Inverting the local geodesic X-ray transform on tensors
IINVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ONTENSORS
PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDR ´AS VASY
Abstract.
We prove the local invertibility, up to potential fields, and stabilityof the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictlyconvex boundary point, on manifolds with boundary of dimension n ≥
3. Wealso present an inversion formula. Under the condition that the manifold canbe foliated with a continuous family of strictly convex surfaces, we prove aglobal result which also implies a lens rigidity result near such a metric. Theclass of manifolds satisfying the foliation condition includes manifolds with nofocal points, and does not exclude existence of conjugate points. Introduction
Let (
M, g ) be a compact Riemannian manifold with boundary. The X-ray trans-form of symmetric covector fields of order m is given by(1.1) If ( γ ) = (cid:90) (cid:104) f ( γ ( t )) , ˙ γ m ( t ) (cid:105) dt, where, in local coordinates, (cid:104) f, v m (cid:105) = f i ...i m v i . . . v i m , and γ runs over all (finitelength) geodesics with endpoints on ∂M . When m = 0, we integrate functions;when m = 1, f is a covector field, in local coordinates, f j dx j ; when m = 2, f is a symmetric 2-tensor field f ij dx i dx j , etc. The problem is of interest by itselfbut it also appears as a linearization of boundary and lens rigidity problems, see,e.g., [19, 18, 23, 24, 26, 6, 5, 4]. Indeed, when m = 0, f can be interpreted asthe infinitesimal difference of two conformal factors, and when m = 2, f ij can bethought of as an infinitesimal difference of two metrics. The m = 1 problem arisesas a linearization of recovery a velocity fields from the time of fly. The m = 4problem appears in linearized elasticity.The problem we study is the invertibility of I . It is well known that potential vector fields, i.e., f which are a symmetric differential d s v of a symmetric fieldof order m − ∂M (when m ≥ I . When m = 0, there are no potential fields; when m = 1, potential fields are just ordinarydifferentials dv of functions vanishing at the boundary; for m = 2, potential fieldsare given by d s v = ( v i,j + v j,i ), with v one form, v = 0 on ∂M ; etc. The naturalinvertibility question is then whether If = 0 implies that f is potential; we callthat property s-injectivity below. Date : October 18, 2014.1991
Mathematics Subject Classification. a r X i v : . [ m a t h . DG ] O c t PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRAS VASY
This problem has been studied extensively for simple manifolds , i.e., when ∂M isstrictly convex and any two points are connected by a unique minimizing geodesicsmoothly depending on the endpoints. For simple metrics, in case of functions( m = 0), uniqueness and a non-sharp stability estimate was established in [13, 12, 2]using the energy method initiated by Mukhometov, and for m = 1, in [1]. Sharpstability follows from [23]. The case m ≥ m = 2 one already contains all the difficulties. In two dimensions, uniquenessfor simple metrics and m = 2 has been proven in [21] following the boundary rigidityproof in [16]. For any m , this was done in [14].In dimensions n ≥
3, the problem still remains open for m ≥
2. Under an explicitupper bound of the curvature, uniqueness and a non-sharp stability was proved bySharafutdinov, see [18, 19] and the references there, using a suitable version ofthe energy method developed in [15]. Convexity of ∂M is not essential for thosekind of results and the curvature assumption can be replaced by an assumptionstronger than requiring no conjugate points, see [20, 7]. This still does not answerthe uniqueness question for metrics without conjugate points however. The firstand the second author proved in [23, 24], using microlocal and analytic microlocaltechniques, that for simple metrics, the problem is Fredholm (modulo potentialfields) with a finitely dimensional smooth kernel. For analytic simple metrics, thereis uniqueness; and in fact, the uniqueness extends to an open and dense set ofsimple metrics in C k , k (cid:29)
1. Moreover, there is a sharp stability L ( M ) → H ( ˜ M )estimate for f (cid:55)→ I ∗ If , where ˜ M is some extension of M , see [22]. We study the m = 2 case there for simplicity of the exposition but the methods extend to any m ≥ m ≥ m = 1 and the m = 0 cases can beseen from the analysis in [23, 24]. When m = 0, the presence of the boundary ∂M is not essential — we can extend ( M, g ) to a complete ( ˜
M , ˜ g ) and just restrict I tofunctions supported in a fixed compact set. When f is an one-form ( m = 1), wehave to deal with non-uniqueness due to exact one-forms but then the symmetricdifferential is d s just the ordinary one d . When n ≥ d s is an elliptic operatorbut recovery of df from d s f is not a local operator. One way to deal with the non-uniqueness due to potential fields is to project on solenoidal ones (orthogonal to thepotential fields). This involves solving an elliptic boundary value problem and thepresence of the boundary ∂M becomes an essential factor. The standard pseudo-differential calculus is not suited naturally to work on manifolds with boundary.In [25], the first two authors study manifolds with possible conjugate pointsof dimension n ≥
3. The geodesic manifold (when it is a smooth manifold) hasdimension 2 n − n when n ≥
3. We restrict I there to an openset Γ of geodesics. Assuming that Γ consists of geodesics without conjugate pointsso that the conormal bundle { T ∗ γ | γ ∈ Γ } covers T ∗ M \
0, we show uniquenessand stability for analytic metrics, and moreover for an open and dense set of suchmetrics. In this case, even though conjugate points are allowed, the analysis is doneon the geodesics in Γ assumed to have no such points.A significant progress is done in the recent work [28], where the second and thethird author prove the following local result: if ∂M is strictly convex at p ∈ ∂M and n ≥
3, then If , acting on functions ( m = 0), known for all geodesics closeenough to the tangent ones to ∂M at p , determine f near p in a stable way. Thenew idea in [28] was to introduce an artificial boundary near p cutting off a small NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 3 part of M including p and to apply the scattering calculus in the new domain Ω c ,treating the artificial boundary as infinity, see Figure 1. Then Ω c is small enough,then a suitable “filtered” backprojection operator is not only Fredholm, but alsoinvertible. We use this idea in the present work, as well. The authors used thislinear results in a recent work [27] to prove local boundary and lens rigidity near aconvex boundary point.The purpose of this paper is to invert the geodesic X-ray transform f (cid:55)→ If onone forms and symmetric 2-tensors ( m = 1 and m = 2) for n ≥ f on suitable opensets Ω ⊂ M from the knowledge of If ( γ ) for Ω-local geodesics γ , i.e. γ contained inΩ with endpoints on ∂M ∩ Ω. More precisely, there is an obstacle to the inversionexplained above: one-forms or tensors which are potential, i.e. of the form d s v ,where v is scalar or a one-form, vanishing at ∂M ∩ Ω, have vanishing integrals alongall the geodesics with endpoints there, so one may always add a potential (exact)form or a potential two-tensor to f and obtain the same localized transform If .Our result is thus the local recovery of f from If up to this gauge freedom ; in astable way. Further, under an additional global convex foliation assumption we alsogive a global counterpart to this result.We now state our main results more concretely. Let ρ be a local boundarydefining function, so that ρ ≥ M . It is convenient to also consider a manifoldwithout boundary ( ˜ M , g ) extending M . First, as in [28], the main local result isobtained for sufficiently small regions Ω = Ω c = { x ≥ , ρ ≥ } , x = x c ; seeFigure 1. Here x = 0 is an ‘artificial boundary’ which is strictly concave as viewedfrom the region Ω between it and the actual boundary ∂M ; this (rather than ∂M )is the boundary that plays a role in the analysis below.We set this up in the same way as in [28] by considering a function ˜ x with strictlyconcave level sets from the super-level set side for levels c , | c | < c , and letting x c = ˜ x + c, Ω c = { x c ≥ , ρ ≥ } . (A convenient normalization is that there is a point p ∈ ∂M such that ˜ x ( p ) = 0 andsuch that d ˜ x ( p ) = − dρ ( p ); then one can take e.g. ˜ x ( z ) = − ρ ( z ) − (cid:15) | z − p | for small (cid:15) >
0, which localizes in a lens shaped region near p , or indeed ˜ x = − ρ which onlylocalizes near ∂ Ω.) Here the requirement on ˜ x is, if we assume that M is compact,that there is a continuous function F such that F (0) = 0 and such thatΩ c ⊂ { ˜ x < − c + F ( c ) } , i.e. as c →
0, Ω c is a thinner and thinner shell in terms of ˜ x . As in [28], ourconstructions are uniform in c for | c | < c . We drop the subscript c from Ω c , i.e.simply write Ω, again as in [28], to avoid overburdening the notation.A weaker version, in terms of function spaces, of the main local theorem, pre-sented in Corollaries 4.17-4.18, is then the following. The notation here is thatlocal spaces mean that the condition is satisfied on compact subsets of Ω \ { x = 0 } ,i.e. the conclusions are not stated uniformly up to the artificial boundary (but areuniform up to the original boundary); this is due to our efforts to minimize the an-alytic and geometric background in the introduction. The dot denotes supporteddistributions in the sense of H¨ormander relative to the actual boundary ρ = 0, i.e.distributions in x > M ) whose support lies in ρ ≥
0, i.e.for ˙ H , this is the H space. PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRAS VASY . ......................................... ...................................... ................................... ................................ ............................... .............................. ............................. ............................ ........................... .......................... ......................... ......................... .......................... ........................... ............................ ............................. .............................. ............................... ................................ ................................... ...................................... .......................................... .......................................................... ......................................................... ........................................................ ....................................................... ...................................................... ...................................................... ....................................................... ........................................................ ......................................................... .......................................................... ⇢ = 0 ˜ x = c r p ˜ M M ˜ x ................................ ? ⇢ . ....................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................ .............................................................................. z ⌦ c Figure 1.
