A Vector Field Method for Non-Trapping, Radiating Spacetimes
aa r X i v : . [ m a t h . A P ] D ec A VECTOR FIELD METHOD FOR NON-TRAPPING, RADIATING SPACETIMES
JES ´US OLIVER
Abstract.
We study the global decay properties of solutions to the linear wave equation in 1+3 dimensionson time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and alocal energy decay estimate, we prove that sufficiently regular solutions to this equation have boundedconformal energy. As an application we also show a conformal energy estimate with vector fields applied tothe solution as well as a global L ∞ decay bound in terms of a weighted norm on initial data. For solutionsto the wave equation in these dynamical backgrounds, our results reduce the problem of establishing theclassical pointwise decay rate t − / in the interior and t − along outgoing null cones to simply proving thatlocal energy decay holds. Contents
1. Introduction 21.1. Decay assumptions on the metric 31.2. Non-trapping and local energy decay 41.3. Weighted norms 51.4. Vector fields and associated norms 61.5. Statement of the main results 61.6. Comparison with previous literature 71.7. Organization of the paper 71.8. Basic notation 82. Preliminary Setup 92.1. Energy formalism 92.2. Conformal changes for vector field multipliers 102.3. No superradiance 113. Bondi Coordinates 143.1. Algebraic formulas involving Bondi coordinates 143.2. Asymptotic estimates involving Bondi coordinates 154. Additional Notation and Preliminary Reduction 185. Conformal Energy Estimate 195.1. Some preliminary estimates 205.2. Core multiplier estimates 235.3. Proof of the main estimates 266. Commutators 276.1. Preliminary estimates 286.2. Interior L -estimates for two derivatives 306.3. Proof of the main estimates 327. Global L ∞ Decay 338. Appendix: Weighted L -Elliptic Estimates 349. Acknowledgements 36References 36 The author was supported in part by NSF grant DMS-1001675 through Jacob Sterbenz. . Introduction
Let ( M , g ) be a 4-dimensional, smooth, asymptotically flat, Lorentzian manifold. Assume M is of theform R × R and that there exist global coordinates ( t, x i ) such that the level sets of t are uniformly space-like.In this work we study the dispersive properties of solutions to the wave equation: ✷ g φ = F ( t, x ) , ( φ, ∂ t φ )(0 , x ) = ( φ , φ ) , (1)where ( φ , φ ) ∈ C ∞ c ( R ) and, ✷ g = | det g | − ∂ α g αβ | det g | ∂ β , in local coordinates. One can think of problem (1) with F = N ( φ, ∇ φ ) and N a non-linear function as a toymodel for many important systems of hyperbolic PDE including: Maxwell-Klein-Gordon, Yang-Mills, andWave Maps. A necessary first step in understanding the stability properties of these systems is to prove thatsmooth solutions launched from sufficiently small, well-localized initial data can be extended for all time.One particularly fruitful strategy for proving this type of result is to show that solutions to the linear waveequation propagating in a fixed background, evolving from C ∞ c data, have pointwise rates similar to those inflat space. Provided that the method of proof is robust enough, one can often leverage the linear bound intoa small-data global existence result for some perturbative non-linear problems. In this work we focus on onesuch method for proving L ∞ linear decay: the vector field method first introduced by S. Klainerman in [16].In the case of 1 + 3 dimensions, this L -based method has a proven record of success in dealing with manysmall-data semilinear and quasilinear problems – at least when the metric g is uniformly close to Minkowskiand the nonlinearities have special structure. The most well-known demonstration the power of this methodis the proof of the global non-linear stability of the Minkowski space by Christodoulou-Klainerman [11].Since the appearance of that work there has been a concerted effort to extend the vector field method tobackgrounds far from Minkowski space. In many cases these efforts have been successful and have yieldedsmall data global existence results for non-linear problems in spacetimes such as: Minkowski space withobstacles [26], exterior Kerr with | a | ≪ M ([20], [24]) and time-dependent, inhomogeneous media ([39], [38])In light of this connection with nonlinear stability results, the problem of pointwise decay via vectorfield method for solutions to the linear wave equation on large, asymptotically flat backgrounds has beenintensely studied. In the last few years the main focus of research activity has been on the Schwarzschildand Kerr metrics. This is due to the fact that it is widely believed that this type of strategy has thebest chance of proving the non-linear stability of the (exterior) subextremal Kerr family. For cutting-edgeresults for the linear problem in black hole spacetimes we mention, without being exhaustive, the work ofDafermos-Rodnianski-Shlapentokh Rothman [14] and Metcalfe-Tataru-Tohaneanu [28] (see also [3] for thecase of Maxwell field). In our work we will turn away from black hole spacetimes to focus on a different,but related, problem for which vector field methods have not yet been developed: pointwise linear decay forsolutions to Eq. (1) on radiating, non-trapping spacetimes. The motivation to look at this problem comesfrom the fact that these spacetimes provide a model for the far exterior portion of a dynamic perturbationof a body emitting gravitational waves. Heuristically speaking, one can think of gravitational waves aslocal disturbances to the spacetime geometry propagating along (characteristic) null hypersurfaces. Theirpresence perturbs the metric g and leads it to decay towards flat space in the null outgoing region at slowerrates than the Schwarzschild or Kerr spacetimes. This, in turn, precludes us from just repeating the proofsof the vector field methods that are available near Minkowski space and forces us to produce a new methodfor dealing with these weak asymptotics. In short, weak decay for the metric takes away the classical VFproofs of L ∞ decay for solutions to Eq. (1).In order to quantitatively define our weak decay for the metric we take the work of Klainerman-Nicolo[18] as a starting point (see also [9] and [34]). In that work, the authors use a double null foliation toderive a hierarchy of decay for all connection coefficients and curvature components near null infinity forradiating spacetimes satisfying the Einstein vacuum equations. In our work, instead of relying on a nullframe, we assume the existence a coordinate system such that metric coefficients, after subtracting theMinkowski metric, obey symbol bounds in the null outgoing region which include those of [18] as a specialcase (see assumption 1.2 below). We chose a coordinate-dependent condition mostly to simplify matterssince the extensive geometric computations associated to null frames are unnecessary for this problem.We also mention that our decay conditions in the null outgoing region are general enough to include, as xamples, the spacetimes of Lindblad-Rodniaski [21] and Bieri [6]. In the timelike region, i.e. within thedomain of dependence of a compact set, our metric is allowed to be time-dependent and may remain farfrom Minkowski space for all time as long as a local energy decay estimate and non-trapping of null geodesicshold. For spacetimes satisfying all these assumptions, we are able to produce a novel vector field methodyielding weighted L and pointwise decay bounds that are analogous to what is available near Minkowskispace via classical vector field method. Furthermore, the norms we impose on the initial data are suitablefor non-linear applications – a topic which we will explore in subsequent work. We now give a detaileddescription of the spacetimes we work with.1.1. Decay assumptions on the metric.
There are mainly two regions that need to be considered sepa-rately: a sufficiently large compact set and its exterior. In general what happens outside of a compact setonly needs a detailed description where r ∼ t . In view of this, we make the following: Definition . Let r = P i =1 p ( x i ) . Define the wave zone to be the set { ( t, x ) | t < r < t } and the interior region to be the wedge { ( t, x ) | r < t } .We let i, j = 1 , , α, β = 0 , , , Assumption 1.2 (Existence of normalized coordinates) . There exists u ( t, x i ) ∈ C ∞ ( M ) with du = 0satisfying the following conditions:i) u = t − r on the set { t < } ∪ { r < t } ∪ { r > t } .ii) (Asymptotics). There exists δ > { t > } ∩ { t < r < t } the function u ≈ t − r in the following sense:(2) | ∂ Jt,x ( ∂ t u − , ∂ i u + ω i ) | . r − δ −| J | τ −| J | , where: τ − = h u i , τ + = C + u + 2 r , τ = τ − τ − , and C is chosen large enough so that τ + > t >
0. In particular τ + ≈ h t + r i .iii) (Renormalization). There exists γ > J the inverse metric coefficients g αβ in ( u, x i ) coordinates satisfy: | ∂ ku e ∂ Jx ( g ij − δ ij ) | . h r i − k −| J |− δ τ − k τ − γk , (3a) | ∂ ku e ∂ Jx ( d g ui + ω i ) | . h r i − k −| J |− δ τ − k τ − γk , (3b) | ∂ ku e ∂ Jx ( g ui − ω i ω j g uj ) | . h r i − k −| J |− δ τ − k τ − γk , (3c) | ∂ ku e ∂ Jx g uu | . h r i − k −| J |− δ τ − k τ − γk , (3d) where d = | det( g αβ ) | is in ( u, x i ) coordinates and: e ∂ x = ∂ x | u = const , ω i = x i r , τ = τ + h r i . In the sequel we also refer to the ( u, x i ) coordinates as Bondi Coordinates . Remark . Let g αβ be in ( t, x i ) coordinates. In the interior region acomputation using the chain rule yields:(4) | ∂ kt ∂ Jx ( g αβ − m αβ ) | . h r i − k −| J |− δ τ − γk , with m − = m = diag ( − , , , { t < r < t } ∪ { t < r } we have:(5) | ∂ kt ∂ Jx ( g αβ − m αβ ) | . h t + r i − k −| J |− δ . In practice we use (3) within the wave zone, we switch to (4) for the interior, and rely on (5) outside thesetwo regions. emark . This hierarchy of decay for different components of the metric along outgoing null directionsis consistent with the radiation of gravitational waves and with the peeling estimates in [18]. These decayrates are also consistent with the metrics constructed in the stability of Minkowski space in wave coordinates[21]. Note, in particular, that the ‘shear’ terms g ui − ω i ω j g uj decay only slightly faster than a solution tothe wave equation on any fixed hypersurface u = const . Remark . (Examples). In Schwarzschild one can define r ∗ = r + 2 M log | r M − | and let u = t − r ∗ . Themetric in ( u, x i ) coordinates then satisfies assumption 1.2 in the wave zone with δ = 1. The subextremalKerr family also satisfies this assumption in the wave zone (see the upcoming work [31] for details). Remark . Estimates (2) and (3) also hold if we trade u for e u = χu + (1 − χ )( t − r ) , where χ ≡ τ c for any 0 < c < | ( τ + ∂ ) k χ | .
1. Thus the assumption that u ≡ t − r in the set { t < } ∪ { r < t } ∪ { r > t } is not the weakest condition we can impose. We chose this conditionmostly to simplify matters dealing with estimates in the interior region. However, the reader should keep inmind that one only needs to use the exact form of u in a narrow wedge | t − r | ≪ r . Remark . By estimates (3) the u = const hypersurfaces are approximately null. We also note that, byconstruction, the e ∂ x derivatives are tangential to these hypersurfaces. Remark . Let R >
0. Inside sets of the form r R , our assumptions allow for g to be a largeperturbation of the Minkowski metric for all t >
0. In particular, the metric g does not have to converge toa stationary metric as t → + ∞ in this region.1.2. Non-trapping and local energy decay.
Next we introduce our non-trapping and local energy decayassumptions. These are the two key ingredients that allow us to handle large deformation errors insideregions of the form r R . Let us start by making the non-trapping assumption precise: Assumption 1.9 (Quantitative non-trapping for null geodesics) . Let R > γ ( s ) be a forward, affinelyparametrized null geodesic satisfying: γ (0) ∈ { r R } , ˙ γs ≡ , ˙ γt (cid:12)(cid:12) s =0 = 1 . Then, for any such γ , there exists a uniform constant C = C ( R , g ) such that γ ( s ) ∈ { r > R } for all s ≥ C .For any norm B [ t , t ] we set up the notation: ℓ pr B [ t , t ] = k f k pℓ pr B [ t ,t ] = X i > k χ i ( r ) f k pB [ t ,t ] ,ℓ pt B [ t , t ] = k f k pℓ pt B [ t ,t ] = X k > k χ k ( t ) f k pB [ t ,t ] ,ℓ pu B [ t , t ] = k f k pℓ pu B [ t ,t ] = X j > k χ j ( u ) f k pB [ t ,t ] , where χ i ( r ) is a series of dyadic cutoffs on h r i ≈ i , χ k ( t ) is a series of dyadic cutoffs on t ≈ k covering[ t , t ], and χ j ( u ) is a series of dyadic cutoffs on h u i ≈ j . We also make the obvious modification for p = ∞ .Using this we can define the Local Energy Decay (LED) norms: k φ k LE [ t ,t ] = k h r i − ∇ φ k ℓ ∞ r L [ t ,t ] + k h r i − φ k ℓ ∞ r L [ t ,t ] , k F k LE ∗ [ t ,t ] = k h r i F k ℓ r L [ t ,t ] . These norms allow us to state the final decay assumption on the metric:
Assumption 1.10 (Local energy decay estimate) . For all values 0 t < t the evolution (1) satisfies:(6) k φ k LE [ t ,t ] . k ∇ φ ( t ) k L x + k ✷ g φ k LE ∗ [ t ,t ] . on-trapping of null geodesics is a necessary condition for (6) to hold in this form. If there’s trapping, thework of Ralston [32] and some additional geometric optics considerations can be used to show that the LEDestimate, if it holds at all, must lose derivatives in a neighborhood of the trapping set. In the non-trappingcase, estimates such as (6) date back to work of Morawetz [29], [30] and are known to hold in a variety ofsettings. In the case of Minkowski space, Keel-Smith-Sogge proved a limiting version of this estimate [15] (seealso [36]). For uniformly small, time-dependent perturbations of Minkowski, Alinhac [2] and Metcalfe-Tataru[27] both established this result. The work of Bony-Hafner [10] extended the validity of this estimate to thecase of large, stationary, non-trapping metrics of the form ds = − dt + h ij ( x ) dx i dx j with h Riemannian(see also [33], [35]). As the Schwarzschild and Kerr solutions have trapped null geodesics, this work will notapply to full perturbations of such spacetimes. However, a suitable modification in the upcoming work [31]will do so.Since the LED estimate is generally expected to hold for a large class of spacetimes, it is a naturalassumption to include in our problem. In particular, the LED estimate should hold for time-dependent,non-trapping, asymptotically flat spacetimes satisfying a smallness condition for ∂ t g and it should also hold(with loss of derivatives) for the domain of outer communications of a small time-dependent perturbationof the sub-extremal Kerr family. In the non-trapping, time-dependent case, some work is already underway to prove this result (upcoming work of Sterbenz-Tataru). In the trapping case, the work [28] alreadyestablished the result for fast decaying perturbations of Kerr with | a | ≪ M . In view of this, we will takethe estimate as given and focus on developing a precise understanding of the asymptotic properties of thesolution via vector fields.1.3. Weighted norms.
