Morawetz estimates for the wave equation at low frequency
aa r X i v : . [ m a t h . A P ] D ec MORAWETZ ESTIMATES FOR THE WAVE EQUATION ATLOW FREQUENCY
ANDR ´AS VASY AND JARED WUNSCH
Abstract.
We consider Morawetz estimates for weighted energy decay of so-lutions to the wave equation on scattering manifolds (i.e., those with largeconic ends). We show that a Morawetz estimate persists for solutions thatare localized at low frequencies, independent of the geometry of the compactpart of the manifold. We further prove a new type of Morawetz estimate inthis context, with both hypotheses and conclusion localized inside the forwardlight cone. This result allows us to gain a 1 / t decay relative towhat would be dictated by energy estimates, in a small part of spacetime. Introduction
In this paper, we show that the celebrated Morawetz estimate [14], expressingdispersion of solutions to the wave equation, holds for low-frequency solutions ona wide class of manifolds with asymptotically flat ends. We also generalize theestimate to a local-in-spacetime estimate inside the forward light cone.It is well known that the decay of energy of a solution to (cid:3) u = 0 at high frequen-cies is closely tied to the geometry of geodesic rays; in particular, the existenceof trapped geodesics is an obstruction to the uniform decay of local energy (seeRalston [18]). Many subsequent results have demonstrated that the decay of highfrequency components of the solution persists in a wide variety of geometric settingsin which there is no trapping of rays (see e.g. [15], [20], [21]).On the other hand, the low-frequency behavior of the solutions is both morerobust and less studied. Intuitively, long wavelengths should be sensitive only tothe crudest aspects of the geometric setting, and in particular, trapping shouldnot be an obstacle to decay. In this paper, we work in the setting of scatteringmanifolds, i.e., we assume that the manifold X has ends resembling the large endsof cones, with the metric taking the form(1.1) g = dr + r h ( r − , dy ) Date : October 28, 2018.The authors gratefully acknowledge partial support from the NSF under grant numbers DMS-0801226 (AV) and DMS-0700318, DMS-1001463 (JW). A.V. is also grateful for support from aChambers Fellowship at Stanford University. The authors are grateful to Daniel Tataru, and toan anonymous referee, for many helpful comments, and in particular for suggesting the refined ℓ ∞ – ℓ version of the estimates in Theorem 1.1. We remark that in the body of the paper, X will refer to the manifold with boundary givenby compactifying such a space. where the noncompact ends are diffeomorphic to (1 , ∞ ) r × Y, with Y a smoothmanifold, and where h is a family (in r ) of metrics on Y. Our two main theoremsdemonstrate the insensitivity of the low-energy behavior of the wave equation togeometry on all but the largest scales. We also allow long-range perturbations ofthe metrics (1.1); see Section 2 for the precise definitions.The results in this paper are as follows. To begin with, we state a result holdingfor low frequency solutions in spatial dimension n ≥ ψ ∈ C ∞ c ((0 , ∞ )) . Let Ψ H denote thefrequency-localization operatorΨ H = ψ ( H (∆ g + V )) . We assume that the function r, appearing in the end structure of the metric (1.1),is extended to be a globally defined, smooth function on X, with min( r ) ≥ r with impunity). Let χ , χ be a partition of unity with χ supportedin r > R ≫ , hence in the ends in which the product decomposition (1.1) applies.Here and throughout the paper, k•k will refer to the norm in the space L ( X, dg ) . When we employ spacetime norms in § k•k M where M is (the compactification of) the spacetime R × X. Theorem 1.1.
Let X be a scattering manifold of dimension n ≥ with a long-range metric, as in (2.1) – (2.2) , and let V ≥ be a symbol of order − − ν, ν > .Let u be a solution to the inhomogeneous wave equation ( (cid:3) + V ) u = f, f ∈ L ( R ; L ( X )) on R t × X, with initial data u | t =0 = u ∈ H , ∂ t u | t =0 = v ∈ L . Then for H sufficiently large, (1.2) Z ∞ (cid:16)(cid:13)(cid:13)(cid:13) r − / Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) χ (log r ) − r − / ∂ r Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) χ r − / ∇ Y Ψ H u (cid:13)(cid:13)(cid:13) + k χ ∇ g Ψ H u k (cid:17) dt . E (0) + (cid:0) Z ∞ k f k dt (cid:1) , where E (0) is the initial energy E (0) = k Ψ H u k H + k Ψ H v k L . Here H and L denote the appropriately defined spaces on X (and k•k denotesthe L ( X ) norm): L is the space of functions square integrable with respect tothe metric, and H , which will henceforth be written H (the “scattering” Sobolevspace, associated to the geometry of large conic ends) denotes the space of functionssatisfying Z | u | + |∇ g u | g dg < ∞ . We may a priori assume that h is in fact a smooth tensor including dr components; that wemay then change variables to remove these components is a result of Joshi-S´a Barreto [9], Section2. ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 3
The Laplacian ∆ g is the nonnegative Laplacian, and ψ ( H ∆ g ) is localizing at lowenergies. The result thus shows that the dispersive effects of the large scale scat-tering geometry always hold for low-frequency solutions, regardless of trapping orother local features. We further remark that we could replace the L ( R ; L ( X ))norm on f on the RHS of the estimate by(1.3) Z ∞ (cid:13)(cid:13)(cid:13) ( r + t ) / log( r + t ) f (cid:13)(cid:13)(cid:13) dt, if desired, to obtain a weighted L spacetime norm instead.Following the suggestion of an anonymous referee, we are in fact able to prove arefined version of this result, with logarithmic losses replaced by estimates in ℓ and ℓ ∞ norms on energy in dyadic spatial shells. Let Υ k denote a spatial decompositionin radial dyadic shells (to be discussed further in §
3, see (3.18)), so r ∼ k on Υ k . Theorem 1.1’.
Let X be a scattering manifold of dimension n ≥ with a long-range metric, as in (2.1) – (2.2) , and let V ≥ be a symbol of order − − ν, ν > .Let u be a solution to the inhomogeneous wave equation ( (cid:3) + V ) u = f, f ∈ L ( R ; L ( X )) ∩ ℓ ( N k ; L ( R × Υ k )) on R t × X, with initial data u | t =0 = u ∈ H , ∂ t u | t =0 = v ∈ L . Then for H sufficiently large, (1.4) Z ∞ (cid:16)(cid:13)(cid:13)(cid:13) r − / Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) χ r − / ∇ Y Ψ H u (cid:13)(cid:13)(cid:13) + k χ ∇ g Ψ H u k (cid:17) dt + (cid:13)(cid:13)(cid:13) χ r − / ∂ r Ψ H u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N k ; L ([0 , ∞ )) × Υ k )) . E (0) + k f k L ([0 , ∞ ); L ( X )) + (cid:13)(cid:13)(cid:13) r / f (cid:13)(cid:13)(cid:13) ℓ ( N k ; L ([0 , ∞ ) × Υ k )) where E (0) is the initial energy E (0) = k Ψ H u k H + k Ψ H v k L . Here we could drop R ∞ (cid:13)(cid:13) χ (log r ) − r − / ∂ r Ψ H u (cid:13)(cid:13) dt from the left hand side of(1.4) as compared to (1.2) without any loss, since it can be estimated by Z ∞ (cid:13)(cid:13)(cid:13) χ (log r ) − r − / ∂ r Ψ H u (cid:13)(cid:13)(cid:13) dt = X k Z ∞ Z Υ k χ (log r ) − r − | ∂ r Ψ H u | dg dt . (cid:16) X k k − (cid:17)(cid:13)(cid:13)(cid:13) χ r − / ∂ r Ψ H u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 , ∞ )) × Υ k )) , so the quantity we estimate on the left hand side in Theorem 1.1’ indeed strongerthan that in Theorem 1.1. A similar argument shows that (1.3) also dominates thethird term on the right hand side of (1.4).Our second main result is a version of the Morawetz estimate that is localizedin spacetime, with both hypotheses and conclusion localized in a sub-cone of theforward light cone. It holds in all dimensions n ≥ . For δ > δ = { t > /δ, r/t < δ } ⊂ R × X. This is the (asymptotic) cone in which we will localize.
ANDRAS VASY AND JARED WUNSCH
In the statement of the following theorem, ∇ denotes the space-time gradient. Theorem 1.2.
Let X be a scattering manifold of dimension n ≥ with a long-range metric as in (2.1) – (2.2) , V ≥ a symbol of order − − ν, ν > . Let ( (cid:3) + V ) u = f. Suppose that δ < and ∇ Ψ H u ∈ t κ L (Ω) , f ∈ r − / t κ − / L (Ω) , κ ∈ R . Then for H sufficiently large, ∇ Ψ H u ∈ t κ ( r/t ) σ L (Ω) , Ψ H u ∈ t κ r ( r/t ) σ L (Ω) for < σ < / .In particular, if K is compact in X ◦ , then on R × K , Ψ H u, ∇ Ψ H u are in t κ − / ǫ L ( R × X ) for every ǫ > (for H sufficiently large). This theorem gives almost half an order of decay as compared to the a prioriassumption, but only in a small part of space-time. Indeed, for any δ >
0, theconclusion in r/t > δ is the same as hypothesis, so in particular δ in the statementof the theorem can be taken arbitrarily small without losing the strength of theconclusion. Thus, Theorem 1.2 really amounts to a regularity statement; it is aclose analogue of the statement for hyperbolic PDE that over compact sets, onehas extra regularity over a priori space-time Sobolev regularity (square integrabilitycan be replaced by continuity along the flow). Indeed, an analogous statement isthat locally L solutions of the wave equation in space time, near t = 0, are in t σ L locally for σ ∈ (0 , / L andthe fact that t − σ is in L ( R ) locally.One virtue of this version of the Morawetz estimate, and even more of the ap-proach to obtaining this estimate, is that they hold the promise of applying quitebroadly, in non-product spacetimes. Even in the present inhomogeneous form theycan be used to study the wave equation for metrics which arise by perturbing theMinkowski metric near infinity in the following way: pick p j , j = 1 , . . . , N , on thesphere at infinity, S n = ∂ R n +1 t,z , in the radial compactification of R n +1 = R t × R nz which lie in the interior of the forward light cone. Blow up the p j ; if one of the p j isthe “north pole,” corresponding to z = 0, t = + ∞ , a neighborhood of the front facecan be identified with a neighborhood of “temporal infinity” in the product asymp-totically Euclidean space-time (see Section 4); the other p j can be transformed tothis via isometries of Minkowski space-time. Thus, if the Minkowski metric is per-turbed in a way corresponding to the local product structure these transformationsinduce, modulo decaying terms in time, the local Morawetz estimate is applicableby treating the decaying terms as inhomogeneities. In future work, we plan toaddress such non-product geometries more systematically.By the standard energy estimate, if ( (cid:3) + V ) u = 0 and u is in the energy space,then for φ ∈ C ∞ c ( R ), φ ( r/t ) ∇ u ∈ t / δ L ( R × X ) , δ > . This observation immediately yields
Corollary 1.3.
