W L |L ψ | ≥ Cλ Z D ω ′ W N |∇ ¯ N ψ | + W T | ¯ ∇ ¯ T ψ | + W T n − X a =1 |∇ ¯ E a ψ | ! (3.35) + Cλ Z D ω ′ W · ψ + Z D ω ′ E ,where D ω ′ denotes the region { < f < ω ′ } for some sufficiently small ω ′ >
0, andwhere L is the conjugated wave operator L = e − λF ( f ) ¯ (cid:3) e λF ( f ) .Moreover, W L , W N , W T , and W are positive weights that depend on the pseudo-convexity of the level sets of f and the reparametrization F . The only term in (3.35)that is not positive is the integral over the “error” E , which must be absorbed intothe remaining positive terms. That this is possible depends largely on the specificcommon forms for ¯ g and f in the inverted settings.By expressing ψ back in terms of φ , and by appropriately controlling the errorterms, we obtain the desired Carleman estimates. The exact estimate depends onthe amount of pseudo-convexity in the level sets of f , as well as on the chosenreparametrization F . For example, in the case of wave equations on Schwarzschildspace-times with decaying potential, we have the following: Z D ω ′ f − λ +1 e λf p | ¯ (cid:3) φ | & λ Z D ω ′ f − λ +1 e λf p · f − p | ¯ ∇ ¯ N φ | + λ Z D ω ′ f − λ +1 e λf p · log rf r | ¯ ∇ ¯ T φ | + n − X a =1 | ¯ ∇ ¯ E a φ | ! + λ Z D ω ′ f − λ +1 e λf p · f − p φ .Here, p is a small positive constant; see (4.20).Finally, we remark that the vanishing assumption required on φ in order for therelevant boundary terms to vanish depends again on the choice of F . In the case F ( f ) = log f + correction (for Theorems 2.3 and 2.5), the requirement is that φ vanishes at a superpolynomial rate in terms of f . Similarly, when F ( f ) = − f − / (for Theorems 2.4 and 2.6), then φ must vanish at a superexponential rate.4. Carleman Estimates
In this section, we establish the general Carleman estimates that will be used toprove our results for all the spacetimes under consideration in this paper.Although the setting we introduce is abstract, in order to prove all our results ina uniform way, the reader should keep in mind that the space-times we consider willbe conformal inversions of the original, physical space-times to which Theorems2.3, 2.4, 2.5, and 2.6 refer. The key point behind the general estimates in thissection is that the machinery outlined in Section 3 is robust , in the sense that theanalysis goes through for sufficiently mild perturbations of ¯ g of the form (3.33).The closeness properties are expressed in terms of specially adapted frames. Preliminaries.
Our estimates will be for the wave operator (cid:3) = (cid:3) g , for an in-complete ( n + 1)-dimensional Lorentz manifold ( D , g ). As discussed earlier, theseare weighted L -estimates; a key ingredient of the weight will be a pseudo-convexfunction f ∈ C ∞ ( D ). We assume that the level sets of f are timelike, i.e.,(4.1) ∇ α f ∇ α f = g αβ ∇ α f ∇ β f > This is encoded in the function Ψ in Propositions 4.1 and 4.2.
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 21
Moreover, we assume at each point of D , there is a local frame, ( E , . . . , E n ),which is “adapted to f ”, that is: • The frame is orthonormal, with(4.2) g ( E α , E β ) = m αβ , [ m αβ ] nα,β =0 := diag( − , , . . . , • E , . . . , E n − are tangent to the level sets of f , and E n (which is normal tothe level sets of f ), satisfies E n f > T , ¯ E , . . . , ¯ E n − , ¯ N ) in Section 3.They also provide a natural way to measure tensor fields on ( D , g ): • Given a vector field X on D , we define(4.3) | X | := n X α =0 [ g ( X, E α )] . • Similarly, for a covariant k -tensor A on D , we define(4.4) | A | := n X α ,...,α k =0 | A ( E α , . . . , E α k ) | .For technical reasons related to the vanishing assumptions required for our Car-leman estimates, we make the following definition: a sequence ( D k ) of compactsubsets of D is called an exhaustion of D if the following conditions hold: • The D k ’s are increasing: D k ⊂ D k +1 for each k . • For each n , the boundary ∂D k can be written as a finite union of smoothspace-like and time-like hypersurfaces of D .Finally, given a function w ∈ C ∞ ( D ), we define the corresponding quantity(4.5) h ( w ) = h := w + 12 (cid:3) f − n − ∈ C ∞ ( D ),In particular, h will be the factor connected to the pseudo-convexity of the levelsets of f , in that we wish for the restriction of h · g − ∇ f to the level sets of f tobe nonnegative-definite (see the discussion in Section 3). The Main Estimates.
With the above background and definitions in place, we arenow prepared to state our two main Carleman estimates. In the statements below,and also in the upcoming proofs, we will use the notation A ≃ B to mean that A ≤ cB and B ≤ cA for some constant c > f ; this is used for Theorems 2.3, and 2.5. Proposition 4.1.
Let ( D , g ) and f be as above, with f sufficiently small on D ,and fix orthonormal frames ( E , . . . , E n ) adapted to f , in the above sense, whichcover all of D . Furthermore, fix a constant p > , and fix Ψ , w ∈ C ∞ ( D ) , with Ψ satisfying (4.6) 0 ≤ Ψ ≪ f p .Assume the following conditions hold: • For any vector field X tangent to the level sets of f , (4.7) ( −∇ f + h · g )( X, X ) ≃ Ψ · | X | . • The following bounds hold for f : | f − ∇ E n f − | + n − X i =0 |∇ E n E a f | ≪ Ψ , (4.8) (cid:12)(cid:12)(cid:12)(cid:12) (cid:3) f − n + 12 (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ E n E n f − (cid:12)(cid:12)(cid:12)(cid:12) ≪ f p . (4.9) • w satisfies the following estimates: (4.10) | w | ≪ f p , | (cid:3) w | . f p − .Let φ ∈ C ∞ ( D ) , such that it satisfies the following vanishing condition: • For each
N > , there exists an exhaustion ( D k ) of D such that, if ν k is aunit normal for the components of ∂D k , then (4.11) lim k ր∞ Z ∂D k f − N e Nf p ( | ν k | + |∇ ν k w | )( φ + |∇ φ | ) = 0 .Then, for sufficiently large λ > , the following estimate holds for φ : (4.12) Z D f − λ +1 e λf p · | (cid:3) φ | & λ Z D f − λ +1 e λf p · f p − |∇ E n φ | + λ Z D f − λ +1 e λf p · f − Ψ n − X i =0 |∇ E i φ | + λ Z D f − λ +1 e λf p · f − p φ . The next Carleman estimate corresponds to functions vanishing superexponen-tially with respect to f ; this is used for Theorems 2.4 and 2.6. Proposition 4.2.
Let ( D , g ) and f be as above, with f sufficiently small on D ,and fix orthonormal frames ( E , . . . , E n ) adapted to f , in the above sense, whichcover all of D . Furthermore, fix a constant q > , and fix Ψ , w ∈ C ∞ ( D ) , with Ψ satisfying (4.13) 0 ≤ Ψ ≪ ,Assume the following conditions hold: • For any vector field X tangent to the level sets of f , (4.14) ( −∇ f + h · g )( X, X ) ≃ Ψ · | X | . • The following bounds hold for f : f − q | f − ∇ E n f − | + n − X i =0 |∇ E n E a f | ≪ Ψ , (4.15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:3) f − n + 12 (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ E n E n f − (cid:12)(cid:12)(cid:12)(cid:12) ≪ . (4.16) • w satisfies the following estimates: (4.17) | w | ≪ , | (cid:3) w | . f − q − . In particular, for the model spacetimes of Section 3, the left-hand side of (4.8) vanishesentirely, as does the quantity |∇ E n E n f − | in (4.9). Here, the integral is with respect to the volume forms of the components of ∂D k . NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 23
Let φ ∈ C ∞ ( D ) , such that it satisfies the following vanishing condition: • For each
N > , there exists an exhaustion ( D k ) of the boundary of D suchthat, if ν k is a unit normal for the components of ∂D k , then (4.18) lim k ր∞ Z ∂D k e − Nf − q ( | ν k | + |∇ ν k w | )( φ + |∇ φ | ) = 0 .Then, for sufficiently large λ > , the following estimate holds for φ : (4.19) Z D f q +1 e λf − q · | (cid:3) φ | & λ Z D f q +1 e λf − q · f − q − |∇ E n φ | + λ Z D f q +1 e λf − q · f − q − Ψ n − X i =0 |∇ E i φ | + λ Z D f q +1 e λf − q · f − q − φ . Proof of the Estimates I: Preliminary Bounds.
The remainder of thissection is focused on the proofs of Propositions 4.1 and 4.2. Here, we establish somepreliminary estimates that will be essential later. The actual Carleman estimatesthemselves will be derived in Section 4.2.We will prove both propositions simultaneously. This can be accomplished byworking with general reparametrizations of f . To extract Propositions 4.1 and 4.2,we need only consider the specific reparametrizations F = F ( f ) of f correspondingto those settings, which we describe below.4.1.1. Reparametrizations.
To prove Proposition 4.1, we define(4.20) F = F := log f − f p .Letting ′ denote differentiation with respect to f , then(4.21) F ≃ log f , F ′ = f − − pf p − , F ′ ≃ f − ,as long as f is sufficiently small. On the other hand, for Proposition 4.2, we define(4.22) F = F := − f − q .Observe that(4.23) F ′ = qf − q − .For the rest of this section, we let F be either F or F , corresponding to the proofof Proposition 4.1 or 4.2, respectively. Observe: Lemma 4.3.