We define the
Weighted LED norms by: k φ k LE a,b [ t ,t ] = k τ a + τ b φ k LE [ t ,t ] , k φ k LE a [ t ,t ] = k φ k LE a, [ t ,t ] , with analogous definitions for k F k LE ∗ ,a,b . Next we set up the fixed-time and null energies that we will use inthe sequel. Define the vector ∇ φ = ( ∂ φ, ..., ∂ φ ) where ∂ , ..., ∂ denotes any basis which can be written asa bounded linear combination of ( t, x ) coordinate derivatives. For t ≥ ConformalEnergy : k φ ( t ) k CE = k τ + τ ∇ φ ( t ) k L x + k τ + ( e ∇ x φ, r − φ )( t ) k L x . For a smooth, positive weight function Ω we also have the
Conjugated Conformal Energy : k φ ( t ) k Ω CE = k τ + τ Ω − ∇ (Ω φ )( t ) k L x + k τ + Ω − e ∇ x (Ω φ )( t ) k L x . (7)In this work we only use the conformal weights I Ω = h r i , II Ω = τ − τ + . We set up the notation: k · k I CE = k · k I Ω CE , k · k II CE = k · k II Ω CE . Note that k · k CE and k · k Ω CE are fixed-time norms and do not contain null energies. To introduce thesewe define the scale of spaces: k φ k NLE [ t ,t ] = k h r i − φ k ℓ ∞ u ℓ ∞ r L [ t ,t ] , k F k NLE ∗ [ t ,t ] = k h r i F k ℓ u ℓ r L [ t ,t ] . Our (weighted) null energies are defined to be: k φ k Ω CH [ t ,t ] = k h r i Ω − e ∂ r (Ω φ ) k NLE , − [ t ,t ] , k · k I CH [ t ,t ] = k · k I Ω CH [ t ,t ] , k · k II CH [ t ,t ] = k · k II Ω CH [ t ,t ] . Using the norms above as building blocks we then define the S norm: k φ k S [ t ,t ] = k φ k I CH [ t ,t ] + k φ k II CH [ t ,t ] + k χ r< t φ k ℓ ∞ t LE [ t ,t ] . Associated to this is the source term norm N : k F k N [ t ,t ] = k ( χ r< t + χ r> t ) F k LE ∗ , , [ t ,t ] + k χ t We now define versions of the previous norms which incorporateadditional decay. These will be stated in terms of modifications of the usual Lorentz vector fields which areadapted to the null geometry of our spacetime as dictated by the function u ( t, x ). First we define the Liealgebra in ( t, x i ) coordinates (see also Eq. (37)): L = (cid:8) T = 1 u t ∂ t , S = u − ru r u t ∂ t + r∂ r , Ω ij = Ω ij,mink − Ω ij,mink uu t ∂ t (cid:9) , where Ω ij,mink = x i ∂ j − x j ∂ i . With Γ ∈ L we define the higher order norms to be: k φ k LE [ t ,t ] = X | J | k Γ J φ k LE [ t ,t ] , k φ ( t ) k CE [ t ,t ] = X | J | k Γ J φ ( t ) k CE [ t ,t ] , with analogous definitions for the weighted LE and CH norms respectively. Using these as building blockswe define the higher order S norm by: k φ k S [ t ,t ] = k φ k I CH [ t ,t ] + k φ k II CH [ t ,t ] + k χ r< t φ k ℓ ∞ t LE [ t ,t ] . Associated to this are the higher-order source term norms: k F k N [ t ,t ] = X | J | k Γ J F k N [ t ,t ] + k χ r< t F k LE − γ [ t ,t ] , k F k M [ t ,t ] = X k + | J | k ( τ + τ ∂ u ) k ( r e ∂ x ) J F k N [ t ,t ] + k χ r< t F k LE − γ [ t ,t ] , Finally, we have the following initial data spaces which will applied to ∇ φ (0): k f k H s,ak = X | I | s, | J | k k h r i a + | J | ∇ I + Jx f k L x . Statement of the main results.Theorem 1.11 (Main Theorem) . Assume that M is of the form R × R and that the metric g satisfiesthe decay assumption 1.2 with < γ < δ , the non-trapping assumption 1.9, and the local energy decayassumption 1.10. We then have the following uniform estimates for all T > :I) (Conformal Energy Estimate) sup t T k φ ( t ) k CE + k φ k S [0 ,T ] . k ∇ φ (0) k H , + k ✷ g φ k ℓ t N [0 ,T ] . (8) II) (Conformal Energy Estimate With Vector fields) sup t T k φ ( t ) k CE + k φ k S [0 ,T ] . k ∇ φ (0) k H , + k ✷ g φ k ℓ t N [0 ,T ] . (9) III) (Global Pointwise Decay) k τ + τ φ k L ∞ [0 ,T ] . k ∇ φ (0) k H , + k ✷ g φ k ℓ t M [0 ,T ] . (10)One can think of this of this as a conditional linear stability result for the wave equation in non-trappingbackgrounds satisfying our weak asymptotics. In other words: for such backgrounds, as long as the LEDestimate (6) holds, then the pointwise decay rates t − in the interior and t − along light cones for solutions tothe linear wave equation (1) will also hold (see [1], [37] for a proof of LED estimates in curved backgrounds).In principle, the decay estimates above should be suitable for non-linear applications since the norm N onthe source term ✷ g φ should be able to handle quadratic derivative non-linearities with a null condition. Wewill explore this in future work.For our results below we will assume the initial data are smooth and compactly supported. However, thiscondition can be relaxed so that the only regularity requirements for the data are that they belong to theweighted Sobolev spaces discussed in the statement of the main theorem. This relaxation can be achievedby standard approximation arguments which we omit. .6. Comparison with previous literature. To the best of the author’s knowledge, this is the firstwork dealing with outgoing metrics via vector fields since the stability of Minkowski space [11], [18] and itsextensions [6]. We also note that not only is our metric large and time-dependent, but we actually relaxthe conditions on the causal structure of our spacetime significantly. In particular, the function u ( t, x ),can deviate appreciably from being a true optical function for ( M , g ) and from the Minkowski analogue u mink = t − r . We believe this type of setup may have some useful applications since in practice constructingan exact optical function is a laborious process and using a suitable replacement might be desirable. Ourresult also differs from previous work in one crucial way: we build our estimates using conformal energyinstead of the Dafermos-Rodnianski p-weighted estimates as in [13], [38], [39] or the fundamental solutionof Minkoswki space as in [28]. Boundedness of conformal energy is crucial since it is precisely the reasonwe are able to prove that solutions in the wave zone decay pointwise at the sharp rate of t − . Another keydifference, at least with respect to the work of S. Yang above, is that we commute once with the scalingvector field S . As mentioned in that work, it is a commonly held belief that one needs to have t∂ t g = O (1)in the set r | t∂ t g | . t − γ will suffice and that therefore the classical methods involving commuting the scaling vectorfield into conformal energy still apply in this general setting. Commuting with S is desirable since it leads tothe higher interior decay rate of t − . Since this rate of decay is integrable in time, we hope that it is usefulfor some non-linear applications.For pointwise decay via vector fields on large, time-dependent, asymptotically flat spacetimes the onlyprevious results are those mentioned above: [38], [39] and [28] (see also [4] for a non-vector field proof ofradiation field asymptotics for weakly decaying metrics with a full asymptotic expansion, as well as [7],[8], [12], [23], [24], [25] for related applications in Black Hole spacetimes). The work [39] establishes an L ∞ decay rate of h t + r i − for compactly supported metrics satisfying a non-sharp version of (6) for largeperturbations. One of the main differences with our results is that ∂ t g only needs to be small in the interiorleading to more general metrics. However, the L ∞ decay proved in that work is weaker than ours in theinterior and the metric equals Minkowski in the exterior. As an application of his method the author alsoshows a small data global existence result for semilinear equations satisfying the null condition. In the morerecent work [38] the same author proves a pointwise decay rate of h r i − h t − r i − for time-dependent metricswhich are uniformly close to Minkowski and decay weakly in the null outgoing region. The main differencewith our work again is that both the interior decay and the wave zone decay achieved for the solution areweaker. Additionally, the outgoing conditions assumed for the metric are inhomogeneous and demand moredecay on the undifferentiated terms g αβ as well as Ω ij g αβ . On the other hand, we point out that once againthe assumptions on the metric in the interior are slightly more general than ours and that global existencefor quasilinear equations satisfying a null condition is again shown as an application. Lastly, in [28] theauthors prove a decay rate of h t + r i − h t − r i − (Price Law) for non-trapping spacetimes with t∂ t g = O (1).The authors assume a sharp LED estimate with norms similar to ours as well as wave zone decay rates | ∂ k ( g − m ) | . h r i − − k – which are more restrictive than ours. Despite the fact that a lot of decay is achievedfor the linear problem the norms for the source term ✷ g φ involved in getting that decay do not allow forapplications to non-linear problems. However, we mention that [28] also proves the Price Law for the blackhole case.1.7. Organization of the paper. In section 2.1 we recall the standard energy formalism for the waveequation. In section 2.2 we set up and prove a generalization of the conformal method of Lindblad-Sterbenz[22] (see also [5]). This method is a general framework for proving weighted energy estimates arising fromasymptotically conformal Killing vector fields in curved spacetimes. This framework is central to our worksince it is the foundation upon which our exterior proof of conformal energy is built. In a curved spacetimethere’s three advantages to using this method versus the classical proof of conformal energy : firstly, sincethe identities are already in divergence form we avoid having to perform several integrations by parts inorder to take advantage of special cancellations for the boundary terms. Secondly, this method is robustenough to prove other useful weighted energy estimates such as the fractional conformal energy bounds wesee in [22]. Lastly, the method is capable of handling the weak decay of our metrics in the wave zone. To the See [17] for the classical proof of the conformal energy in a curved background close to Minkowski. est of the author’s knowledge, no other method has proved capable of proving conformal energy boundswith these types of conditions.In the case of Minkowski space, which is the only case covered in [22], this method is motivated by theobservation that the Morawetz vector field K mink = ( t + r ) ∂ t + 2 tr∂ r is conformal Killing. Therefore it isdesirable to understand how the energy formalism for multipliers changes under conformal maps: g → Ω − g .The choices I Ω = r and II Ω = t − r make K mink a Killing field in these backgrounds. Since the deformationerrors vanish, it is a simple matter to then use the energy formalism corresponding to the conformal metricsto prove two conjugated Morawetz estimates which, together, combine to yield the conformal energy bound.To