Let
X, g, and V be as in Theorem 1.2. Suppose that (cid:3) u = 0 , withinitial conditions u | t =0 = u ∈ H , ∂ t u | t =0 = v ∈ L . ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 5
There exists δ > small such that (1.6) Z Z Ω δ (cid:12)(cid:12)(cid:12) t σ − / − δ r − − σ Ψ H u (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) t σ − / − δ r − σ ∇ Ψ H u (cid:12)(cid:12)(cid:12) dt dg . E (0) whenever σ ∈ (0 , / , and H ≫ is sufficiently large. The result is of special interest in the case n = 3 , where Theorem 1.1 does notapply, but where Corollary 1.3 yields a spatially localized estimate. Let K ⊂ X ◦ be compact. Then Corollary 1.3 specializes to show: Corollary 1.4.
For all ǫ > , Z ∞ (cid:13)(cid:13) t − ǫ Ψ H u (cid:13)(cid:13) + (cid:13)(cid:13) t − ǫ ∇ Ψ H u (cid:13)(cid:13) dt . E (0) . for H sufficiently large. Note, by contrast, that conservation of energy would give the same estimate with t − ǫ replaced by t − / − ǫ , so this estimate represents an improvement of t − / indecay relative to energy estimates; by contrast, it loses t − ǫ relative to Theorem 1.2(which of course only holds for n ≥ globally , and in particular this is the case on small perturbationsof Euclidean space.The proof of Theorem 1.1 is, as with the usual Morawetz inequality, a “multi-plier” argument based on the first order differential operator A = (1 / ∂ r − ∂ ∗ r ) . The subtlety is that the crucial commutator term[∆ , A ]is positive for r ≫ , in the ends of the manifold, but certainly not in the interiorwhere there is indeed no natural definition of the radial function r or of ∂ r . Wemust thus introduce cutoffs, which in turn introduce terms in the commutator thathave to be treated as errors. At high frequency, these errors are disastrous, but atlow frequency, we may employ a Poincar´e/Hardy inequality to control them. Fur-ther technical subtleties arise in estimating remainder terms from the commutator,applied to Ψ H u. These estimates are among the principal technical innovationshere—see Proposition 2.1 below.The proof of Theorem 1.2 uses a slightly different commutator argument (for-mally a “commutator” rather than a “multiplier” in the usual parlance) that occursin space-time. The localizer in the cone Ω multiplies a Morawetz-like commutantwith a positive weight in r but decay in t ; derivatives of the cutoff are controlledby the a priori decay assumptions, namely ∇ Ψ H u | Ω ∈ t κ L ( R × X ), which in thecorollaries are implied by energy estimates.Related results have recently been pursued by a number of authors in the settingof asymptotically Euclidean manifolds. Bony-H¨afner [1] explored Mourre estimateson asymptotically Euclidean spaces at low energy, while both Bouclet [3] and Bony-H¨afner [2] have recently obtain iterated resolvent estimates in the low frequencyregime sufficient to prove strong weighted decay estimates for asymptotically Eu-clidean metrics. These results yield stronger decay than what is obtained here, butin a narrower class of geometries and at the cost of a more intricate argument which ANDRAS VASY AND JARED WUNSCH involves first proving resolvent estimates; one of the virtues of the techniques usedhere is the direct use of the hyperbolic equation, apart from the estimates neces-sary to localize at low energy. More in the spirit of our approach, Metcalfe-Tataruhave proved an analogous theorem to our Theorem 1.1 for small perturbations ofEuclidean space, and Tataru [19] has also proved strong energy decay estimates ina class of 3 + 1-dimensional Lorentz metrics that includes the Schwarzschild andKerr spacetimes; see also the more recent [13]. (There has been considerable workon these specific Lorentzian examples motivated by problems in general relativity;we will not review that literature here.) The only prior results on low energy esti-mates in the setting of general scattering manifolds, in addition to the authors’ [22],is the recent work of Guillarmou-Hassell-Sikora [6], which employs a sophisticatedlow-energy parametrix construction to obtain strong decay estimates. Due to thenature of the construction, the requirements on the metric in [6] are more stringentthan the long-range assumptions used here.Many of the low-frequency estimates used in this paper were first developed bythe authors in [22] following the results of Bony-H¨afner [1] in the asymptoticallyEuclidean setting, using rather different methods.We have discussed both low- and high-frequency estimates above; for complete-ness, we remark that intermediate frequencies (i.e. frequencies in any compact sub-set of (0 , ∞ )) are always well-behaved in that the solution of the wave equation local-ized to these frequencies decays rapidly inside the forward light cone, in r/t < c < Conjugated spectral cutoffs
Notation and setting.
Before stating our results on conjugated spectralcutoffs, we very briefly recall the basic definitions of the b- and scattering structureson a compact n -dimensional manifolds with boundary, denoted X ; we refer to [11]for more detail. A boundary defining function x on X is a non-negative C ∞ functionon X whose zero set is exactly ∂X , and whose differential does not vanish there; auseful example to keep in mind is x = r − (modified to be smooth at the origin) inthe compactification of an asymptotically Euclidean space. We recall that ˙ C ∞ ( X ),which may also be called the set of Schwartz functions, is the subset of C ∞ ( X )consisting of functions vanishing at the boundary with all derivatives, the dual of˙ C ∞ ( X ) is tempered distributional densities C −∞ ( X ; Ω X ); tempered distributions C −∞ ( X ) are elements of the dual of Schwartz densities, ˙ C ∞ ( X ; Ω X ).Let V ( X ) be the Lie algebra of all C ∞ vector fields on X ; thus V ( X ) is the setof all C ∞ sections of T X . In local coordinates ( x, y , . . . , y n − ), ∂ x , ∂ y , . . . , ∂ y n − form a local basis for V ( X ), i.e. restrictions of elements of V ( X ) to the coordinatechart can be expressed uniquely as a linear combination of these vector fields with C ∞ coefficients. We next define V b ( X ) to be the Lie algebra of C ∞ vector fieldstangent to ∂X ; in local coordinates x∂ x , ∂ y , . . . , ∂ y n − ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 7 form a local basis in the same sense. Thus, V b ( X ) is the set of all C ∞ sections of abundle, called the b-tangent bundle of X , denoted b T X . Finally, V sc ( X ) = x V b ( X )is the Lie algebra of scattering vector fields; x ∂ x , x∂ y , . . . , x∂ y n − form a local basis now. Again, V sc ( X ) is the set of all C ∞ sections of a bundle,called the scattering tangent bundle of X , denoted sc T X . We write Diff b ( X ),resp. Diff sc ( X ), for the algebra of differential operators generated by V b ( X ), resp. V sc ( X ), over C ∞ ( X ); these are thus finite sums of finite products of vector fieldsin V b ( X ), resp. V sc ( X ), with C ∞ ( X ) coefficients. The dual bundles of b T X , resp. sc T X are b T ∗ X and sc T ∗ X , and are called the b- and the scattering cotangentbundles, respectively; locally they are spanned by dxx , d y , . . . , dy n − , resp. dxx , d y x , . . . , dy n − x . We let S k ( X ) , the space of symbols of order k , consist of functions f such that x k Lf ∈ L ∞ ( X ) for all L ∈ Diff b ( X ) . We note, in particular, that x ρ C ∞ ( X ) ⊂ S − ρ ( X )since Diff b ( X ) ⊂ Diff( X ). As Diff b ( X ) (a priori acting, say, on tempered distribu-tions) preserves S k ( X ), and one can extend Diff b ( X ) and Diff sc ( X ) by “generalizingthe coefficients:” S k Diff m b ( X ) = { X j a j Q j : a j ∈ S k ( X ) , Q j ∈ Diff m b ( X ) } , with the sum being locally finite, and defining S k Diff m sc ( X ) similarly. In particular, x k Diff m b ( X ) ⊂ S − k Diff m b ( X ) , x k Diff m sc ( X ) ⊂ S − k Diff m sc ( X ) . Then Q ∈ S k Diff m sc ( X ), Q ′ ∈ S k ′ Diff m ′ sc ( X ) gives QQ ′ ∈ S k + k ′ Diff m + m ′ sc ( X ), andthe analogous statement for S k Diff m b ( X ) also holds.In this paper, we are concerned with scattering metrics, that is to say, smoothmetrics on X ◦ that, near the boundary, take the form(2.1) g = dx x + hx + g with h = h ( x, y, dy ) a family of metrics on ∂X, and(2.2) g ∈ S − ν ( X ; sc T ∗ X ⊗ sc T ∗ X )for some ν > g is a positive definite inner product onthe fibers of sc T X ; long-range metrics like this were considered in [22]. We referto Section 2 and the beginning of Section 3 of [22] for more details. Note thatsubstituting r = x − gives form 1.1 described above when g = 0. We let L ( X )denote the space of functions square integrable with respect to the metric density;locally near a point in the boundary, this is equivalent to using the density x − ( n +1) dx dy . . . dy n − . Note that the vector fields in V sc ( X ) are precisely those with bounded length withrespect to g. Correspondingly, a typical example of an element of Diff ( X ) is ANDRAS VASY AND JARED WUNSCH the Laplace-Beltrami operator, ∆ g = d ∗ d . We also permit short range potentialperturbations of ∆ g , namely(2.3) V ∈ S − − ν ( X ) , ν > , V ≥ V ≥ , though the arguments also work for potentials with small negative parts; see [22,Footnote 5].2.2. Estimates.
In this section we employ some of the results of [22] to obtain thefollowing. Let ψ ∈ C ∞ c ( R ) andΨ H = ψ ( H (∆ g + V )) , ∆ = ∆ g . Proposition 2.1.