Both choices of F satisfy (4.24) f F ′ & , f | F ′′ | . F ′ .Furthermore, for sufficiently large λ > , (4.25) e − F ≥ , f − . F ′ . e − λF . Throughout our proof, we will also refer to the auxiliary function(4.26) G := − ( f F ′ ) ′ .In particular, when F is either F or F , then G is, respectively,(4.27) G = p f p − , G = q f − q − .Note that in both cases G = G or G = G , we have the following properties: Lemma 4.4.
Both choices of G satisfy (4.28) 0 < G . F ′ .Furthermore, Ψ is related to F and G (in both cases) via the following estimates: (4.29) F ′ Ψ ≪ G , Ψ ≪ , Ψ ≪ f G . f F ′ . In addition, in terms of the above language, the assumptions (4.7)-(4.10) and(4.14)-(4.17) for Propositions 4.1 and 4.2 imply the following:
Proposition 4.5.
Assuming the hypotheses of Proposition 4.1 and 4.2, and setting F and G accordingly as above, then: ( −∇ f + h · g )( X, X ) ≃ Ψ | X | , (4.30) F ′ | f ∇ E n f − f | + n − X i =0 |∇ E n E i f | ≪ Ψ , (4.31) F ′ (cid:12)(cid:12)(cid:12)(cid:12) (cid:3) f − n + 12 (cid:12)(cid:12)(cid:12)(cid:12) + F ′ (cid:12)(cid:12)(cid:12)(cid:12) ∇ E n E n f − (cid:12)(cid:12)(cid:12)(cid:12) ≪ G , (4.32) F ′ | w | ≪ G , | (cid:3) w | . f F ′ G . (4.33) Furthermore, the vanishing conditions (4.11) and (4.18) can be aggregated as (4.34) lim k ր∞ Z ∂D k e − NF ( | ν k | + |∇ ν k w | )( φ + |∇ φ | ) = 0 , N > .Remark. Note that metrically equivalent tensor fields—e.g., a vector field T α andthe corresponding one-form T α := g αβ T β —have the same tensor norm (as definedin (4.3) and (4.4)). In particular, this is true for the differential ∇ α ψ = ∂ α ψ of ascalar ψ ∈ C ∞ ( D ) and its g -gradient ∇ α ψ = g αβ ∇ β ψ .4.1.2. Estimates for f . The first task is to obtain preliminary estimates for f . Forconvenience, we will use throughout the proof the abbreviations(4.35) ℓ := ∇ α f ∇ α f , ¯ ℓ := 12 ∇ α f ∇ α ℓ = ∇ α f ∇ β f ∇ αβ f .Note in particular that E n = ℓ − · grad f ,while ¯ ℓ encodes the normal component of ∇ f . Lemma 4.6.
The following estimates hold: (4.36) |∇ f | ≃ f , ℓ ≃ f , F ′ | ℓ − f | ≪ Ψ .Proof. The last inequality in (4.29) and (4.31) imply that(4.37) |∇ E n f − f | ≪ f − ( F ′ ) − Ψ ≪ f .Since |∇ f | = ∇ E n f = ℓ by definition, the first two estimates in (4.36) followimmediately. Finally, we apply (4.37) and the comparison ℓ ≃ f to obtain | ℓ − f | ≤ | ℓ − f || ℓ + f | ≪ f · f − ( F ′ ) − Ψ = ( F ′ ) − Ψ,completing the proof of the final estimate in (4.36). (cid:3)
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 25
Lemma 4.7.
With F and G as before, the following estimates hold: (4.38) F ′ (cid:12)(cid:12)(cid:12)(cid:12) h − (cid:12)(cid:12)(cid:12)(cid:12) ≪ G , |∇ f | . , F ′ (cid:12)(cid:12)(cid:12)(cid:12) ¯ ℓ − f (cid:12)(cid:12)(cid:12)(cid:12) ≪ f G .Proof. First, by rewriting (4.5) as h = 12 + w + 12 (cid:18) (cid:3) f − n + 12 (cid:19) .and by applying (4.28), (4.32), and (4.33), we obtain the first inequality in (4.38).For |∇ f | , we begin by applying (4.28), (4.29), (4.31), and (4.32), which yields(4.39) |∇ E n E n f | .
12 + ( F ′ ) − G . n − X i =0 |∇ E n E i f | . Ψ . ∇ f , we note the algebraic identity ∇ E a E a f + ∇ E b E b f + 2 ∇ E a E b f = ∇ E a + E b ,E a + E b f , 1 ≤ a, b ≤ n − n − X a,b =1 |∇ E a E b f | . Ψ + h . (cid:12)(cid:12)(cid:12)(cid:12) ¯ ℓ − f (cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ (cid:12)(cid:12)(cid:12)(cid:12) ∇ E n E n f − (cid:12)(cid:12)(cid:12)(cid:12) + 12 | ℓ − f | ≪ f ( F ′ ) − G + ( F ′ ) − Ψ.Since Ψ . f G by (4.29), this proves the last inequality in (4.38). (cid:3) Proof of the Estimates II: The Main Derivation.
We are now preparedto derive the Carleman estimates, (4.12) and (4.19), in earnest. Here we will followsome of the nomenclature of [24]. Throughout, we fix φ ∈ C ∞ ( D ) and λ > • Define ψ ∈ C ∞ ( D ) and the operator L by(4.41) ψ := e − λF φ , L ψ := e − λF (cid:3) ( e λF ψ ). • Define the auxiliary function w ′ ∈ C ∞ ( D ) by(4.42) w ′ := w − n −
14 = h − (cid:3) f . • In addition, we define the shorthands(4.43) Sψ := ∇ α f ∇ α ψ , S w ψ := Sψ − w ′ ψ . • Let Q denote the stress-energy tensor for the wave equation, applied to ψ :(4.44) Q αβ := ∇ α ψ ∇ β ψ − g αβ ∇ µ ψ ∇ µ ψ . Algebraic Expansions.
The first step is an algebraic expansion of the expres-sion L ψS w ψ . From this, we can obtain a pointwise lower bound for |L ψ | . Lemma 4.8.
Recall the notations (4.41) - (4.44) , and define also the following: Λ = − ( F ′ ) ¯ ℓ − F ′ F ′′ ℓ − ( F ′ ) ℓh , E = 2 F ′ h + F ′′ ℓ , (4.45) π αβ = −∇ αβ f + hg αβ , P β = Q αβ ∇ α f + P ♭β + P ♯β , P ♯β = − w ′ · ψ ∇ β ψ + 12 ∇ β w ′ · ψ , P ♭β = 12 λ ℓ ( F ′ ) ∇ β f · ψ .Then, the following inequality holds: ( F ′ ) − |L ψ | ≥ λ F ′ · | S w ψ | + 2 λπ αβ · ∇ α ψ ∇ β ψ + 2 λ Λ · ψ (4.46) + 2 λ E · ψS w ψ − λ (cid:3) w · ψ + 2 λ ∇ β P β .Proof. We begin by expanding L ψ as follows: L ψ = (cid:3) ψ + 2 λ · ∇ α F ∇ α ψ + e − λF (cid:3) e λF · ψ = (cid:3) ψ + 2 λF ′ · Sψ + λ ∇ α F ∇ α F · ψ + λ (cid:3) F · ψ = (cid:3) ψ + 2 λF ′ · Sψ + λ ( F ′ ) ℓ · ψ + λF ′′ ℓ · ψ + λF ′ (cid:3) f · ψ = (cid:3) ψ + 2 λF ′ · S w ψ + λ ( F ′ ) ℓ · ψ + λ E · ψ ,Multiplying the above by S w ψ yields(4.47) L ψS w ψ = (cid:3) ψS w ψ + 2 λF ′ · | S w ψ | + λ ( F ′ ) ℓ · ψS w ψ + λ E · ψS w ψ .Letting A = λ ( F ′ ) ℓ , then the product rule implies A · ψS w ψ = 12 A · ∇ β f ∇ β ( ψ ) − A w ′ · ψ (4.48) = 12 ∇ β ( A∇ β f · ψ ) − ∇ β f ∇ β A · ψ − A h · ψ = ∇ β P ♭β + λ Λ · ψ ,where in the last step, we recalled (4.45), and we observed that − ∇ β f ∇ β A − A h = − λ ∇ β f ∇ β ℓ · ( F ′ ) − λ F ′ ℓ · ∇ β f ∇ β ( F ′ ) − λ ( F ′ ) ℓh = − λ ( F ′ ) ¯ ℓ − λ F ′ F ′′ ℓ − λ ( F ′ ) ℓh .Combining (4.47) and (4.48), we see that(4.49) L ψS w ψ = (cid:3) ψS w ψ + 2 λF ′ · | S w ψ | + λ Λ · ψ + λ E · ψS w ψ + ∇ β P ♭β .Next, recalling the stress-energy tensor Q in (4.44), we compute ∇ β ( Q αβ ∇ α f ) = (cid:3) ψSψ + ∇ αβ f · ∇ α ψ ∇ β ψ − (cid:3) f · ∇ β ψ ∇ β ψ , ∇ β P ♯β = − w ′ · ψ (cid:3) ψ − w ′ · ∇ β ψ ∇ β ψ + 12 (cid:3) w · ψ .Summing the above identities, we obtain(4.50) ∇ β ( Q αβ ∇ α f + P ♯β ) = (cid:3) ψS w ψ − π αβ · ∇ α ψ ∇ β ψ + 12 (cid:3) w · ψ .Combining (4.49) and (4.50) yields L ψS w ψ = 2 λF ′ · | S w ψ | + π αβ · ∇ α ψ ∇ β ψ + λ Λ · ψ (4.51) NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 27 + λ E · ψS w ψ − (cid:3) w · ψ + ∇ β P β .Finally, (4.46) follows immediately from (4.51) and the following basic inequality: L ψS w ψ ≤ λ − ( F ′ ) − · |L ψ | + 12 λF ′ · | S w ψ | . (cid:3) Positivity and Error Estimates.