The functions ρ and ˜ x when the background is flatspace ˜ M . The intersection of ρ ≥ x c > x c =˜ x + c , so this is the region ˜ x > − c ) is the lens shaped region O p .Note that, as viewed from the superlevel sets, thus from O p , ˜ x hasconcave level sets. At the point z , L integrates over geodesics inthe indicated small angle. As z moves to the artificial boundary x c = 0, the angle of this cone shrinks like Cx c so that in the limitthe geodesics taken into account become tangent to x c = 0. Theorem 1.1. (See Corollaries 4.17-4.18.) With
Ω = Ω c as above, there is c > such that for c ∈ (0 , c ) , if f ∈ L (Ω) then f = u + d s v , where v ∈ ˙ H (Ω \{ x = 0 } ) ,while u ∈ L (Ω \ { x = 0 } ) can be stably determined from If restricted to Ω -local geodesics in the following sense. There is a continuous map If (cid:55)→ u , wherefor s ≥ , f in H s (Ω) , the H s − norm of u restricted to any compact subset of Ω \ { x = 0 } is controlled by the H s norm of If restricted to the set of Ω -localgeodesics.Replacing Ω c = { ˜ x > − c } ∩ M by Ω τ,c = { τ > ˜ x > − c + τ } ∩ M , c can be takenuniform in τ for τ in a compact set on which the strict concavity assumption onlevel sets of ˜ x holds. The uniqueness part of the theorem generalizes Helgason’s type of support the-orems for tensors fields for analytic metrics [9, 10, 3]. In those works however,analyticity plays a crucial role and the proof is a form of a microlocal analyticcontinuation. In contrast, no analyticity is assumed here.As in [28], this theorem can be applied in a manner to obtain a global conclusion.To state this, assume that ˜ x is a globally defined function with level sets Σ t whichare strictly concave from the super-level set for t ∈ ( − T, x ≤ M . Then we have: Theorem 1.2. (See Theorem 4.19.) Suppose M is compact. Then the geodesicX-ray transform is injective and stable modulo potentials on the restriction of one-forms and symmetric 2-tensors f to ˜ x − (( − T, in the following sense. For all τ > − T there is v ∈ ˙ H (˜ x − (( τ, such that f − d s v ∈ L (˜ x − (( τ, can be stablyrecovered from If in the sense that for s ≥ and f ∈ H s locally on ˜ x − (( τ, , the H s − norm of v restricted to compact subsets of ˜ x − (( τ, is controlled by the H s norm of If on local geodesics.Remark . This theorem, combined with Theorem 2 in [26] (with a minor change— the no-conjugate condition there is only needed to guarantee a stability estimate,
NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 5 and we have it in our situation), implies a local, in terms of a perturbation of themetric, lens rigidity uniqueness result near metric satisfying the foliation condition.Manifolds satisfying the foliation condition include manifolds without focal points[17]. Subdomains M of R n with the metric c − ( r ) dx , r = | x | satisfying the Her-glotz [8] and Wiechert and Zoeppritz [29] condition ddr rc ( r ) > M satisfy it aswell since then the Euclidean spheres | x | = r form a strictly convex foliation. Con-jugate points in that case may exist, and small perturbations of such metrics satisfythe condition, as well. We can also formulate semi-global results: if we can foliate M \ K with K ⊂ M compact, then we can recover f up to a potential field therein a stable way, with stability degenerating near ∂M . This can be considered as alinearized model of the seismology problem for anisotropic speeds of propagation.One such example is metrics c − ( r ) dx (and close to them) for which ddr rc ( r ) > a ≤ r ≤ b and M ⊂ {| x | ≤ b } . Then f can be stably recovered for | x | > a up to a potential field.Similarly to our work [27], this paper, and its methods, will have applications tothe boundary rigidity problem; in this case without the conformal class restriction.This paper is forthcoming.The plan of the paper is the following. In Section 2 we sketch the idea of theproof, and state the main technical result. In Section 3 we show the ellipticity of themodified version of LI , modified by the addition of gauge terms. This essentiallyproves the main result if one can satisfy the gauge condition. In Section 4 weanalyze the gauge condition and complete the proof of our main results.2. The idea of the proof and the scattering algebra
We now explain the basic ideas of the paper.The usual approach in dealing with the gauge freedom is to add a gauge con-dition, which typically, see e.g. the work of the first two authors [24], is of the solenoidal gauge condition form, δ sg f = 0, where δ sg is the adjoint of d s with respectto the Riemannian metric on M . Notice that actually the particular choice of theadjoint is irrelevant; once one recovers f in one gauge, one could always expressit in terms of another gauge, e.g. in this case relative to a different Riemannianmetric.In order to motivate our gauge condition, we need to recall the method intro-duced by the last two authors in [28] to analyze the geodesic X-ray transform onfunctions: the underlying analysis strongly suggests the form the gauge conditionshould take.As in [28] we consider an operator L that integrates over geodesics in a smallcone at each point, now multiplying with a one form or symmetric 2-tensor, inthe direction of the geodesic, mapping (locally defined) functions on the space ofgeodesics to (locally defined) one forms or tensors. The choice of the operator,or more concretely the angle, plays a big role; we choose it to be comparable tothe distance to the artificial boundary, x = 0. In this case LI ends up beingin Melrose’s scattering pseudodifferential algebra, at least once conjugated by anexponential weight. (The effect of this weight is that we get exponentially weakestimates as we approach the artificial boundary.) The main analytic problem onefaces then is that, corresponding to the gauge freedom mentioned above, LI is notelliptic, unlike in the scalar (function) setting. PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRAS VASY
Concretely L is defined as follows. Near ∂ Ω, one can use coordinates ( x, y ), with x = x c = ˜ x + c as before, y coordinates on ∂ Ω. Correspondingly, elements of T p M can be written as λ ∂ x + η ∂ y . The unit speed geodesics which are close to beingtangential to level sets of ˜ x (with the tangential ones being given by λ = 0) througha point p can be parameterized by say ( λ, ω ) (with the actual unit speed being apositive multiple of this) where ω is unit length with respect to say a Euclideanmetric. The concavity of the level sets of ˜ x , as viewed from the super-level sets,means that d dt ˜ x ◦ γ is bounded below by a positive constant along geodesics inΩ c , as long as c is small, which in turn means that, for sufficiently small C > | λ | < C √ x indeed remain in x ≥ M ). Thus,if If is known along Ω-local geodesics, it is known for geodesics ( x, y, λ, ω ) in thisrange. As in [28] we use a smaller range | λ | < C x because of analytic advantages,namely the ability work in the well-behaved scattering algebra. Thus, for χ smooth,even, non-negative, of compact support, to be specified, in the function case [28]considered the operator Lv ( z ) = x − (cid:90) χ ( λ/x ) v ( γ x,y,λ,ω ) dλ dω, where v is a (locally, i.e. on supp χ , defined) function on the space of geodesics,here parameterized by ( x, y, λ, ω ). (In fact, L had a factor x − only in [28], withanother x − placed elsewhere; here we simply combine these, as was also donein [27, Section 3]. Also, the particular measure dλ dω is irrelevant; any smoothpositive multiple would work equally well.) In this paper, with v still a locallydefined function on the space of geodesics, for one-forms we consider the map L (2.1) Lv ( z ) = (cid:90) χ ( λ/x ) v ( γ x,y,λ,ω ) g sc ( λ ∂ x + ω ∂ y ) dλ dω, while for 2-tensors(2.2) Lv ( z ) = x (cid:90) χ ( λ/x ) v ( γ x,y,λ,ω ) g sc ( λ ∂ x + ω ∂ y ) ⊗ g sc ( λ ∂ x + ω ∂ y ) dλ dω, so in the two cases L maps into one-forms, resp. symmetric 2-cotensors, where g sc is a scattering metric used to convert vectors into covectors — this is discussed indetail below.Since it plays a crucial role even in the setup, by giving the bundles of which ourtensors are sections of, as well as the gauge condition, we need to discuss scatteringgeometry and the scattering pseudodifferential algebra, introduced by Melrose in[11], at least briefly. There is a more thorough discussion in [28, Section 2], thoughthe cotangent bundle, which is crucial here, is suppressed there. Briefly, the scat-tering pseudodifferential algebra Ψ m,l sc ( X ) on a manifold with boundary X is thegeneralization of the standard pseudodifferential algebra given by quantizations ofsymbols a ∈ S m,l , i.e. a ∈ C ∞ ( R n × R n ) satisfying(2.3) | D αz D βζ a ( z, ζ ) | ≤ C αβ (cid:104) z (cid:105) l −| α | (cid:104) ζ (cid:105) −| β | for all multiindices α, β in the same way that on a compact manifold withoutboundary ˜ X , Ψ m ( ˜ X ) arises from (localized) pseudodifferential operators on R n via considering coordinate charts. More precisely, R n can be compactified to aball R n , by gluing a sphere at infinity, with the gluing done via ‘reciprocal polarcoordinates’; see [28, Section 2]. One then writes Ψ m,l sc ( R n ) for the quantizations ofthe symbols (2.3). Then Ψ m,l sc ( X ) is defined by requiring that locally in coordinate NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 7 charts, including charts intersecting with ∂X , the algebra arises from Ψ m,l sc ( R n ).(One also has to allow smooth Schwartz kernels on X × X which are vanishing toinfinite order at ∂ ( X × X ), in analogy with the smooth Schwartz kernels on ˜ X × ˜ X .)Thus, while the compactification is extremely useful to package information, thereader should keep in mind that ultimately almost all of the analysis reduces touniform analysis on R n . Since we are working with bundles, we also mentionthat scattering pseudodifferential operators acting on sections of vector bundles aredefined via local trivializations, in which these operators are given by matrices ofscalar scattering pseudodifferential operators (i.e. are given by the R n definitionabove if in addition these trivializations are made to be coordinate charts), up tothe same smooth, infinite order vanishing at ∂ ( X × X ) Schwartz kernels as in thescalar case.Concretely, the compactification R n , away from 0 ∈ R n ⊂ R n , is just [0 , ∞ ) x × S n − ω , where the identification with R n \{ } is just the ‘inverse polar coordinate’ map( x, ω ) (cid:55)→ x − ω , with r = x − the standard radial variable. Then a straightforwardcomputation shows that translation invariant vector fields ∂ z j on R nz lift to thecompactification (via this identification) to generate, over C ∞ ( R n ), the Lie algebra V sc ( R n ) = x V b ( R n ) of vector fields, where on a manifold with boundary V b ( X ) is theLie algebra of smooth vector fields tangent to the boundary of X . In general, if x is aboundary defining function of X , we let V sc ( X ) = x V b ( X ). Then Ψ , ( X ) contains V sc ( X ), corresponding to the analogous inclusion on Euclidean space, and the vectorfields in Ψ , ( X ) are essentially the elements of V sc ( X ), after a slight generalizationof coefficients (since above a does not have an asymptotic expansion at infinity in z , only symbolic estimates; the expansion would correspond to smoothness of thecoefficients).Now, a local basis for V sc ( X ), in a coordinate chart ( x, y , . . . , y n − ), is x ∂ x , x∂ y , . . . , x∂ y n − directly from the definition, i.e. V ∈ V sc ( X ) means exactly that locally, on U ⊂ XV = a ( x ∂ x ) + (cid:88) a j ( x∂ y j ) , a j ∈ C ∞ ( U ) . This gives that elements of V sc ( X ) are exactly smooth sections of a vector bundle, sc T X , with local basis x ∂ x , x∂ y , . . . , x∂ y n − . In the case of X = R n , this simplymeans that one is using the local basis x ∂ x = − ∂ r , x∂ y j = r − ∂ ω j , where the ω j arelocal coordinates on the sphere. An equivalent global basis is just ∂ z j , j = 1 , . . . , n ,i.e. sc T R n = R nz × R n is a trivial bundle with this identification.The dual bundle sc T ∗ X of sc T X correspondingly has a local basis dxx , dy x , . . . , dy n − x ,which in case of X = R n becomes − dr, r dω j , with local coordinates ω j on thesphere. A global version is given by using the basis dz j , with covectors writtenas (cid:80) ζ j dz j ; thus sc T ∗ R n = R nz × R nζ ; this is exactly the same notation as in thedescription of the symbol class (2.3), i.e. one should think of this class as livingon sc T ∗ R n . Thus, smooth scattering one-forms on R n , i.e. sections of sc T ∗ R n , aresimply smooth one-forms on R n with an expansion at infinity. Similar statementsapply to natural bundles, such as the higher degree differential forms sc Λ k X , aswell as symmetric tensors, such as Sym T ∗ X . The latter give rise to scatteringmetrics g sc , which are positive definite inner products on the fibers of sc T X (i.e.positive definite sections of Sym T ∗ X ) of the form g sc = x − dx + x − ˜ h , ˜ h astandard smooth 2-cotensor on X (i.e. a section of Sym T ∗ X ). For instance, one PLAMEN STEFANOV, GUNTHER UHLMANN AND ANDRAS VASY can take, in a product decomposition near ∂X , g sc = x − dx + x − h , h a metricon the level sets of x .The principal symbol of a pseudodifferential operator is the equivalence class of a as in (2.3) modulo S m − ,l − , i.e. modulo additional decay both in z and in ζ on R n × R n . In particular, full ellipticity is ellipticity in this sense, modulo S m − ,l − ,i.e. for a scalar operator lower bounds | a ( z, ζ ) | ≥ c (cid:104) z (cid:105) l (cid:104) ζ (cid:105) m for | z | + | ζ | > R , where R is suitably large. This contrasts with (uniform) ellipticity in the standard sense,which is a similar lower bound, but only for | ζ | > R . Fully elliptic operators areFredholm between the appropriate Sobolev spaces H s,r sc ( X ) corresponding to thescattering structure, see [28, Section 2]; full ellipticity is needed for this (as showne.g. by taking ∆ − R n , ∆ the flat positive Laplacian). If a is matrix valued,ellipticity can be stated as invertibility for large ( z, ζ ), together with upper boundsfor the inverse: | a ( z, ζ ) − | ≤ c − (cid:104) z (cid:105) − l (cid:104) ζ (cid:105) − m ; this coincides with the above definitionfor scalars.We mention also that the exterior derivative d ∈ Diff ( X ; sc Λ k , sc Λ k +1 ) for all k . Explicitly, for k = 0, in local coordinates, this is the statement that df = ( ∂ x f ) dx + (cid:88) j ( ∂ y j f ) dy j = ( x ∂ x f ) dxx + (cid:88) j ( x∂ y j ) dy j x , with x ∂ x , x∂ y j ∈ Diff ( X ), while dxx , dy j x are smooth sections of sc T ∗ X (locally,where this formula makes sense). Such a computation also shows that the principalsymbol, in both senses, of d , at any point ξ dxx + (cid:80) j η j dy j x , is wedge product with ξ dxx + (cid:80) j η j dy j x . A similar computation shows that the gradient with respect toa scattering metric g sc is a scattering differential operator (on any of the naturalbundles), with principal symbol given by tensor product with ξ dxx + (cid:80) j η j dy j x ,hence so is the symmetric gradient on one forms, with principal symbol given bythe symmetrized tensor product with ξ dxx + (cid:80) j η j dy j x . Note that all of these prin-cipal symbols are actually independent of the metric g sc , and d itself is completelyindependent of any choice of a metric (scattering or otherwise).If we instead consider the symmetric differential d s with respect to a smoothmetric g on X , as we are obliged to use in our problem since its image is what isannihilated by the ( g -geodesic) X-ray transform I , it is a first order differential op-erator between sections of bundles T ∗ X and Sym T ∗ X . Writing dx, dy j , resp., dx , dx dy j and dy i dy j for the corresponding bases, this means that we have a matrix offirst order differential operators. Now, as the standard principal symbol of d s is justtensoring with the covector at which the principal symbol is evaluated, the first or-der terms are the same, modulo zeroth order terms, as when one considers d sg sc , andin particular they correspond to a scattering differential operator acting betweensection of sc T ∗ X and Sym T ∗ X . (This can also be checked explicitly using thecalculation done below for zeroth order term, but the above is the conceptual reasonfor this.) On the other hand, with dx = dx ⊗ dx , dx dy i = ( dx ⊗ dy i + dy i ⊗ dx ),etc., these zeroth order terms form a matrix with smooth coefficients in the localbasis dx ⊗ ∂ x , dx ⊗ ∂ y j , dx dy i ⊗ ∂ x , dx dy i ⊗ ∂ y j , dy k dy i ⊗ ∂ x , dy k dy i ⊗ ∂ y j NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 9 of the homomorphism bundle hom( T ∗ X, Sym T ∗ X ). In terms of the local basis dx x ⊗ ( x ∂ x ) , dx x ⊗ ( x∂ y j ) , dxx dy i x ⊗ ( x ∂ x ) ,dxx dy i x ⊗ ( x∂ y j ) , dy k dy i x ⊗ ( x ∂ x ) , dy k dy i x ⊗ ( x∂ y j )of hom( sc T ∗ X, Sym T ∗ X ), these are all smooth, and vanish at ∂X to order2 , , , , , d s ∈ Diff ( X, sc T ∗ X, Sym T ∗ X ), andthat the only non-trivial contribution of these zeroth order terms to the princi-pal symbol is via the entry corresponding to dy k dy i x ⊗ ( x ∂ x ) = dy k dy i ⊗ ∂ x , whichhowever is rather arbitrary.Returning to the choice of gauge, in our case the solenoidal gauge relative to g would not be a good idea: the metric on M is an incomplete metric as viewedat the artificial boundary, and does not interact well with LI . We circumventthis difficulty by considering instead the adjoint δ s relative to a scattering metric,i.e. one of the form x − dx + x − h , h a metric on the level sets of x . While δ s , d s are then scattering differential operators, unfortunately δ s d s on functions,or one forms, is not fully elliptic in the scattering sense (full ellipticity is neededto guarantee Fredholm properties on Sobolev spaces in a compact setting), withthe problem being at finite points of sc T ∗ X , X = { x ≥ } . For instance, in thecase of X being the radial compactification of R n , we would be trying to invert theLaplacian on functions or one-forms, which has issues at the 0-section. However, ifwe instead use an exponential weight, which already arose when LI was discussed,we can make the resulting operator fully elliptic, and indeed invertible for suitableweights.Thus, we introduce a Witten-type (in the sense of the Witten Laplacian) solenoidalgauge on the scattering cotangent bundle, sc T ∗ X or its second symmetric power,Sym T ∗ X . Fixing (cid:122) >
0, our gauge is e (cid:122) /x δ s e − (cid:122) /x f s = 0 , or the e − (cid:122) /x -solenoidal gauge . (Keep in mind here that δ s is the adjoint of d s relative to a scattering metric.) We are actually working with f (cid:122) = e − (cid:122) /x f throughout; in terms of this the gauge is δ s (cid:122) f s (cid:122) = 0 , δ s (cid:122) = e (cid:122) /x δ s e − (cid:122) /x . Theorem 2.1. (See Theorem 4.15 for the proof and the formula.) There exists (cid:122) > such that for (cid:122) ≥ (cid:122) the following holds.For Ω = Ω c , c > small, the geodesic X-ray transform on e (cid:122) /x -solenoidal one-forms and symmetric 2-tensors f ∈ e (cid:122) /x L (Ω) , i.e. ones satisfying δ s ( e − (cid:122) /x f ) =0 , is injective, with a stability estimate and a reconstruction formula.In addition, replacing Ω c = { ˜ x > − c } ∩ M by Ω τ,c = { τ > ˜ x > − c + τ } ∩ M , c can be taken uniform in τ for τ in a compact set on which the strict concavityassumption on level sets of ˜ x holds. Ellipticity up to gauge
With L defined in (2.1)-(2.2), the main analytic points are that, first, LI is (aftera suitable exponential conjugation) a scattering pseudodifferential operator of order −
1, and second, by choosing an additional appropriate gauge-related summand, thisoperator LI is elliptic (again, after the exponential conjugation). These results arestated in the next two propositions, with the intermediate Lemma 3.2 describingthe gauge related summand. Proposition 3.1.
On one forms, resp. symmetric 2-cotensors, the operators N (cid:122) = e − (cid:122) /x LIe (cid:122) /x , lie in Ψ − , ( X ; sc T ∗ X, sc T ∗ X ) , resp. Ψ − , ( X ; Sym T ∗ X, Sym T ∗ X ) , for (cid:122) > .Proof. The proof of this proposition follows that of the scalar case given in [28,Proposition 3.3] and in a modified version of the scalar case in [27, Proposition 3.2].For convenience of the reader, we follow the latter proof very closely, except thatwe do not emphasize the continuity statements in terms of the underlying metricitself, indicating the modifications.Thus, recall that the map(3.1) Γ + : S ˜ M × [0 , ∞ ) → [ ˜ M × ˜ M ; diag] , Γ + ( x, y, λ, ω, t ) = (( x, y ) , γ x,y,λ,ω ( t ))is a local diffeomorphism, and similarly for Γ − in which ( −∞ ,
0] takes the place of[0 , ∞ ); see the discussion around [28, Equation (3.2)-(3.3)]; indeed this is true formore general curve families. Here [ ˜ M × ˜ M ; diag] is the blow-up of ˜ M at the diagonal z = z (cid:48) , which essentially means the introduction of spherical/polar coordinates, oroften more conveniently projective coordinates, about it. Concretely, writing the(local) coordinates from the two factors of ˜ M as ( z, z (cid:48) ),(3.2) z, | z − z (cid:48) | , z − z (cid:48) | z − z (cid:48) | give (local) coordinates on this space. Since the statement regarding the pseudodif-ferential property of LI is standard away from x = 0, we concentrate on the latterregion. Correspondingly, in our coordinates ( x, y, λ, ω ), we write( γ x,y,λ,ω ( t ) , γ (cid:48) x,y,λ,ω ( t )) = ( X x,y,λ,ω ( t ) , Y x,y,λ,ω ( t ) , Λ (cid:91)x,y,λ,ω ( t ) , Ω (cid:91)x,y,λ,ω ( t ))for the lifted geodesic γ x,y,λ,ω ( t ).Recall from [28, Section 2] that coordinates on Melrose’s scattering double space,on which the Schwartz kernels of elements of Ψ s,r sc ( X ) are conormal to the diagonal,near the lifted scattering diagonal, are (with x ≥ x, y, X = x (cid:48) − xx , Y = y (cid:48) − yx . Note that here