Suppose n ≥ , g and V as in (2.1) , (2.2) , (2.3) . For ≤ s , ≤ ρ , s + ρ < min(2 , n/ , L ∈ Diff ( X ) , Lx s + ρ Ψ H x − s ≤ CH − ρ , H > . While the spectral cutoff is in the “(anti-)semi-classical” operator H (∆ g + V )with large parameter, we emphasize that the operator L in the above propositionis simply a differential operator without parameter. As first-order scattering differ-ential operators are spanned over C ∞ ( X ) by x ∂ x , ∂ y , and the constant function,to prove the proposition, it suffices to check for these particular values of L. Equiv-alently, we can simply use the vector-valued L = ∇ g , the gradient with respect tothe scattering metric, as well as L = 1 . Note that for ρ = 0 and s = 0 Proposition 2.1 follows automatically from thefunctional calculus and elliptic regularity; for ρ = 0 and s = 1 this is proved in [22],hence it follows by interpolation for ρ = 0, 0 ≤ s ≤
1. Thus, if ρ = 0, we need todeal with 1 < s < n ≥
4, and 1 < s < / n = 3.Now, with ˜ ψ ∈ C ∞ c ( C ) an almost analytic extension of ψ , and R ( z ) = ( H (∆ g + V ) − z ) − we have Lx s + ρ Ψ H x − s = 12 π Z ∂ ˜ ψ ( z ) Lx s + ρ R ( z ) x − s dz d ¯ z. (2.4)Thus, we only need to obtain uniform bounds on Lx s + ρ R ( z ) x − s to prove the propo-sition.We begin with a preliminary lemma, which is an analogue of [22, Proposition 4.3],where it was proved in the range 0 ≤ s < / n ≥ Lemma 2.2.
For ≤ s < min(1 , ( n − / , k x s ∇ g u k ≤ C k (∆ g + V ) u k (1+ s ) / k u k (1 − s ) / Proof.
By [22, Proposition 4.3], we may assume n ≥
4; then the range in the lemmais 0 ≤ s <
1. By [22, Equation (A.5)], we have for 0 ≤ s < ( n − / k x s ∇ g u k ≤ C k (∆ g + V ) u k / k x s u k / . Moreover, by [22, Corollary 3.5], we have for 0 < s < ( n − /
2, 0 ≤ θ ≤ k x s + θ u k ≤ C k x s ∇ u k θ k x s u k − θ . ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 9
Applying this with 0 < θ = s <
1, we obtain that k x s u k ≤ C k x s ∇ g u k s k x s u k − s . Substituting into (2.5) yields(2.7) k x s ∇ g u k − s/ ≤ C k (∆ g + V ) u k / k x s u k (1 − s ) / . Now using (2.6) with 0 in place of s and s in place of θ yields k x s u k ≤ C k∇ g u k s k u k − s ≤ C k (∆ g + V ) u k s/ k u k − s/ , with the second inequality arising from k∇ g u k ≤ k (∆ g + V ) u k k u k , which is (2.5) with s = 0 . Substitution into (2.7) yields(2.8) k x s ∇ g u k − s/ ≤ C k (∆ g + V ) u k / s (1 − s ) / k u k (1 − s )(1 − s/ / . Since 1 / s (1 − s ) / − s/ s ) /
2, raising both sides to the power (1 − s/ − yields k x s ∇ g u k ≤ C k (∆ g + V ) u k (1+ s ) / k u k (1 − s ) / . This finishes the proof. (cid:3)
Now, letting u = R ( z ) f , and writing(∆ g + V ) u = H − ( H (∆ g + V ) − z ) u + H − zu, we deduce from Lemma 2.2 that for 0 ≤ s < min(2 , ( n − / k x s ∇ g R ( z ) f k ≤ C k H − ( f + zR ( z ) f ) k (1+ s ) / k R ( z ) f k (1 − s ) / . Since k R ( z ) f k ≤ | Im z | − k f k , this gives k x s ∇ g R ( z ) f k ≤ CH − (1+ s ) (1 + | z | / | Im z | ) (1+ s ) / | Im z | − (1 − s ) / k f k (2.9) ≤ CH − (1+ s ) | z | (1+ s ) / | Im z | − k f k . (2.10)Using the Hardy/Poincar´e inequality [22, Proposition 3.4], we deduce the followingextension of [22, Proposition 4.5] (where it was proved for 0 ≤ s < /
2, i.e. theresult below is only an improvement if n ≥ Corollary 2.3.
Suppose ≤ s < min(1 , ( n − / , L ∈ S − − s Diff ( X ) . Then k LR ( z ) f k ≤ CH − (1+ s ) | z | (1+ s ) / | Im z | − k f k . We can extend the range of s when L is zero’th order by interpolating the case s = 0 above, namely k xR ( z ) f k ≤ CH − | z | / | Im z | − k f k , with the estimate k R ( z ) f k ≤ | Im z | − k f k to obtain the following result (in whichthe range 1 ≤ s < min(2 , n/
2) is a special case of Corollary 2.3, and it is the range0 ≤ s ≤ Corollary 2.4.
Suppose ≤ s < min(2 , n/ . Then k x s R ( z ) f k ≤ CH − s | z | s/ | Im z | − k f k . As we are interested in the functional calculus for compactly supported func-tions, we suppress the large z behavior, and work with z in a compact set. As aconsequence of the preceding two corollaries we conclude: Corollary 2.5.
For ≤ s < min(2 , n/ , L ∈ S − s Diff ( X ) , z in a compact set, (2.11) k LR ( z ) f k ≤ CH − s | Im z | − k f k , uniformly in z and H ≥ .Proof. If 1 ≤ s , then L ∈ S − s Diff ( X ) ⊂ S − s Diff ( X ) and Corollary 2.3 provesthe result.On the other hand, if 0 ≤ s <
1, one can write L = aV + b , a, b ∈ S − s ( X ), V ∈ V sc ( X ) = x V b ( X ), so aV ∈ S − − s Diff ( X ), so Corollary 2.3 gives k aV R ( z ) f k ≤ CH − − s | Im z | − k f k . Furthermore, Corollary 2.4 yields k bR ( z ) f k ≤ CH − s | Im z | − k f k . Combining these two estimates proves the corollary. (cid:3)
Now we consider conjugates of R ( z ). Below sometimes n = 3 and n ≥ n = 3 imposes in the preced-ing results: min(2 , n/
2) = 3 /
2, rather than 2, in that case. First we prove a resultfor conjugation by small powers of x (namely, x s , 0 ≤ s ≤ Lemma 2.6.
Suppose n ≥ . Suppose ≤ s ≤ , ≤ ρ , s + ρ < min(2 , n/ , andif n = 3 then in addition ρ < . For L ∈ Diff ( X ) , and z in a compact set, (2.12) k Lx ρ + s R ( z ) x − s k ≤ CH − ρ | Im z | − uniformly as H → + ∞ .Proof. The case s = 0 follows from Corollary 2.5. Note that the conditions imply0 ≤ ρ < n ≥ n = 3, assume that 1 / < s ≤
1; if n ≥ Lx s + ρ R ( z ) x − s = Lx ρ R ( z ) + Lx s + ρ [ R ( z ) , x − s ]= Lx ρ R ( z ) − Lx s + ρ R ( z )[ H (∆ + V ) , x − s ] R ( z ) . For L ∈ Diff ( X ), by Corollary 2.5, as 0 ≤ ρ < min(2 , n/ k Lx ρ R ( z ) k ≤ CH − ρ | Im z | − As for the second term on the right hand side, we note that [∆ +
V, x − s ] ∈ x − s Diff ( X ) . Consequently, we may employ Corollary 2.3, using 0 < s ≤ ≤ − s <
2) if n ≥
4, resp. 1 / < s ≤ ≤ − s < /
2) if n = 3. This yields k [∆ + V, x − s ] R ( z ) k ≤ CH − (2 − s ) | Im z | − . On the other hand, as 0 ≤ s + ρ < min(2 , n/ k Lx s + ρ R ( z ) k ≤ CH − ( s + ρ ) | Im z | − , so k Lx s + ρ R ( z )[ H (∆ + V ) , x − s ] R ( z ) k ≤ CH − ρ | Im z | − . ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 11
Combining these results proves (2.12) if 0 < s ≤ n ≥
4, and also if 1 / < s ≤ n = 3, and completes the proof in these cases.It remains to deal with n = 3, and 0 < s ≤ /
2. This follows by interpolationbetween s = 0 and s > /
2, noting that given 0 ≤ ρ < s ′ ∈ (1 / , ρ + s ′ < /
2; we then interpolate between 0 and this value s ′ . This laststep is the reason for the restriction ρ <
1; if ρ ≥
1, the desired s ′ does not exist.This completes the proof of the Lemma. (cid:3) We now extend the range of allowable exponents s by an inductive argument: Proposition 2.7.