We next show that, except for the divergenceterm, the right-hand side of (4.46) is positive. The first step of this process is toshow that Λ is positive and that E is appropriately bounded. More specifically,Λ represents the weight of the zero-order terms in the Carleman estimates, and itmust absorb all the other zero-order weights in our derivation (including E ). Lemma 4.9.
The following estimates hold: (4.52) Λ ≃ f F ′ G , |E| . G .Moreover, for sufficiently large λ , the following inequality holds: (4.53) ( F ′ ) − |L ψ | ≥ λ F ′ · | S w ψ | + 2 λπ αβ · ∇ α ψ ∇ β ψ + λ Λ · ψ + 2 λ ∇ β P β ,Proof. We begin with the bound for E . From its definition in (4.45), we can write E = F ′ + f F ′′ + F ′ (2 h −
1) + F ′′ ( ℓ − f )(4.54) = − G + F ′ (2 h −
1) + F ′′ ( ℓ − f ).By (4.24), (4.29), (4.36), and (4.38), we estimate F ′ | h − | ≪ G , F ′′ | ℓ − f | . f − F ′ | ℓ − f | ≪ f − Ψ ≪ G .Combining the above inequalities with (4.54) results in the inequality for E in (4.52).Next, for Λ, we expand its definition in (4.45):(4.55) Λ = −
12 ( F ′ ) f − F ′ F ′′ f −
12 ( F ′ ) f + E Λ = f F ′ G + E Λ ,where the error terms E Λ are given by E Λ = − F ′ F ′′ ( ℓ − f ) − ( F ′ ) (cid:18) ¯ ℓ − f (cid:19) − ( F ′ ) (cid:18) ℓh − f (cid:19) .We can then estimate E Λ using (4.24), (4.36), and (4.38): | E Λ | . F ′ · f F ′′ · | ℓ − f | + ( F ′ ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ¯ ℓ − f (cid:12)(cid:12)(cid:12)(cid:12) + | ℓ − f | + f (cid:12)(cid:12)(cid:12)(cid:12) h − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≪ F ′ Ψ + f F ′ G .Thus, (4.29) yields | E Λ | ≪ f F ′ G , which with (4.55) implies the first part of (4.52).Finally, for (4.53), we first observe that(4.56) 2 λ E · ψS w ψ ≤ λ F ′ · | S w ψ | + λ ( F ′ ) − E · ψ .Moreover, (4.24), (4.28), (4.33), and (4.52), we have λ | (cid:3) w | · ψ . λf F ′ G · ψ , λ ( F ′ ) − E · ψ . λ ( F ′ ) − G · ψ . λ f F ′ G · ψ .Thus, applying the above, along with the first bound in (4.52), we see that(4.57) λ | (cid:3) w | · ψ + λ ( F ′ ) − E · ψ ≤ λ Λ · ψ ,for sufficiently large λ . Applying (4.56) and (4.57) to (4.46) results in (4.53). (cid:3) The remaining estimate deals with terms in (4.53) that are quadratic in ∇ ψ . Lemma 4.10.
There exist constants C , C > such that F ′ · | S w ψ | + Λ · ψ ≥ C G · | Sψ | , (4.58) G · | Sψ | + π αβ · ∇ α ψ ∇ β ψ ≥ C f G · |∇ E n ψ | + C Ψ · n − X i =0 |∇ E i ψ | .In particular, if λ is sufficiently large, then there exists C > such that ( F ′ ) − |L ψ | ≥ C λf G · |∇ E n ψ | + C λ Ψ · n − X i =0 |∇ E i ψ | (4.59) + C λ Λ · ψ + 2 λ ∇ β P β .Proof. First, note that (4.59) follows by combining (4.53) and (4.58) and by takinglarge enough λ . Moreover, observe that, by (4.24) and (4.28), we obtain G · | Sψ | . G · | S w ψ | + G | w ′ | · ψ . F ′ · | S w ψ | + f F ′ G | w ′ | · ψ .Since | w ′ | . π αβ · ∇ α ψ ∇ β ψ = π E n E n |∇ E n ψ | + n − X i,j =0 π E i E j ∇ E i ψ ∇ E j ψ (4.60) + 2 n − X i =0 π E n E i ∇ E n ψ ∇ E i ψ = K ⊥⊥ + K kk + K ⊥k ,By the assumption (4.30), we have the lower bound K kk & Ψ · n − X i =0 |∇ E i ψ | .Thus, by (4.60) and the above, we obtain, for some C ′ >
0, that(4.61) π αβ · ∇ α ψ ∇ β ψ ≥ C ′ Ψ · n − X i =0 |∇ E i ψ | − |K ⊥⊥ | − |K ⊥k | ,Moreover, recalling (4.36), we see that(4.62) G · | Sψ | = ℓG · |∇ E n ψ | .To control K ⊥k , we expand and bound using (4.31):(4.63) |K ⊥k | . n − X i =0 |∇ E n E i f ||∇ E n ψ ||∇ E i ψ | ≪ Ψ · n X α =0 |∇ E α ψ | .Furthermore, for K ⊥⊥ , we expand K ⊥⊥ = (cid:18) − ∇ E n E n f (cid:19) · |∇ E n ψ | + (cid:18) h − (cid:19) · |∇ E n ψ | .Applying (4.32) and (4.38), we then estimate(4.64) |K ⊥⊥ | ≪ ( F ′ ) − G · |∇ E n ψ | . f G · |∇ E n ψ | . NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 29
Combining (4.36) and (4.61)-(4.64) results in the second inequality of (4.58). (cid:3)
Integral and Boundary Estimates.
Having obtained a pointwise lower boundfor |L ψ | from (4.59), it remains to integrate this over D . The only term on theright-hand side of (4.59) which needs not be nonnegative is the divergence of P ,which contributes boundary terms after integration. Here, we show, by integratingover D k and passing to the limit, that this boundary contribution vanishes. Lemma 4.11.
The following limit holds: (4.65) lim k ր∞ Z D k ∇ β P β = 0 .As a result, for sufficiently large λ , we have that Z D ( F ′ ) − |L ψ | & λ Z D f G · |∇ E n ψ | + Ψ · n − X i =0 |∇ E i ψ | ! (4.66) + λ Z D f F ′ G · ψ .Proof. Recalling the definition of P in (4.45), the divergence theorem yields Z D k ∇ β P β = Z ∂D k Q αβ ν βk ∇ α f + Z ∂D k P ♭β ν βk + Z ∂D k P ♯β ν βk = I k + I k + I k .Thus, to prove (4.65), it suffices to show that each I lk vanishes as k ր ∞ .Recalling (4.41) and applying (4.36), we obtain |∇ ψ | . e − λF n X α =0 |∇ E α φ | + λe − λF |∇ E n F | · | φ | . e − λF |∇ φ | + λe − λF f F ′ · | φ | ,By (4.25), we see that for sufficiently large λ ,(4.67) |∇ ψ | . e − λF ( |∇ φ | + | φ | ).For I k , we expand Q using (4.44) to obtain | I k | ≤ Z ∂D k |∇ α f ∇ α ψ ||∇ ν k ψ | + 12 Z ∂D k |∇ ν k f ||∇ α ψ ∇ α ψ | .Applying (4.67), the above becomes(4.68) | I k | ≤ Z ∂D k |∇ f || ν k ||∇ ψ | . Z ∂D k e − λF | ν k | ( |∇ φ | + φ ).Similarly, for I k , we expand P ♭β using (4.45) and estimate(4.69) | I k | . λ Z ∂D k f ( F ′ ) | ν k ||∇ f | · ψ . λ Z ∂D k e − λF | ν k | · φ ,where we used (4.25) to control F ′ . Finally, for I k , we expand P ♯β , and use thetrivial bound | w ′ | .
1, which can be obtained from (4.33): | I k | . Z ∂D k | ν k | · | ψ ||∇ ψ | + Z ∂D k |∇ ν k w | · ψ . Z ∂D k | ν k | · |∇ ψ | + Z ∂D k ( | ν k | + |∇ ν k w | ) · ψ . Recalling (4.67), we obtain the estimate(4.70) | I k | . Z ∂D k e − λF ( | ν k | + |∇ ν k w | ) · ( |∇ φ | + φ ).Recalling the vanishing condition (4.34), then (4.68)-(4.70) imply thatlim k ր∞ ( | I k | + | I k | + | I k | ) = 0,which completes the proof of (4.65). For (4.66), we integrate (4.59) over D k : Z D k ( F ′ ) − | Lψ | ≥ C λ Z D k f G · |∇ E n ψ | + C λ Z D k Ψ · n − X i =0 |∇ E i ψ | + C λ Z D k Λ · ψ + 2 λ Z D k ∇ β P β .Taking a limit of this as k ր ∞ , the last term on the right-hand side vanishes by(4.65). Moreover, by applying the monotone convergence theorem on the remaining(nonnegative) terms, we obtain the desired inequality (4.66). (cid:3) Completion of the Proof.