extend this method to curved spacetimes we once again look at the conformal wave equation and use itto develop a general formalism for multipliers. Inspired by the Minkowski case, we choose smooth positiveweights I Ω = h r i , II Ω = τ − τ + which asymptotically behave like r and t − r . We use these in combinationwith a modified Morawetz vector field K = u ∂ u + 2( u + r ) e ∂ r which is asymptotically Killing with respectto these conformal backgrounds. Given this input, the generalized Lindblad-Sterbenz machinery establishedin our work effectively reduces the bulk of the proof of (8) to a multiplier bound modulo error terms. Itis important to use both of these weights in in our method since I Ω degenerates where r ∼ II Ω iswell-behaved there and (locally) controls the bulk of the conformal energy. In the wave zone the oppositebehavior takes place and it is the weight I Ω that is responsible for the bound on conformal energy.The preceding method requires a positive-definite energy density associated to ∂ t . This is addressed insection 2.3 by proving a general result stating that non-trapping, plus smallness of ∂ t g , plus asymptoticflatness implies that the vector field ∂ t is uniformly timelike. For our types of metrics this implies that ∂ t isuniformly timelike in the asymptotic region t ≫ 1. The proof of this fact is by contradiction: if ∂ t g is smalland ∂ t is close to null then the inner product h ∂ t , ∂ t i ≈ h ∂ t , ∂ t i ≈ − t ≫ ∂ t is timelikeplus the dominant energy condition gives us a coercive bound for the weighted energies we wish to control.Thirdly, in the wave zone the K field is set-up so that the deformation tensors yield better spacetime errorscompared to the standard Minkowski Morawetz field K mink . In short, the Lindblad-Sterbenz formalismcoupled with the hierarchy of decay (3) suffices to control these error terms.In section 6 we prove the higher conformal bound (9). We do this by commuting the equation once withthe Lie algebra L – in particular we avoid the use of the Lorentz boosts Ω i = t∂ i − x i ∂ t . In the wavezone the desired estimates follow from the fact that the modified vector fields have favorable errors thatwork well with the renormalization (3). In the interior the main problem is that (4) implies | t∂ t g | . t − γ ,thus commuting with the scaling vector field is non-trivial. We fix this in stages: we start by proving somecommutator estimates. After applying Hardy estimates to the ensuing lower order terms, the main errorsarising from commuting with S consist of T-weighted L terms with two derivatives supported inside r < ct with c ≪ . We control these by proving a t-weighted LED estimate with vector fields and use it to trade t∂ t φ for Sφ plus small errors. This leaves only terms with two spatial derivatives t ∇ x φ to be bounded.Thanks to the global weighted L elliptic estimate (138), we are able to trade two space derivatives for theelliptic part of the wave operator P = | g | − ∂ i | g | g ij ∂ j . We then trade tP for ∇ Γ φ + t ✷ g φ + t ∇ ∂ t φ . Thismethod is somewhat reminiscent of the work of Klainerman-Sideris [19] and relies crucially on the globalweighted L elliptic estimate (138) which is shown in the appendix. In this procedure it is convenient to usethe L norm (instead of the dyadic LED norms) when commuting with S, Ω ij because we ultimately needto resort to an L Hardy estimate in order to deal with the lower order terms. Only then can we applythe elliptic estimate (138) to close the argument outlined above. Additionally, using the L norms for theseterms is advantageous since it also sets up the estimates so we can re-use them in the proof of the globalpointwise decay in section 6 which follows by a similar type of argument.1.8. Basic notation. The following notation will be used in the sequel: We denote A . B (resp. “ A ≪ B ”; “ A ≈ B ”) if A CB for some fixed C > A ǫB for a small ǫ ; both A . B and B . A ). • By default, any norms involving a range for the t variable have t ∈ [ t , t ] with 0 t < t . • Given norms k · k A , k · k B and a weight ω , the notation B ⊆ ωA means k ωf k A . k f k B . • The notation m = m − = diag ( − , , , 1) denotes the Minkowski metric in ( t, x ) coordinates. • The notation η − denotes the inverse Minkowski metric in ( u mink , x i ) coordinates with u = u mink = t − r : η uu = 0 , η ui = − ω i , η ij = δ ij , ω i = x i r . • We denote e ∂ x = ∂ x | u = const in Bondi coordinates. We also denote e ∂ r = ω i e ∂ i . • Ω ij := { Ω ij } i Energy formalism. Here we recall the basic energy setup for vector fields multipliers and commutatorsfor problem (1).2.1.1. Vector field multipliers. Define the Energy-Momentum Tensor associated to ( g, φ ) by: T αβ = ∂ α φ∂ β φ − g αβ ∂ µ φ∂ µ φ .T αβ is related to the wave operator ✷ g φ by the identity D α T αβ = ✷ g φ · ∂ β φ , with D denoting the covariantderivative. Given a smooth vector field X we define the 1-form ( X ) P α = T αβ X β . Taking the divergence ofthis we arrive at the well-known formula:(12) D α ( X ) P α = ✷ g φ · Xφ + 12 ( X ) π αβ T αβ , with L X g αβ = ( X ) π αβ . The symmetric 2-tensor ( X ) π is the Deformation Tensor of g with respect to X and measures the change of g under the flow generated by X . Integrating (12) over the time slab { ( t, x ) | t ≤ t ≤ t } and using Stokes’ theorem we get the Multiplier Identity :(13) Z t = t ( X ) P α N α d dx − Z t = t ( X ) P α N α d dx = Z t t Z R (cid:0) ✷ g φ · Xφ + 12 ( X ) π αβ T αβ (cid:1) dV g , where dV g = d dtdx and N is the vector defined in Eq. (11). The integrand on the left hand side is the Energy Density associated to X through the foliation by spacelike hypersurfaces t = const . We also recallthat T αβ obeys: The Dominant Energy Condition : For any two timelike, future-directed vector fields X, Y the energymomentum tensor T αβ satisfies |∇ φ | . T ( X, Y ).2.1.2. Formulae for commutators and multipliers. Given a vector field X , we define the Normalized Defor-mation Tensor of X to be: ( X ) b π = ( X ) π − g · trace ( ( X ) π ) . This tensor is present in some of the most important formulas dealing with vector fields for the wave equation. emma 2.1 (Basic formulas involving ( X ) b π ) . Let φ be a smooth function and X a vector field. The followingidentities hold: ( X ) b π αβ = − d − X ( d g αβ ) − g αβ ∂ γ X γ + g αγ ∂ γ X β + g βγ ∂ γ X α , (14a) [ ✷ g , X ] φ = D α ( X ) b π αβ D β φ + ( D γ X γ ) ✷ g φ , (14b) D α ( X ) P α = ✷ g φ · Xφ + 12 ( X ) b π αβ ∂ α φ∂ β φ , (14c) where (14a) is computed in local coordinates.Proof. We’ll prove each of these formulas separately. Part 1: ( The identity (14a)) In local coordinates: ( X ) π αβ = − X ( g αβ ) + g αγ ∂ γ X β + g βγ ∂ γ X α . (15)Subtracting the expression g αβ ( trace ( X ) π ) = g αβ d − ∂ γ ( X γ d ) from both sides gives the result. Part 2: ( The identity (14b)) In local coordinates we have the following well-known formula (see [2]):[ ✷ g , X ] φ = ( X ) π αβ D αβ φ + ( D α ( X ) π αβ ) ∂ β φ − ∂ α ( trace ( X ) π ) ∂ α φ . Using the definition of ( X ) b π : ( X ) π αβ D αβ φ = ( X ) b π αβ D αβ φ + 12 ( tr ( X ) b π ) ✷ g φ , ( D α ( X ) π αβ ) ∂ β φ = ( D α ( X ) b π αβ ) ∂ β φ + 12 ∂ α ( tr ( X ) b π ) ∂ α φ . Identity (14b) follows by combining these two statements. Part 3: ( The identity (14c)) Combining (12) with the identity: T αβ ( X ) π αβ = ∂ α φ∂ β φ ( X ) π αβ − g αβ ∂ α φ∂ β φ · trace ( ( X ) π ) = ∂ α φ∂ β φ ( X ) b π αβ , gives the result. (cid:3) Conformal changes for vector field multipliers. As mentioned in the introduction, our goal inthis section is to record how all the formulae associated with the vector field multiplier method change underconformal deformations of the metric.2.2.1. The conformal wave equation. Let g αβ be a Lorentzian metric on an 1 + 3 dimensional spacetime. Weconsider a conformally equivalent metric e g αβ where Ω e g = g for some smooth weight function Ω > 0. Let e D denote the covariant derivative of e g and ✷ e g = e D α e D α the corresponding wave operator. We then have thefollowing standard formula from geometry: Lemma 2.2 (Conformal wave equation) . Let ψ = Ω φ and ✷ g φ = F . Then, for the wave operator ✷ e g of theconformal metric e g = Ω − g we have: ✷ e g ψ + V ψ = Ω F , V = Ω ✷ g Ω − . (16) Proof. Since ✷ e g = Ω ( ✷ g − g αβ ∂ α (ln Ω) ∂ β ), rescaling ✷ g ψ = ✷ g Ω φ yields: ✷ g ψ = Ω F + 2 g αβ ∂ α (ln Ω) ∂ β ψ − W φ , (17)with: W = − ✷ g Ω + 2Ω ∂ α (ln Ω) ∂ α (ln Ω) = Ω ✷ g Ω − . (cid:3) .2.2. Conformal vector field multipliers. Let χ ( t, x ) a smooth cutoff function and ψ = Ω φ . Using equation(16) we define the Conformal Energy-Momentum Tensor for ( ψ, χ ):(18) e T χαβ = ∂ α ψ∂ β ψ − e g αβ ( e g µν ∂ µ ψ∂ ν ψ − χV ψ ) , V = Ω ✷ g Ω − . This satisfies the divergence law:(19) e D α e T χαβ = (( χ − V ψ + Ω F ) ∂ β ψ + 12 ∂ β ( χV ) ψ . Let ( X ) e P χα = e T χαβ X β . We have the following identities: Lemma 2.3 (Conformal multiplier identity) . Let X be a vector field, χ ( t, x ) be a smooth cutoff functionand ✷ g φ = F . Then, ( X ) e P χα satisfies the identity: Z t = t ( X ) e P χα N α Ω − d dx − Z t = t ( X ) e P χα N α Ω − d dx = Z t t Z R e D α ( X ) e P χα Ω − dV g , (20) where dV g = d dtdx and N given by Eq. (11) . Additionally, for the divergence on the RHS we have theidentity: (21) e D α ( X ) e P χα Ω − = F · Ω − Xψ + Ω − A αβ ∂ α ψ∂ β ψ + B χ φ + C χ φ Ω − Xψ , with: (Ω ,X ) A = 12 (cid:0) ( X ) b π + 2 X ln(Ω) g − (cid:1) , (22) (Ω ,X ) B χ = 12Ω (cid:0) X ( χV ) − trace ( A ) χV (cid:1) , (23) (Ω ,X ) C χ = 1Ω ( χ − V . (24) On the RHS of the last four lines above all contractions are computed with respect to g .Proof. Identity (20) follows immediately by integrating e D α ( X ) e P χα with respect to the volume form dV e g = p | e g | dtdx = Ω − dV g over the set t t t and applying Stokes’ theorem. It remains to compute thedivergence e D α ( X ) e P χα . Using Eq. (14c) we have:(25) e D α ( X ) e P χα = 12 ( d L X e g ) αβ ∂ α ψ∂ β ψ − 14 trace( d L X e g ) χV ψ + (cid:0) ( χ − V ψ + Ω F (cid:1) Xψ + 12 X ( χV ) ψ , where all the contractions are computed with respect to e g . To compute the first two RHS terms we use theidentities:(26) L X e g = Ω − (cid:0) L X g − X ln(Ω) g (cid:1) , d L X e g = Ω − (cid:0) d L X g + 2 X ln(Ω) g (cid:1) , Substituting the last line into RHS (25) gives us (21) and (22) – (24). (cid:3) Remark . Choosing Ω = 1 and χ ≡ g = m , χ ≡ I Ω = r, II Ω = t − r gives L X e g ≡ V ≡ Remark . The quantity e P α N α will denote e P χα N α with χ ≡ No superradiance. In order to produce the coercive bound |∇ ψ | . ( ∂ t ) e P α N α inside the set r ∂ t to be uniformly timelike. Since ( g − m )could remain large inside this set as t → + ∞ , we have no reason, a priori, to expect this condition to hold.To address this issue, we will prove below that given some mild conditions on the metric, the vector field ∂ t is uniformly timelike everywhere. We start with some preliminary lemmas: Lemma 2.6 (Coercive bound for energy on null-geodesics) . Let T be a past-directed, uniformly timelikevector field and L be a future-directed null vector field with L α = g αβ ξ β . The following uniform bound holds: h T, L i ≈ k ξ k ℓ ( R ) . (27) roof. It suffices to show h T, L i & k ξ k ℓ ( R ) . Since T is uniformly time-like we can construct a (local) systemof coordinates { ∂ , ∂ i } such that: ∂ = T , g = − , g i = 0 , g ij Y i Y j ≈ δ ij Y i Y j = k Y k ℓ ( R ) . By our hypotheses we have h T, L i > 0. Therefore, in this system of coordinates h T, L i = g β ξ β = − ξ > L is null we have: g ( ξ ) + g ij ξ i ξ j = 0 ⇒ − ξ = ( g ij ξ i ξ j ) ≈ ( X i ξ i ) . The lemma follows. (cid:3) Lemma 2.7 (Exponential bounds for null-geodesic coefficients) . Let ( M , g αβ ) be asymptotically flat andsuppose the vector field N given by Eq. (11) , is uniformly timelike and future-directed. Additionally, assumethat g αβ satisfies the quantitative non-trapping assumption 1.9. Let ˙ γ = ξ be an outgoing, future-directednull-geodesic given in ( t, x i ) coordinates by ξ with affine parameter s satisfying ˙ γt (cid:12)(cid:12) s =0 = 1 with ˙ γs ≡ , γ (0) ∈ {| x | R } . Then, there exists a constant A ( R ) > such that for all t > : k ξ (0) k ℓ ( R ) A − . k ξ ( t ) k ℓ ( R ) . k ξ (0) k ℓ ( R ) A . (28) Proof. Let λ > k ξ (0) k ℓ ( R ) = 1 and s be an affine parameter. By assumption 1.9 there exists C λ > s > C λ , γ ( s ) ∈ {| g − m | < λ } . Therefore, by choosing λ small it suffices to prove the resultfor the range s ∈ [0 , C λ ]. For this we will use ( t, x i ) coordinates. We claim that there exists a constant k ( R ) > t > k ξ (0) k ℓ ( R ) exp( − kt ) . k ξ ( t ) k ℓ ( R ) . k ξ (0) k ℓ ( R ) exp( kt ) . (29)To prove the claim we consider the Hamiltonian formulation for the geodesic flow. Let p ( x, ξ ) = g αβ ξ α ξ β bethe principal symbol for ✷ g . The Hamiltonian flow obeys the equations: dx α ds = ∂ ξ α p , dξ α ds = − ∂ x α p . Since − N is uniformly timelike, past-directed and L α = g αβ ξ β is null, line (27) implies: dtds = ∂ ξ p = 2 g β ξ β = 2 h− N, L i ≈ k ξ k ℓ ( R ) . By chain rule: dsdt = (2 h− N, L i ) − ≈ k ξ k − ℓ ( R ) . This leads to: dx dt = 1 , dx i dt = dx i ds dsdt = ∂ ξ i p g β ξ β = O (1) ,dξ α dt = − ∂ x α p g β ξ β ≈ O ( k ξ k ℓ ( R ) ) . Therefore the x α change with constant speed. For the frequency ξ α , since the last statement holds for all α ,there exists a k > − k k ξ k ℓ ( R ) ddt k ξ k ℓ ( R ) k k ξ k ℓ ( R ) . Integrating these bounds finishes the proof of the claim. Combining with our initial remarks yields thelemma. (cid:3) Lemma 2.8. Let ( M , g ) satisfy all the hypotheses of Lemma 2.7. Choose R > and denote ξ , λ , A ( R ) as above. Then, for λ sufficiently small, there exists C > such that in the exterior region {| g − m | < λ } : |h ∂ t , ξ i| > C A − > . (30) Proof. Since ξ is a null vector, choosing λ sufficiently small and using asymptotic flatness and non-trappinggives us ξ = (cid:0) ∂ t + ω i ∂ i + O ( λ ) (cid:1) k ξ k ℓ ( R ) . By lemma 2.7 this implies |h ∂ t , ξ i| > C k ξ k ℓ ( R ) > C A − forsome constant C > (cid:3) Remark . In general we have C A − ≪ A . e now prove the main result of this section: Proposition 2.10 ( ∂ t is uniformly timelike) . Let ( M , g ) satisfy all the hypotheses of Lemma 2.7 togetherwith |L ∂ t g | < ǫ . Choose R > and denote A, C > as above. Then there exists ǫ ntrap ( A ) > sufficientlysmall such that for all < ǫ < ǫ ntrap the bound h ∂ t , ∂ t i − C A − holds everywhere.Proof. Let ǫ > 0, choose λ sufficiently small (as in the previous lemma) and let C λ be the constant given inassumption 1.9. The proof is by contradiction and breaks up into two cases: Case 1: ( ∂ t becomes null ) Assume for a contradiction that there exists a p ∈ M such that h ∂ t , ∂ t i p = 0. Let γ ( s ) be the unique, affinely parametrized forward null geodesic with ˙ γ (0) = ∂ t ∈ T p M and ˙ γt (cid:12)(cid:12) s =0 = 1. Forsufficiently small τ ∈ ( − τ , τ ) we can define a smooth 1-parameter family of curves γ τ ( s ) with γ ( s ) = γ ( s ).Let dds = ξ , then along the null geodesic γ we have: ξ h ∂ t , ξ i = h∇ ξ ∂ t , ξ i + h ∂ t , ∇ ξ ξ i = −h [ ∂ t , ξ ] , ξ i + h∇ ∂ t ξ, ξ i = 12 L ∂ t g ( ξ, ξ ) . Integrating this along γ from s = 0 to s = s > h ∂ t , ξ i s = s = h ∂ t , ξ i s =0 + 12 Z s L ∂ t g ( ξ, ξ ) ds . (31)Since ξ (0) = ∂ t , the first term on the RHS (31) satisfies h ∂ t , ξ i s =0 = 0. Using lemma 2.7 together with thehypotheses yields the almost conservation law: |h ∂ t , ξ i s | A ǫ · s . (32)Choosing ǫ ntrap < C A − ( A · C λ ) − contradicts line (30) when s = 2 C λ . Case 2: ( ∂ t is close to null ) By asymptotic flatness, together with continuity of g αβ and the results in case1, we may assume that h ∂ t , ∂ t i < 0. Suppose for a contradiction that there exists a point p ∈ M such that0 > h ∂ t , ∂ t i p > − C A − . By computing in an appropriate local coordinate system near p we can find anoutgoing null vector ξ ∈ T p M such that h ∂ t , ξ i p = h ∂ t , ∂ t i p . Proceeding in the same manner as before wethen get: h ∂ t , ξ i s = s = h ∂ t , ξ i s =0 + 12 Z s L ∂ t g ( ξ, ξ ) ds . (33)Since ξ (0) = ξ , our initial remarks imply: |h ∂ t , ξ i s =0 | = |h ∂ t , ξ i p | C A − . Combining this with line (33) and using lemma 2.7 together with our hypotheses: |h ∂ t , ξ i s | C A − + 12 A ǫ · s . (34)Choosing: ǫ ntrap < C A − ( A · C λ ) − contradicts (30) when s = 2 C λ . (cid:3) This proposition leads to two important consequences: Corollary 2.11. Let ( M , g ) satisfy the conditions of the main theorem 1.11. Then:I) There exists a T ntrap ≫ such that ∂ t is uniformly timelike for all t ∈ [ T ntrap , + ∞ ) .II) In ( t, x i ) coordinates, the operator P ( t, x, ∇ x ) = ∂ i h ij ∂ j with h ij = d g ij is uniformly elliptic. Fur-thermore, inside the region { r < t } ∩ [ T ntrap , + ∞ ) , P ( t, x, ∇ x ) satisfies the uniform estimate: (35) | P φ | . | ✷ g φ | + | ∂ t φ | + h r i − δ | ∂ t ∇ x φ | + h r i − − δ (cid:0) | ∂ t φ | + τ − γ |∇ x φ | (cid:1) , with h ij = δ ij + O ( h r i − δ ) . roof. Part 1: ( Statement for ∂ t ) Choose T ntrap ≫ h T ntrap i − γ C < ǫ ntrap holdswith C = the maximum of the implicit constants in estimates (3). In the interior this yields |L ∂ t g | < ǫ with ǫ = ǫ ntrap . In the exterior this bound follows by (3) and the fact that γ < δ . An application ofproposition 2.10 then yields the result. Part 2: ( Ellipticity ) By Cramer’s rule we have det( M ) = h ∂ t , ∂ t i det( g − ) with M = ( g ij ) the 3 × g − . By part I we have g < C < g − ) < C < M ) is always positive and has uniformlower bounds. Since M is a 3 × g has Lorentzian signature it follows that M has threepositive eigenvectors. This proves the first claim. For (35) we note that:(36) ✷ g = g ∂ t + d − (cid:0) ∂ i g i d ∂ t + ∂ t g i d ∂ i + P (cid:1) . This, together with the asymptotic form (4) of the metric inside r < t , combine to give the result. (cid:3) Bondi Coordinates Algebraic formulas involving Bondi coordinates. The vector fields X ∈ L written in ( u, x i )coordinates have a very simple form: Lemma 3.1 (Lie algebra property) . In ( u, x i ) coordinates the vector fields defined in Eq. (1.4) are given by: (37) L = (cid:8) T = ∂ u , S = u∂ u + r e ∂ r , Ω ij = x i e ∂ j − x j e ∂ i } , and L forms a Lie algebra on M .Proof. By chain rule: ∂ u = ( u t ) − ∂ t , e ∂ i = ∂ i − ( u t ) − u i ∂ t , e ∂ r = ∂ r − ( u t ) − u r ∂ t . (38)Line (2) implies u t ≈ ∂ u , S ] = ∂ u , [ ∂ u , Ω ij ] = 0 , [ S, Ω ij ] = 0 , [Ω ij , Ω kl ] = − δ ( ik Ω jl ) . (39)The Lie algebra property follows. (cid:3) Remark . Since u t ≈ ∂ t ≈ ∂ u . This will be used often in thesequel.Next we compute the key quantities from lemmas 2.1 and 2.3 of the previous section. Lemma 3.3 (Formulas for deformation tensors) . Let Ω be a smooth function and X be a vector field inBondi coordinates. We have the following formula for the contravariant tensor A in line (22) : ( X ) b π + 2 X ln(Ω) g − = − d − (cid:0) L X η − + ( ∂ u X u + e ∂ r X r + e ∂ i X i + 2( X r r − X ln Ω)) η − (cid:1) + R X , (40) where X i = X i − ω i ω j X j denotes the angular portion of X and X r = ω i X i the radial portion. The remaindertensor R is given by the covariant formula: (41) R X = − d − (cid:0) L X ( d g − − η − ) + ( ∂ u X u + e ∂ r X r + e ∂ i X i + 2( r − X r − X ln Ω))( d g − − η − ) (cid:1) . Proof. We start with the identities: ( X ) b π αβ = − d − L X ( d g − ) − g αβ ∂ γ X γ , (42) e ∂ γ X γ = 2 r − X r + ∂ u X u + e ∂ r X r + e ∂ i X i . (43)Applying this to formula (14a) in Bondi coordinates:(44) ( X ) b π + 2 X ln(Ω) g − = − d − (cid:0) L X ( d g − ) + ( ∂ u X u + e ∂ r X r + e ∂ i X i + 2( X r r − X ln Ω)) d g − (cid:1) . Adding and subtracting d − ( L X η − + η − ) on the last line gives Eq. (40) and Eq. (41). (cid:3) emma 3.4 (Formulas for commutators) . Let d = | det( g αβ ) | and X ∈ { ∂ u , Ω ij } be in Bondi coordinates.The following identities hold: (45) [ ✷ g , X ] = D α R αβX D β + 12 X ln( d ) ✷ g , where: (46) R X = − d − L X ( d g − − η − ) . For S in Bondi coordinates we have: (47) [ ✷ g , S ] = D α R αβS D β + 12 (4 + S ln( d )) ✷ g , where R S is given by formula: (48) R S = − d − (cid:0) L X ( d g − − η − ) + ( ∂ u X u + e ∂ r X r + e ∂ i X i )( d g − − η − ) (cid:1) . Proof. For X ∈ { ∂ u , Ω ij } we use formula (40) with Ω = 1:(49) ( X ) b π = − d − (cid:0) L X η − + ( e ∂ α X α ) η − + L X ( d g − − η − ) + ( e ∂ α X α )( d g − − η − ) (cid:1) . For these two vector fields we have L X η − = 0 and ∂ u X u = e ∂ i X i = 0. Applying Eq. (14b) gives us (45)and (46). For X = S one can compute: L S η − + ( ∂ u S u + e ∂ r S r + e ∂ i S i ) η − = 0 , r − S r = 1 , ∂ u S u + e ∂ i S i = 4 . Substituting this together with (43) into (49) then applying (14b) finishes the proof of (45) – (47). (cid:3) Asymptotic estimates involving Bondi coordinates. Our first task here is to compute the decayrates for the Lie derivatives L X g with X ∈ L . Lemma 3.5 (Basic Lie derivative estimates) . Let X = X α ∂ α be in Bondi coordinates.I) Suppose that X satisfies the symbol-type bounds: (50) (cid:12)(cid:12) ( τ − ∂ u ) k ( h r i e ∂ x ) J X u (cid:12)(cid:12) . τ − , (cid:12)(cid:12) ( τ − ∂ u ) k ( h r i e ∂ x ) J X i (cid:12)(cid:12) . h r i , and obeys the conditions: (51) e ∂ i X u = ∂ u X i = e ∂ r r − ( X i − ω i ω j X j ) ≡ . Let R αβ be a contravariant two tensor satisfying the bounds: | ∂ ku e ∂ Jx R ij | . h r i − k −| J |− δ τ − k τ − γk , (52a) | ∂ ku e ∂ Jx R ui | . h r i − k −| J |− δ τ − k τ − γk , (52b) | ∂ ku ∂ Jx ( R ui − ω i ω j R uj ) | . h r i − k −| J |− δ τ − k τ − γk , (52c) | ∂ ku e ∂ Jx R uu | . h r i − k −| J |− δ τ − k τ − γk , (52d) with similar estimates for R iu . Then, the Lie derivative L X R satisfies the bounds (52) with theexponent − γk above replaced by − γ (1 + k ) .II) Alternatively, if we substitute the condition ∂ u X r = 0 with ∂ u ( X i − ω i X r ) = 0 and keep the rest of (50) and (51) the same, then the result of part I holds with the bound on line (52a) replaced by: | ∂ ku e ∂ Jx ( ω i ω j R ij ) | . h r i − k −| J |− δ τ − − k τ − γ (1+ k ) , (53) | ∂ ku e ∂ Jx ( R ij − ω i ω j ω k ω l R kl ) | . h r i − k −| J |− δ τ − k τ − γ (1+ k ) . (54) III) Let X = ∂ u and suppose R satisfies (52) . Then h r i τ L X R satisfies (52) as well. roof of Lemma 3.5. Part 1: ( The R bounds involving condition (50)) We begin with the proof of estimates(52) for L X R assuming conditions (50) and (51) or the alternative listed in item II above. The formula forthe Lie derivative is: L X R αβ = X ( R αβ ) − ∂ γ ( X α ) R γβ − ∂ γ ( X β ) R αγ . We check each component: Case 1: ( The uu component ) Here we have: L X R uu = X ( R uu ) − ∂ u ( X u ) R uu − e ∂ i ( X u )( R ui ) − e ∂ i ( X u )( R iu ) . Since e ∂ i X u = 0, the estimate on line (52d) for L X R uu is immediate from estimates (50)–(52). Case 2: ( The ui and iu components ) By symmetry of the estimates on lines (52) it suffices to treat the ui case. We have: L X R ui = X ( R ui ) − ∂ u ( X u ) R ui − ∂ u ( X i ) R uu − e ∂ j ( X u ) R ji − e ∂ j ( X i ) R uj . Using estimates (50)–(52) we get a symbol bound on the order of h r i − δ τ τ − γ for this term. In additionone sees that for all parts of the above formula save the expression X ( ω i R ur ) − e ∂ r ( X i ) R ur the bound is onthe order of h r i − δ τ τ − γ . To see the improvement for L X R ui − ω i ω j L X R uj we note that the worst term isabsent once one subtracts off the radial part since:( X ( ω i ) − ω i ω j X ( ω j )) R ur − e ∂ r ( X i − ω i ω j X j ) R ur = − r e ∂ r r − ( X i − ω i ω j X j ) R ur = 0 , where we used (51) for the last identity above. Case 3a: ( The ij components assuming ∂ u X i = 0) Here we have: L X R ij = X ( R ij ) − e ∂ k ( X i ) R kj − e ∂ k ( X j ) R ik . The bound on line (52a) for L X R ij follows by multiplying together (50) and (52a). Case 3b: ( The ij components assuming ∂ u ( X r ) = 0) In this case we are still assuming ∂ u ( X i − ω i ω j X j ) = 0.Therefore: L X R ij = X ( R ij ) − ω i ∂ u ( X r ) R uj − ω j ∂ u ( X r ) R iu − e ∂ k ( X i ) R kj − e ∂ k ( X j ) R ik . By (50)–(52) this has a symbol bound of order h r i − δ τ − τ − γ . On the other hand all but the second andthird terms above yield a bound of order h r i − δ τ − γ . Subtracting the radial part yields: L X R ij − ω i ω j L X R rr = − ω i ∂ u ( X r )( R uj − ω j R ur ) − ω j ∂ u ( X r )( R iu − ω i R ru ) + O ( h r i − δ τ − γ ) . By line (52c) we have O ( h r i − δ τ − γ ) symbol bounds for the first two terms on the RHS above as well. Part 2: ( Estimates involving ∂ u ) Fix a dyadic region rτ ≈ k . Then if X = ∂ u the vector field 2 k X satisfiesconditions (50) and all of (51). From the above calculations one immediately has all of (52) for rτ L X R . (cid:3) Along a similar vein we can derive the following set of bounds which will be needed when employing theconformal multiplier method. Lemma 3.6. Let g αβ be in Bondi coordinates.(1) (Estimates for the determinant) Let d = | det( g αβ ) | be computed in Bondi coordinates ( u, x i ) . Wehave the symbol bounds: | ( h r i τ ∂ u ) k ( h r i e ∂ x ) J ( d − | . h r i − δ . (55) (2) (Estimates for the conformal potential) Let I Ω = h r i , II Ω = τ − τ + and Ω V = Ω ✷ g Ω − . The poten-tials I V , II V satisfy the following symbol bounds: (cid:12)(cid:12) ( h r i τ ∂ u ) k ( h r i e ∂ x ) J ( I V ) (cid:12)(cid:12) . h r i − δ τ − , (56a) (cid:12)(cid:12) ( h r i τ ∂ u ) k ( h r i e ∂ x ) J ( II V ) (cid:12)(cid:12) . ( τ − τ + ) h r i − − δ . (56b) roof. Part 1: ( Determinant bounds ) Follows from estimates (3) since the determinant is a continuous func-tion of the metric components g αβ . Part 2: ( Potential bounds ) Let R = d g − − η − and write the wave operator in Bondi coordinates as: ✷ g = d − ( ✷ η + e ∂ α R αβ e ∂ β ), where ✷ η is the Minkowski wave equation in Bondi coordinates. Expanding ✷ η yields:(57) ✷ η = − ∂ u e ∂ r + e ∂ r − r − ∂ u + 2 r − e ∂ r + r − X i Choose e A to be an orthonormal basis for the spacelike two plane spanned by Y A and let c BA be thechange of basis. Then | c AB − δ AB | ≪ 1. Let L, L be, respectively, the outgoing and incoming null generatorsover span e A with h L, L i = − 1. We have: X = θL + c AX e A + γL for some set of coefficients θ, c AX , γ .From h X, Y A i = O ( ω ) we have h X, e A i = O ( ω ) and so c AX = O ( ω ). Thus h X, X i = − θγ + O ( ω ) and so θγ = O ( ω ) follows from h X, X i = O ( ω ). (cid:3) Proof of (59) . By choosing C sufficiently large it suffices to prove the result in the wave zone t ≈ r with r ≫ 1. Consider the local basis { e ∂ r , Y A } where Y A is a (local) euclidean ONB on the spheres r = const, u = const . We now check the hypotheses of the preceding lemma. Since the metric g is asymptotically flat |h Y A , Y B i − δ AB | ≪ 1. On the other hand by the asymptotic formulas (3) and Cramer’s rule we have h e ∂ r , e ∂ r i = O ( h r i − δ τ ) and h e ∂ r , Y A i = O ( h r i − δ τ ). Additionally, inside this region, assumption 1.2 together ith Prop. 2.10 imply |h e ∂ r , L i| = | θ | ≈ 1. An application of the previous lemma then gives us, with ω = h r i − δ τ :(61) e ∂ r = L + O ( h r i − δ τ ) f / ∇ x + O ( h r i − δ τ ) ∇ , where L is (outgoing) null and f / ∇ x denotes derivatives tangent to u = const, r = const which also lie inthe null plane generated by L . Let e T , ψ be as in Eq. (18) with χ ≡ 0. Since N is uniformly timelike andfuture-directed: e T ( L, N ) ≈ | Lψ | + | f / ∇ x ψ | , | e T ( f / ∇ x , N ) | . | f / ∇ x ψ | · |∇ ψ | + | g αβ ∂ α ψ∂ β ψ | , | e T ( ∇ , N ) | . |∇ ψ | . Using the bounds (3) and Young’s inequality with c > h r i − δ τ | g αβ ∂ α ψ∂ β ψ | . c | e ∇ x ψ | + c − h r i − δ τ |∇ ψ | . Choosing c ≪ 1, applying Eq. (61) together with the last three inequalities and absorbing the small | e ∇ x ψ | term gives: e T ( e ∂ r , N ) & | e ∇ x ψ | + O ( h r i − δ τ ) |∇ ψ | . Adding the undifferentiated terms and using the definition of e T χ finishes the proof. (cid:3) Proof of (60) . The vector field N is uniformly timelike and future-directed by our initial assumptions. Thevector field ∂ t is uniformly timelike and future-directed by proposition 2.10. The proof of inequality (60)then follows from ∂ u ≈ ∂ t and the dominant energy condition. (cid:3) Additional Notation and Preliminary Reduction In this section we reduce to the asymptotic region where t ≫ 1. We first set up some notation which willbe used below to absorb small errors inside this region. Definition µ , ǫ , T ∗ and I ∗ ) . We make the following definitions:a) Let 0 < µ < be sufficiently small so that µ · sup j C j ≪ where C j are the implicit constants in all theestimates in Lemma 6.7 and lines (135), (136).b) Choose 0 < ǫ ≪ min { γ, µ } satisfying the following property: for any estimate in the sequel, of the form A C ( B + ǫA ) with absolute constant C > 0, the number ǫ is small enough that we can absorb C ǫA on the LHS to yield the bound A C B .c) Let T ∗ ( ǫ ) > T ntrap ≫ < ( ǫT ∗ ) − γ < ǫ . (63)d) Let I ∗ = [ T ∗ , ∞ ) with T ∗ as above.The constant ǫ depends only on {k g k C , γ, δ, µ } and on the implicit constants in the assumptions of themain theorem. In principle, we can choose an explicit ǫ satisfying the property above. However, as we onlyuse this constant to close a finite number of estimates below, it is neither necessary nor particularly usefulto keep track of its size. We also note that ǫ will usually arise from a small gain in t power in our estimates,with the only exception being the small interior wedge in the proof of estimate (80). On the other hand, thepurpose of T ∗ is to give us an explicit lower bound for t which help us produce ǫ via inequality (63) when t ≫ t ∈ I ∗ . In thesequel it suffices to show that estimates (8), (9), and (10) hold for all t ∈ I ∗ with constants that do notdepend on t . This is a straightforward consequence of local energy estimates. Lemma 4.2 (Reduction to the asymptotic region t ∈ I ∗ ) . Estimates (8) , (9) , and (10) hold for all t ∈ [0 , T ∗ ] Proof. By local energy estimates both (8) and (9) follow in the range t ∈ [0 , T ∗ ] with constants that dependon T ∗ . Estimate (10) follows similarly after an application of the L ∞ − L Sobolev embedding. (cid:3) . Conformal Energy Estimate In this section we prove the conformal energy estimate (8). This bound will form the basis for the higherregularity estimate (9) as well as the global L ∞ decay (10). In the asymptotic region t ∈ I ∗ the conformalenergy estimate will follow from: Theorem 5.1. Assume the hypotheses of the main theorem hold. Then, for all [ t , t ] ⊂ I ∗ the followinginequalities hold:I) (T-weighted LED estimate in timelike regions) k χ r< t φ k ℓ ∞ t LE . k χ r< t ✷ g φ k ℓ ∞ t N + sup t t t k φ ( t ) k CE . (64) II) (Uniform boundedness) (65) sup t t t k ∇ φ ( t ) k L x + k e ∇ x φ k NLE , − + k φ k LE . k ∇ φ ( t ) k L x + k ✷ g φ k LE ∗ . III) (Conformal energy estimate with interior error) For any < c < : (66) sup t t t k φ ( t ) k CE + k φ k I CH + k φ k II CH . k φ ( t ) k CE + c − k ✷ g φ k ℓ t N + c k χ r< t φ k ℓ ∞ t LE . Let’s show how the conformal energy estimate follows from this: Proof of estimate (8) . By definition 4.1 we can choose a c small enough satisfying ǫ ≪ c , yet smaller thanthe implicit constants in estimates (64) – (66). Taking an appropriate linear combination of (64) and (66):(67) sup t t t k φ ( t ) k CE + k φ k I CH + k φ k II CH + c k χ r< t φ k ℓ ∞ t LE . k φ ( t ) k CE + c − k ✷ g φ k ℓ t N + c sup t t t k φ ( t ) k CE . Since c ≪ , T ∗ ] theresult follows from Lemma 4.2. (cid:3) The proof of the t-weighted LED estimate is modular and does not depend on anything other thanassumption 1.10. Let’s prove this estimate right now. Proof of estimate (64) . We apply the LED