X, Y are as in [28] around Equation (3.10), not as in [28, Section 2](where the signs are switched), which means that we need to replace ( ξ, η ) by( − ξ, − η ) in the Fourier transform when computing principal symbols. Further, itis convenient to write coordinates on [ ˜ M × ˜ M ; diag] in the region of interest (seethe beginning of the paragraph of Equation (3.10) in [28]), namely (the lift of) | x − x (cid:48) | < C | y − y (cid:48) | , as x, y, | y − y (cid:48) | , x (cid:48) − x | y − y (cid:48) | , y (cid:48) − y | y − y (cid:48) | , with the norms being Euclidean norms, instead of (3.2); we write Γ ± in terms ofthese. Note that these are x, y, x | Y | , xX | Y | , ˆ Y . Moreover, by [28, Equation(3.10)] and NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 11 the subsequent equations, combined also with Equations (3.14)-(3.15) there, λ, ω, t are given in terms of x, x (cid:48) , y, y (cid:48) as(Λ ◦ Γ − ± ) (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) = x X − α ( x, y, x | Y | , xX | Y | , ˆ Y )) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) with ˜Λ ± smooth,(Ω ◦ Γ − ± ) (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) = ˆ Y + x | Y | ˜Ω ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) with ˜Ω ± smooth and ± ( T ◦ Γ − ± ) (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) = x | Y | + x | Y | ˜ T ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17) with ˜ T smooth.In particular,(Λ ◦ Γ − ± ) ∂ x + (Ω ◦ Γ − ± ) ∂ y = (cid:16) x X − α ( x, y, x | Y | , xX | Y | , ˆ Y )) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) ∂ x + (cid:16) ˆ Y + x | Y | ˜Ω ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) ∂ y . Thus, a smooth metric g = dx + h applied to this yields(3.3) (Λ ◦ Γ − ± ) dx + (Ω ◦ Γ − ± ) h ( ∂ y ) = x (cid:16) x (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x (cid:17) = x (cid:16) x (cid:16) X − α ( x, y, x | Y | , xX | Y | , ˆ Y )) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) dxx + (cid:16) ˆ Y + x | Y | ˜Ω ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) h ( ∂ y ) x (cid:17) , while so g sc applied to this yields(3.4) x − (cid:16) x − (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x (cid:17) = x − (cid:16)(cid:16) X − α ( x, y, x | Y | , xX | Y | , ˆ Y )) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) dxx + (cid:16) ˆ Y + x | Y | ˜Ω ± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) h ( ∂ y ) x (cid:17) . Notice that on the right hand side of (3.4) the singular factor of x − in frontof dxx disappears due to the factor x in Λ, while on the right hand side of (3.3)correspondingly dxx has a vanishing factor x . This means, as we see below, thatthe dxx component behaves trivially at the level of the boundary principal symbolof the operator N (cid:122) , defined like N (cid:122) but with g in place of g sc , so in fact onecan never have full ellipticity in this case; this is the reason we must use g sc in thedefinition of N (cid:122) . One also needs to have Λ (cid:91)x,y,λ,ω ( t ) , Ω (cid:91)x,y,λ,ω ( t ) evaluated at ( x (cid:48) , y (cid:48) ), since this isthe tangent vector λ (cid:48) ∂x (cid:48) + ω (cid:48) ∂ y (cid:48) with which our tensors are contracted as they arebeing integrated along the geodesic. In order to compute this efficiently, we recallfrom [28, Equation (3.14)] that x (cid:48) = x + λt + α ( x, y, λ, ω ) t + O ( t ) , y (cid:48) = y + ωt + O ( t ) , with the O ( t ), resp. O ( t ) terms having smooth coefficients in terms of ( x, y, λ, ω ).Correspondingly, λ (cid:48) = dxdt = λ + 2 α ( x, y, λ, ω ) t + O ( t ) , ω (cid:48) = dydt = ω + O ( t ) . This gives that in terms of x, y, x (cid:48) , y (cid:48) , λ (cid:48) is given byΛ (cid:48) ◦ Γ − ± = Λ ◦ Γ − ± + 2 α ( x, y, Λ ◦ Γ − ± , Ω ◦ Γ − ± )( T ◦ Γ − ± ) + ( T ◦ Γ − ± ) ˜Λ (cid:48) ◦ Γ − ± , with ˜Λ (cid:48) smooth in terms of x, y, Λ ◦ Γ − ± , Ω ◦ Γ − ± , T ◦ Γ − ± . Substituting these inyieldsΛ (cid:48) ◦ Γ − ± = x X − α ( x, y, x | Y | , xX | Y | , ˆ Y ) | Y | | Y | + 2 x | Y | α ( x, y, x | Y | , xX | Y | , ˆ Y )+ x | Y | ˜Λ (cid:48) ( x, y, x | Y | , xX | Y | , ˆ Y )= x X + α ( x, y, x | Y | , xX | Y | , ˆ Y ) | Y | | Y | + x | Y | ˜Λ (cid:48) ( x, y, x | Y | , xX | Y | , ˆ Y )while Ω (cid:48) ◦ Γ − ± = ˆ Y + x ˜Ω (cid:48) ( x, y, x | Y | , xX | Y | , ˆ Y ) . Correspondingly,(Λ (cid:48) ◦ Γ − ± ) ∂ x + (Ω (cid:48) ◦ Γ − ± ) ∂ y = x − (cid:16) x − (Λ (cid:48) ◦ Γ − ± ) x ∂ x + (Ω (cid:48) ◦ Γ − ± ) x∂ y (cid:17) = x − (cid:16)(cid:16) X + α ( x, y, x | Y | , xX | Y | , ˆ Y ) | Y | | Y | + x | Y | ˜Λ (cid:48)± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) x ∂ x + (cid:16) ˆ Y + x | Y | ˜Ω (cid:48)± (cid:16) x, y, x | Y | , xX | Y | , ˆ Y (cid:17)(cid:17) x∂ y (cid:17) . Then, similarly, near the boundary as in [28, Equation (3.13)], one obtains theSchwartz kernel of N (cid:122) on one forms:(3.5) K (cid:91) ( x, y, X, Y )= (cid:88) ± e − (cid:122) X/ (1+ xX ) χ (cid:16) X − α ( x, y, x | Y | , xX | Y | , ˆ Y ) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , x | X || Y | , ˆ Y (cid:17)(cid:17)(cid:16) x − (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x (cid:17)(cid:16) x − (Λ (cid:48) ◦ Γ − ± ) x ∂ x + (Ω (cid:48) ◦ Γ − ± ) x∂ y (cid:17) | Y | − n +1 J ± (cid:16) x, y, X | Y | , | Y | , ˆ Y (cid:17) , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 13 with the density factor J smooth, positive, = 1 at x = 0; there is a similar formulafor 2-tensors. Note that the factor x − in (3.4), as well as another x − from writing(Λ (cid:48) ◦ Γ − ± ) ∂ x + (Ω (cid:48) ◦ Γ − ± ) ∂ y = x − (cid:16) x − (Λ (cid:48) ◦ Γ − ± ) x ∂ x + (Ω (cid:48) ◦ Γ − ± ) x∂ y (cid:17) are absorbed into the definition of L , (2.1)-(2.2), hence the different powers ( − x, y, | Y | , X | Y | , ˆ Y are valid coordinates on the blow-up of the scattering diagonal in | Y | > (cid:15) | X | , (cid:15) > χ , cf. the discussion after [28, Equation(3.12)], so the argument of χ is smooth on this blown up space . In addition, due to the order x vanishing of Λ, x − (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x , resp. x − (Λ ◦ Γ − ± ) x ∂ x + (Ω ◦ Γ − ± ) x∂ y are smooth sections of sc T ∗ X , resp. sc T X , pulled back from the left, resp. right,factor of X , thus their product defines a smooth section of the endomorphismbundle of sc T ∗ X .Since this homomorphism factor is the only difference from [28, Proposition 3.3],and we have shown its smoothness properties as a bundle endomorphism, this provesthe proposition as in [28, Proposition 3.3].If we defined N (cid:122) , as N (cid:122) but using a smooth metric g in place of g sc , we wouldhave the Schwartz kernel(3.6) K (cid:91) ( x, y, X, Y )= (cid:88) ± e − (cid:122) X/ (1+ xX ) χ (cid:16) X − α ( x, y, x | Y | , xX | Y | , ˆ Y ) | Y | | Y | + x ˜Λ ± (cid:16) x, y, x | Y | , x | X || Y | , ˆ Y (cid:17)(cid:17)(cid:16) x (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x (cid:17)(cid:16) x − (Λ (cid:48) ◦ Γ − ± ) x ∂ x + (Ω (cid:48) ◦ Γ − ± ) x∂ y (cid:17) | Y | − n +1 J ± (cid:16) x, y, X | Y | , | Y | , ˆ Y (cid:17) , and x (Λ ◦ Γ − ± ) dxx + (Ω ◦ Γ − ± ) h ( ∂ y ) x , resp. x − (Λ ◦ Γ − ± ) x ∂ x + (Ω ◦ Γ − ± ) x∂ y are again smooth sections of sc T ∗ X , resp. sc T X , pulled back from the left, resp.right, factor of X , but, as pointed out earlier, with the coefficient of dxx vanishing,thus eliminating the possibility of ellipticity in this case. (cid:3) Before proceeding, we compute the principal symbol of the gauge term d s (cid:122) δ s (cid:122) .For this recall that d s (cid:122) = e − (cid:122) /x d s e (cid:122) /x , with d s defined using the background metric g (on one forms; the metric is irrelevant for functions), and δ s (cid:122) is its adjoint withrespect to the scattering metric g sc (not g ). In order to give the principal symbols,we use the basis dxx , dyx for one forms, with dyx understood as a short hand for dy x , . . . , dy n − x , while for2-tensors, we use a decomposition dxx ⊗ dxx , dxx ⊗ dyx , dyx ⊗ dxx , dyx ⊗ dyx . Note that symmetry of a 2-tensor is the statement that the 2nd and 3rd (block)entries are the same (up to the standard identification), so for symmetric 2-tensorswe can also use dxx ⊗ s dxx , dxx ⊗ s dyx , dyx ⊗ s dyx , where the middle component is the common dxx ⊗ dyx and dyx ⊗ dxx component. Lemma 3.2.
On one forms, the operator d s (cid:122) δ s (cid:122) ∈ Diff , ( X ; sc T ∗ X, sc T ∗ X ) hasprincipal symbol (cid:18) ξ + i (cid:122) η ⊗ (cid:19) (cid:0) ξ − i (cid:122) ι η (cid:1) = (cid:18) ξ + (cid:122) ( ξ + i (cid:122) ) ι η ( ξ − i (cid:122) ) η ⊗ η ⊗ ι η (cid:19) . On the other hand, on symmetric 2-tensors d s (cid:122) δ s (cid:122) ∈ Diff , ( X ; Sym T ∗ X, Sym T ∗ X ) has principal symbol ξ + i (cid:122) η ⊗ ( ξ + i (cid:122) ) a η ⊗ s (cid:18) ξ − i (cid:122) ι η (cid:104) a, . (cid:105) ( ξ − i (cid:122) ) ι sη (cid:19) = ξ + (cid:122) ( ξ + i (cid:122) ) ι η ( ξ + i (cid:122) ) (cid:104) a, . (cid:105) ( ξ − i (cid:122) ) η ⊗ ( η ⊗ ) ι η + ( ξ + (cid:122) ) η ⊗ (cid:104) a, . (cid:105) + ( ξ + i (cid:122) ) ι sη ( ξ − i (cid:122) ) a aι η + ( ξ − i (cid:122) ) η ⊗ a (cid:104) a, . (cid:105) + η ⊗ s ι η , where a is a suitable symmetric 2-tensor.Proof. This is an algebraic symbolic computation, so in particular it can be donepointwise. Since one can arrange that the metric g sc used to compute adjoints is ofthe form x − dx + x − dy , where dy is the flat metric, at the point in question,one can simply use this in the computation. With our coordinates at the point inquestion, trivializing the inner product, g sc , the inner product on one-forms is givenby the matrix (cid:18) (cid:19) while on 2-tensors by . First consider one-forms. Recall from Section 2 that the full principal symbol of d , in Diff ( X ; C , sc T ∗ X ), with C the trivial bundle, is, as a map from functions toone-forms,(3.7) (cid:18) ξη ⊗ (cid:19) . NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 15
Thus the symbol of d s (cid:122) = e − (cid:122) /x d s e (cid:122) /x , which conjugation effectively replaces ξ by ξ + i (cid:122) (as e − (cid:122) /x x D x e (cid:122) /x = x D x + i (cid:122) ), is (cid:18) ξ + i (cid:122) η ⊗ (cid:19) . Hence δ s (cid:122) has symbol given by the adjoint of that of d s (cid:122) with respect to the innerproduct of g sc , which is (cid:0) ξ − i (cid:122) ι η (cid:1) . Thus, the principal symbol of d s (cid:122) δ s (cid:122) is the product, (cid:18) ξ + (cid:122) ( ξ + i (cid:122) ) ι η ( ξ − i (cid:122) ) η ⊗ η ⊗ ι η (cid:19) , proving the lemma for one forms.We now turn to symmetric 2-tensors. Again, recall from Section 2 that the fullprincipal symbol of the gradient relative to g , in Diff ( X ; sc T ∗ X ; sc T ∗ X ⊗ sc T ∗ X ),is, as a map from one-forms to 2-tensors (which we write in the four block form asbefore) is(3.8) ξ η ⊗ ξb η ⊗ , where b is a 2-tensor on Y = ∂X , and thus that of d s (with symmetric 2-tensorsconsidered as a subspace of 2-tensors) is ξ η ⊗ ξ η ⊗ ξa η ⊗ s , with a a symmetric 2-tensor (the symmetrization of b ). (Notice that a, b only play arole in the principal symbol at the boundary, not in the standard principal symbol,i.e. as ( ξ, η ) → ∞ .) Here a arises due to the treatment of d s , which is defined usinga standard metric g , as an element of Diff sc ( X ; sc T ∗ X, Sym T ∗ X ); it is acting onthe one-dimensional space Span { dxx } by multiplying the coefficient of dxx to producea symmetric 2-tensor on Y . Note that here the lower right block has ( ijk ) entry(corresponding to the ( ij ) entry of the symmetric 2-tensor and the k entry of theone-form) given by ( η i δ jk + η j δ ik ). Thus the symbol of d s (cid:122) = e − (cid:122) /x d s e (cid:122) /x , whichconjugation effectively replaces ξ by ξ + i (cid:122) (as e − (cid:122) /x x D x e (cid:122) /x = x D x + i (cid:122) ), is ξ + i (cid:122) η ⊗ ( ξ + i (cid:122) ) η ⊗ ( ξ + i (cid:122) ) a η ⊗ s . Thus, δ s (cid:122) has symbol given by the adjoint of that of d s (cid:122) with respect to this innerproduct, which is (cid:18) ξ − i (cid:122) ι η ι η (cid:104) a, . (cid:105) ( ξ − i (cid:122) ) ( ξ − i (cid:122) ) ι sη (cid:19) . Here the lower right block has ( (cid:96)ij ) entry given by ( η i δ (cid:96)j + η j δ i(cid:96) ). Here theinner product (cid:104) a, . (cid:105) as well as ι η are with respect to the identity because of the trivialization of the inner product; invariantly they with respect to the inner productinduced by h . Correspondingly, the product, in the more concise notation forsymmetric tensors, has the symbol as stated, proving the lemma. (cid:3) The proof of the next proposition, on ellipticity, relies on the subsequently statedtwo lemmas, whose proofs in turn take up the rest of this section.
Proposition 3.3.