Recall now from (4.41) that(4.71) ψ = e − λF φ , L ψ = e − λF (cid:3) φ , ∇ E i ψ = e − λF ∇ E i φ ,for any 0 ≤ i < n . Moreover, ∇ E n ψ = e − λF · ∇ E n φ − λe − λF F ′ ∇ E n f · φ ,so that, recalling (4.31) and (4.36), we can estimate e − λF · |∇ E n φ | . |∇ E n ψ | + λ ( F ′ |∇ E n f | ) · ψ . |∇ E n ψ | + λ f ( F ′ ) · ψ .Multiplying both sides by ( F ′ ) − G and applying (4.24) yields(4.72) e − λF ( F ′ ) − G · |∇ E n φ | . f G · |∇ E n ψ | + λ f F ′ G · ψ .Finally, combining (4.66) with (4.71) and (4.72), and letting W λ = e − λF ( F ′ ) − ,we obtain the generalized Carleman inequality Z D W λ · | (cid:3) φ | & λ Z D W λ G · |∇ E n φ | + F ′ Ψ · n − X i =0 |∇ E i φ | ! (4.73) + λ Z D W λ · f ( F ′ ) G · ψ .To recover the final inequalities (4.12) and (4.19) from (4.73), we do the following: • Replace F = F in the case of (4.12), or F = F for (4.19). • Similarly, substitute G by either G or G . • The weights W λ corresponding to F = F and F = F , respectively, are W λ, = f − λ +1 e λf p , W λ, = f q +1 e λf − q .This completes the proofs of Propositions 4.1 and 4.2. Remark.
Note that up to and including (4.73), the preceding proof made no ref-erences to the explicit definitions of F and F . In particular, the entirety of ourproof relied only on the following assumptions: NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 31 • Conditions characterizing F and G : (4.24), (4.25), (4.28), and (4.29). • Conditions (4.30)-(4.34), which guarantee that the underlying space-timeand f are sufficiently close to the special cases considered in Section 3.If the above assumptions hold for our setting and for our choice of F , then thepreceding proof implies that the more general Carleman estimate (4.73) holds. Sucha generalized estimate allows for different unique continuation results for operators L g as in (1.1), but with the lower-order terms allowed to have different asymptoticbehavior from those considered in Theorems 2.3-2.6.5. Proof of Theorems 2.3-2.6.
Preliminaries and Notation.
We now consider the class of space-times( D , g ) and operators L g = (cid:3) g + a α ∂ α + V addressed in Theorems 2.3-2.6. Theresults will be proven together , by reducing them to Propositions 4.1, 4.2. Thereare three key steps:We first consider certain special conformal rescalings of the underlying metrics g , g ¯ g = Ω g ,which transform the boundaries at infinity I + ǫ ∪ I − ǫ and I + r ∪ I − r into complete double null cones emanating from a point; we denote these cones by I for thepurpose of this discussion. The domains D ǫω , D ω are then mapped to the exterior domains of these cones; see Figure 4. This is a generalization of the warped inversion discussed in Section 3.1. Now, L g transforms to a new wave operator ¯ L g (whoseprincipal symbol is (cid:3) g ) defined over the manifolds ( D , ¯ g ) such that the solutions φ to L g φ = 0 yield new solutions ¯ φ to ¯ L ¯ g ¯ φ = 0; cf. Section 5.2.Then, we prove that the inverted metrics ¯ g and the operators ¯ L ¯ g fulfill the re-quirements of Propositions 4.1, 4.2; cf. Section 5.3. In particular, we can deriveCarleman estimates for (cid:3) ¯ g , for functions defined over the domains D ǫω ′ , D ω ′ (de-pending on the setting).In the last step, we show that these Carleman estimates directly imply that¯ φ = 0 on the domains D ǫω ′ , D ω ′ ; cf. Section 5.4. This implies that φ , which solvesthe original equation L g φ = 0, vanishes on the same domain.In order to give a unified proof of Theorems 2.3-2.6, we introduce some uni-form language and notation. We will refer to the settings of Theorems 2.3, 2.4 asthe “zero mass” ( M ) settings and to Theorems 2.5, 2.6 as the “positive mass”( M + ) settings. We collectively refer to the domains D ǫω , D ω as D ω ; our metrics g below will be defined on those domains. Recall that D is covered by coordinates { u, v, y , . . . , y n − } with − u, v ∈ (0 , + ∞ ) and y , . . . y n − which cover the sphere S n − . We will be denoting ∂ A := ∂ y A . Definition 5.1.
Let ˜ u := − u + ǫ , ˜ v := v + ǫ in the zero mass settings, and ˜ u := − u ,˜ v := v in the positive mass settings. We also set r ∗ := v − u for brevity.Note that in all settings of Section 2, the functions f ǫ , f concide with:(5.1) f = 1˜ u ˜ v . These essentially require that F grows at least as fast as log f , and also that F should notbe log f itself. The latter point accounts for the correction term added to F . Moreover the metric in all theorems considered is of the general form: g := µ d u − K d u d v + ν d v + n − X A,B =1 r γ AB d y A d y B (5.2) + n − X A =1 c Au d y A d u + n − X A =1 c Av d y A d v ,where r is a smooth function that:( M ) is given by r = r ∗ = v − u in the zero mass settings,( M + ) obeys the differential inequality (2.25) in the positive mass setting: (cid:16) O (cid:0) r ∗− (cid:1)(cid:17) d v − (cid:16) O (cid:0) r ∗− (cid:1)(cid:17) d u = (cid:16) mr (cid:17) d r + n − X A =1 O (cid:0) r ∗− (cid:1) d y A Moreover, r is required to be both large and bounded above on a fixed levelset r ∗ = M ; see (2.24).The metric coefficients belong to the classes K = 1 − mr , γ AB = ˚ γ AB + O δ (cid:18) r ∗ (cid:19) ,(5.3) c Au , c Av = O δ (cid:18) r ∗ (cid:19) , µ = O δ (cid:18) r ∗ (cid:19) , ν = O δ (cid:18) r ∗ (cid:19) ,where ˚ γ AB is the round metric on S u,v with respect to the coordinates y A , and δ > m a smooth function such that( M ) 0 < δ ≪ ǫ , and m satisfies m = O δ ( r ∗ ).( M + ) m is a smooth positive function on the spacetime satisfying a uniform lowerbound m ≥ m min > D (see (2.22)), and δ is any positiveconstant. Furthermore, from (2.23), m satisfies, for some η > | ∂ u m | , | ∂ v m | ≤ C ( v − u ) − , | ∂ A m | ≤ C ( v − u ) f η . On the level of second derivatives, we always require (2.26) for some η > (cid:3) g (cid:16) mr (cid:17) = O δ ( f η ).With this notation in place, we can define the conformal inversion of the met-rics g . The inverted metric g is now defined by:(5.5) g = K − f g. Transformation of the Equations to the Inverted Picture.
Let us nextsee how solutions to a wave equation over ( D , g ) correspond to solutions of a new wave equation over ( D , ¯ g ). Consider on D the equation(5.6) L g φ := (cid:3) g φ + a α ∂ α φ + V φ = 0.Recall the conformal Laplacian, P g := (cid:3) g − n − n R g , See the statement of Theorems 2.3, 2.4.
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 33 where R g is the scalar curvature for g . This operator enjoys the following conformaltransformation law for all functions Ω > φ :(5.7) P Ω g (Ω − n − φ ) = Ω − n +32 P g φ .Therefore, letting ¯ g = Ω g , ¯ φ = Ω − n − φ , we derive:(5.8) (cid:3) g φ = Ω n +32 (cid:3) ¯ g ¯ φ + n − n (Ω n +32 R ¯ g − Ω n − R g ) ¯ φ .Thus, a direct computation shows that L g φ = Ω n +32 ¯ L ¯ g ¯ φ , where ¯ L ¯ g is the corre-sponding operator¯ L ¯ g = (cid:3) ¯ g + ¯ a α ∂ α + ¯ V , ¯ a α = Ω − a α ,(5.9) ¯ V = Ω − V + n −
14 Ω − a α ∂ α (Ω ) + n − n (Ω − R g − R ¯ g ).In particular, it holds that L g φ ≡ if and only of ¯ L ¯ g ¯ φ ≡ − R g − R ¯ g = 4 nn − − n +32 (cid:3) g (Ω n − )(5.10) = 2 n Ω − (cid:3) g Ω + n ( n − − · g ( ∇ Ω , ∇ Ω).5.3.
Verification of the Key Assumptions for the Carleman Estimates inthe Inverted Picture.