bound (6) to 2 k χ k ( t ) χ r< t φ where χ k are a series of dyadiccutoffs supported where t ≈ k and χ k ( t ) = 0. Commuting with ✷ g gives us:(68) k τ + χ k χ r< t φ k LE . k e χ k e χ r< t ( ∇ φ, h r i − φ ) k LE ∗ + k τ + χ k χ r< t ✷ g φ k LE ∗ , where e χ denote cutoffs with slightly larger support. For the first RHS term and for fixed N, C > M = M ( C, N ) > k e χ k e χ r< t ( ∇ φ, h r i − φ ) k LE ∗ . M k e χ k h r i + N τ − N + e χ r< t ( ∇ φ, h r i − φ ) k L + X j The rest of this section is devoted to the proof of the estimates (65) and (66). This will be done over thecourse of the next three subsections. .1. Some preliminary estimates. Here we establish a number of technical estimates needed in the proofof Theorem 5.1. Each argument below is self-contained. Lemma 5.2 (Hardy estimates) . For test functions φ we have the following fixed-time bound: k r a − φ ( t ) k L x a + 1 k r a ∂ r φ ( t ) k L x , − < a < ∞ . (69) Additionally let χ h r i < , χ h u i < be smooth cutoff functions supported on the sets h r i < , h u i < respectively.For all t ∈ I ∗ one has the fixed-time bounds: k χ h r i < τ + φ ( t ) r k L x . k φ ( t ) k II CE + k ∇ φ ( t ) k L x + k χ h r i∼ τ + φ ( t ) r k L x , (70) k χ h u i < φ ( t ) k L x . k φ ( t ) k I CE + k ∇ φ ( t ) k L x + k χ h u i∼ φ ( t ) k L x . (71) Proof of estimate (69) . For a fixed value of the angular variable we have the integral identity:(72) (2 a + 1) Z ∞ r a φ dr = − Z ∞ r a +1 φ∂ r φdr . As long as 2 a + 1 > (cid:3) Proof of estimate (70) . We apply (69) with a = 0 to k χ h r i < r − τ + φ k L x . Since t ≈ τ + τ inside this set weget: k χ h r i < τ + φ ( t ) r k L x . k χ h r i < ( τ + τ ∂ r φ )( t ) k L x + k χ h r i∼ τ + φ ( t ) r k L x . (73)Within this region we have (cid:12)(cid:12) ( II Ω) − ∂ r ( II Ω) (cid:12)(cid:12) τ + τ ) − . Combining this with Young’s inequality: | τ + τ ∂ r φ | (cid:0) | τ + τ ( II Ω) − ∂ r ( II Ω φ ) | + φ (cid:1) . (74)Applying this inequality to the first term on the RHS(73) yields: k χ h r i < ( τ + τ ∂ r φ )( t ) k L x . k χ h r i < τ + τ ∂ r ( II Ω φ )( t ) II Ω k L x + k χ h r i < φ ( t ) k L x . (75)For the last term on RHS(75) we use the support property followed by (69) with a = 0 to get: k χ h r i < φ k L x . k ∇ φ k L x + k χ h r i∼ r − φ k L x . This finishes the proof of estimate (70). (cid:3) Proof of estimate (71) . For a fixed value of the angular variable we have the integral identity:(76) − Z ∞ u r φ χ h u i < r dr = Z ∞ (cid:0) u∂ r φ · φχ h u i < + 2 ur − φ χ h u i < + φ χ h u i∼ (cid:1) r dr = E + E + E . Integrating Eq. (76) in the angular variable and using inequality (2) on the LHS above gives us:(77) (1+ O ( r − δ )) Z S Z ∞ φ χ h u i < r drdω = − Z S Z ∞ u r φ χ h u i < r drdω = Z S |E | dω + Z S |E | dω + Z S |E | dω Since r ≫ O ( r − δ ) as a small bootstrap error. Next we takeabsolute values on RHS(77) and bound each term separately.For R |E | dω we go back to Bondi derivatives via ∂ r = e ∂ r + u r ∂ u , then apply (cid:12)(cid:12) ( I Ω) − ∂ r ( I Ω) (cid:12)(cid:12) . h r i − toconjugate by I Ω. Using Young’s inequality then gives us, with 0 < c ≪ Z S |E | dω . c − (cid:0) k χ h u i < τ + τ I Ω − ( ∂ u ( I Ω φ ) , e ∂ r ( I Ω φ )) k L x + k χ h u i < r − τ + τ φ k L x (cid:1) + c k χ h u i < φ k L x . he last term above can be bootstrapped to LHS(77) by choosing c sufficiently small. For the next-to-lastterm on the RHS above we observe that inside the set where h u i < 2, the condition τ + τ = O (1) holds. Afteran application of estimate (69) with a = 0 we get:(78) k χ h u i < r − τ + τ φ k L x . k ∇ φ k L x + k χ h u i∼ φ k L x . For R |E | dω we apply Young’s inequality and get: Z S |E | dω . c − k χ h u i < r − τ − φ k L x + c k χ h u i < φ k L x . Choosing c sufficiently small allows us to bootstrap the small error term to LHS(77), while the other termis addressed directly via (78).Since the term R |E | dω is acceptable as part of the RHS, we combine the last few lines and take squareroots to finish the proof of estimate (71). (cid:3) Lemma 5.3 (Estimates for undifferentiated boundary terms) . Let χ r< t ( r/t ) be a smooth cutoff supportedon the wedge r < t . Let I V , II V denote the potentials in Lemma 3.6.2. For test functions φ and for allvalues t ∈ I ∗ one has the fixed-time bounds: k τ − χ r< t ( I Ω) − · I V φ ( t ) k L x . k φ ( t ) k II CE + ǫ · k φ ( t ) k CE , (79) k τ − χ r< t ( II Ω) − · II V φ ( t ) k L x . ǫ · k φ ( t ) k CE . (80) Proof. Part 1: ( Proof of estimate (79)) Let χ r 2. Inside r < ǫt we use the bound (56b) and r/t < ǫ to get: k τ − χ r<ǫt ( II Ω) − · II V φ ( t ) k L x . k ( r/t ) χ r<ǫt r − − δ tφ k L x . ǫ k φ k CE . For the term supported where ǫt/ < r < t/ r − δ . ( ǫt ) − δ . ǫ . (cid:3) Lemma 5.4 (Conjugation removal) . For all t ∈ I ∗ we have the fixed-time bounds: k φ ( t ) k CE ≈ k φ ( t ) k I CE + k φ ( t ) k II CE + k ∇ φ ( t ) k L x . (82) roof. It suffices to show k φ ( t ) k CE . k φ ( t ) k I CE + k φ ( t ) k II CE + k ∇ φ ( t ) k L x . This reduces to proving: k τ + ( e ∂ r φ, τ ∂ u φ )( t ) k L x . k τ + (cid:0) e ∂ r ( I Ω φ ) I Ω , τ ∂ u ( I Ω φ ) I Ω , φr (cid:1) ( t ) k L x , (83) k τ + φ ( t ) r k L x . X J = I,II k τ + ( e ∂ r ( J Ω φ ) J Ω , τ ∂ u ( J Ω φ ) J Ω )( t ) k L x + k ∇ φ ( t ) k L x . (84)To prove (83) we start with the identities: I Ω − e ∂ r ( I Ω φ ) = e ∂ r φ + h r i − rφ , II Ω − e ∂ r ( II Ω φ ) = e ∂ r φ + 2 τ − φ . (85)We multiply the first identity by τ + then square. Applying Young’s inequality with c ≪ | τ + τ ∂ u φ | yields: | τ + e ∂ r φ | + | τ + τ ∂ u φ | . | τ + I Ω − e ∂ r ( I Ω φ ) | + | τ + τ I Ω − ∂ u ( I Ω φ ) | + | r − τ + φ | . Integrating the last line gives us (83). Estimate (84) will follow directly from the claim: k r − τ − φ ( t ) k L x + k φ ( t ) k L x + k r − τ + φ ( t ) k L x . RHS (84) . (86)To prove the claim we bound each of the terms in the LHS above from left-to-right. For the first term wesubtract the two identities in line (85) then multiply by τ + to get: τ + ( II Ω − e ∂ r ( II Ω φ ) − I Ω − e ∂ r ( I Ω φ )) = −h r i − r ( C + u ) φ + 2 h r i − φ , (87)where we have used τ + = C + u + 2 r . Next observe that (2 r ) − h r i − r r − holds in the set where 1 r .Therefore re-arranging (87), squaring and using Young’s inequality yields: | r − τ − φ | . | τ + II Ω − e ∂ r ( II Ω φ ) | + | τ + I Ω − e ∂ r ( I Ω φ ) | + | r − φ | . (88)which is valid for 1 r . Multiplying this by a smooth cutoff χ h r i > , integrating, and taking the re-sulting bound in a linear combination with estimate (70) then applying the Hardy bound (69) gives us k r − τ − φ ( t ) k L x . RHS (84). This bounds the first term on LHS (86).To control k φ ( t ) k L x we define the linear combination of ∂ u , e ∂ r derivatives: ∂ = 2 ∂ u − e ∂ r . This satisfies: ∂ ( τ + ) = 0 , ∂ ( u ) = 2 , ∂ ( r ) = − . Collectively these imply:(89) u ( II Ω − ∂ ( II Ω φ ) − I Ω − ∂ ( I Ω φ )) = 2(1 −h u i − ) φ + (1 −h r i − ) r − uφ = ( u +2 r ) r − φ − (2 h u i − + h r i − r − u ) φ . Rearranging the first equation above, squaring and using Young’s inequality gives us: | (cid:0) − h u i − (cid:1) φ | . u (cid:0) | ∂ ( II Ω φ ) II Ω | + | ∂ ( I Ω φ ) I Ω | (cid:1) + | (1 − h r i − ) r − τ − φ | . (90)Next we apply estimate (88) to the last term on the RHS(90), we multiply the resulting bound by χ h u i > ,then integrate and use estimate (69) for the undifferentiated term. This gives us control of k φ ( t ) k L x withinthe region h u i > 1. Taking this resulting bound in a linear combination with estimate (71) then yields k φ ( t ) k L x . RHS (84). Finally, the result for k r − τ + φ ( t ) k L x . RHS (84) follows by using the last line inequation (89) and applying the bounds above. (cid:3) .2. Core multiplier estimates. In this section we list and prove two multiplier bounds which will be thecore constituents of estimates (65) and (66). Proposition 5.5 (Output of ∂ u ) . For any interval [ t , t ] ⊂ I ∗ we have the uniform estimate: (91) sup t t t k ∇ φ ( t ) k L x + k e ∇ x φ k NLE , − . k χ t 0, identities (40) and (41) get us: d ( Y j ) b π αβ = 2 − j χ j ( u )( η αβ − η αu δ βu − η βu δ αu ) + R αβ , where: R αβ = −L Y j ( d g − − η − ) + 2 − j χ j ( u )( d g − − η − ) . The vector field h r i τ Y j satisfies all of the assumptions (50) and (51). So by the results of part I of lemma3.5, R αβ satisfies the pointwise bounds: | ( R rr , r R rA , r R AB ) | . h r i − δ τ − − τ − γ , |R ru | . h r i − δ τ − − τ τ − γ , | r R uA | . h r i − δ τ − − τ τ − γ , |R uu | . h r i − δ τ − − τ τ − γ . Thus a little bit of additional calculation involving the previous formulas shows that A αβ satisfies thepointwise bounds: ( Y j ) A rr & τ − − χ j ( u ) − C h r i − − δ τ − τ − γ , | ( Y j ) A rA | . r − h r i − − δ τ − τ − γ ,r · ( Y j ) A AB & τ − − χ j ( u ) δ AB − C h r i − − δ τ − τ − γ , | ( Y j ) A uu | . h r i − − δ τ − γ , | ( Y j ) A ur | . h r i − − δ τ − τ − γ | r · ( Y j ) A uA | . h r i − − δ τ − γ , here δ AB denotes the standard inverse metric on S . Integrating the resulting contraction over the timeslab [ t , t ] and taking sup j gives us, with suitable constants 0 < c ≪ C :(93) c k e ∇ x φ k NLE , − C k χ t Here we use the conformal multiplier setup of Lemma 2.2 with weights Ω = I Ω , II Ω.For j ∈ Z we define the multiplier vector fields: X j = (1 + χ These terms are O ( τ − ) and O (1), respectively, this allows us to treat them as lower order errors below.To bound R we use the fact that on a dyadic scale τ + ≈ k the vector field 2 − k K satisfies the symbolbounds (50) and all the conditions on line (51) except we have ∂ u X r = 0. Therefore we are in case II oflemma 3.5 and so we have estimates (52) with the modification (53) and (54). In particular thanks to (3)the error R satisfies: |R rr | . h r i − δ τ + τ − τ − γ , | ( r R rA , r R AB ) | . h r i − δ τ + τ − γ , |R ru | . h r i − δ τ + τ τ − γ , | r R uA | . h r i − δ τ + τ τ − γ , |R uu | . h r i − δ τ + τ τ − γ . Combining the last few lines we get: | A uu | . h r i − δ τ + τ τ − γ , A rr & τ τ − − χ j ( u ) − C (cid:0) h r i − δ τ + τ − τ − γ (cid:1) ,r A AB & τ − χ j ( u ) δ AB − C (cid:0) h r i − δ τ + τ − γ (cid:1) , | A ur | . h r i − δ τ + τ τ − γ , | rA rA | . h r i − δ τ + τ − γ , | rA uA | . h r i − δ τ + τ τ − γ , here δ AB again denotes the standard inverse metric on S . Integrating the resulting contraction over thetime slab [ t , t ] and taking sup j then gives us, with suitable constants 0 < c ≪ C :(96) c k φ k Ω CH [ t ,t ] C k χ t 0. This clearly holds by choosing c sufficiently small. Since u = t − r here, thesecond