Suppose n ≥ . Suppose ρ ≥ , L ∈ Diff ( X ) . For ≤ s ≤ s + ρ < min(2 , n/ , (2.13) k Lx s + ρ R ( z ) x − s k ≤ CH − ρ | Im z | − . Proof. If s = 0, the result follows from Corollary 2.5. If n ≥
4, 0 < s <
1, it followsfrom Lemma 2.6.Assume now that 1 ≤ s < min(2 , n/ n = 3, let ρ ′ = s/
2; if n ≥
4, let ρ ′ = s −
1. In either case, 0 ≤ ρ ′ <
1; if n = 3 then in addition ρ ′ > /
2. Let L ′ = Lxρ ∈ x ρ Diff ( X ) . Write L ′ x s R ( z ) x − s = L ′ R ( z ) + L ′ x s [ R ( z ) , x − s ]= L ′ R ( z ) − (cid:16) L ′ x s R ( z ) x − ρ ′ (cid:17) (cid:16) x ρ ′ [ H (∆ + V ) , x − s ] R ( z ) (cid:17) . First, by Corollary 2.5 since ρ ′ < min( n/ , , k L ′ R ( z ) k ≤ CH − ρ ′ | Im z | − . Next, we apply (2.12), with s = ρ ′ and ρ replaced by ρ + s − ρ ′ . The hypothesesare satisfied as ρ ′ + ( ρ + s − ρ ′ ) = ρ + s < min( n/ ,
2) and as 0 ≤ ρ + s − ρ ′ < n = 3 since ρ + s < / ρ ′ = s/ s ≥
1, resp. 0 ≤ ρ + s − ρ ′ = ρ + 1 if n ≥ k L ′ x s R ( z ) x − ρ ′ k ≤ CH − s − ρ + ρ ′ | Im z | − . On the other hand, x ρ ′ [∆ + V, x − s ] ∈ x − s + ρ ′ Diff ( X ), so by Corollary 2.3 (takinginto account that if n ≥
4, 2 − s + ρ ′ = 1, and if n = 3 then 1 ≤ − s + ρ ′ < / k x ρ ′ [(∆ + V ) , x − s ] R ( z ) k ≤ CH − s − ρ ′ | Im z | − . Combining these results, we have shown that k L ′ x s R ( z ) x − s k ≤ CH − ρ | Im z | − , completing the proof if 1 ≤ s < min(2 , n/ n ≥
4, we already covered the caseof s < n ≥
4, the proof is complete.So suppose that n = 3. If 0 < s <
1, and 0 ≤ ρ < /
2, then one can interpo-late between s = 0 and s = 1 with the same (fixed) value of ρ to obtain (2.13),completing the proof if 0 ≤ ρ < / s , 0 ≤ s < / s + ρ ′ < / ≤ s + ρ ′ < / s = 0 and ρ = 0 to obtain the full result. (cid:3) In view of (2.4), Proposition 2.7 proves Proposition 2.1.We need a second, much less delicate, property of spectral cutoffs. Note thathere we do not need to work with low energies: the cutoff ψ is fixed, and there is no H in the statement of the proposition. (If we worked with ψ ( H (∆ + V )), H could be traced through the argument given below to yield polynomially growingbounds in H , i.e. one can gain decay in t at the cost of losing powers of H .) Proposition 2.8.
Suppose that φ, χ ∈ C ∞ c ( R ) and supp(1 − χ ) ∩ supp φ = ∅ , ψ ∈ C ∞ c ( R ) . Also let r = x − . Then for L ∈ Diff ( X ) , N ∈ N , k Lφ ( r/t ) ψ (∆ + V )(1 − χ ( r/t )) k L ( L ,L ) ≤ C N t − N , t ≥ . Proof.
We first remark that[ x ∂ x , φ (1 / ( xt ))] = − t − φ ′ (1 / ( xt )) , while xD y j and elements of C ∞ ( X ) commute with the multiplication operator by φ (1 / ( xt )). Thus, by an inductive argument, for Q ∈ Diff m sc ( X ), we have(2.14) [ Q, φ (1 / ( xt ))] = X | α | + j ≤ m − X k f j,α,k ( t ) φ j,α,k (1 / ( xt )) a j,α,k ( x D x ) j ( xD y ) α where the sum over k is finite, f j,α,k ∈ S − ([1 , ∞ )) (i.e. is C ∞ and is a symbol oforder − a j,α,k ∈ C ∞ ( X ), φ j,α,k ∈ C ∞ c ( R ) and supp φ j,α,k ⊂ supp dφ .Equivalently, we may put all of the factors φ j,α (1 / ( xt )) on the right (at the cost ofchanging these factors as well as the other coefficients), so(2.15) [ Q, φ (1 / ( xt ))] = X | α | + j ≤ m − X k ˜ f j,α,k ( t )˜ a j,α,k ( x D x ) j ( xD y ) α ˜ φ j,α,k (1 / ( xt )) , with the tilded objects having the same properties as the untilded ones above.For each t ≥ Lφ ( r/t ) = φ ( r/t ) L + [ L, φ ( r/t )] and use that forany Q ∈ Diff m sc ( X ), Qψ (∆ + V ) is bounded on L (by elliptic regularity), so animmediate consequence of (2.14) is that Lφ ( r/t ) ψ (∆ + V )(1 − χ ( r/t )) is a boundedoperator on L which is uniformly bounded in t ≥
1; we now must obtain improved(decaying) bounds in t .To do so, we work with the resolvent R ( z ) = (∆ + V − z ) − , Im z = 0, and notethat if ρ ∈ C ∞ c ( R ) and 1 − χ have disjoint support, so ρ (1 − χ ) = 0, then commuting ρ through the resolvent, ρ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) = − [ R ( z ) , ρ (1 / ( xt ))](1 − χ (1 / ( xt )))= R ( z )[∆ + V, ρ (1 / ( xt ))] R ( z )(1 − χ (1 / ( xt ))) . (2.16)By (2.15), this has the form(2.17) X | α | + j ≤ X k f j,α,k ( t ) R ( z ) a j,α,k ( x D x ) j ( xD y ) α ρ j,α,k (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) , with supp ρ j,α,k ⊂ supp dρ . Thus, ρ j,α,k (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) is of the sameform as the left hand side of (2.16), so (2.16) can be further expanded by expandingthese last three factors in (2.17) as in (2.16). Note that as we expand, each timewe obtain a factor like f j,α,k ∈ S − ([1 , ∞ )); these commute with all other factors,and after N iterations, the product of these is in S − N ([1 , ∞ )). Inductively, doing N iterations, we deduce that φ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) is a sum of terms of theform(2.18) f ( t ) R ( z ) A R ( z ) A . . . R ( z ) A N ˜ φ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) , ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 13 with A j ∈ Diff ( X ), ˜ φ ∈ C ∞ c ( R ) with supp ˜ φ ⊂ supp dφ , and f ∈ S − N ([1 , ∞ )), andone can also interchange A N and ˜ φ (1 / ( xt )) if convenient (at the cost of obtainingdifferent operators in the same class).But k A j R ( z ) k ≤ C | Im z | − for z in a compact set, and similarly for LR ( z )when L ∈ Diff ( X ) (see Corollary 2.5 with H fixed) so using that ˜ φ and 1 − χ are uniformly bounded in sup norm, hence as bounded operators on L , we deducethat for L ∈ Diff ( X ), k Lφ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) k ≤ Ct − N | Im z | − N − . The Cauchy-Stokes formula, with ˜ ψ ∈ C ∞ c ( C ) an almost analytic extension of ψ , Lφ (1 / ( xt )) ψ (∆ + V )(1 − χ (1 / ( xt )))= 12 π Z ∂ ˜ ψ ( z ) Lφ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) dz d ¯ z, (2.19)now immediately proves the proposition. (cid:3) In fact, it is useful to add a weight in x as well: Proposition 2.9.
Suppose that φ, χ ∈ C ∞ c ( R ) and supp(1 − χ ) ∩ supp φ = ∅ , ψ ∈ C ∞ c ( R ) . Also let r = x − . Then for L ∈ Diff ( X ) , N ∈ N , m ∈ N k Lr m φ ( r/t ) ψ (∆ + V )(1 − χ ( r/t )) r m k L ( L ,L ) ≤ C N t − N , t ≥ . Proof.
First, the r m = x − m in front of ψ (∆ + V ) is harmless since on the supportof φ it is bounded by t m , so it can be absorbed into t − N on the right hand sideif we increase N. Moreover, for each t , Qx m ψ (∆ + V ) x − m is bounded for Q ∈ Diff ( X ), as is immediate from the scattering calculus of Melrose as ψ (∆ + V )is a pseudodifferential operator of order −∞ . However, there is a simple directargument: via the Cauchy-Stokes formula this reduces to a boundedness statementfor Qx m R ( z ) x − m . This in turn is shown by commuting x − m through R ( z ), muchas we commuted φ ( r/t ) above, in this case gaining a power of x (instead of t − )each time we do a commutation, so in m steps we reduce to a product of m + 1resolvents and bounded functions, giving a bound(2.20) k Qx m R ( z ) x − m k ≤ C | Im z | − m − . Thus, as in the previous proposition, we only need to prove the decaying uni-form estimate in t . (Strictly speaking, one has to regularize, replacing x − m by(1 + ǫx − ) − m x − m , which is bounded for ǫ >
0, and let ǫ →
0, in which case thecommutator argument presented above gives uniform control of the terms.)We thus proceed as above, so by (2.18) we only need to prove that A N ˜ φ (1 / ( xt )) R ( z )(1 − χ (1 / ( xt ))) x − m is L bounded. As in the fixed t setting, it is convenient to insert a factor of x m onthe left, using that x − m . t m on the support of ˜ φ , and thus, commuting ˜ φ through A N , and using the uniform boundedness of ˜ φ and 1 − χ in the sup norm, we arereduced to showing that A ′ N x m R ( z ) x − m is bounded (with a polynomial bound in | Im z | − ). But this is exactly the content of (2.20), completing the proof. (cid:3) Global Morawetz estimate