We shall now prepare the application of the Carlemanestimates of Section 4 to the wave operators (cid:3) g corresponding to g , and functionssupported in D ω ′ that vanish in a suitable weighted sense on the (semi-compactified)boundary I . We recall that this boundary is (in an intrinsic sense) a complete doublenull cone, emanating from a point that corresponds to spatial infinity. In this section, we show that with our choice (5.1) for the function f , we canfind a function w such that the requirements of Propositions 4.1, 4.2 are fulfilledfor the metrics g . In particular, we show that the level sets of f are pseudo-convexfor g ; in fact, ¯ π = − ¯ ∇ f + h ( w ) ¯ g ,restricted to the level sets of f , is a positive tensor. We demonstrate this propertyusing an explicit asymptotically orthonormal frame (with respect to ¯ g ) adapted to f . In Section 5.4, we shall then apply Propositions 4.1, 4.2 to functions that satisfythe appropriate vanishing conditions.The model spacetimes discussed in Section 3 provide a guide for the choice ofthese functions and the construction of the frames. We recall at this point thatthe term mr in the metric component − K d u d v of g S and the positivity of m yield extra positivity for the tensor ¯ π in tangential directions. The underlying cause ofthe improved pseudo-convexity is the discrepancy between the functions r ∗ := v − u and r . The same type of positivity is present for all our “positive mass space-times”.More specifically, it is captured by the components (5.27), (5.24) of the Hessian of f below. This leads us to derive certain lower bounds for the difference r ∗ − r for allspace-times under consideration, which as we shall see highlights the importance ofthe mass term when it is positive rather than zero. We note that if we had considered the Penrose compactification instead of (5.5) then thesegments I + ǫ , I − ǫ would have been converted to an incomplete double null cone; in that settingthe boundary spheres { ˜ v = + ∞ , ˜ u = 0 } , { ˜ u = ∞ , ˜ v = 0 } would have caused significant difficulties. Bounds on r ∗ − r . Recall that in the zero mass setting we have simply r ∗ = r .In the positive mass setting, we remark that (2.25) together with the boundednessbelow of r on the set { r ∗ = M } imply that for r ∗ large enough,(5.11) r ≃ r ∗ . Here we have only used the trivial bound 1 ≤ m/r ≤
2, whereas taking thelower order terms into account, the relation (2.25) along with the bounds on thederivatives of m imply: Lemma 5.2.
In the positive mass setting, there exists
C > so that for r ∗ largeenough: (5.12) r ∗ − r ≥ m min log r − C. Proof.
Recall that by assumption | r ∗ − r | ≤ C on { r ∗ = M } . The inequality (5.12)can be proven separately for v sufficiently large and − u sufficiently large; we hereshow the first case. We derive from (2.25):d r ∗ = d( r + 2 m log r ) − O (cid:18) r ∗ (cid:19) d v − ∂ v m log r d v − ∂ u m log r d u (5.13) + O (cid:18) r ∗ (cid:19) d u + n − X A =1 (cid:20) O (cid:18) r ∗ (cid:19) − ∂ A m ) log r (cid:21) d y A .Now, fixing any values y A ∗ , u ∗ , consider the curve { y A = y A ∗ , u = u ∗ } . We thenintegrate (5.13) on this curve (the integrals of the 1-forms containing d u, d y A thenvanish). In view of (5.11), the assumed boundedness (2.24) of r on { r ∗ = M } , andthe bound (2.23) on | ∂ v m | , the bound (5.12) follows. (cid:3) Note that with the same proof we also have the corresponding upper bound,(5.14) r ∗ − r ≤ C m log r , for r ∗ large enough,where C is constant that depends on m . Let us also note here for future referencethat (in all settings)(5.15) v ≤ r ∗ ≃ r , − u . r , r − . p f ,as long as r ∗ is large enough.5.3.2. Main properties of f as Carleman weight function on ( D , ¯ g ) . The main taskin this section is to show that f , defined by (5.1) as a function on the confor-mally inverted spacetime ( D , g ), where g := K − f g , fulfills the requirements ofPropositions 4.1, 4.2, with suitable choices of w , h ( w ) , Ψ, and for a suitable frame.The main proposition of this section is then: Proposition 5.3.
Consider a metric g of the form (5.2). Let the inverted metric g be defined by (5.5) and the function f by (5.1) . Let w be the function ( M ) w := − ǫr ∗ in the zero-mass setting, ( M + ) w := − m min r ∗ log r ∗ in the positive mass setting,and moreover h := w + (cid:3) g f − n − , as in (4.5) .Then, there exist orthonormal frames ( ˜ E , . . . , ˜ E n ) adapted to f (as defined inSection 4) such that the conditions (4.6) - (4.10) of Proposition 4.1 and (4.13) - (4.17) of Propositions 4.2 are fulfilled (for p = η/ and q = 2 / , respectively), with ( M ) Ψ := ǫr ∗ in the zero mass setting, NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 35 ( M + ) Ψ := m min r ∗ log r ∗ in the positive-mass setting,and with w as above. Furthermore, the frame elements ˜ E α satisfy (5.16) f − ˜ u · ∂ u , f − ˜ v · ∂ v , r − f − X = n X α =0 h O δ (cid:16) r ∗ (cid:17)i ˜ E α .for any vector field X tangent to the S u,v ’s which is unit with respect to ˚ γ .Proof. It is convenient to start our calculations in the physical metric g ; we thenobtain the bounds for the relevant quantities in the inverted metric g via standardformulae for conformal transformations. In fact, for convenience we consider first a“mild” conformal rescaling of the physical space-time metric g :(5.17) g ♯ := K − g. We begin with deriving bounds for the inverse of the metric g ♯ and its Christoffelsymbols. Convention:
For the components of g ♯ , Γ ♯ below, we use the entries u, v, A = 1 , . . . , n −
1, to refer to the coordinates u, v, y , . . . , y n − .Given the assumptions on the metric (5.2), the rescaling (5.17), and the bounds(5.21), we derive for the inverse of the metric g ♯ : g ♯uu = O δ (cid:16) r ∗ (cid:17) , g ♯uv = −
12 + O δ (cid:16) r ∗ (cid:17) , g ♯vv = O δ (cid:16) r ∗ (cid:17) ,(5.18) g ♯Au , g ♯Av = O δ (cid:16) r ∗ (cid:17) , g ♯AB = r − h ˚ γ AB + O δ (cid:16) r ∗ (cid:17)i .(5.19)We also derive the following bounds for certain Christoffel symbols of g ♯ relevantfor the conclusions below:Γ ♯uuu , Γ ♯vvv , Γ ♯uuv , Γ ♯vuv , Γ ♯vuu , Γ ♯uvv = O δ (cid:16) r ∗ (cid:17) ,(5.20a) Γ ♯uuI , Γ ♯vvI , Γ ♯vuI , Γ ♯uvI = O δ (cid:16) r ∗ (cid:17) ,(5.20b) Γ ♯uAB = 12 r ˚ γ AB + O δ (1) , Γ ♯vAB = − r ˚ γ AB + O δ (1) . (5.20c)In deriving the above formulas we used that ∂r∂u = − mr + O δ (cid:16) r ∗ (cid:17) , ∂r∂v = 1 − mr + O δ (cid:16) r ∗ (cid:17) ,which follows from (2.25), as well as the estimates ∂ u (cid:16) mr (cid:17) , ∂ v (cid:16) mr (cid:17) = O δ (cid:18) r ∗ (cid:19) , ∂ A (cid:16) mr (cid:17) = O δ ( f η ),(5.21) g αβ · ∂ α (cid:16) mr (cid:17) ∂ β (cid:16) mr (cid:17) = O δ ( f η ),which follow from our assumptions on m and r .We now consider the frame ( T, E , . . . , E n − , N ), where E , . . . , E n − is an or-thonormal frame for r ˚ γ and T , N are defined by: T := 12 p f (˜ u∂ u + ˜ v∂ v ) = 12 r ˜ u ˜ v ∂ u + r ˜ v ˜ u ∂ v ! ,(5.22a) N := 12 p f (˜ u∂ u − ˜ v∂ v ) = 12 r ˜ u ˜ v ∂ u − r ˜ v ˜ u ∂ v ! .