First consider the case of one forms. Let (cid:122) > . Given ˜Ω , aneighborhood of X ∩ M = { x ≥ , ρ ≥ } in X , for suitable choice of the cutoff χ ∈ C ∞ c ( R ) and of M ∈ Ψ − , ( X ) , the operator A (cid:122) = N (cid:122) + d s (cid:122) M δ s (cid:122) , N (cid:122) = e − (cid:122) /x LIe (cid:122) /x , d s (cid:122) = e − (cid:122) /x d s e (cid:122) /x , is elliptic in Ψ − , ( X ; sc T ∗ X, sc T ∗ X ) in ˜Ω .On the other hand, consider the case of symmetric 2-tensors. Then there exists (cid:122) > such that for (cid:122) > (cid:122) the following holds. Given ˜Ω , a neighborhood of X ∩ M = { x ≥ , ρ ≥ } in X , for suitable choice of the cutoff χ ∈ C ∞ c ( R ) and of M ∈ Ψ − , ( X ; sc T ∗ X, sc T ∗ X ) , the operator A (cid:122) = N (cid:122) + d s (cid:122) M δ s (cid:122) , N (cid:122) = e − (cid:122) /x LIe (cid:122) /x , d s (cid:122) = e − (cid:122) /x d s e (cid:122) /x , is elliptic in Ψ − , ( X ; Sym T ∗ X, Sym T ∗ X ) in ˜Ω .Proof. The proof of this proposition is straightforward given the two lemmas weprove below. Indeed, as we prove below in Lemma 3.4, provided χ ≥ χ (0) > e − (cid:122) /x LIe (cid:122) /x has positive definite principal symbol at fiber infinityin the scattering cotangent bundle when restricted to the subspace of sc T ∗ X or Sym T ∗ X given by the kernel of the symbol of δ s (cid:122) , where the inner product isthat of the scattering metric we consider (with respect to which δ s is computed);in Lemma 3.5 we show a similar statement for the principal symbol at finite pointsunder the assumption that χ is sufficiently close, in a suitable sense, to an evenpositive Gaussian, with the complication that for 2-tensors we need to assume (cid:122) > d s (cid:122) M δ s (cid:122) to it, where M has positive principalsymbol, and is of the correct order, we obtain an elliptic operator, completing theproof of Proposition 3.3. (cid:3) We are thus reduced to proving the two lemmas we used.
Lemma 3.4.
Both on one-forms and on symmetric 2-tensors, N (cid:122) is elliptic atfiber infinity in sc T ∗ X when restricted to the kernel of the principal symbol of δ s (cid:122) .Proof. This is very similar to the scalar setting. With S = X − α ( ˆ Y ) | Y | | Y | , ˆ Y = Y | Y | , the Schwartz kernel of N (cid:122) at the scattering front face x = 0 is, by (3.5), given by e − (cid:122) X | Y | − n +1 χ ( S ) (cid:16)(cid:16) S dxx + ˆ Y · dyx (cid:17)(cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17)(cid:17) on one forms, respectively e − (cid:122) X | Y | − n +1 χ ( S ) (cid:16)(cid:16)(cid:16) S dxx + ˆ Y · dyx (cid:17) ⊗ (cid:16)(cid:16) S dxx + ˆ Y · dyx (cid:17)(cid:17)(cid:17)(cid:17)(cid:16)(cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17) ⊗ (cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17)(cid:17) NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 17 on 2-tensors, where ˆ Y is regarded as a tangent vector which acts on covectors, andwhere ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) maps one forms to scalars, thus (cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17) ⊗ (cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17) maps symmetric 2-tensors to scalars, while S dxx + ˆ Y · dyx maps scalars to one forms,so (cid:16) S dxx + ˆ Y · dyx (cid:17) ⊗ (cid:16) S dxx + ˆ Y · dyx (cid:17) maps scalars to symmetric 2-tensors. In order to make the notation less confusing,we employ a matrix notation, (cid:16) S dxx + ˆ Y · dyx (cid:17)(cid:16) ( S + 2 α | Y | )( x ∂ x ) + ˆ Y · ( x∂ y ) (cid:17) = (cid:18) S ( S + 2 α | Y | ) S (cid:104) ˆ Y , ·(cid:105) ˆ Y ( S + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) , with the first column and row corresponding to dxx , resp. x ∂ x , and the secondcolumn and row to the (co)normal vectors. For 2-tensors, as before, we use adecomposition dxx ⊗ dxx , dxx ⊗ dyx , dyx ⊗ dxx , dyx ⊗ dyx , where the symmetry of the 2-tensor is the statement that the 2nd and 3rd (block)entries are the same. For the actual endomorphism we write(3.9) S S (cid:104) ˆ Y , ·(cid:105) S (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) (cid:0) ( S + 2 α | Y | ) ˆ Y ˆ Y ( S + 2 α | Y | ) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) ( S + 2 α | Y | ) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) (cid:1) = S ( S + 2 α | Y | ) S ( S + 2 α | Y | ) (cid:104) ˆ Y , ·(cid:105) S ( S + 2 α | Y | ) (cid:104) ˆ Y , ·(cid:105) S (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) S ( S + 2 α | Y | ) Y S ( S + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , . (cid:105) S ( S + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , . (cid:105) S ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) S ( S + 2 α | Y | ) Y S ( S + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , . (cid:105) S ( S + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , . (cid:105) S ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) ( S + 2 α | Y | ) ˆ Y ˆ Y ( S + 2 α | Y | ) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) ( S + 2 α | Y | ) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) . Here we write subscripts 1 and 2 for clarity on ˆ Y to denote whether it is acting onthe first or the second factor, though this also immediately follows from its positionwithin the matrix.Now, the standard principal symbol is that of the conormal singularity at thediagonal, i.e. X = 0, Y = 0. Writing ( X, Y ) = Z , ( ξ, η ) = ζ , we would need toevaluate the Z -Fourier transform as | ζ | → ∞ . This was discussed in [28] aroundEquation (3.8); the leading order behavior of the Fourier transform as | ζ | → ∞ canbe obtained by working on the blown-up space of the diagonal, with coordinates | Z | , ˆ Z = Z | Z | (as well as z = ( x, y )), and integrating the restriction of the Schwartzkernel to the front face, | Z | − = 0, after removing the singular factor | Z | − n +1 , alongthe equatorial sphere corresponding to ζ , and given by ˆ Z · ζ = 0. Now, concretely inour setting, in view of the infinite order vanishing, indeed compact support, of theSchwartz kernel as X/ | Y | → ∞ (and Y bounded), we may work in semi-projectivecoordinates, i.e. in spherical coordinates in Y , but X/ | Y | as the normal variable;the equatorial sphere then becomes ( X/ | Y | ) ξ + ˆ Y · η = 0 (with the integral of courserelative to an appropriate positive density). With ˜ S = X/ | Y | , keeping in mind that terms with extra vanishing factors at the front face, | Y | = 0 can be dropped, wethus need to integrate(3.10) (cid:18) ˜ S ˜ S (cid:104) ˆ Y , ·(cid:105) ˜ S ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) χ ( ˜ S ) = (cid:18) ˜ S ˆ Y (cid:19) ⊗ (cid:0) ˜ S ˆ Y (cid:1) χ ( ˜ S ) , on this equatorial sphere in the case of one-forms, and the analogous expression inthe case of symmetric 2-tensors. Now, for χ ≥ S, ˆ Y ). As ( ˜ S, ˆ Y ) runs through the ( ξ, η )-equatorialsphere, we are taking a positive (in the sense of non-negative) linear combinationof the projections to the span of the vectors in this orthocomplement, with theweight being strictly positive as long as χ ( ˜ S ) > δ s (cid:122) consists of covectorsof the form v = ( v , v (cid:48) ) with ξv + η · v (cid:48) = 0. Hence, if we show that for each suchnon-zero vector ( v , v (cid:48) ) there is at least one ( ˜ S, ˆ Y ) with χ ( ˜ S ) > ξ ˜ S + η · ˆ Y = 0and ˜ Sv + ˆ Y · v (cid:48) (cid:54) = 0, we conclude that the integral of the projections is positive,thus the principal symbol of our operator is elliptic, on the kernel of the standardprincipal symbol of δ s (cid:122) . But this is straightforward if χ (0) > v (cid:48) = 0 then ξ = 0 (since v (cid:54) = 0), one may take ˜ S (cid:54) = 0 small, ˆ Y orthogonalto η (such ˆ Y exists as η ∈ R n − , n ≥ v (cid:48) (cid:54) = 0 and v (cid:48) is not a multiple of η , then take ˆ Y orthogonal to η but notto v (cid:48) , ˜ S = 0,(3) if v (cid:48) = cη with v (cid:48) (cid:54) = 0 (so c and η do not vanish) then ξv + c | η | = 0 so withˆ Y still to be chosen if we let ˜ S = − η · ˆ Yξ , then ˜ Sv + ˆ Y · v (cid:48) = c ( ˆ Y · η )(1 + | η | ξ )which is non-zero as long as ˆ Y · η (cid:54) = 0; this can be again arranged, togetherwith ˆ Y · η being sufficiently small (such ˆ Y exists again as η ∈ R n − , n ≥ S is small enough in order to ensure χ ( ˜ S ) > δ s (cid:122) .In the case of symmetric 2-tensors, the matrix (3.10) is replaced by(3.11) ˜ S ˜ S ˆ Y ˜ S ˆ Y ˆ Y ⊗ ˆ Y ⊗ (cid:0) ˜ S ˜ S (cid:104) ˆ Y , ·(cid:105) ˜ S (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y ⊗ ˆ Y , ·(cid:105) (cid:1) χ ( ˜ S ) , which again is a non-negative multiple of a projection. For a symmetric 2-tensor ofthe form v = ( v NN , v NT , v NT , v T T ) in the kernel of the principal symbol of δ s (cid:122) , wehave by Lemma 3.2 that(3.12) ξv NN + η · v NT = 0 ,ξv NT + 12 ( η + η ) · v T T = 0 , where η resp. η denoting that the inner product is taken in the first, resp. second,slots. Taking the inner product of the second equation with η gives ξη · v NT + ( η ⊗ η ) v T T = 0 . Substituting this into the first equation yields ξ v NN = ( η ⊗ η ) v T T . NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 19
We now consider two cases, ξ = 0 and ξ (cid:54) = 0.If ξ (cid:54) = 0, then for a symmetric 2-tensor being in the kernel of the principalsymbol of δ s (cid:122) at fiber infinity and of (3.11) for ( ˜ S, ˆ Y ) satisfying ξ ˜ S + η · ˆ Y = 0, i.e.˜ S = − ηξ · ˆ Y is equivalent to(3.13) v NN = ξ − ( η ⊗ η ) v T T ,v NT = − ξ ( η + η ) · v T T (cid:16)(cid:16) η · ˆ Yξ (cid:17) η ⊗ ηξ + η · ˆ Yξ (cid:16) ηξ ⊗ ˆ Y + ˆ Y ⊗ ηξ (cid:17) + ˆ Y ⊗ ˆ Y (cid:17) · v T T = 0 , and the last equation is equivalent to (cid:16)(cid:16) η · ˆ Yξ ηξ + ˆ Y (cid:17) ⊗ (cid:16) η · ˆ Yξ ηξ + ˆ Y (cid:17)(cid:17) · v T T = 0 . If η = 0, the first two equations say directly that v NN and v NT vanish, while thelast one states that ( ˆ Y ⊗ ˆ Y ) · v T T = 0 for all ˆ Y (we may simply take ˜ S = 0); butsymmetric 2-tensors of the form ˆ Y ⊗ ˆ Y span the space of all symmetric 2-tensors(as w ⊗ w + w ⊗ w = ( w + w ) ⊗ ( w + w ) − w ⊗ w − w ⊗ w ), so we concludethat v T T = 0, and thus v = 0 in this case. On the other hand, if η (cid:54) = 0 then takingˆ Y = (cid:15) ˆ η + (1 − (cid:15) ) / ˆ Y ⊥ and substituting into this equation yields (cid:16)(cid:16) | η | ξ (cid:17) (cid:15) ˆ η ⊗ ˆ η + (cid:16) | η | ξ (cid:17) (cid:15) (1 − (cid:15) ) / (ˆ η ⊗ ˆ Y ⊥ + ˆ Y ⊥ ⊗ ˆ η )+(1 − (cid:15) ) ˆ Y ⊥ ⊗ ˆ Y ⊥ (cid:17) · v T T = 0 . Note that ˜ S = − (cid:15) | η | ξ , so | ˜ S | is small when | (cid:15) | is sufficiently small. Substitutingin (cid:15) = 0 yields ( ˆ Y ⊥ ⊗ ˆ Y ⊥ ) · v T T = 0; since cotensors of the form ˆ Y ⊥ ⊗ ˆ Y ⊥ span η ⊥ ⊗ η ⊥ ( η ⊥ being the orthocomplement of η ), we conclude that v T T is orthogonalto every element of η ⊥ ⊗ η ⊥ . Next, taking the derivative in (cid:15) at (cid:15) = 0 yields(ˆ η ⊗ ˆ Y ⊥ + ˆ Y ⊥ ⊗ ˆ η ) · v T T = 0 for all ˆ Y ⊥ ; symmetric tensors of this form, togetherwith η ⊥ ⊗ η ⊥ , span all tensors in ( η ⊗ η ) ⊥ . Finally taking the second derivative at (cid:15) = 0 shows that (ˆ η ⊗ ˆ η ) · v T T = 0, this in conclusion v T T = 0. Combined with thefirst two equations of (3.13), one concludes that v = 0, thus the desired ellipticityfollows.On the other hand, if ξ = 0 (and so η (cid:54) = 0), then for a symmetric 2-tensor beingin the kernel of the principal symbol of δ s (cid:122) at fiber infinity and of (3.11) for ( ˜ S, ˆ Y )satisfying ξ ˜ S + η · ˆ Y = 0, i.e. η · ˆ Y = 0 is equivalent to(3.14) η · v NT = 0 , ( η + η ) · v T T = 0 , ˜ S v NN + 2 ˜ S ˆ Y · v NT + ( ˆ Y ⊗ ˆ Y ) · v T T = 0 . Since there are no constraints on ˜ S (apart from | ˜ S | small), we can differentiate thelast equation up to two times and evaluate the result at 0 to conclude that v NN = 0,ˆ Y · v NT = 0 and ( ˆ Y ⊗ ˆ Y ) · v T T = 0. Combined with the first two equations of (3.14),this shows v = 0, so again the desired ellipticity follows.Thus, in summary, both on one forms and on symmetric 2-tensors the principalsymbol at fiber infinity is elliptic on the kernel of that of δ s (cid:122) , proving the lemma. (cid:3) Lemma 3.5.