In this section, we return to the notation r = 1 /x to facilitate comparisons withthe usual method of Morawetz estimates. To simplify the computation, we assume g = 0 with the notation of (2.1)–(2.2). Then, in the paragraph of (3.7) we reinstatearbitrary g (with ν > g = g = dr + r h (1 /r, y, dy )with r ∈ ( r , ∞ ) . Then we compute(3.2) ∂ ∗ r = − ∂ r − ∂ r (log( r n − √ h )) = − ∂ r − n − r − ∂ r log h (1 /r )= − ∂ r − n − r + e ∈ r − Diff ( X )with e ∈ S − ( X ) . We as usual may write, in the ends of X where r ≫ , ∆ g = ∂ ∗ r ∂ r + r − ∆ Y ∈ r − Diff ( X )where ∆ Y is the family of Laplacians on the boundary at infinity defined by thefamily of metrics h (1 /r, • ) . Now in general, for F ( r ) ∈ C ∞ ([0 , ∞ )) we let(3.3) A F = (1 / F ( r ) ∂ r − ∂ ∗ r F ( r )) , which is thus skew-adjoint. The principal symbol of the commutator of ∆ g with A F is straightforward to compute, and one deduces that [∆ , A F ] and 2 ∂ ∗ r F ′ ( r ) ∂ r +2 r − F ( r )∆ Y have the same principal symbol, so their difference is first order. Sincethey are both formally self-adjoint and real, the same holds for the difference, whichis thus zero’th order, and can be computed by applying it to the constant function1: (cid:0) [∆ , A F ] − (2 ∂ ∗ r F ′ ( r ) ∂ r + 2 r − F ( r )∆ Y ) (cid:1) g A F −
12 ∆ g ∂ ∗ r ( F ( r ))We note some particular instances of this computation. Let ϑ ( r ) be a functionon R such that ϑ ( r ) = ( r ≤ / r ≥ , and ϑ ′ ≥ , ϑ ′ ( r ) ≥ / ≤ r ≤ . Let Λ ≫ κ be a small constant. We compute the following as zero’th orderterms for these various cases: F ( r ) = r s = ⇒ −
12 ∆ g ∂ ∗ r ( F ( r )) = n −
12 ( s − n + s − r s − + O ( r s − ) ,F ( r ) = (1 − / log r ) = ⇒ −
12 ∆ g ∂ ∗ r ( F ( r )) = (1 + O (1 / log r )) ( n − n − r F ( r ) = κ ϑ ( r/ Λ) = ⇒ −
12 ∆ g ∂ ∗ r ( F ( r )) = κ O ( r − )(with the “big-Oh” term in the last case being uniformly bounded as Λ → ∞ ). ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 15
Thus in particular, we may let F ( r ) = r s and (abusing notation) denote A F by A s in this case. Then(3.4)[∆ g + V, A s ] = 2 s∂ ∗ r r s − ∂ r + 2 r s − ∆ Y − n −
12 ( s − n + s − r s − + r s − Q Y + f, where Q Y denotes an operator of second order involving only derivatives in Y withcoefficients in S ( X ) and f ∈ S s − − ν ( X ) . The two principal terms have the same(non-negative!) sign if s ≥
0, with the sign being definite if s >
0; if n ≥
3, thezero’th order term has the same sign if in addition s ≤
1; the sign is definite if s <
1, and in case n = 3, s > A F with F ( r ) = (1 − / log r ) . We let A ′ denote the operator in this case, and compute A ′ = (1 / (cid:18)(cid:0) − r (cid:1) A + A (cid:0) − r (cid:1)(cid:19) , where A ≡ A = 12 (cid:0) ∂ r − ∂ ∗ r (cid:1) . Then we have(3.5)[∆ g + V, A ′ ] = 2 ∂ ∗ r r (log r ) ∂ r + (1 + e )2 r − ∆ Y + (1 + e ) ( n − n − r + r − Q Y , with e i = O ((log r ) − ) a symbol in S ( X ) . (We have recycled the notation Q Y todenote different operators of the same form.)Finally, in obtaining ℓ ∞ – ℓ bounds, we will employ F ( r ) = 1 − r + κ ϑ (cid:0) r Λ (cid:1) ;with κ a (small) positive constant. Let A ′′ denote the corresponding operator.Thus [∆ g + V, A ′′ ] =2 ∂ ∗ r (cid:0) κ Λ ϑ ′ ( r/ Λ) + 1 r (log r ) (cid:1) ∂ r + (1 + e )2 r − ∆ Y + (1 + e ) ( n − n − r + r − Q Y , (3.6)with e = O ((log r ) − ) a symbol in S ( X ) and e = O ((log r ) − ) + κ O (1) a symbolin S ( X ) , bounded by 1 / r ≫ κ ≪ . We now return to arbitrary metric g = g + g , with g of the warped productform (3.1). We write ∆ g for the Laplacian of g and B ∗ , for the adjoint of anoperator B with respect to g , while B ∗ is written for the adjoint with respect to g . Thus, in (3.2)–(3.5), all adjoints are of the ∗ , type, and all Laplacians are thatof g . Since A s and A ′ depend on the metric via the adjoints, we write A s, and A ′ for the g versions. Now, as(3.7) g ∈ S − ν ( X ; sc T ∗ X ⊗ sc T ∗ X ) = S − ν ( X ; b T ∗ X ⊗ b T ∗ X ) , we have √ g = √ g (1 + ˜ g ) , ˜ g ∈ S − ν ( X ) , and thus ∂ ∗ r − ∂ ∗ , r ∈ S − − ν ( X ) , i.e. the effect on (3.2) is that e is replaced by a slightly less decaying symbol.Correspondingly, A s − A s, ∈ S s − − ν Diff ( X ) , A ′ − A ′ ∈ S − − ν Diff ( X ) , A ′′ − A ′′ ∈ S − − ν Diff ( X ) . Also, ∆ g − ∆ g ∈ S − − ν Diff ( X ) . Combining these, and expanding the commutators,(3.8) [∆ g + V, A s ] − [∆ g + V, A s, ] ∈ S − s − ν Diff ( X ) , [∆ g + V, A ′ ] − [∆ g + V, A ′ ] ∈ S − − ν Diff ( X )[∆ g + V, A ′′ ] − [∆ g + V, A ′′ ] ∈ S − − ν Diff ( X ) , as always with uniform estimates as Λ → ∞ in the A ′′ case.In order to make the foregoing into a global computation, we need to add a cutoffthat localizes near infinity. So let χ ( r ) be a cutoff function equal to 0 for r < r > . Let χ ( r ) = χ ( r/R ) with R ≫ . Then(3.9)[∆ g + V, χ ( r ) A s ] = 2 s∂ ∗ r χ ( r ) r s − ∂ r + 2 r s − χ ( r )∆ Y + n −
12 (1 − s )( n + s − r s − + ∇ ∗ X (1 − χ ( r )) ∇ X + r s − E s + r s − − ν ˜ E s , where E s ∈ Diff ( X ), ˜ E s ∈ Diff ( X ), and(3.10)[∆ g + V, χ ( r ) A ′ ] = 2 ∂ ∗ r χ ( r ) r (log r ) ∂ r + (1 + e )2 r − χ ( r )∆ Y + (1 + e ) ( n − n − r + ∇ ∗ X (1 − χ ( r )) ∇ X + r − E + r − − ν ˜ E where E ∈ Diff ( X ), ˜ E ∈ Diff ( X ) (and are different from the E s , ˜ E s in (3.9)),and e i = O ((log r ) − ) satisfy symbol estimates. Since on any compact set, theterms with e i can be absorbed into E s , we may assume here that | e i | < / e i > / . For the same reason, by shifting a large compactly supported part of the ˜ E s termin (3.9) into the faster-decaying E s we may assume that ˜ E s can be absorbed intothe first three terms of (3.9) without compromising their positivity: for any δ > |h r s − − ν ˜ E s v, v i|≤ δ (cid:16) s k p χ ( r ) r ( s − / ∂ r v k + 2 k r ( s − / p χ ( r ) ∇ Y v k + n −
12 (1 − s )( n + s − k r ( s − / v k (cid:17) (3.11)for all v ∈ ˙ C ∞ ( X ) (hence by density whenever the right hand side is finite), with.We will take δ = 1 /
2. Likewise, we may assume that the terms in (3.10) satisfy: |h r − − ν ˜ E v, v i|≤ δ (cid:16) k p χ ( r ) r / (log r ) ∂ r v k + 2 k r − / p χ ( r ) ∇ Y v k + ( n − n − k r − / v k (cid:17) . (3.12) ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 17
We will take δ = 1 / A ′′ , but with the additionof an extra ∂ r term:(3.13) [∆ g + V, χ ( r ) A ′′ ] =2 ∂ ∗ r (cid:0) χ ( r ) r (log r ) + κ Λ ϑ ′ ( r/ Λ) (cid:1) ∂ r + (1 + e )2 r − χ ( r )∆ Y + (1 + e ) ( n − n − r + ∇ ∗ X (1 − χ ( r )) ∇ X + r − E + r − − ν ˜ E , with error terms estimated as above (provided κ is taken sufficiently small).We now consider the pairing(3.14) Z T h χ ( r ) A ′ ( (cid:3) + V ) u, u i dt. While we will manipulate this expression for an arbitrary solution u, we remarkthat the calculations in the following paragraph are unchanged if we replace u byΨ H u ≡ ψ ( H (∆ g + V )) u, which indeed we will do below.We first remark that moving χ ( r ) and A ′ to the right slot of the pairing (3.14)and inserting the weight r / (log r ), resp. its reciprocal, in the two slots, we obtain,for γ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T h χ ( r ) A ′ ( (cid:3) + V ) u, u i dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T k r / (log r )( (cid:3) + V ) u k dt ! / Z T k r − / (log r ) − ( A ′ ) ∗ χ ( r ) u k dt ! / ≤ γ − Z T k r / (log r )( (cid:3) + V ) u k dt + γ Z T k r − / (log r ) − ( A ′ ) ∗ χ ( r ) u k dt, (3.15)where h· , ·i denotes the spatial inner product. On the other hand, integrating (3.14)twice by parts yields(3.16) − h χ ( r ) A ′ u t , u i| T + h χ ( r ) A ′ u, u t i T − Z T h [ χ ( r ) A ′ , ∆ g + V ] u, u i dt. Let E ( s ) = (cid:0) k∇ g u k + k ∂ t u k + (cid:13)(cid:13)(cid:13) √ V u (cid:13)(cid:13)(cid:13) (cid:1) | t = s denote the energy norm at fixed time. By the Hardy/Poincar´e inequality, we have k χA ′ u ( t ) k . E ( t ) . We further compute, as usual, that E ′ ( t ) = 2 Re h u t , ( (cid:3) + V ) u i ≤ E ( t ) / k ( (cid:3) + V ) u k . Hence E ( T ) / ≤ E (0) / + Z T k ( (cid:3) + V ) u k dt . E (0) / + Z T k ( (cid:3) + V ) u k dt Consequently, the first two terms in (3.16) are bounded by a multiple of E (0) + Z T k ( (cid:3) + V ) u k dt ! uniformly in t. Thus, (3.10), (3.12) and (3.15) now allow us to estimate (absorbing lower-orderterms into positive ones, and choosing γ small so that the second term on the righthand side of (3.15) can be absorbed into the left hand side below):(3.17) Z T (cid:10) ∂ ∗ r r − χ ( r ) log( r ) − ∂ r u, u (cid:11) + (cid:10) r − u, u (cid:11) + (cid:10) r − χ ( r )∆ Y u, u (cid:11) + h∇ ∗ X χ ( r ) ∇ X u, u i dt . Z T (cid:10) r − E u, u (cid:11) dt + E (0) + (cid:0) Z T k ( (cid:3) + V ) u k dt (cid:1) . We apply this estimate to spectrally localized data, i.e. replace u by Ψ H u , asindicated at the beginning of the previous paragraph. We state the following lemmamore generally than is immediately necessary, in the n ≥ Lemma 3.1.