(5.22b) This frame is asymptotically orthonormal; in fact, from (5.15), we have g ♯ ( T, T ) = − O δ (cid:16) fr ∗ (cid:17) , g ♯ ( T, N ) = O δ (cid:16) fr ∗ (cid:17) , g ♯ ( N, N ) = 1 + O δ (cid:16) fr ∗ (cid:17) ,(5.23a) g ♯ ( E i , E j ) = δ ij + O δ (cid:16) r ∗ (cid:17) , i, j = 1 , . . . , n − g ♯ ( T, E i ) = O δ (cid:18) √ fr ∗ (cid:19) , g ♯ ( N, E i ) = O δ (cid:18) √ fr ∗ (cid:19) , i = 1 , . . . , n − T and E , . . . E n − are tangent to the level sets of f .Next, we calculate the components of the g ♯ -Hessian of f relative to the aboveframe. To keep notation simple, we write by convention ∇ ♯ij f = ( ∇ ♯ f )( E i , E j ).Note we may assume without loss of generality that E i = r ∂ y i . First,(5.24) ∇ ♯T T f = 14 h ˜ u ˜ v ∇ ♯uu f + 2 ∇ ♯uv f + ˜ v ˜ u ∇ ♯vv f i = 12 f + O δ (cid:16) f r ∗ (cid:17) Next,(5.25) ∇ ♯ij f = − f r (˜ v + ˜ u ) δ ij + O δ (cid:16) f r ∗ (cid:17) . In particular, in view of ˜ v + ˜ u = v − u + 2 ǫ ( M ) and (5.12) we have( M ) ˜ v + ˜ ur = v − u + 2 ǫr = 1 + 2 ǫr (5.26a) ( M + ) ˜ v + ˜ ur ≥ m min r log r − Cr (5.26b)and thus for all i ∈ { , . . . , n − } :( M ) − ∇ ♯ii f ≥ f + ǫr f + O δ (cid:16) f r ∗ (cid:17) (5.27a) ( M + ) − ∇ ♯ii f ≥ f + m min r f log r + O δ (cid:16) f r ∗ (cid:17) (5.27b)in the zero mass and positive mass settings respectively, for r large enough. Also,(5.28) ∇ ♯NN f = 14 (cid:16) ˜ u ˜ v ∇ uu f + ˜ v ˜ u ∇ ♯vv f − ∇ ♯vu f (cid:17) = 3 f O δ (cid:18) f r ∗ (cid:19) .Finally, we calculate the off-diagonal terms in the g ♯ -Hessian of f (recall (5.15)):(5.29) ∇ ♯T i f = 12 r "r ˜ v ˜ u ∇ ♯uI f + r ˜ u ˜ v ∇ ♯vI f = O δ (cid:18) f r ∗ (cid:19) ,(5.30) ∇ ♯T N f = 14 h ˜ u ˜ v ∇ ♯uu f − ˜ v ˜ u ∇ ♯vv f i = O δ (cid:16) f r ∗ (cid:17) .Following the same calculations as in (5.29) we derive:(5.31) ∇ Ni f = O δ (cid:16) f r ∗ (cid:17) . At any given point we may choose the coordinates y A : A = 1 , . . . , n − NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 37
We have thus derived bounds for all components of ( ∇ ♯ ) f with respect to theframe T, N, E , . . . , E n − . For future reference, we note that the above implies: (cid:3) g ♯ f = ( g ♯ ) NN ∇ ♯NN f + 2( g ♯ ) NT ∇ ♯T N f + 2( g ♯ ) Ni ∇ ♯Ni f (5.32) + ( g ♯ ) T T ∇ ♯T T f + 2( g ♯ ) T i ∇ ♯T i f + ( g ♯ ) ij ∇ ♯ij f = 2 − ( n − f − ( n −
1) ˜ v + ˜ u − rr f + O δ (cid:16) f r ∗ (cid:17) = − n − f + O δ (Ψ f ),where in the last step we have used that by (5.14), in the case ( M + ), | ˜ v + ˜ u − r | = | r ∗ − r | ≤ C log r . We next consider the inverted metric(5.33) ¯ g = f g ♯ , and seek to calculate the components of ¯ ∇ f in a frame adapted to f . Thesecomponents tranform under the conformal change (5.33) according to the rule:(5.34) ∇ µν f = ∇ ♯ µν f − ∂ µ log f )( ∂ ν f ) + ( g ♯ ) αβ ( ∂ α log f )( ∂ β f ) g ♯µν Consider the rescaled frame(5.35) ¯ E := ¯ T := f − T , ¯ E n := ¯ N := f − N , ¯ E i := f − E i .We apply the Gram-Schmidt process to ¯ E , . . . , ¯ E n − , ¯ T , ¯ N (in that order) to obtaina frame ˜ E := ˜ T , ˜ E , . . . , ˜ E n − , ˜ E n := ˜ N which is exactly g -orthonormal, (5.36) ¯ g ˜ i ˜ j := ¯ g ( ˜ E i , ˜ E j ) = m ij . Then, by construction, this frame is adapted to f , in the sense of Section 4. Inparticular, ˜ E i f = 0 for all 0 ≤ i ≤ n −
1, while (5.23) implies:˜ E i − ¯ E i = X ≤ j ≤ i O δ (cid:16) r ∗ (cid:17) ¯ E j , ≤ i ≤ n − , (5.37a) ˜ T − ¯ T = O δ (cid:18) fr ∗ (cid:19) ¯ T + X ≤ j ≤ n − O δ (cid:18) √ fr ∗ (cid:19) ¯ E j , (5.37b) ˜ N − ¯ N = O δ (cid:16) fr ∗ (cid:17) ¯ N + O δ (cid:16) fr ∗ (cid:17) ¯ T + X ≤ j ≤ n − O δ (cid:18) √ fr ∗ (cid:19) ¯ E j . (5.37c)First, we compute and estimate ¯ ∇ f , where ¯ ∇ refers to the Levi-Civita connectionof g . Using (5.22), (5.35), and (5.37), we obtain¯ ∇ ˜ N f = (cid:16) O δ (cid:16) fr ∗ (cid:17)(cid:17) ¯ ∇ ¯ N f = f (cid:16) O δ (cid:16) fr ∗ (cid:17)(cid:17) .In particular, this implies part of the estimates in (4.8) and (4.15):(5.38) | f − ¯ ∇ ˜ N f − | = O δ (cid:16) fr ∗ (cid:17) = O δ (Ψ).We also note here that(5.39) ( g ♯ ) αβ ( ∂ α f )( ∂ β f ) = ( g ♯ ) NN ( ∇ N f ) = f + O δ (cid:16) f r ∗ (cid:17) . We let the indices ˜ i correspond to the frame ( ˜ E i ); the barred indices i refer to the frame ( ¯ E i ). We next wish to calculate the components of ¯ ∇ f . For this, we use the transfor-mation law (5.34) for the Hessian of f under conformal rescalings (5.33). Setting(5.40) h ′ := w + 12 ,and using (5.39), we find, for 1 ≤ i, j ≤ n − −∇ ij f + h ′ g ij == − f − h ∇ ♯ij f + ( g ♯ ) NN ∇ N (log f )( ∇ N f ) g ♯ij − h ′ f (cid:16) δ ij + O δ (cid:16) r ∗ (cid:17)(cid:17)i = − f − h ∇ ♯ij f + f δ ij − f δ ij − wf δ ij + O δ (cid:16) f r ∗ (cid:17)i ,where in applying (5.34), we have used that E i f = 0. Now, recalling the definitionof w in the statement of the proposition, we see that( M ) w = − ǫ r ∗ = − ǫ r ,( M + ) w = − m min r ∗ log r ∗ ≥ − m min r log r .where we recalled that r ∗ = r in the ( M )-case and used (5.12) in the ( M + )-case.Now, combining the above and applying (5.27), we obtain −∇ ij f + h ′ g ij ( M ) ≥ ǫr δ ij + O δ (cid:16) r ∗ (cid:17) ,(5.41) ( M + ) ≥ m min r log r δ ij + O (cid:16) r ∗ (cid:17) ,Similarly, using (5.25) and (5.14),(5.42) − ∇ ij f + h ′ g ij = h ˜ v + ˜ u − r r + w i δ ij + O δ (cid:16) r ∗ (cid:17) = O δ (Ψ).Therefore, using (5.37), we obtain, for r ∗ sufficiently large,(5.43) − ∇ ˜ i ˜ j f + h ′ g ˜ i ˜ j = −∇ ij f + h ′ g ij + O δ (cid:16) Ψ r ∗ (cid:17) ≃ Ψ δ ij + O δ (cid:16) Ψ r ∗ (cid:17) .Next, using (5.29), (5.34), and (5.37), we find that for 1 ≤ i ≤ n − −∇ ¯ T ¯ i f + h ′ ¯ g ¯ T ¯ i = − f − [ ∇ ♯T i f + ( g ♯ ) NN ∇ N (log f )( ∇ N f ) g ♯T i ] + h ′ g ¯ T ¯ i (5.44) = − f − ∇ ♯T i f + O δ (cid:16) r ∗ (cid:17) = O δ (cid:16) r ∗ (cid:17) , −∇ ˜ T ˜ i f + h ′ ¯ g ˜ T ˜ i = h O δ (cid:16) r ∗ (cid:17)i X ≤ k ≤ n − [ −∇ ¯ T ¯ k f + h ′ ¯ g ¯ T ¯ k ]+ X ≤ k,l ≤ n − O δ (cid:18) √ fr ∗ (cid:19) ∇ ¯ k ¯ l f = O δ (Ψ).Analogously, from (5.31), (5.34), and (5.37), we also see that(5.45) ∇ ¯ N ¯ i f = f − ∇ ♯Ni f + O δ (cid:16) r ∗ (cid:17) = O δ (cid:16) r ∗ (cid:17) , ∇ ˜ N ˜ i f = O δ (Ψ).Applying (5.24), we can compute −∇ T T f + h ′ ¯ g T T
NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 39 = − f − h ∇ ♯T T f + ( g ♯ ) NN ∇ N (log f )( ∇ N f ) g ♯T T − h ′ f g ♯T T i = − f − h f − f + 12 f + wf + O δ (cid:16) f r (cid:17)i = − w + O δ (cid:16) r ∗ (cid:17) ,and henceforth(5.46) − ∇ ˜ T ˜ T f + h ′ g ˜ T ˜ T = −∇ T T f + h ′ g T T + O δ (cid:18) √ f wr ∗ (cid:19) ≃ Ψ | g ˜ T ˜ T | . Furthermore, from (5.30), we have ∇ ¯ T ¯ N f = f − [ ∇ ♯T N f + ( g ♯ ) NN ∇ N (log f )( ∇ N f ) g ♯T N ] = O δ (cid:16) r ∗ (cid:17) , ∇ ˜ T ˜ N f = O δ (Ψ).(5.47)Combining (5.38), (5.45), and (5.47), we obtain the requirements (4.8) and (4.15).From (5.28) and (5.34), we find ∇ NN f = f − [ ∇ ♯NN f − f − ( ∇ N f ) + f − ( g ♯ ) NN ( ∇ N f ) g ♯NN ]= 12 + O δ (cid:16) r ∗ (cid:17) .Thus, we derive, using (5.37),(5.48) (cid:12)(cid:12)(cid:12) ∇ ˜ N ˜ N f − (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ∇ ¯ N ¯ N f − (cid:12)(cid:12)(cid:12) + O δ (cid:16) fr ∗ (cid:17) = O δ (cid:16) r ∗ (cid:17) .Now, combining (5.43), (5.46), and (5.48), we also obtain(5.49) (cid:3) g f = n + 12 + O δ (Ψ),which along with (5.48) proves (4.9) and (4.16). Moreover, setting h := w + 12 (cid:3) g f − n −
14 ,and applying (5.49), we find that −∇ f + h g satisfies, as desired, (4.7) and (4.14).It remains to check the requirements (4.10) and (4.17) on the function w . Clearly,by definition, w = O δ (Ψ) (this has already been used earlier). Moreover, using theChristoffel symbols (5.20), we calculate (cid:3) g w : (cid:3) ¯ g w = f − [ (cid:3) g ♯ w + ( n − f − ( g ♯ ) αβ ∇ ♯α f ∇ ♯β w ]( M ) = f − h O δ (cid:16) r ∗ (cid:17) + O δ (cid:16) fr ∗ (cid:17)i = O δ ( f − ) , ( M + ) = f − h O δ (cid:16) log r ∗ r ∗ (cid:17) + O δ (cid:16) f log r ∗ r ∗ (cid:17)i = O δ ( f − − η ) , η > . (5.50)This completes the proof that w , Ψ, f , and the frame ˜ E , . . . , ˜ E n have therequired properties (4.6)-(4.10) of Proposition 4.1 and (4.13)-(4.17) of Proposition4.2, on ( D , g ). Finally, the estimate (5.16) follows by unwinding the definitions ofthe ˜ E α ’s—in particular, this is a consequence of (5.22), (5.35), and (5.37). (cid:3) Carleman Estimates and Uniqueness.