bound in line (100) holds trivially. Case 2: ( Exterior region ct < r ) The first bound in (100) holds since K r τ ≈ τ − and the term containingthe gain r − δ is small. The second bound only needs to be proved inside a neighborhood of the wave zone.Integrating estimate (2) yields K r = 2 tr + O ( r − δ ) so we can bootstrap the term with the gain r − δ here.This finishes the proof of the estimates in line (100). e now use this claim to control the boundary terms: integrating (99) and using line (100) gives us:(101) sup j (cid:0) Z t = t ( X j ) e P χα N α Ω − d dx − Z t = t ( X j ) e P χα N α Ω − d dx (cid:1) . k φ ( t ) k Ω CE − k φ ( t ) k Ω CE + sup t t t k τ − χ r< t Ω − V φ ( t ) k L x . We add equations (96) + (97) + (98) + (101) and choose c sufficiently small to bootstrap the Ω CH energyterm on RHS(98). Re-arranging terms, taking absolute values, sup in t then square roots finishes the proofof estimate (92). (cid:3) Remark . We require the full decay for g ui − ω i ω j g uj and g uu given by (3) in order to produce estimate(101). If we assume weaker decay for these components, estimate (59) would be also be weaker and the termscontaining the gain K r r − δ in the exterior could no longer be bootstrapped. In other words, the exteriorproof of (101) above would no longer work.5.3. Proof of the main estimates. We now prove parts II and III of Theorem 5.1. Proof of Proposition 5.5. We bound each of the error terms on RHS(91). Step 1: ( Bounding the source term ) By Young’s inequality:sup j (cid:12)(cid:12)(cid:12)ZZ t t t ✷ g φ · Y j φ dV g (cid:12)(cid:12)(cid:12) . c − k ✷ g φ k LE ∗ + c k φ k LE . (102) Step 2: ( Bounding the spacetime error terms ) To control ∇ φ in the exterior we use r − δ . ǫ together with LE ⊆ r − − δ L . For τ − e ∇ x φ in the exterior we use r − δ . ǫ together with NLE ⊆ r − δ τ − − L . For theinterior we use γ < δ so that LE ⊆ r − + γ − δ L and combine this with t − γ . ǫ . This yields:(103) k χ t We work on each error term on RHS(92). Step 1: ( Bounding the source term ) Let c , c ≪ ℓ t → ℓ t × ℓ ∞ t Holder in time to get:sup j k χ t In this section we commute the equation once with the Lie algebra L = { ∂ u , S, Ω ij } and use our conformalenergy estimate (8) to produce the higher order bound (9). By Lemma 4.2 it suffices to prove our estimatefor the asymptotic region t ∈ I ∗ . Inside I ∗ the corresponding estimate (9) will follow from: Theorem 6.1. Assume the hypotheses of the main theorem and estimate (8) hold. Then for all [ t , t ] ⊂ I ∗ we have the following uniform bounds:I) (CE estimate for ∂ u φ with error) (108) sup t t t k ∂ u φ ( t ) k CE + k ∂ u φ k S . ǫ (cid:0) sup t t t k φ ( t ) k CE + k χ r< t φ k ℓ ∞ t LE (cid:1) + k ∂ u φ ( t ) k CE + k ✷ g φ k ℓ t N . II) (CE estimate for Sφ and Ω ij φ with error) (109) sup t t t X Γ= S, Ω ij k Γ φ ( t ) k CE + k Γ φ k S . ǫ (cid:0) sup t t t k φ ( t ) k CE + k χ r< t φ k ℓ ∞ t LE (cid:1) + X Γ= S, Ω ij k Γ φ ( t ) k CE + k ✷ g φ k ℓ t N . Let’s show how the conformal energy estimate with vector fields follows from this theorem: Proof of estimate (9) . Adding estimates (108) + (109) + (8) yields:sup t t t k φ ( t ) k CE + k φ k S . ǫ (cid:0) sup t t t k φ ( t ) k CE + k χ r< t φ k ℓ ∞ t LE (cid:1) + k φ ( t ) k CE + k ✷ g φ k ℓ t N . We bootstrap the small error terms and close the estimate. Inside [0 , T ∗ ], lemma (4.2) gives us the analogousbound. A combination of these two estimates then yields (9). (cid:3) The rest of this section is devoted to proving Theorem 6.1. .1. Preliminary estimates. Our goal in this first part is to establish some commutator bounds for allvector fields Γ ∈ L . We start with the following: Lemma 6.2. (Pointwise bounds) Let Γ ∈ L be in rectangular Bondi coordinates ( u, x i ) . The followinguniform estimate holds: (110) X Γ= S, Ω ij | [ ✷ g , Γ] φ | . h r i − − δ τ − γ (cid:0) X k + | J | k =2 , | J |6 =2 | τ − ( h r i τ ∂ u ) k ( h r i e ∂ x ) J φ | + | ( h r i τ ∂ u ) φ | + | ( h r i e ∂ x ) φ | + h r i | ✷ g φ | (cid:1) . In the case of [ ✷ g , ∂ u ] the same estimate holds with the exponent − γ above replaced with − γ .Proof. By (45), (46), and (47) it suffices to bound the quantities: E ( u, α ) = d − ∂ u d R uα Γ ∂ α φ , E ( α, u ) = d − ∂ α d R uα Γ ∂ u φ , E ( i, j ) = d − e ∂ i d R ij Γ e ∂ j φ , E ( s ) = d − Γ( d ) , where: d R Ω ij = −L Γ ( d g − − η − ) ,d R S = −L S ( d g − − η − ) − ( ∂ u S u + e ∂ r S r + e ∂ i S i )( d g − − η − ) . To bound these quantities we use part I of lemma 3.5 together with estimate (55) for d = det( g αβ ). Thisyields: (cid:12)(cid:12) E ( u, α ) | + (cid:12)(cid:12) E ( α, u ) (cid:12)(cid:12) . h r i − δ τ − γ (cid:0) τ | e ∂ x ∂ u φ | + τ | ∂ u φ | + h r i − τ | ∂ u φ | + h r i − τ − | e ∂ x φ | (cid:1) , (cid:12)(cid:12) E ( i, j ) (cid:12)(cid:12) . h r i − δ τ − γ (cid:0) | e ∂ x φ | + h r i − | e ∂ x φ | (cid:1) , (cid:12)(cid:12) E ( s ) (cid:12)(cid:12) . h r i − δ τ − γ | ✷ g φ | . In the case of Γ = ∂ u we use part III of lemma 3.5 which yields the improvement in the interior by the sameargument. (cid:3) Remark . A quick computation using (3) shows that the estimate derived in lemma 6.2 above for thevector field ∂ u is missing a weight τ − − in the exterior. We chose to omit this weight since there’s no realimprovement in the estimates below if we were to include it. Ultimately, this is a consequence of our weakdecay for the g ij metric components which, essentially, force us to treat ∂ u at the same level of decay as S and Ω ij in the exterior. Remark . From this point on we will often make use of the parameter µ ≪ { µt < r } ∩ I ∗ we control all weighted combinations of two derivatives in RHS(110) via: Lemma 6.5 (Exterior Klainerman-Sideris identity) . Let Γ ∈ L be in rectangular Bondi coordinates ( u, x i ) .In the exterior region { µt < r } ∩ I ∗ one has the following uniform estimate: X k + | J | | ( rτ ∂ u ) k ( r e ∂ x ) J φ | . X k + | J | =1 , | I | | ( rτ ∂ u ) k ( r e ∂ x ) J Γ I φ | + r τ | ✷ g φ | . (111) Proof. Let R = d g − η . Inside { µt < r } ∩ I ∗ the estimates (3) imply the uniform bound: r τ | ∂ α R αβ ∂ β φ | . r − δ X k + | J | | ( rτ ∂ u ) k ( r e ∂ x ) J φ | , where all quantities are computed with respect to Bondi coordinates. By the support property and estimate(63) we have r − δ . ( µt ) − δ . ǫ . Thus, we may replace ✷ g by the Minkowski wave operator ✷ η in estimate r τ + e ∂ r S = r uτ + ∂ u e ∂ r + r τ − e ∂ r + r τ + e ∂ r , r uτ + ✷ η = − r uτ + ∂ u e ∂ r + 12 r uτ + e ∂ r − ruτ + ∂ u + ruτ + e ∂ r + 12 uτ + X i As a consequence of these two lemmas we get: Proposition 6.6 (Global commutator bounds) . Let Γ = S, Ω ij . The following uniform estimates hold on I ∗ . (112) k [ ✷ g , Γ] φ k ℓ t N . k χ r<µt h r i − τ − γ + ( ∇ φ, h r i − ∇ φ ) k L + ǫ sup t t t k φ ( t ) k CE + k ✷ g φ k ℓ t N , we also have the following (interior) improvement in the case of Γ = ∂ u : (113) k [ ✷ g , ∂ u ] φ k ℓ t N . k χ r<µt h r i − τ − γ + ( ∇ φ, h r i − ∇ φ ) k ℓ ∞ r L + ǫ sup t t t k φ ( t ) k CE + k h r i − − δ τ − τ − γ ✷ g φ ) k ℓ t N . Proof of estimate (112) . We treat the regions r < µt and µt < r separately. Case 1: ( The interior r < µt ) We multiply estimate (110) by:2 j k χ t ≈ k χ h r i≈ j χ r<µt , square, integrate then use γ < δ together with the inclusion L ⊆ h r i γ − δ τ − γ + ℓ t,r L . For the source term weapply estimate (69) with a = 1 / 2. Combining all this:(114) X Γ= S, Ω ij k χ r<µt [ ✷ g , Γ] φ k ℓ t N . k χ r<µt h r i − τ − γ + ( ∇ φ, h r i − ∇ φ ) k L + k χ r<µt ✷ g φ k ℓ t N . Case 2: ( The exterior region µt < r ) We multiply estimate (110) by: h r i τ + τ χ t ≈ k χ h r i≈ j χ h u i≈ l χ µt 2, a number which nolonger allows us to commute with cutoffs. To resolve this issue, we will use a Klainerman-Sideris type boundtogether with the elliptic estimate (138) to establish the necessary inequalities in this case. In light of thisdiscussion, the ellipticity of the operator P ( t, x, ∇ x ) (see Corollary 2.11.II) is of fundamental importance forthe proof of the next lemma. Consequently, all computations are in ( t, x i ) coordinates here. Lemma 6.7 (Klainerman-Sideris type estimates for the interior) . The following uniform bounds hold insidethe region I ∗ :I) (Bounds for ∇ ∂ t φ ) k χ r<µt τ − γ + h r i ∇ ∂ t φ k ℓ ∞ r L . ǫ k χ r< t ∂ t φ k ℓ ∞ t LE , (115) k χ r<µt τ − γ + h r i − γ ∇ ∂ t φ k L . ǫ k χ r< t φ k ℓ ∞ t LE + µ k χ r<µt τ − γ + h r i − γ ∇ x φ k L . (116) II) (Bounds for ∇ x φ ) k χ r<µt τ − γ + h r i ∇ x φ k ℓ ∞ r L . ǫ k χ r< t ( ∂ t φ, φ ) k ℓ ∞ t LE + k ✷ g φ k N , (117) k χ r<µt τ − γ + h r i − γ ∇ x φ k L . ǫ (cid:0) sup t t t k φ ( t ) k CE + k φ k S (cid:1) + k ✷ g φ k N + k χ r<µt τ − γ + h r i − γ ∇ ∂ t φ k L . (118) III) (Bounds for ∇ φ ) k χ r<µt τ − γ + h r i ∇ φ k ℓ ∞ r L . ǫ k χ r< t ( ∂ t φ, φ ) k ℓ ∞ t LE , (119) k χ r<µt τ − γ + h r i − γ ∇ φ k L . ǫ (cid:0) sup t t t k φ ( t ) k CE + k φ k S (cid:1) + k ✷ g φ k N (120) Proof of Lemma 6.7. Part 1: ( Bounds for ∇ ∂ t φ ) For estimate (115) we split the gain t − γ : half goes to t − γ . ǫ and the other half is used in the inclusion ℓ ∞ t,r L ⊆ t − γ ℓ ∞ r L . The bound follows by using thedefinition of the norms and the support property.For estimate (116) we start with the pointwise inequality: | ∂ t φ | . | t − Sφ | + ( r/t ) | ∂ r φ | , (121)which is valid inside r < t . Applying this to ∂ t φ and ∇ x φ yields, respectively: | ∂ t φ | . | t − ∂ t Sφ | + | t − ∂ t φ | + ( r/t ) | ∂ r ∂ t φ | , | ∂ t ∇ x φ | . | t − ∇ x Sφ | + | t − ∇ x φ | + ( r/t ) | ∂ r ∇ x φ | . ultiplying these two bounds by h r i − + γ τ − γ + χ r<µt , squaring, integrating and using the support property( r/t ) < µ we get: k χ r<µt τ − γ + h r i − γ ∂ t φ k L . ǫ k χ r< t φ k ℓ ∞ t LE + µ k χ r<µt τ − γ + h r i − γ ∇ x ∂ t φ k L , (122) k χ r<µt τ − γ + h r i − γ ∇ x ∂ t φ k L . ǫ k χ r< t φ k ℓ ∞ t LE + µ k χ r<µt τ − γ + h r i − γ ∇ x φ k L , (123)where we have split the gain t γ − γ as in the previous proof to produce the ǫ · ℓ ∞ t LE terms. By definition4.1.a, the constant µ is small enough that we can add (122) + (123), bootstrap the term ∇ x ∂ t φ , and getestimate (116). Part 2: ( Bounds for ∇ x φ ) To prove estimate (117) we let P ( t, x, ∇ x ) be as in Corollary 2.11.II. By thatresult, the operator P is uniformly elliptic in ( t, x i ) coordinates. Thus, commuting χ r<µt with ∇ x and usingbasic elliptic estimates: k χ r<µt h r i − τ − γ + ∇ x φ k ℓ ∞ r L . k χ r<µt h r i − τ − γ + P φ k ℓ ∞ r L + µ − k χ r ∼ µt h r i − τ − γ + ( r − ∇ φ, r − φ ) k ℓ ∞ r L . k χ r<µt h r i − τ − γ + P φ k ℓ ∞ r L + ǫ k χ r< t φ k ℓ ∞ t LE , where µ − comes from terms where derivatives land on the cutoff χ r<µt . We also used µ − t − γ . ǫ on thelast line which follows since definition 4.1 implies ǫ ≪ µ . Applying estimate (35) to the first term on the lastline above: k χ r<µt h r i − τ − γ + P φ k ℓ ∞ r L . k χ r<µt h r i − τ − γ + (cid:0) ✷ g φ, h r i − δ ∂ t ∂ x φ, h r i − − δ τ − γ ∂ x φ (cid:1) k ℓ ∞ r L + k χ r<µt h r i − τ − γ + (cid:0) h r i − − δ ∂ t φ, ∂ t φ (cid:1) k ℓ ∞ r L . k ✷ g φ k ℓ t N + ǫ (cid:0) k χ r< t ∂ t φ k ℓ ∞ t LE + k χ r< t φ k ℓ ∞ t LE (cid:1) , on the last line we used the gain τ − γ + to put the ✷ g φ term in the form above. For all other terms we split t − γ as we did before and use the definition of the norms to finish the proof of (117).For estimate (118) we can apply the weighted L estimate (138) since the exponent of h r i − + γ is abovethe threshold value a = 1 / 2. This introduces a term supported where µt < r which we also need to control.After an application of estimate (35) we get: k χ r<µt h r i − + γ τ − γ + (cid:0) ∇ x φ, h r i − ∇ x φ (cid:1) k L . k χ r<µt h r i − + γ τ − γ + P φ k L + k χ µt We now prove Theorem 6.1. Proof. Part 1: ( Proof of estimate (108)) We commute equation (1) with the vector field ∂ u then use theconformal energy estimate (8) and apply the commutator bound (113) to get:(126) sup t t t k ∂ u φ ( t ) k CE + k ∂ u φ k S . k χ r<µt h r i − τ − γ + ( ∇ φ, h r i − ∇ φ ) k ℓ ∞ r L + ǫ sup t t t k φ ( t ) k CE + k ∂ u φ ( t ) k CE + k ∂ u ( ✷ g φ ) k ℓ t N + k h r i − − δ τ − τ − γ ✷ g φ ) k ℓ t N . The last source term above is obviously bounded by k ✷ g φ k ℓ t N . Next we bound the terms containing χ r<µt :for the terms with only one derivative, we drop the weight h r i − and use t − γ . ǫ together with the inclusion ℓ ∞ t LE ⊆ h r i − t − γ ℓ ∞ r L . For the terms containing χ r<µt with two derivatives, we apply (119) directly.Using these two bounds in succession on the RHS (126) yields: k χ r<µt h r i − τ − γ + ( ∇ φ, h r i − ∇ φ ) k ℓ ∞ r L . ǫ k χ r< t ( ∂ t φ, φ ) k ℓ ∞ t LE . Bootstrapping the highest order term above to LHS(126) finishes the proof of estimate (108). Part 2: ( Proof of estimate (109)) We commute equation (1) with the vector fields { S, Ω ij } , use the conformalenergy estimate (8), then apply estimate (112) to the commutators. This yields:(127) sup t t t X Γ= S, Ω ij k Γ φ ( t ) k CE + k Γ φ k S . k h r i − τ − γ + χ r<µt (cid:0) ∇ φ, h r i − ∇ φ (cid:1) k L + ǫ sup t t t k φ ( t ) k CE + X Γ= S, Ω ij k Γ φ ( t ) k CE + k ✷ g φ k ℓ t N . We multiply the terms containing χ r<µt above times the weight h r i γ . For the ensuing terms with onlyone derivative, the exponent of h r i a is now above the a = 3 / k χ r<µt h r i − + γ τ − γ + ∇ φ k L . k χ r<µt h r i − + γ τ − γ + ∇ x ∇ φ k L . (128)Thus all terms containing χ r<µt in RHS(127) are now of the form LHS(120). An application of the latterestimate finishes the proof of Theorem 6.1. (cid:3) . Global L ∞ Decay In this section we prove the pointwise bound (10). Let’s begin by showing some preliminary estimates. Lemma 7.1 (Preliminary estimates) . For test functions φ ( t, x ) , f ( t, x ) the following uniform bounds hold:I) (Global L ∞ estimate ) (129) k h r i τ φ k L ∞ x . X k + | J | k ( h r i τ ∂ u ) k ( h r i e ∂ x ) J φ k L x . II) (Interior L ∞ estimate) Assume that φ is supported in r < t . Then one has: (130) k φ k L ∞ x . k h r i ( ∇ φ, h r i − ∇ φ, h r i − φ ) k ℓ ∞ r L x . III) (Average to uniform bounds via scalings) Let f be supported on the time interval [ T ∗ , ∞ ) . Then for all a ∈ R and T > T ∗ : (131) sup t T k τ a + f ( t ) k L x . k τ a − + ( f, r e ∂ r f, Sf ) k L [0 ,T ] . Proof. Step 1: ( Proof of Estimate (129)) Inside the set r L ∞ − L Sobolevestimate. For the complement it suffices to consider the region t < r < t as the remainder is easier tohandle because we have u = t − r there. Using dyadic cutoffs we may assume φ is supported where τ − ≈ k and r ≈ j . Using angular sector cutoffs in the x variable we may further assume without loss of generalitythat φ is supported in a π wedge about the x axis. Now introduce new variables on t = const : y = 2 − k u , y = 2 − j x , y = 2 − j x . There exists vector fields e α , α = 0 , , , ∂ y i | t = const = P α c iα e α where c iα are uniformly boundedand such that: 2 k + j X | I | k e I φ k L ( dy ) . X k + | J | k ( rτ ∂ u ) k ( r e ∂ x ) J φ k L ( dx ) . Estimate (129) follows from this last line by concatenating the Sobolev embeddings H ⊆ L and W , ⊆ L ∞ . Step 2: ( Proof of (130)) Once again by the L ∞ − L Sobolev estimate it suffices to prove the result outside r 1. By the support property we have τ ≈ 1, therefore applying estimate (129) to r − χ h r i≈ k φ , takingsup k and using the fact that ∂ u , e ∂ x are bounded linear combinations of ∂ t , ∂ x derivatives yields the claim. Step 3: ( Proof of (131)) Integrating the time derivative of ( t + r ) a f over 0 < t < T we have:( T + r ) a f ( T ) = r a f (0) + 2 a Z T ( t + r ) a − f dt + 2 Z T ( t + r ) a f ∂ t f dt . Using Cauchy-Schwartz on the last RHS term, integrating in x over 0 < r < ∞ , then using the supportproperty of f we get: k τ a + f ( T ) k L x . k τ a − + ( f, r e ∂ r f, Sf ) k L [0 ,T ] . This yields (131). (cid:3) Now we demonstrate the main L ∞ bound. Proof of estimate (10) . By Lemma 4.2 and by using an appropriate cutoff function we may assume φ issupported on [ T ∗ , ∞ ) and that T > T ∗ . We then estimate the timelike and null/spacelike regions separately.Let µ be as in definition 4.1.a. Step 1: ( Estimate for µt < r ) Applying (129) to χ µt Theorem 8.1 (Global elliptic estimate) . Let the operator P ( t, x, ∇ x ) = ∂ i h ij ∂ j with h ij = d g ij be uni-formly elliptic. Suppose that for all t > , the h ij satisfy the uniform bounds: (137) | ( h r i ∂ x ) α ( h ij − δ ij ) | . h r i − δ . Then for all t > the operator P satisfies the fixed-time estimates: (138) k h r i a (cid:0) ∇ x φ, h r i − ∇ x φ (cid:1) ( t ) k L x . k h r i a P φ ( t ) k L x , − < a < , where the implicit constants are independent of t .Proof. Let ∆ , ∆ − denote the standard 3D Laplacian and its inverse, respectively. Write P φ = F . Weapproximately solve for F in terms of a Neumann series: e φ = ∆ − k X i =0 R i F , R = I − P ∆ − . Then we have: P ( e φ − φ ) = R k +1 F . Therefore, setting L ,ax for the norms on line (138) it suffices to show:(139) ∇ ∆ − : L ,ax → L ,ax , h r i − ∇ ∆ − : L ,ax → L ,ax , R : L ,ax → L ,a + δx , for the range < a and a + δ < , followed by the non-perturbative estimate:(140) k h r i∇ x φ ( t ) k L x + k ∇ x φ ( t ) k L x . k h r i P φ ( t ) k L x . (cid:3) roof of (139) . We decompose into dyadic scales | x | ∼ i , | y | ∼ j with | x | . i = 0 since the weightsare non-singular. Step 1: ( ∇ ∆ − : L ,ax → L ,ax is bounded ) To establish this it suffices to show: X i,j (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . k F k L , − ax k G k L ,ax , (141)where K is the convolution kernel for ∇ ∆ − . We break up the proof into cases: Case 1: ( | i − j | = O (1)) The operator defined above is a singular integral operator. In this case the weights2 − aj ≈ − ai ≈ X i + j = O (1) (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X i,j k χ i F k L , − a k χ j G k L ,a , with χ i , χ j smooth cutoff functions supported where | x | ∼ i , | y | ∼ j , respectively. Case 2: ( i > j + c ) We now have | K ( x − y ) | = O ( | x | − ) and since convolution with an L function is abounded operator in any L p space with p ≥ X i>j + c (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X i>j + c − ( i − j ) · a ( i − j ) k χ i F k L , − a k χ j G k L ,a , which forces − + a Case 3: ( j > i + c ) by switching the roles of x and y and using the same argument as case 2: X j>i + c (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X j>i + c ( i − j ) · a ( i − j ) k χ i F k L , − a k χ j G k L ,a , which is convergent for − − a < 0. This proves (141). Step 2: ( h x i − ∇ ∆ − : L ,ax → L ,ax is bounded ) We again aim to show:(142) X i,j (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . k F k L , − ax k G k L ,ax , with the kernel K ( x, y ) the convolution kernel for h r i − ∇ ∆ − . Case 1: ( | i − j | = O (1)) Here we have K ( x, y ) = O ( h x i − | x − y | − ). By the Hardy-Littlewood-Sobolevinequality: X i + j = O (1) (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X i + j = O (1) (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) G ( y ) h x i| x − y | dxdy (cid:12)(cid:12) . X i,j − i · i · j k χ i F k L k χ j G k L . X i,j k χ i F k L , − a k χ j G k L ,a , where we’ve used 2 − i ≈ i · j ≈ Case 2: ( i > j + c ) Here we have | K ( x − y ) | = O ( h x i − | x | − ). Therefore: X i>j + c (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X i>j + c − i · − ( i − j ) · a ( i − j ) k χ i F k L , − a k χ j G k L ,a , and since i > j + c the extra 2 − i helps us get − + a < Case 3: ( j > i + c ) Here we have | K ( x − y ) | = O ( h x i − | y | − ). Hence: X j>i + c (cid:12)(cid:12)ZZ | x |∼ i , | y |∼ j F ( x ) K ( x − y ) G ( y ) dxdy (cid:12)(cid:12) . X j>i + c i · ( i − j ) · a ( i − j ) k χ i F k L , − a k χ j G k L ,a , and since j > i + c the extra 2 i gives us the restriction − − a < Step 3: ( R : L ,a → L ,a + δ is bounded ) By estimate (137) we have: P − ∆ = ∂ i ( h ij − δ ij ) ∂ j = O ( h r i − δ ) ∂ x + O ( h r i − − δ ) ∂ x . This observation together with the results above finish the proof. (cid:3) roof of (140) . Let D denote the Levi-Civita connection for h and let dV h be the corresponding volumeform. We have the estimate: Z R D i φD i φ dV h . Z R h r i | P φ | dV h , (143)which follows from Green’s identity: − Z R D i φD i φ dV h = Z R P φ · φ dV h , by taking absolute value, applying Young’s inequality and using the Hardy estimate: Z R (cid:12)(cid:12) r − φ (cid:12)(cid:12) dV h . Z R | Dφ | dV h . To prove the estimate for two derivatives we integrate by parts twice then take absolute value, apply (137)together with estimate (143) and Young’s inequality to produce: Z R (cid:12)(cid:12) h r i ( D i D j φ )( D i D j φ ) (cid:12)(cid:12) dV h . Z R (cid:0) h r i | P φ | + h r i| P φ || Dφ | + h r i − − δ | φ || Dφ | + D i φD i φ (cid:1) dV h . Z R h r i | P φ | dV h . This finishes the proof of (140). (cid:3) Acknowledgements The author would like to thank his advisor Jacob Sterbenz for suggesting the problem, for sharing manyvaluable insights, and for providing unconditional support throughout the writing of this work. References [1] S. Alinhac. 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