Suppose n ≥ . If E ∈ Diff ( X ) and u is as above, then for ≤ ρ , ≤ s , ≤ ρ + s < min(2 , n/ , (cid:12)(cid:12)(cid:12)D r − s + ρ ) E Ψ H u, Ψ H u E(cid:12)(cid:12)(cid:12) . H − ρ (cid:13)(cid:13) r − s Ψ H u (cid:13)(cid:13) . Proof.
It suffices, by rearranging the LHS (as commutator terms require the sameform of estimates) to show that for L ∈ Diff ( X ) , (cid:13)(cid:13)(cid:13) r − ( s + ρ ) L Ψ H u (cid:13)(cid:13)(cid:13) . H − ρ (cid:13)(cid:13) r − s Ψ H u (cid:13)(cid:13) . Let φ ∈ C ∞ c ( R ) be identically 1 on supp ψ . By Proposition 2.1, H ρ r − ( s + ρ ) Lφ ( H (∆ g + V )) r s is uniformly bounded in H , so for all v , (cid:13)(cid:13)(cid:13) H ρ r − ( s + ρ ) Lφ ( H (∆ + V )) v (cid:13)(cid:13)(cid:13) . (cid:13)(cid:13) r − s v (cid:13)(cid:13) . Applying this with v = Ψ H u completes the proof of the lemma. (cid:3) Taking s = 3 / ρ = (1 − ǫ ) / , < ǫ <
1, which we may if n ≥
4, we deduce: If preferred, we could of course replace the norm on the inhomogeneity by the weighted L spacetime norm Z T (cid:13)(cid:13)(cid:13) (2 + t ) / log(2 + t )( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) dt. ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 19
Lemma 3.2. If n ≥ , E ∈ Diff ( X ) and u is as above, then for < ǫ < , (cid:12)(cid:12)(cid:10) r − ǫ E Ψ H u, Ψ H u (cid:11)(cid:12)(cid:12) . H − ǫ (cid:13)(cid:13)(cid:13) r − / Ψ H u (cid:13)(cid:13)(cid:13) . As a consequence, we can estimate, a fortiori, (cid:10) r − E Ψ H u, Ψ H u (cid:11) . H − ǫ (cid:13)(cid:13)(cid:13) r − / Ψ H u (cid:13)(cid:13)(cid:13) so that for H sufficiently large, this term may be absorbed in the main term on theleft of (3.17). Thus, for H sufficiently large, Z T (cid:13)(cid:13)(cid:13) r − / (log r ) − √ χ ( r ) ∂ r Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) r − / Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) r − / √ χ ( r ) ∇ Y Ψ H u (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)p − χ ∇ X Ψ H u (cid:13)(cid:13)(cid:13) dt . E (0) + (cid:0) Z T k ( (cid:3) + V ) u k dt (cid:1) , with constants independent of T. This concludes the proof of Theorem 1.1. (cid:3)
To prove Theorem 1.1’ we proceed similarly, but with A ′′ replacing A ′ . We nowlet(3.18) Υ k = { r ∈ [2 k , k +1 ] } denote a dyadic decomposition in radial shells. Then we obtain by the same argu-ment(3.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T h χ ( r ) A ′′ ( (cid:3) + V ) u, u i dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T D r / ( (cid:3) + V ) u, r − / ( A ′′ ) ∗ χ ( r ) u E dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T X k Z Υ k (cid:12)(cid:12)(cid:12) r / ( (cid:3) + V ) u (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) r − / ( A ′′ ) ∗ χ ( r ) u (cid:12)(cid:12)(cid:12) dg dt ≤ X k (cid:13)(cid:13)(cid:13) r / ( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) L ([0 ,T ] × Υ k ) (cid:13)(cid:13)(cid:13) r − / ( A ′′ ) ∗ χ ( r ) u (cid:13)(cid:13)(cid:13) L ([0 ,T ] × Υ k ) ≤ (cid:13)(cid:13)(cid:13) r / ( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) ℓ ( N ; L ([0 ,T ] × Υ k )) (cid:13)(cid:13)(cid:13) r − / ( A ′′ ) ∗ χ ( r ) u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 ,T ] × Υ k )) ≤ γ − (cid:13)(cid:13)(cid:13) r / ( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) ℓ ( N ; L ([0 ,T ] × Υ k )) + γ (cid:13)(cid:13)(cid:13) r − / ( A ′′ ) ∗ χ ( r ) u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 ,T ] × Υ k ))0 ANDRAS VASY AND JARED WUNSCH Thus, for γ > Z T (cid:16)(cid:10) ∂ ∗ r r − χ ( r ) log( r ) − ∂ r u, u (cid:11) + D ∂ ∗ r κ Λ ϑ ′ ( r/ Λ) χ ( r ) ∂ r u, u E + (cid:10) r − u, u (cid:11) + (cid:10) r − χ ( r )∆ Y u, u (cid:11) + h∇ ∗ X χ ( r ) ∇ X u, u i (cid:17) dt . Z T (cid:10) r − E u, u (cid:11) dt + E (0) + Z T k ( (cid:3) + V ) u k dt ! + γ − (cid:13)(cid:13)(cid:13) r / ( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) ℓ ( N ; L ([0 ,T ] × Υ k )) + γ (cid:13)(cid:13)(cid:13) r − / ( A ′′ ) ∗ χ ( r ) u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 ,T ] × Υ k )) . Now we take the supremum of the LHS over Λ ∈ { k , k ∈ N } to obtain(3.21) Z T (cid:16)(cid:10) ∂ ∗ r r − χ ( r ) log( r ) − ∂ r u, u (cid:11) + (cid:10) r − u, u (cid:11) + (cid:10) r − χ ( r )∆ Y u, u (cid:11) + h∇ ∗ X χ ( r ) ∇ X u, u i (cid:17) dt + (cid:13)(cid:13)(cid:13) χ ( r ) r − / ∂ r u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 ,T ] × Υ k )) . Z T (cid:10) r − E u, u (cid:11) dt + E (0) + Z T k ( (cid:3) + V ) u k dt ! + γ − (cid:13)(cid:13)(cid:13) r / ( (cid:3) + V ) u (cid:13)(cid:13)(cid:13) ℓ ( N ; L ([0 ,T ] × Υ k )) + γ (cid:13)(cid:13)(cid:13) r − / ( A ′′ ) ∗ χ ( r ) u (cid:13)(cid:13)(cid:13) ℓ ∞ ( N ; L ([0 ,T ] × Υ k )) . where we have of course used the fact that r ∼ Λ on supp ϑ ′ ( r/ Λ) . Taking γ suffi-ciently small to absorb the last term in the LHS yields and applying the resultingestimate to spectrally localized data as above yields Theorem 1.1’. (cid:3) A local Morawetz estimate
In this section, we prove Theorem 1.2. We fix some ˜ ψ ∈ C ∞ c ( R ) which is identi-cally 1 on supp ψ . We write u simply in place ofΨ H u = ψ ( H (∆ + V )) u throughout this section to simplify the notation.As our estimates will be in spacetime, we also define the manifold with cornersgiven by its compactification, M = R × X, with R denoting the compactification of R to an interval and X our scatteringmanifold, regarded as usual as a manifold with boundary endowed with the singularmetric (2.1). In this section we will employ the notation k•k M to denote thespacetime L norm, and for the sake of emphasis, we will use k•k X for the spatialnorm, previously denoted simply k•k . We will likewise let ∇ X denote the gradientin the spatial directions only, previous denoted ∇ g , while letting ∇ denote thespacetime gradient. ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 21
We may assume that the δ in (1.5) is sufficiently small for convenience (as theresult is stronger then); e.g. take δ < /
4. Let φ ∈ C ∞ c ( R ), supported in [ − , − , < c < δ < φ ( µ ) = φ ( µ/c ), andlet ˜ φ ( µ ) = φ ( µ/ (3 δ )). Also, we let χ be as in Section 3, see the definition just above(3.9), so χ ≡ k = 2 κ − , s = 1 − σ, let B k,s = 12 (cid:16) φ ( r/t ) ˜ φ ( t ) ( r/t ) s t − k χ ( r ) ∂ r − ∂ ∗ r φ ( r/t ) ˜ φ ( t ) ( r/t ) s t − k χ ( r ) (cid:17) , so on the complement of supp(1 − φ ˜ φ ) (where φ ˜ φ ≡
1) we actually have B k,s = t − k − s A s with A s given by (3.3). Note that supp d ˜ φ ∩ supp φ is a compact subset of M ◦ , so wecan effectively ignore terms in which ˜ φ is commuted through differential operatorsbelow.Before proceeding, we recall that in { r > r , t > } ⊂ M , V b ( M ) is spanned by r∂ r , t∂ t and vector fields on Y , over C ∞ ( M ). We now blow up the corner ∂ M ,where both x = 0 and t − = 0, to obtain the manifold˜ M = [ M ; ∂ M ] . The blowup procedure (cf. [11, Appendix]; the simple setting of [10, Section 4.1]where a codimension 2 corner is blown up is analogous to the present case, withour t − taking the place of x ′ in [10]) serves to replace the corner t − = r − = 0 of M by its inward-pointing spherical normal bundle, in this case diffeomorphic to Y times an interval. This new boundary hypersurface will be denoted mf . We let tfdenote the “temporal infinity” face given by the lift of the set t − = 0 in M to ˜ M .
The main consequence of the blowup procedure is that it introduces r/t as a smoothfunction where it is bounded, and its differential is non-vanishing on the interior ofmf (where it is thus one of the standard coordinates, namely the variable along theinterval referred to above; others being t − and y, coordinates on Y = ∂X ; we use t − as a boundary defining function for mf). However, V b ( ˜ M ) is still spanned bythe lift of the same vector fields, except that the smooth structure is replaced by C ∞ ( ˜ M ). Correspondingly, the symbol spaces are unaffected by the blow-up. Now,the support of dφ only intersects the front face of ˜ M among all boundary faces,and r/t is bounded from both above and below by positive constants there. Thus,on the support of dφ , t − V b ( ˜ M ) is locally generated by ∂ r , ∂ t and r − ∂ Y , i.e. byvector fields corresponding to the energy space. Thus, on supp dφ, ∇ v ∈ t κ L ( M )implies v ∈ t κ +1 L ( M ), k < ( n − /
2, by a Hardy inequality, and so we obtainmore generally: The notation is short for “main face,” as for X Euclidean (diffeomorphically, not metrically),this can be identified with an open dense subset of the boundary of the radial compactification ofMinkowski space. Note that the light cone, r/t = 1, hits the boundary in the interior of mf. If r/t = 1 in mf were blown up inside ˜ M , the resulting front face is where Friedlander’s radiationfield [4] can be defined by rescaling a solution to the wave equation. Indeed, in the interior of thisfront face (in t > r − t = ( r/t − /t − becomes a smooth variable along the fibers of the frontface. ˜ M tfmfspf yr/tt − r − t − M Figure 1.
The compactified spacetime M and the blown-upspacetime ˜ M .
Lemma 4.1.
Let Υ be an open set in ˜ M with Υ ∩ ∂ ˜ M ⊂ mf ◦ . If supp v ⊂ Υ , ∇ v ∈ t κ L ( M ) = ⇒ v ∈ t κ +1 H ( ˜ M ) . Here the Sobolev space is still relative to the metric density (so L ( M ) and L ( ˜ M ) are the same, as only the smooth structure at infinity changed).Before proceeding, we caution the reader that near temporal infinity , tf, i.e. thelift of t = + ∞ on M to ˜ M , weighted versions of V b ( ˜ M ) do not give rise to thefinite length vector fields relative to dt + g (i.e. the energy space). Indeed, in theinterior of tf, one is in a compact set of the spatial slice X , and V b ( ˜ M ) is spannedby t∂ t and smooth vector fields on X , while the energy space corresponds to ∂ t andsmooth vector fields on X . In order to emphasize the structure when we stay awayfrom tf and also from spatial infinity spf, i.e. the lift of ∂X × R , we write˜ M ′ = ˜ M ◦ ∪ mf ◦ = ˜ M \ (tf ∪ spf) . Our convention below is that inner products and norms are on M unless other-wise specified by a subscript X (in which case they are on X ). The inner product on M on functions is given by integration against the density of dt − g , or equivalentlythat of dt + g . The inner product on vectors on M uses the positive definite innerproduct, based on dt + g , unless otherwise specified. The one case where we takeadvantage of the “otherwise specified” disclaimer is when we consider the indefiniteDirichlet form in (4.4), where we use the indefinite form induced by dt − g (so thedensity is positive definite, but not the pointwise pairing between vectors) in orderto use the PDE.Now,(4.1)[ ∂ t , t − k − s ] = − k + s ) t − k − s − ∂ t + ( k + s )( k + s + 1) t − k − s − ∈ t − k − s − Diff ( R t ) . Moreover, on ˜ M ′ (hence on supp dφ ), B k,s ∈ t − k − Diff ( ˜ M ′ ) , (cid:3) ∈ t − Diff ( ˜ M ′ ) = ⇒ [ (cid:3) , B k,s ] ∈ t − k − Diff ( ˜ M ′ ) . ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 23
In particular, from (3.9),[ ∂ t + ∆ g + V, B k,s ]= t − k − s φ ( r/t ) ˜ φ ( t ) (cid:16) X i Q ∗ i r s t − R i + (2 s∂ ∗ r r s − χ ( r ) ∂ r + 2 r s − χ ( r )∆ Y + n −
12 (1 − s )( n + s − r s − + ∇ ∗ X (1 − χ ( r )) ∇ X + r s − E s + r s − − ν ˜ E s ) (cid:17) + F k,s , (4.2)where Q i ∈ t − Diff ( R t ), R i ∈ r − Diff ( X ) contains no y -derivatives and is sup-ported on supp χ , E s ∈ Diff ( X ), ˜ E s ∈ Diff ( X ), F k,s ∈ t − k − Diff ( ˜ M ) and F k,s issupported on supp dφ . Note that the term with ∇ X has spatially compact support,and could be absorbed into E s , but writing the commutator in the stated manner isconvenient for it gives a positive definite result modulo E s , which we later controlseparately, by low energy techniques. Moreover, for 0 < s < < σ < / E s term can be absorbed into the first threeterms on the right hand side by (3.11). Thus, for 0 < s < E s term, while the terms Q ∗ i r s t − R i arising fromthe commutator with ∂ t , are indefinite. However, we can estimate k t − ( k + s ) / r ( s − / ˜ φφR i u k M ≤ C ( k t − ( k + s ) / r ( s − / ˜ φφ∂ r u k M + k t − ( k + s ) / ˜ φφr ( s − / u k M ) , with C independent of c in the definition of φ . We also have a similar estimate forthe time derivatives, using the PDE and the fact that t − . r − on supp φ : k t − ( k + s ) / r ( s − / ˜ φφQ i u k M ≤ C (cid:0) k t − ( k + s ) / r ( s − / ˜ φφ∂ t u k M + k t − ( k + s ) / r ( s − / ˜ φφu k M (cid:1) ≤ C (cid:16) k t − ( k + s ) / r ( s − / ˜ φφ ∇ X u k M + k t − ( k + s ) / r ( s − / ˜ φφu k M + k t − ( k + s ) / r ( s +1) / ˜ φφ ( (cid:3) + V ) u k M + k t − ( k + s ) / r ( s − / ∇ X u k supp d ( φ ˜ φ ) + k t − ( k + s ) / r ( s − / ˜ φφu k supp d ( φ ˜ φ ) (cid:17) . (4.3)To show the validity of the last step, consider the (indefinite!) Dirichlet form relativeto dt − g , and use that h t α r β ˜ φφ ∇ u, t α r β ˜ φφ ∇ u i M,dt − g − h t α r β +1 ˜ φφ (cid:3) u, t α r β − ˜ φφu i M = h f t α − r β ˜ φφu, t α r β ˜ φφ∂ t u i M + h f t α r β − ˜ φφu, t α r β ˜ φφ ∇ X u i M + X j h t α − r β Q ′ j u, t α r β R ′ j u i M , (4.4)where f , f and Q ′ j are zero’th order symbols on ˜ M , R ′ j ∈ t − V b ( ˜ M ), and Q ′ j issupported on supp d ( φ ˜ φ ) (so R ′ j can be taken supported nearby). Indeed, the termson the right hand side can be controlled: the second term directly by the positivespatial term via Cauchy-Schwarz,(4.5) |h f t α r β − ˜ φφu, t α r β ˜ φφ ∇ X u i M | . k t α r β − ˜ φφu k M + k t α r β ˜ φφ ∇ X u k M , while the first can be Cauchy-Schwarzed with a small constant in front of the ∂ t factor in the pairing, which then can be reabsorbed in the Dirichlet form, while theother factor can be controlled directly from the spatial positive term using that r/t is bounded, and indeed small:(4.6) |h f t α − r β ˜ φφu, t α r β ˜ φφ∂ t u i M | . ǫ k t α r β ˜ φφ∂ t u k M + ǫ − k t α − r β ˜ φφu k M . In addition, the (cid:3) u term can be controlled using Cauchy-Schwarz:(4.7) h t α r β +1 ˜ φφ (cid:3) u, t α r β − ˜ φφu i M ≤ k t α r β − ˜ φφu k M + k t α r β +1 ˜ φφ (cid:3) u k M . Finally, we change from (cid:3) to (cid:3) + V using that V ∈ S − ( X ), so(4.8) k t α r β +1 ˜ φφV u k M . k t α r β − ˜ φφu k M . Combining (4.4)–(4.8),(1 − ˜ Cǫ ) k t α r β ˜ φφ∂ t u k M . k t α r β ˜ φφ ∇ X u k M + ǫ − k t α − r β ˜ φφu k M + (cid:0) k t α r β − ˜ φφu k M + k t α r β ˜ φφ ∇ X u k M (cid:1) + (cid:0) k t α − r β u k d ( φ ˜ φ ) + k t α r β ˜ φφ ∇ X u k d ( φ ˜ φ ) (cid:1) + k t α r β +1 ˜ φφ ( (cid:3) + V ) u k M , (4.9)where ǫ > C , so in particular we mayassume ˜ Cǫ < /
2. This proves the second inequality in (4.3).Since |h Q ∗ i ˜ φφr s t − t − ( k + s ) R i u, u i M |≤ c k t − ( k + s ) / r ( s − / ˜ φφR i u k M k t − ( k + s ) / r ( s − / ˜ φφQ i u k M ≤ c k t − ( k + s ) / r ( s − / ˜ φφR i u k M + c k t − ( k + s ) / r ( s − / ˜ φφQ i u k M , (4.10)with c arising from the extra factor of r/t, which is ≤ c on supp φ , in the pairingover what is included in the two factors on the right, for sufficiently small c > d ( φ ˜ φ ) and modulo terms involving ( (cid:3) + V ) u .It remains to check that the “quantum” term (in that we need global estimateson tf to control it), E s , in (4.2) can be controlled. With ρ ∈ C ∞ c ( R ) identically 1on a neighborhood of supp φ , supported in ( − , ρ ( µ ) = ρ ( µ/c ), c asabove, and consider ρ = ρ ( r/t ). We write |h r s − φ ( r/t ) E s φ ( r/t ) u, u i X | . k r ( s − / φ ( r/t ) ∇ X u k X + k r ( s − / φ ( r/t ) u k X . We remark that the left hand side rearranges the factors from (4.2) by commuting afactor of φ through E s ; the difference is supported in supp dφ , and when multipliedby t − k − s , it is in t − k − Diff ( ˜ M ), i.e. can be controlled the same way F k,s was (andis indeed better behaved). Now write u = ˜ ψ ( H (∆ + V )) ρ ( r/t ) u + ˜ ψ ( H (∆ + V ))(1 − ρ ( r/t )) u, and estimate the summands (and their X -gradients) individually.First, to deal with the ˜ ψ ( H (∆ + V ))(1 − ρ ( r/t )) u term, note that by Proposi-tion 2.9, for any N , k φ ( r/t ) ˜ ψ ( H (∆ + V ))(1 − ρ ( r/t )) r N k L ( L ( X ) ,L ( X )) + k∇ X φ ( r/t ) ˜ ψ ( H (∆ + V ))(1 − ρ ( r/t )) r N k L ( L ( X ) ,L ( X )) ≤ C H,N t − N , ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 25 where the constant C H,N depends on H (and N ). Since u is tempered, for suf-ficiently large N , the space-time norm of r − N t − N u is bounded (with the bounddepending on H ), controlling this term.Next, to deal with the ˜ u = ˜ ψ ( H (∆ + V )) ρ ( r/t ) u term, we use the low energy localization, i.e. we take H large. First, for ǫ ′ ∈ (0 , / k r ( s − / φ ( r/t )˜ u k X ≤ k r ( s − / ˜ u k X ≤ k r ( s − / − ǫ ′ ˜ u k X , k r ( s − / φ ( r/t ) ∇ X ˜ u k X ≤ k r ( s − / − ǫ ′ ∇ X ˜ u k X . By the Hardy/Poincar´e lemma, Lemma 3.1, using 0 < s <