Recall we have shown in Propo-sition 5.3 that the inverted metrics ¯ g , along with f , w , Ψ, satisfy on D ω ′ therequirements (4.6)-(4.10) of Propositions 4.1 and (4.13)-(4.17) of Propositon 4.2, if ω ′ is sufficiently small. To complete the proofs of our main theorems, it remains toapply the Carleman estimates (4.12) and (4.19) to show that φ vanishes on D ω ′ .To be more specific, we will apply the Carleman estimates as follows:( P d ) To prove Theorems 2.3 and 2.5 (i.e., wave equations with decaying poten-tials), we will apply Proposition 4.1, with p = 2 η .( P b ) To prove Theorems 2.4 and 2.6 (i.e., wave equations with bounded poten-tials), we will apply Proposition 4.2, with q = 2 / P d ) in the main discussion. Thelatter path ( P b ) is proved entirely analogously; differences in the computations willbe addressed in remarks and footnotes.5.4.1. Bounds for ¯ L ¯ g . The first step is to obtain the asymptotic bounds for thelower-order terms ¯ a α and ¯ V of the operator ¯ L ¯ g , defined in (5.9) and correspondingto L g in the inverted setting. Recall once again that ¯ g is obtained from g by¯ g = K − f g = Ω g .Here, it will also be useful to decompose the above into two steps, g ♯ = Ω g , ¯ g = Ω g ♯ , Ω = Ω · Ω ,where Ω − = 1 − mr , Ω = f .Note that from (5.9), we immediately obtain(5.51) | ¯ a α | . f − | a α | , | Ω − V | . f − | V | . f − η .Furthermore, another term in the expansion of ¯ V can be controlled by | Ω − a α ∂ α Ω | . f − | a α ∂ α ( K − f ) | . f − | a α ∂ α f | + f − (cid:12)(cid:12)(cid:12) a α ∂ α (cid:16) mr (cid:17)(cid:12)(cid:12)(cid:12) .Recalling the assumptions ((2.8) or (2.28)) for the a α ’s, bounding ∂ α ( m/r ) using(5.21), and recalling (5.15), we see that(5.52) | Ω − a α ∂ α Ω | . f − · f r − + f − · f + η r − . f − η .The only remaining term in (5.9) is the difference of scalar curvatures. Recallingthe general identity (5.10) and the bounds (5.21) and (5.4), we see that(5.53) | Ω − R g ♯ − R g | . (cid:12)(cid:12)(cid:12) (cid:3) g (cid:16) mr (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) g αβ ∂ α (cid:16) mr (cid:17) ∂ β (cid:16) mr (cid:17)(cid:12)(cid:12)(cid:12) . f − η .Moreover, by (5.10), along with (5.32) and (5.39), we see thatΩ − R ¯ g − R g ♯ = 2 nf − (cid:3) g f + n ( n − f − · g ( ∇ f, ∇ f )(5.54) = − n ( n − f − + n ( n − f − + O ( f − Ψ) = O ( f − η ).Combining (5.53) and (5.54), we obtain the estimate(5.55) | Ω − R g − R ¯ g | . Ω − | Ω − R ¯ g − R g ♯ | + | Ω − R g ♯ − R g | . f − η .From (5.51) and our assumptions for the a α ’s (e.g., (2.8)), we have(5.56) ¯ a u = O (˜ uf − r − ), ¯ a v = O (˜ vf − r − ), ¯ a A = O ( f − r − ). NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 41
Similarly, for ¯ V , we have from (5.51), (5.52), and (5.55) that(5.57) ¯ V = O ( f − η ).The derivations of bounds for ¯ a α and ¯ V in setting of ( P b ) is analogous.5.4.2. The Vanishing Condition.
The next step is to apply the Carleman estimate(4.12). Let φ satisfy L g φ ≡
0, and let ¯ φ = Ω − n − φ , so that ¯ L ¯ g ¯ φ ≡ χ ∈ C ∞ (0 ,
1) satisfying χ ( x ) = 1 for x ≤ − κ (0 < κ < χ ( x ) = 0 near x = 1, and consider ˜ χ, ˜ φ ∈ C ∞ ( D ω ′ ), given by˜ χ := χ ◦ ( ω ′− f ), ˜ φ := ˜ χ ¯ φ ,We claim that ˜ φ fulfills the vanishing requirement (4.11) in Proposition 4.1. Since ˜ χ is bounded, the assumptions (2.9) and (2.29) imply that Z D ω ′ ( ˜ φ + | ∂ u ˜ φ | + | ∂ v ˜ φ | + | ∂ I ˜ φ | ) r N d V g < ∞ , N ∈ N ,where d V g is the volume form with respect to the physical metric g . From (5.15)and (5.16), we see that each ˜ E α , 0 ≤ α ≤ n , can be expanded as˜ E α = c u ∂ u + c v ∂ v + n − X I =1 c I ∂ I , | c u | + | c v | + | c I | . r N ,for some N ∈ N . As a result, we have the following integral vanishing condition on¯ φ , with respect to inverted metric ¯ g and its associated volume form dV ¯ g ,(5.58) Z D ω ′ ( ˜ φ + | ¯ ∇ ˜ φ | g ) r N d V ¯ g < ∞ , N ∈ N ,where | ¯ ∇ ¯ φ | ¯ g denotes the frame-based (¯ g -)tensor norm defined in (4.3), (4.4).Given λ >
0, we construct an exhaustion of D ω ′ (as defined in Section 4) by D k = { τ k ≤ f ≤ ω ′ , σ − k ≤ v + u ≤ σ + k } ,where τ k ց σ − k ց −∞ , σ + k ր + ∞ are to be chosen. Note that ∂ D k = H k ∪ H ∪ Σ + k ∪ Σ − k , H k = { f = τ k , σ − k < v + u < σ + k } , G k = { f = ω ′ , σ − k < v + u < σ + k } , Σ ± k = { τ − k < f < τ + k , v + u = σ ± k } , H k and G k are timelike, and the Σ ± k ’s are spacelike. Also, let ν k denote the boundaryof ∂ D k . Due to the support of ˜ φ , we have for any N ∈ N that(5.59) lim k ր∞ Z G k ( | ν k | ¯ g + | ¯ ∇ ν k w | )( ˜ φ + | ¯ ∇ ˜ φ | g ) f − N e Nf p d ¯ V H = 0,where d ¯ V G k is the volume form of the induced metric ¯ g | G k .For H k , we apply the co-area formula to (5.58) and derive that for all N ∈ N ,(5.60) Z ω ′ Z { f = τ } ( ˜ φ + | ¯ ∇ ˜ φ | g ) r N | ¯ ∇ f | − g d ¯ V { f = τ } d τ < ∞ .Note that because of (5.15) and (5.39), we can drop the factor | ¯ ∇ f | − g in (5.60).Moreover, by (5.15) and the observation that f − N e Nf p . f − N . r N , The proof of superexponential vanishing, (4.18), in the setting of ( P b ) is analogous butslightly easier, since the vanishing assumptions (2.14) and (2.31) are stronger. it follows that there exists τ k ց k ր∞ Z H k ( ˜ φ + | ¯ ∇ ˜ φ | g ) f − N e Nf p d ¯ V H k = 0.The unit normal ν k on H k is simply | grad f | − g · grad f , so we can also bound | ν k | by some power of r . Recalling the definition of w (see Proposition 5.3), we can alsosimilarly bound | ¯ ∇ ν k w | . Combining the above points, we hence obtain the limit(5.61) lim k ր∞ Z H k ( | ν k | ¯ g + | ¯ ∇ ν k w | )( ˜ φ + | ¯ ∇ ˜ φ | g ) f − N e Nf p d ¯ V H k = 0.The corresponding limits on Σ ± k are handled similarly. In this case, we have | ν k | ¯ g . r d , | ¯ ∇ ν k w | . r d ,for some d >
0. By another co-area formula argument and (5.15), we obtain(5.62) lim k ր∞ Z Σ + k ∪ Σ − k ( | ν k | ¯ g + | ¯ ∇ ν k w | )( ˜ φ + | ¯ ∇ ˜ φ | g ) f − N e Nf p d ¯ V Σ + k ∪ Σ − k = 0.Thus, we conclude that the vanishing condition (4.11) holds.5.4.3. The Carleman Estimate.