For (cid:122) > on one forms N (cid:122) is elliptic at finite points of sc T ∗ X when restricted to the kernel of the principal symbol of δ s (cid:122) . On the other hand,there exists (cid:122) > such that on symmetric 2-tensors N (cid:122) is elliptic at finite pointsof sc T ∗ X when restricted to the kernel of the principal symbol of δ s (cid:122) .Proof. Again this is similar to, but technically much more involved than, the scalarsetting. We recall from [28] that the kernel is based on using a compactly supported C ∞ localizer, χ , but for the actual computation it is convenient to use a Gaussianinstead χ instead. One recovers the result by taking φ ∈ C ∞ c ( R ), φ ≥
0, identi-cally 1 near 0, and considering an approximating sequence χ k = φ ( ./k ) χ . Thenthe Schwartz kernels at the front face still converge in the space of distributionsconormal to the diagonal, which means that the principal symbols (including atfinite points) also converge, giving the desired ellipticity for sufficiently large k .Recall that the scattering principal symbol is the Fourier transform of the Schwartzkernel at the front face, so we now need to compute this Fourier transform. Westart with the one form case. Taking χ ( s ) = e − s / (2 ν ( ˆ Y )) as in the scalar caseconsidered in [28] for the computation (in the scalar case we took ν = (cid:122) − α ; herewe leave it unspecified for now, except demanding 0 < ν < (cid:122) − α as needed forthe Schwartz kernel to be rapidly decreasing at infinity on the front face), we cancompute the X -Fourier transform exactly as before, keeping in mind that this needsto be evaluated at − ξ (just like the Y Fourier transform needs to be evaluated at − η ) due to our definition of X : | Y | − n e − iα ( − ξ − i (cid:122) ) | Y | (cid:18) D σ − α | Y | D σ − D σ (cid:104) ˆ Y , ·(cid:105) ˆ Y ( − D σ + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) ˆ χ (( − ξ − i (cid:122) ) | Y | )= c √ ν | Y | − n e iα ( ξ + i (cid:122) ) | Y | (cid:18) D σ − α | Y | D σ − D σ (cid:104) ˆ Y , ·(cid:105) ˆ Y ( − D σ + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) e − ν ( ξ + i (cid:122) ) | Y | / with c >
0, and with D σ differentiating the argument of ˆ χ . One is left withcomputing the Y -Fourier transform, which in polar coordinates takes the form (cid:90) S n − (cid:90) [0 , ∞ ) e i | Y | ˆ Y · η | Y | − n e iα ( ξ + i (cid:122) ) | Y | (cid:18) − D σ ( − D σ + 2 α | Y | ) − D σ (cid:104) ˆ Y , ·(cid:105) ˆ Y ( − D σ + 2 α | Y | ) ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) ˆ χ ( − ( ξ + i (cid:122) ) | Y | ) | Y | n − d | Y | d ˆ Y , and the factors | Y | ± ( n − cancel as in the scalar case. Explicitly evaluating thederivatives, writing φ ( ξ, ˆ Y ) = ν ( ˆ Y )( ξ + i (cid:122) ) − iα ( ˆ Y )( ξ + i (cid:122) ) , yields(3.15) (cid:90) S n − (cid:90) ∞ e i | Y | ˆ Y · η (cid:18) iν ( ξ + i (cid:122) )( iν ( ξ + i (cid:122) ) + 2 α ) | Y | + ν iν ( ξ + i (cid:122) ) | Y |(cid:104) ˆ Y , ·(cid:105) ˆ Y ( iν ( ξ + i (cid:122) ) + 2 α ) | Y | ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) × e − φ | Y | / d | Y | d ˆ Y .
We extend the integral in | Y | to R , replacing it by a variable t , and using thatthe integrand is invariant under the joint change of variables t → − t and ˆ Y → − ˆ Y . NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 21
This gives 12 (cid:90) S n − (cid:90) R e it ˆ Y · η (cid:18) iν ( ξ + i (cid:122) )( iν ( ξ + i (cid:122) ) + 2 α ) t + ν iν ( ξ + i (cid:122) ) t (cid:104) ˆ Y , ·(cid:105) ˆ Y ( iν ( ξ + i (cid:122) ) + 2 α ) t ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:19) × e − φt / dt d ˆ Y .
Now the t integral is a Fourier transform evaluated at − ˆ Y · η , under which multipli-cation by t becomes D ˆ Y · η . Since the Fourier transform of e − φ ( ξ, ˆ Y ) t / is a constantmultiple of(3.16) φ ( ξ, ˆ Y ) − / e − ( ˆ Y · η ) / (2 φ ( ξ, ˆ Y )) , we are left with (cid:90) S n − φ ( ξ, ˆ Y ) − / (cid:32) iν ( ξ + i (cid:122) )( iν ( ξ + i (cid:122) ) + 2 α ) D Y · η + ν iν ( ξ + i (cid:122) ) (cid:104) ˆ Y , ·(cid:105) D ˆ Y · η ˆ Y ( iν ( ξ + i (cid:122) ) + 2 α ) D ˆ Y · η ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:33) × e − ( ˆ Y · η ) / (2 φ ( ξ, ˆ Y )) d ˆ Y , which explicitly gives(3.17) (cid:90) S n − φ ( ξ, ˆ Y ) − / iν ( ξ + i (cid:122) )( iν ( ξ + i (cid:122) ) + 2 α ) (cid:16) − ( ˆ Y · η ) φ ( ξ, ˆ Y ) + φ ( ξ, ˆ Y ) (cid:17) + ν iν ( ξ + i (cid:122) ) (cid:104) ˆ Y , ·(cid:105) i ˆ Y · ηφ ( ξ, ˆ Y ) ˆ Y ( iν ( ξ + i (cid:122) ) + 2 α ) i ˆ Y · ηφ ( ξ, ˆ Y ) ˆ Y (cid:104) ˆ Y , ·(cid:105) × e − ( ˆ Y · η ) / (2 φ ( ξ, ˆ Y )) d ˆ Y .
Now observe that the top left entry of the matrix is exactly − νφ ( ξ, ˆ Y ) (cid:16) − ( ˆ Y · η ) φ ( ξ, ˆ Y ) + 1 φ ( ξ, ˆ Y ) (cid:17) + ν = ν ( ˆ Y · η ) φ ( ξ, ˆ Y ) = ν ( ξ + i (cid:122) )( ν ( ξ + i (cid:122) ) − iα ) ( ˆ Y · η ) φ ( ξ, ˆ Y ) . Thus, the matrix in the integrand is (cid:32) − ν ( ξ + i (cid:122) ) φ ( ˆ Y · η )ˆ Y (cid:33) ⊗ (cid:16) − ( ν ( ξ + i (cid:122) ) − iα ) φ ( ˆ Y · η ) (cid:104) ˆ Y , ·(cid:105) (cid:17) . Now, if we take ν = (cid:122) − α as in the scalar case in [28], then ν ( ξ + i (cid:122) ) − iα = ν ( ξ − i (cid:122) ) , while φ = ( ξ + i (cid:122) )( ν ( ξ + i (cid:122) ) − iα ) = ν ( ξ + (cid:122) ) is real, so the matrix, with this choice of ν , is orthogonal projection to the span of( − ν ( ξ + i (cid:122) ) φ ( ˆ Y · η ) , ˆ Y ). The expression (3.17) becomes(3.18)( ξ + (cid:122) ) − / (cid:90) S n − ν − / (cid:32) − ν ( ξ + i (cid:122) ) ξ + (cid:122) ( ˆ Y · η )ˆ Y (cid:33) ⊗ (cid:16) − ν ( ξ − i (cid:122) ) ξ + (cid:122) ( ˆ Y · η ) (cid:104) ˆ Y , ·(cid:105) (cid:17) e − ( ˆ Y · η ) / (2 ν ( ξ + (cid:122) )) d ˆ Y , which is thus a superposition of positive (in the sense of non-negative) operators,which is thus itself positive. Further, if a vector ( v , v (cid:48) ) lies in the kernel of theprincipal symbol of δ s (cid:122) , i.e. ( ξ − i (cid:122) ) v + ι η v (cid:48) = 0, then orthogonality to ( − ν ( ξ + i (cid:122) ) ξ + (cid:122) ( ˆ Y · η ) , ˆ Y ) for any particular ˆ Y would mean0 = − ν ( ξ − i (cid:122) ) ξ + (cid:122) ( ˆ Y · η ) v + ˆ Y · v (cid:48) = νξ + (cid:122) ( η · v (cid:48) )( ˆ Y · η ) + ˆ Y · v (cid:48) . Note that S n − is at least one dimensional (i.e. is the sphere in at least a 2-dimensional vector space). Consider v (cid:48) (cid:54) = 0; this would necessarily be the caseof interest since v = − ( ξ − i (cid:122) ) − ( η · v (cid:48) ). If η = 0, picking ˆ Y parallel to v (cid:48) showsthat there is at least one choice of ˆ Y for which this equality does not hold. If η (cid:54) = 0,and v (cid:48) is not a multiple of η , we can take ˆ Y orthogonal to η and not orthogonal to v (cid:48) , which again gives a choice of ˆ Y for the equality above does not hold. Finally, if v (cid:48) is a multiple of η , the expression at hand is just ν | η | ξ + (cid:122) ( ˆ Y · v (cid:48) ) + ˆ Y · v (cid:48) , so choosingany ˆ Y not orthogonal to v (cid:48) again gives a ˆ Y for which the equality does not hold.Therefore, (3.18) is actually positive definite when restricted to the kernel of thesymbol of δ s (cid:122) , as claimed.We now turn to the 2-tensor version. With B ij corresponding to the terms with i factors of S and j factors of S +2 α | Y | prior to the Fourier transform, the analogueof (3.15) is(3.19) (cid:90) S n − (cid:90) ∞ e i | Y | ˆ Y · η B B (cid:104) ˆ Y , ·(cid:105) B (cid:104) ˆ Y , ·(cid:105) B (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) B Y B ˆ Y (cid:104) ˆ Y , . (cid:105) B ˆ Y (cid:104) ˆ Y , . (cid:105) B ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) B Y B ˆ Y (cid:104) ˆ Y , . (cid:105) B ˆ Y (cid:104) ˆ Y , . (cid:105) B ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) B ˆ Y ˆ Y B ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) B ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) B ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) × e − φ | Y | / d | Y | d ˆ Y , with B = 1 ,B = iν ( ξ + i (cid:122) ) | Y | ,B = − ν ( ξ + i (cid:122) ) | Y | + ν,B = i ( ν ( ξ + i (cid:122) ) − iα ) | Y | ,B = − ν ( ξ + i (cid:122) )( ν ( ξ + i (cid:122) ) − iα ) | Y | + ν,B = − iν ( ξ + i (cid:122) ) ( ν ( ξ + i (cid:122) ) − iα ) | Y | + (3 iν ( ξ + i (cid:122) ) + 2 αν ) | Y | ,B = − ( ν ( ξ + i (cid:122) ) − iα ) | Y | + ν,B = − iν ( ξ + i (cid:122) )( ν ( ξ + i (cid:122) ) − iα ) | Y | ,B = ν ( ξ + i (cid:122) ) ( ν ( ξ + i (cid:122) ) − iα ) | Y | + ν ( − ν ( ξ + i (cid:122) ) + 12 iνα ( ξ + i (cid:122) ) + 4 α ) | Y | + 3 ν . NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 23