1, and choosing ǫ ′ ∈ (0 , /
2) such that − ( s − / ǫ ′ < / k r ( s − / − ǫ ′ ˜ u k X + k r ( s − / − ǫ ′ ∇ X ˜ u k X . H − ǫ ′ k r ( s − / ρ ( r/t ) u k X = H − ǫ ′ (cid:0) k φ ( r/t ) r ( s − / u k X + k ( ρ ( r/t ) − φ ( r/t )) r ( s − / u k X (cid:1) . (4.11)For H sufficiently large, the first term on the right hand side of (4.11) can beabsorbed into the zero’th order positive definite spatial term,(4.12) D φ ( r/t ) (cid:0) s∂ ∗ r r s − χ ( r ) ∂ r + 2 r s − χ ( r )∆ Y + n −
12 (1 − s )( n + s − r s − (cid:1) u, u E X , modulo errors supported on supp d ( ˜ φφ ), which are of the same kind as those in F k,s in (4.2). On the other hand, the second term on the right hand side of (4.11) issupported on supp( ρ − φ ), and is thus similar to the F k,s terms, with slightly largersupport (but still within the region where this proposition gives no improvementupon the a priori assumption). Explicitly, when also integrated in t , on supp( ρ − φ )we have ∇ u ∈ t κ L ( M ) (with ( k + 1) / κ ), and since on supp( ρ − φ ), r ∼ t , Z t − k − s ˜ φ ( t ) k ( ρ ( r/t ) − φ ( r/t )) r ( s − / u k X dt . Z t − k − s ˜ φ ( t ) k t ( s − / ( ρ ( r/t ) − φ ( r/t )) u k X dt . k t − ( k +1) / ˜ φ ( t ) ( ρ ( r/t ) − φ ( r/t )) u k M . (4.13)Now, for γ > |h B k,s u, ( (cid:3) + V ) u i M | = |h B k,s u, ρ ( r/t )( (cid:3) + V ) u i M |≤ k r ( s +1) / t − ( k + s ) / ρ ( r/t )( (cid:3) + V ) u k M k r − ( s +1) / t ( k + s ) / B k,s u k M ≤ γ − k r ( s +1) / t − ( k + s ) / ρ ( r/t )( (cid:3) + V ) u k M + γ k r − ( s +1) / t ( k + s ) / B k,s u k M ≤ γ − k r ( s +1) / t − ( k + s ) / ρ ( r/t )( (cid:3) + V ) u k M + γ (cid:16) k r ( s − / t − ( k + s ) / ρ ( r/t ) χ ( r ) ∂ r u k M + k r ( s − / t − ( k + s ) / ρ ( r/t ) u k M + k t − ( k + s ) / (1 − χ ( r )) ρ ( r/t ) ∇ X u k M (cid:17) . Taking γ > outside supp(1 − φ ˜ φ ), but in that region we have a priori control of these terms.We thus deduce from (4.2) that k t − ( k + s ) / r ( s − / ˜ φφ ∇ X u k M + k t − ( k + s ) / r ( s − / ˜ φφu k M . k r ( s +1) / t − ( k + s ) / ( (cid:3) + V ) u k L (Ω) + k u k t ( k +3) / H (Υ) , (4.14)where Υ = supp dφ ∪ supp( ρ − φ ) and with the last space being H ( ˜ M ) on Υ,provided 0 < s < k∇ u k t ( k +1) / L (Υ) . In view of the Dirichlet form estimate, (4.9), we also get the bound for the timederivative which is analogous to (4.14). Noting that k t − ( k + s ) / r ( s − / ˜ φφ ∇ u k M + k t − ( k + s ) / r ( s − / ˜ φφu k M = k t − ( k +1) / ( t/r ) (1 − s ) / ˜ φφ ∇ u k M + k t − ( k +1) / ( t/r ) (1 − s ) / r − ˜ φφu k M , and 0 < s <
1, so 0 < (1 − s ) / < /
2, while (using r/t < s > k r ( s +1) / t − ( k + s ) / ( (cid:3) + V ) u k L (Ω) . k r / t − k/ ( (cid:3) + V ) u k L (Ω) , this proves the estimate of the theorem, provided the formal pairings employedabove are finite and hence the computation justified.In order to employ the above argument without a priori assumptions on u , weneed to introduce a weight, (1 + αt ) − , α >
0, in B k,s to justify the arguments, andlet α →
0. Roughly speaking, such a weight does not cause problems since we coulddeal with arbitrary weights t − k already (i.e. k did not need to have a sign). Moreprecisely, the contribution of this weight to the commutator of (1 + αt ) − B k,s and (cid:3) is similar to (4.1), namely [ ∂ t , t − k − s (1 + αt ) − ] ∈ t − k − s − Diff ( R t ) for α > t − k − s − Diff ( R t ) for α ∈ [0 , Q ∗ i R i terms of (4.2), which in turn arosefrom (4.1); these are estimated in (4.10), and can be controlled by making c small.Letting α →
0, and using standard functional analytic arguments, completes theproof of the theorem. (cid:3)
We finally remark that a version of this argument also works for solutions ofthe wave equation at intermediate frequencies, via Mourre estimate techniques (seeespecially [5]). While this is not interesting for the homogeneous wave equation,whose middle-frequency solutions will have rapid decay inside the light cone, for theinhomogeneous equation it may be of some interest. The key point then is that oneworks with ψ (∆+ V ), and ˜ ψ (∆+ V ), supported near a fixed energy λ >
0, and onecontrols the analogue of the E s term (that arose above) by shrinking the supportof ˜ ψ , using that ˜ ψ (∆ + V ) E s converges to 0 in norm as the support is shrunk to { λ } by the relative compactness of E s (which follows from elliptic estimates andits decay at spatial infinity). This holds since ˜ ψ (∆ + V ) converges to 0 strongly asthe support is shrunk to { λ } , since λ is not an L eigenvalue of ∆ + V . References [1] J.-F. Bony and D. H¨afner,
The semilinear wave equation on asymptotically Euclidean man-ifolds , preprint, 2008.[2] J.-F. Bony and D. H¨afner,
Local energy decay for several evolution equations on asymptoti-cally euclidean manifolds , arXiv:1008.2357.
ORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY 27 [3] J.-M. Bouclet,
Low frequency estimates and local energy decay for asymptotically euclideanLaplacians , arXiv:1003.6016.[4] F. G. Friedlander,
Notes on the wave equation on asymptotically Euclidean manifolds , J.Funct. Anal. (2001), no. 1, 1–18.[5] R. G. Froese and I. Herbst. A new proof of the Mourre estimate.
Duke Math. J. , 49:1075–1085,1982.[6] C. Guillarmou, A. Hassell, A. Sikora,
Resolvent at low energy III: the spectral measure, arXiv:1009.3084.[7] A. Hassell and A. Vasy. The spectral projections and the resolvent for scattering metrics.
J.d’Analyse Math. , 79:241–298, 1999.[8] A. Hassell and A. Vasy. The resolvent on Laplace-type operators on asymptotically conicspaces.
Ann. Inst. Fourier , 51:1299–1346, 2001.[9] Joshi, Mark S.; S´a Barreto, Antˆonio
Recovering asymptotics of metrics from fixed energyscattering data,
Invent. Math. 137 (1999), no. 1, 127–143.[10] Richard B. Melrose.
The Atiyah-Patodi-Singer index theorem , volume 4 of
Research Notesin Mathematics . A K Peters Ltd., Wellesley, MA, 1993.[11] R. B. Melrose,
Spectral and scattering theory for the Laplacian on asymptotically Euclidianspaces . In
Spectral and scattering theory , M. Ikawa, editor, Marcel Dekker, 1994.[12] Metcalfe, Jason and Tataru, Daniel
Global parametrices and dispersive estimates for variablecoefficient wave equations,
Math. Ann., to appear.[13] Metcalfe, Jason, Tataru, Daniel, and Tohaneau, Mihai
Price’s law on nonstationary space-times , arXiv: 1104.5437.[14] Morawetz, Cathleen S.
The decay of solutions of the exterior initial-boundary value problemfor the wave equation,
Comm. Pure Appl. Math. 14 1961 561–568.[15] Morawetz, Cathleen S., Ralston, James V. and Strauss, Walter A.,
Decay of solutions of thewave equation outside nontrapping obstacles,
Comm. Pure Appl. Math. 30 (1977), no. 4,447-508.[16] E. Mourre. Operateurs conjug´es et propri´et´es de propagation.
Commun. Math. Phys. , 91:279–300, 1983.[17] P. Perry, I. M. Sigal, and B. Simon. Spectral analysis of N-body Schr¨odinger operators.
Ann.Math. , 114:519–567, 1981.[18] Ralston, James V.
Solutions of the wave equation with localized energy,
Comm. Pure Appl.Math. 22 1969 807–823.[19] Tataru, Daniel,
Local decay of waves on asymptotically flat stationary space-times, arXiv:0910.5290.[20] Va˘ınberg, B. R.,
The short-wave asymptotic behavior of the solutions of stationary problems,and the asymptotic behavior as t → ∞ of the solutions of nonstationary problems. (Russian)Uspehi Mat. Nauk 30 (1975), no. 2(182), 3-55.[21] Vodev, Georgi, Local energy decay of solutions to the wave equation for nontrapping metrics,
Ark. Mat. 42 (2004), no. 2, 379-397.[22] A Vasy and J. Wunsch,
Positive commutators at the bottom of the spectrum,
J. Func. Anal.
2, (2010), 503–523.
Department of Mathematics, Stanford UniversityDepartment of Mathematics, Northwestern University
E-mail address : [email protected] E-mail address ::