From now on, we assume all spacetime integrals arewith respect to d V ¯ g . Recalling Proposition 5.3 along with the above, we can nowapply the Carleman estimate (4.12) to ˜ φ . In particular, using (4.12) and boundingthe ˜ E α ˜ φ from below using (5.16), we see that Z D ω ′ f − λ +1 e λf p | (cid:3) ¯ g ˜ φ | & λ Z D ω ′ f − λ +1 e λf p f − Ψ[˜ u ( ∂ u ˜ φ ) + ˜ v ( ∂ v ˜ φ ) ](5.63) + λ Z D ω ′ f − λ +1 e λf p f − · r − Ψ˚ γ AB ∂ A ˜ φ∂ B ˜ φ + λ Z D ω ′ f − λ +1 e λf p · f − p ˜ φ . Remark.
On the other hand, in the setting of ( P b ), we apply Proposition 4.2 instead,with q = 2 /
3. The corresponding estimate is Z D ω ′ f e λf − | (cid:3) ¯ g ˜ φ | & λ Z D ω ′ f e λf − · f − Ψ[˜ u ( ∂ u ˜ φ ) + ˜ v ( ∂ v ˜ φ ) ](5.64) + λ Z D ω ′ f e λf − · f − r − Ψ˚ γ AB ∂ A ˜ φ∂ B ˜ φ + λ Z D ω ′ f e λf − · f − ˜ φ .5.4.4. Unique Continuation.
The proof of the vanishing of ˜ φ in D ω ′ now followsfrom the Carleman estimates via a standard argument. First, observe that(5.65) (cid:3) ¯ g ˜ φ = ( (cid:3) ¯ g ¯ φ ) ˜ χ + ¯ ∇ α ¯ φ ¯ ∇ α ˜ χ + ¯ φ (cid:3) ¯ g ˜ χ = − ¯ a α ∂ α ˜ φ − ¯ V ˜ φ + J .Here, J is a function depending on ˜ φ and ¯ ∇ ˜ φ and supported in { (1 − κ ) ω ′ ≤ f < ω ′ } .Moreover, from the vanishing properties of ¯ φ and ¯ ∇ ¯ φ , we can also see that Z D ω ′ J r N < ∞ , N ∈ N . In particular, ν k will be close to f − ( ∂ u + ∂ v ), which can be expanded in terms of ¯ T and ¯ N . NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 43
Thus, applying (5.63) and (5.65), we derive: λ Z D ω ′ f − λ +1 e λf p · f − Ψ · [˜ u ( ∂ u ˜ φ ) + ˜ v ( ∂ v ˜ φ ) ](5.66) + λ Z D ω ′ f − λ +1 e λf p · f − r − Ψ · ˚ γ AB ∂ A ˜ φ∂ B ˜ φ + λ Z D ω ′ f − λ +1 e λf p · f − p · ˜ φ . Z D ω ′ f − λ +1 e λf p · ( | ¯ a α ∂ α ˜ φ | + | ¯ V ˜ φ | + J ).As long as λ is sufficiently large, in view of the bounds on ¯ a u , ¯ a v , ¯ a I , and V (see(5.56) and (5.57)), we can absorb the first two terms in the right-hand side of(5.66) into the left-hand side. Moreover, recalling the support of J , and droppingthe first-order terms in the left-hand side, we see that λ Z D (1 − κ ) ω ′ f − λ +1 e λf p · f − p · ˜ φ . Z D ω ′ \D (1 − κ ) ω ′ f − λ +1 e λf p · J .(5.67)Since the weight f − λ +1 e λf p (for sufficiently small f ) is bounded from below on D (1 − κ ) ω ′ by its value at f = (1 − κ ) ω ′ and is bounded from above on D ω ′ \ D (1 − κ ) ω ′ by the same value, then we can drop the weight from (5.67): Z D (1 − κ ) ω ′ f − p · ˜ φ . λ − Z D ω ′ \D (1 − κ ) ω ′ J .(5.68)Letting λ ր ∞ , we see that ¯ φ = ˜ φ ≡ D (1 − κ ) ω ′ . Since this holds for all κ >
0, this completes the proofs of the main theorems. Appendix A. Kerr Metric in Comoving Coordinates
We show here that the Kerr space-times of mass m > a fulfill the requirements of Theorem 2.5. This necessitates a non-trivialchoice of “co-moving” coordinates. Their effect is to adapt to the underlying an-gular momentum, and yield a form of the Kerr metiric which asymptotes to theSchwarzschild solution one order faster than in the Boyer-Lindquist coordinates;these coordinates were first introduced by Carter in the context of Kerr de Sittermetrics [11]. We first perform this change of coordinates in Minkowski spacetime. Express the(3 + 1)-dimensional Minkowski space-time in the usual ( t , r , θ , φ )-coordinates:(A.1) g = − d t + d r + r (cid:0) d θ + sin θ d φ (cid:1) We now change to a “comoving” coordinate system given by the transformation:(A.2) ( t , r , θ , φ ) → ( t, r, θ, φ ) , where for some fixed a > t ( t ) = t , φ ( φ ) = φ , (A.3a) Again, the corresponding argument from (5.64) in the ( P b )-case is entirely analogous. We should note here that the Kerr metric has been written in double null cooridnates byIsrael and Pretorius in [35]. However this depends on some implicitly defined functions and itdoes not seem straightforward to check that the requirements of Theorem 2.5 are fulfilled. r ( θ, r ) = r + a sin θ , r ( θ, r ) cos θ ( θ, r ) = r cos θ . (A.3b)Let us also denote by(A.4) ρ = r + a cos θ The metric expressed in these comoving coordinates takes the form:(A.5) g = f + h , where(A.6) f = − d t + ( r + a ) sin θ d φ , h = ρ ( r + a ) d r + ρ d θ . Let us now recall the Kerr spacetime. It has two Killing vectorfields T , Φ, andthe metric also takes the form(A.7a) g = f + h where f in Boyer-Lindquist coordinates ( t, r, θ, φ ) is the following metric on thegroup orbits of T = ∂ t , Φ = ∂ φ ,(A.7b) f = sin θ ρ (cid:16) a d t − ( r + a ) d φ (cid:17) − ∆ r ρ (cid:16) d t − a sin θ d φ (cid:17) , and h is the following metric on the orthogonal surfaces:(A.7c) h = ρ h r d r + d θ i . Here, for positive fixed mass m > r = (cid:0) r + a (cid:1) − mr . Thus we observe(A.9) g (cid:12)(cid:12) m =0 = g , i.e. by setting the mass m = 0 in the Kerr solution we obtain Minkowski space incorotating coordinates. In short, we can express the Kerr metrics as:(A.10) g = g + 2 mrρ (cid:16) d t − a sin θ d φ (cid:17) + 2 mrρ ∆ r | m =0 ∆ r d r . We may view the last two terms as perturbation and express them again in the( t , r , θ , φ )-coordinates. Sinced t − a sin θ d φ = d t − a sin θ d φ (A.11) ρ d r = (cid:16) r ( r − r ) + r + a (cid:17) d r − a r sin θ cos θ d θ (A.12)we obtain in reference to the Schwarzschild metric,(A.13) g m = − (cid:16) − mr (cid:17) d t + 11 − mr d r + r (cid:0) d θ + sin θ d φ (cid:1) that for any Kerr metric g , as 1 /r → (cid:0) g − g m (cid:1) t t = 2 mrρ − mr = O (cid:16) mr a r (cid:17) (A.14) (cid:0) g − g m (cid:1) t φ = − mrρ a sin θ = O (cid:16) mr a (cid:17) (A.15) NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 45 (cid:0) g − g m (cid:1) φ φ = 2 mrρ a sin θ = O (cid:16) mr a (cid:17) (A.16) (cid:0) g − g m (cid:1) r r = 2 mrρ r (cid:0) r ( r − r ) + r + a (cid:1) ( r + a ) − mr − mr = O (cid:16) mr a r (cid:17) (A.17) (cid:0) g − g m (cid:1) r θ = − mrρ r a r sin θ cos θ (cid:16) r ( r − r ) r + a + 1 (cid:17) = O (cid:16) mr a r (cid:17) (A.18) (cid:0) g − g m (cid:1) θ θ = 2 mrρ r a r + a r sin θ cos θ = O (cid:16) mr a r (cid:17) . (A.19)Note that to show (A.17) we have used (A.3b) to infer(A.20) r − r = a sin θr + r , r ( r − r ) = O (cid:0) a (cid:1) . In summary, we have calculated that with respect to the null coordinates of theunderlying Schwarzschild metric g m , u = ( t − r ∗ ) / v = ( t + r ∗ ) /
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NIQUE CONTINUATION FROM INFINITY FOR LINEAR WAVES 47
Department of Mathematics, University of Toronto, 40 St George Street Rm 6290,Toronto ON M5S 2E4, Canada
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E-mail address : [email protected] Department of Mathematics, University of Toronto, 40 St George Street Rm 6290,Toronto ON M5S 2E4, Canada
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