Note that the leading term of B jk , in terms of the power of | Y | involved, is simply( iν ( ξ + i (cid:122) ) | Y | ) j ( i ( ν ( ξ + i (cid:122) ) − iα ) | Y | ) k ; this arises by all derivatives in (3.9) arisingby Fourier transforming in S (which gives a derivative − D σ in the dual variable σ )falling on the exponential, e − νσ / , which is then evaluated at σ = − ( ξ + i (cid:122) ) | Y | .However, for the full scattering principal symbol all terms are relevant.Next, we extend the | Y | integral to R , writing the corresponding variable as t and do the Fourier transform in t (with a minus sign, i.e. evaluated at − ˆ Y · η ) asin the one-form setting. This replaces t by D ˆ Y · η , as above, and in view of (3.16),explicitly evaluating the derivatives, we obtain the following analogue of (3.17)(3.20) (cid:90) S n − C C (cid:104) ˆ Y , ·(cid:105) C (cid:104) ˆ Y , ·(cid:105) C (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) C Y C ˆ Y (cid:104) ˆ Y , . (cid:105) C ˆ Y (cid:104) ˆ Y , . (cid:105) C ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) C Y C ˆ Y (cid:104) ˆ Y , . (cid:105) C ˆ Y (cid:104) ˆ Y , . (cid:105) C ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) C ˆ Y ˆ Y C ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) C ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) C ˆ Y ˆ Y (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) × φ ( ξ, ˆ Y ) − / e − ( ˆ Y · η ) / (2 φ ( ξ, ˆ Y )) d ˆ Y , where, with ρ = ˆ Y · η , C = 1 ,C = − ν ( ξ + i (cid:122) ) φ − ρ,C = ν ( ξ + i (cid:122) ) φ − ρ − iανφ − ( ξ + i (cid:122) ) ,C = − ( ν ( ξ + i (cid:122) ) − iα ) φ − ρ,C = ν ( ξ + i (cid:122) ))( ν ( ξ + i (cid:122) ) − iα ) φ − ρ ,C = − ν ( ξ + i (cid:122) ) ( ν ( ξ + i (cid:122) ) − iα ) φ − ρ + 2 ανiφ − ρ,C = ( ν ( ξ + i (cid:122) ) − iα ) φ − ρ + φ − ( ν ( ξ + i (cid:122) ) − iα )2 iα,C = − ν ( ξ + i (cid:122) )( ν ( ξ + i (cid:122) ) − iα ) φ − ρ − iανφ − ρ,C = ν ( ξ + i (cid:122) ) ( ν ( ξ + i (cid:122) ) − iα ) φ − ρ − α νφ − ρ + 4 α νφ − . Note again that the highest order term, in terms of the power of ρ , of C jk is( ν ( ξ + i (cid:122) )) j ( ν ( ξ + i (cid:122) ) − iα ) k ( − j + k φ − j − k , corresponding to all derivatives D ρ falling on the exponential e − ρ / (2 φ ) , evaluated at ρ = ˆ Y · η .Notice that C is exactly the (1 ,
1) entry in the one-form calculation, (3.17),while C , resp. C , are the factors in the (1 ,
2) and (2 ,
1) entries, for similarreasons. Now, it is easy to check that the matrix in (3.20) is(3.21) C ˆ Y C ˆ Y C ˆ Y ˆ Y ⊗ (cid:0) C C (cid:104) ˆ Y , ·(cid:105) C (cid:104) Y, ·(cid:105) (cid:104) ˆ Y , ·(cid:105) (cid:104) ˆ Y , ·(cid:105) (cid:1) . Letting ν = (cid:122) − α as in the one-form setting, the second factor here is the adjoint(involving of complex conjugates) of the first, in particular (with ρ = ˆ Y · η ) C = − ν ( ξ − i (cid:122) ) φ − ρ, C = ν ( ξ − i (cid:122) ) φ − ρ +2 iαν ( ξ − i (cid:122) ) φ − , φ = ν ( ξ + (cid:122) ) , so (3.21) is just a positive multiple of projection to the span of ( C , ˆ Y C , ˆ Y C , ˆ Y ˆ Y ).Thus, as in the one form setting, we have a superposition of positive (in the sense of non-negative) operators, so it remains to check that as ˆ Y varies, these vectorsspan the kernel of δ s (cid:122) .For a symmetric 2-tensor of the form v = ( v NN , v NT , v NT , v T T ) in the kernel ofthe principal symbol of δ s (cid:122) , we have by Lemma 3.2 that(3.22) ( ξ − i (cid:122) ) v NN + η · v NT + a · v T T = 0 , ( ξ − i (cid:122) ) v NT + 12 ( η + η ) · v T T = 0 , where η resp. η denoting that the inner product is taken in the first, resp. second,slots. Taking the inner product of the second equation with η gives( ξ − i (cid:122) ) η · v NT + ( η ⊗ η ) · v T T = 0 . Substituting this into the first equation yields( ξ − i (cid:122) ) v NN + (( ξ − i (cid:122) ) a − η ⊗ η ) · v T T = 0 , so v NN = ( ξ − i (cid:122) ) − ( η ⊗ η − ( ξ − i (cid:122) ) a ) · v T T , v NT = − − ( ξ − i (cid:122) ) − ( η + η ) · v T T . For a fixed ˆ Y for v in the kernel of the symbol of δ s (cid:122) to be in the kernel of theprojection (3.21) means that (cid:16) C ( ξ − i (cid:122) ) − ( η ⊗ η − ( ξ − i (cid:122) ) a ) − C ( ξ − i (cid:122) ) − ( η ⊗ ˆ Y + ˆ Y ⊗ η )+ ˆ Y ⊗ ˆ Y (cid:17) · v T T = 0 , so recalling ν = (cid:122) − α , φ = ν ( ξ + (cid:122) ), (cid:16) ( ξ + i (cid:122) ) − ( ξ + (cid:122) ) − ( ˆ Y · η ) + 2 iα ( ξ + (cid:122) ) − )(( ξ − i (cid:122) ) − ( η ⊗ η ) − a )+ ( ξ + (cid:122) ) − ( ˆ Y · η )( η ⊗ ˆ Y + ˆ Y ⊗ η ) + ˆ Y ⊗ ˆ Y (cid:17) · v T T = 0 . Now, it is convenient to rewrite this in terms of ‘semiclassical’ (in h = (cid:122) − )variables ξ (cid:122) = ξ/ (cid:122) , η (cid:122) = η/ (cid:122) . It becomes (cid:16) ( ξ (cid:122) + i ) − ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) ) + 2 i (cid:122) − α ( ξ (cid:122) + 1) − )(( ξ (cid:122) − i ) − ( η (cid:122) ⊗ η (cid:122) ) − (cid:122) − a )+ ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) )( η (cid:122) ⊗ ˆ Y + ˆ Y ⊗ η (cid:122) ) + ˆ Y ⊗ ˆ Y (cid:17) · v T T = 0 . Letting (cid:122) − = h →
0, one obtains (cid:16) ( ξ (cid:122) + i ) − ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) ) ( ξ (cid:122) − i ) − ( η (cid:122) ⊗ η (cid:122) )+ ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) )( η (cid:122) ⊗ ˆ Y + ˆ Y ⊗ η (cid:122) ) + ˆ Y ⊗ ˆ Y (cid:17) · v T T = 0 , i.e. (cid:16)(cid:16) ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) ) η (cid:122) + ˆ Y (cid:17) ⊗ (cid:16) ( ξ (cid:122) + 1) − ( ˆ Y · η (cid:122) ) η (cid:122) + ˆ Y (cid:17)(cid:17) · v T T = 0 . One can see that this last equation, when it holds for all ˆ Y , implies the vanishingof v T T just as for the principal symbol at fiber infinity. Indeed, if η (cid:122) = 0 then wehave ( ˆ Y ⊗ ˆ Y ) · v T T = 0 for all ˆ Y , and symmetric 2-tensors of the form ˆ Y ⊗ ˆ Y spanthe space of all symmetric 2-tensors (as w ⊗ w + w ⊗ w = ( w + w ) ⊗ ( w + w ) − w ⊗ w − w ⊗ w ), so we conclude that v T T = 0, and thus v = 0 in this case. NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 25
On the other hand, if η (cid:122) (cid:54) = 0 then taking ˆ Y = (cid:15) ˆ η (cid:122) + (1 − (cid:15) ) / ˆ Y ⊥ and substitutinginto this equation yields (cid:16)(cid:16) | η (cid:122) | ξ (cid:122) + 1 (cid:17) (cid:15) ˆ η (cid:122) ⊗ ˆ η (cid:122) + (cid:16) | η (cid:122) | ξ (cid:122) + 1 (cid:17) (cid:15) (1 − (cid:15) ) / (ˆ η (cid:122) ⊗ ˆ Y ⊥ + ˆ Y ⊥ ⊗ ˆ η (cid:122) )+ (1 − (cid:15) ) ˆ Y ⊥ ⊗ ˆ Y ⊥ (cid:17) · v T T = 0 . Substituting in (cid:15) = 0 yields ( ˆ Y ⊥ ⊗ ˆ Y ⊥ ) · v T T = 0; since cotensors of the formˆ Y ⊥ ⊗ ˆ Y ⊥ span η ⊥ (cid:122) ⊗ η ⊥ (cid:122) ( η ⊥ (cid:122) being the orthocomplement of η (cid:122) ), we conclude that v T T is orthogonal to every element of η ⊥ (cid:122) ⊗ η ⊥ (cid:122) . Next, taking the derivative in (cid:15) at (cid:15) = 0 yields (ˆ η (cid:122) ⊗ ˆ Y ⊥ + ˆ Y ⊥ ⊗ ˆ η (cid:122) ) · v T T = 0 for all ˆ Y ⊥ ; symmetric tensors ofthis form, together with η ⊥ (cid:122) ⊗ η ⊥ (cid:122) , span all tensors in ( η (cid:122) ⊗ η (cid:122) ) ⊥ . Finally takingthe second derivative at (cid:15) = 0 shows that (ˆ η (cid:122) ⊗ ˆ η (cid:122) ) · v T T = 0, this in conclusion v T T = 0. Combined with the first two equations of (3.13), one concludes that v = 0. Correspondingly one concludes that for sufficiently large (cid:122) > (cid:3) As already explained, this lemma completes the proof of Proposition 3.3. The gauge condition and the proof of the main results
The still remaining analytic issue is to check that we can arrange the gaugecondition, δ s (cid:122) f (cid:122) = 0. We do this by considering various regions Ω j , which aremanifolds with corners: they have the artificial boundary, ∂X , which is ‘at infinity’in the scattering calculus sense, as well as the ‘interior’ boundary ∂ int Ω j , whichcould be ∂M , or another (farther away) hypersurface.Recall that our gauge freedom is that we can add to f (without changing If )any tensor of the form d s v , with v vanishing at ∂M or on a hypersurface furtheraway, such as ∂ int Ω j , i.e. to f (cid:122) = e − (cid:122) /x f (without changing Ie (cid:122) /x f (cid:122) ) any ten-sor of the form d s (cid:122) v (cid:122) = e − (cid:122) /x de (cid:122) /x v (cid:122) with a similar vanishing condition. If welet ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) be the ‘solenoidal Witten Laplacian’, and we impose Dirichletboundary condition on ∂ int Ω j (to get the desired vanishing for v (cid:122) ), and we showthat ∆ (cid:122) ,s is invertible (with this boundary condition) on suitable function spaces,then S (cid:122) , Ω j φ = φ s (cid:122) , Ω j = φ − d s (cid:122) ∆ − (cid:122) ,s, Ω j δ s (cid:122) φ, P (cid:122) , Ω j φ = d s (cid:122) Q (cid:122) , Ω j φ, Q (cid:122) , Ω j φ = ∆ − (cid:122) ,s, Ω j δ s (cid:122) φ, are the solenoidal ( S ), resp. potential ( P ) projections of φ on Ω j . Notice that P (cid:122) , Ω j φ is indeed in the range of d s (cid:122) applied to a function or one-form vanishingat ∂ int Ω j thanks to the boundary condition for ∆ (cid:122) ,s , which means that Q (cid:122) , Ω j maps to such functions or tensors. Thus S (cid:122) , Ω j φ differs from φ by such a tensor, so Ie (cid:122) /x f (cid:122) = Ie (cid:122) /x S (cid:122) , Ω j f (cid:122) . Further, δ s S (cid:122) , Ω j φ = δ s (cid:122) φ − δ s (cid:122) d s (cid:122) ∆ − (cid:122) ,s, Ω j δ s (cid:122) φ = 0 , so δ s (cid:122) f (cid:122) = 0, i.e. the gauge condition we want to impose is in fact satisfied.Thus, it remains to check the invertibility of ∆ (cid:122) ,s with the desired boundarycondition. Before doing this we remark: Lemma 4.1.
For (cid:122) > , the operator ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) is (jointly) elliptic in Diff , ( X ) on functions. On the other hand, there exists (cid:122) > such that for (cid:122) ≥ (cid:122) the operator ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) is (jointly) elliptic in Diff , ( X ; sc T ∗ X, sc T ∗ X ) on one forms. Infact, on one forms (for all (cid:122) > ) (4.1) δ s (cid:122) d s (cid:122) = 12 ∇ ∗ (cid:122) ∇ (cid:122) + 12 d (cid:122) δ (cid:122) + A + R, where R ∈ x Diff ( X ; sc T ∗ X, sc T ∗ X ) , A ∈ Diff ( X ; sc T ∗ X ; sc T ∗ X ) is independentof (cid:122) and where ∇ (cid:122) = e − (cid:122) /x ∇ e (cid:122) /x , with ∇ gradient relative to g sc (not g ), d (cid:122) = e − (cid:122) /x de (cid:122) /x the exterior derivative on functions, while δ (cid:122) is its adjoint on one-forms.Proof. Most of the computations for this lemma have been performed in Lemma 3.2.In particular, the symbolic computation is algebraic, and can be done pointwise,where one arranges that g sc is as in Lemma 3.2. Since the function case is simpler,we consider one-forms. Thus the full principal symbol of d s (cid:122) (with symmetric 2-tensors considered as a subspace of 2-tensors) is ξ + i (cid:122) η ⊗ ( ξ + i (cid:122) ) η ⊗ ( ξ + i (cid:122) ) a η ⊗ s , that of δ s (cid:122) is (cid:18) ξ − i (cid:122) ι η ι η (cid:104) a, . (cid:105) ( ξ − i (cid:122) ) ( ξ − i (cid:122) ) ι sη (cid:19) . with the lower right block having ( (cid:96)ij ) entry given by ( η i δ (cid:96)j + η j δ i(cid:96) ). Correspond-ingly, the product, ∆ s (cid:122) , has symbol(4.2) (cid:18) ξ + (cid:122) + | η | ( ξ + i (cid:122) ) ι η ( ξ − i (cid:122) ) η ⊗ ( ξ + (cid:122) ) + ι sη η ⊗ s (cid:19) + (cid:18) (cid:104) a, . (cid:105) a (cid:104) a, . (cid:105) η ⊗ s ι sη a (cid:19) , with the lower right block having (cid:96)k entry ( ξ + (cid:122) ) δ (cid:96)k + | η | δ (cid:96)k + η (cid:96) η k , andwhere we separated out the a terms.Now ellipticity is easy to see if a = 0, with a (cid:122) -dependent lower bound then,and this can be used to absorb the a term by taking (cid:122) > g sc is ξ η ⊗ ξ η ⊗ , with no non-zero entry in the lower left hand corner unlike for the g -gradient in(3.8), and thus the adjoint ∇ ∗ (cid:122) of ∇ (cid:122) has principal symbol (cid:18) ξ − i (cid:122) ι η ξ − i (cid:122) ι sη (cid:19) . Correspondingly, ∇ ∗ (cid:122) ∇ (cid:122) has symbol(4.3) (cid:18) ξ + (cid:122) + | η | ξ + (cid:122) + | η | (cid:19) , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 27 which is certainly elliptic (including at finite points in sc T ∗ ∂X X !), and indeed issimply ξ + (cid:122) + | η | times the identity matrix. Now, d = d s going from functionsto one-forms has symbol (cid:18) ξη (cid:19) , so its conjugate e − (cid:122) /x de (cid:122) /x has symbol (cid:18) ξ + i (cid:122) η (cid:19) ,its adjoint, δ (cid:122) has symbol (cid:0) ξ − i (cid:122) ι η (cid:1) , and now d (cid:122) δ (cid:122) has symbol (cid:18) ξ + (cid:122) ( ξ + i (cid:122) ) ι η ( ξ − i (cid:122) ) η η ⊗ ι η (cid:19) . Combining these, we see that the first term in (4.2), i.e. in the principal symbol of δ s (cid:122) d s (cid:122) , is the same as ∇ ∗ (cid:122) ∇ (cid:122) + d (cid:122) δ (cid:122) , with both terms non-negative, and thefirst actually positive definite, with a lower bound ξ + (cid:122) + | η | times the identity.This proves (4.1), with the principal symbol of A given by the second term in (4.2),which is in particular independent of (cid:122) . Since with C a bound for a , the symbolof A is bounded by C + 2 C | η | ≤ C (1 + (cid:15) − ) + (cid:15) | η | for any (cid:15) >
0, in particular (cid:15) <
1, this shows that the principal symbol of δ s (cid:122) d s (cid:122) is positive definite if (cid:122) > (cid:3) We now turn to the invertibility question. Let ˙ H m,l sc (Ω j ) be the subspace of H m,l sc ( X ) consisting of distributions supported in Ω j , and let ¯ H m,l sc (Ω j ) the space ofrestrictions of elements of H m,l sc ( X ) to Ω j . Thus, ˙ H m,l sc (Ω j ) ∗ = ¯ H − m, − l sc (Ω j ). Herewe shall be mostly interested in m = 1, l = 0; then at ∂ int Ω j , away from ∂X ,˙ H , (Ω j ) is the standard H -space (which is ˙ H in H¨ormander’s notation, whichwe adopt), while ¯ H − , is H − there (which is ¯ H − in H¨ormander’s notation).Further, ˙ C ∞ (Ω j ), with the dot denoting infinite order vanishing at all boundaryhypersurfaces, or indeed C ∞ c (Ω j ) (compact support), are dense in H , , so ˙ H , (Ω j )is the completion of these spaces in the H , ( X )-norm. In addition, the norm on H , ( X ) is equivalent to (cid:107)∇ u (cid:107) L + (cid:107) u (cid:107) L , where the norms are with respect to anyscattering metric, and ∇ is any differential operator with principal symbol givenby d , such as the gradient relative to any (perhaps different from the one givingthe norm) scattering metric. For L = H , , or for the weighted L -spaces H ,l sc ,the dots and bars do not make any difference (do not change the space) as usual.Further, the inclusion map ˙ H , → L (or indeed even ¯ H , → L ) is compact.As usual, all these spaces can be defined for sections of vector bundles, such as sc T ∗ Ω j X , by local trivializations. The norm on ˙ H , (Ω j , sc T ∗ Ω j ) is still induced bya gradient ∇ with respect to any scattering differential operator the same way. Lemma 4.2.
The operator on functions ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) , considered as a map ˙ H , → ( ˙ H , ) ∗ = ¯ H − , is invertible for all (cid:122) > .On the other hand, there exists (cid:122) > such that for (cid:122) ≥ (cid:122) , the operator ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) on one forms is invertible.Remark . The reason for having some (cid:122) >
0, and requiring (cid:122) ≥ (cid:122) , in the oneform case (rather than merely (cid:122) >
0) is that d s is relative to a standard metric g ,not a scattering metric. The proof given below in fact shows that if d s is replacedby d sg sc , relative to any scattering metric g sc , then one may simply assume (cid:122) > Proof.
The following considerations apply to both the function case and the one-form case. Relative to the scattering metric with respect to which δ s is defined, thequadratic form of ∆ (cid:122) ,s is (cid:104) ∆ (cid:122) ,s u, v (cid:105) = (cid:104) d s (cid:122) u, d s (cid:122) v (cid:105) . So in particular (cid:107) d s (cid:122) u (cid:107) L ≤ (cid:107) ∆ (cid:122) ,s u (cid:107) ¯ H − , (cid:107) u (cid:107) ˙ H , ≤ (cid:15) − (cid:107) ∆ (cid:122) ,s u (cid:107) H − , + (cid:15) (cid:107) u (cid:107) H , . Correspondingly, if one has an estimate(4.4) (cid:107) u (cid:107) ˙ H , ≤ C (cid:107) d s (cid:122) u (cid:107) L , or equivalently (for a different C ) (cid:107)∇ u (cid:107) L + (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L , then for small (cid:15) >
0, one can absorb (cid:15) (cid:107) u (cid:107) H , into the left hand side above, giving (cid:107) u (cid:107) ˙ H , ≤ C (cid:107) d s (cid:122) u (cid:107) L ≤ C (cid:48) (cid:107) ∆ (cid:122) ,s u (cid:107) ¯ H − , . This in turn gives invertibility in the sense discussed in the statement of the theoremsince ∆ (cid:122) ,s is formally (and as this shows, actually) self-adjoint, so one has the sameestimates for the formal adjoint.On the other hand, if one has an estimate(4.5) (cid:107) u (cid:107) ˙ H , ≤ C (cid:107) d s (cid:122) u (cid:107) L + C (cid:107) u (cid:107) ˙ H , − , or equivalently (cid:107)∇ u (cid:107) L + (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L + C (cid:107) u (cid:107) ˙ H , − , then for (cid:15) > (cid:107) u (cid:107) ˙ H , ≤ C (cid:107) d s (cid:122) u (cid:107) L + C (cid:107) u (cid:107) ˙ H , ≤ C (cid:48) (cid:107) ∆ (cid:122) ,s u (cid:107) ¯ H − , + C (cid:48) (cid:107) u (cid:107) ˙ H , − . Again, by formal self-adjointness, one gets the same statement for the adjoint,which implies that ∆ (cid:122) ,s is Fredholm (by virtue of the compactness of the inclusion˙ H , → ˙ H , − ), and further that the invertibility is equivalent to the lack of kernelon ˙ H , (since the cokernel statement follows by formal self-adjointness). Note that(4.5) follows quite easily from Lemma 4.1 (and is standard on functions as d s = ∇ then), in the form case using the Dirichlet boundary condition to apply (4.1) to u and pair with u but we discuss invertibility, taking advantage of Lemma 4.1 later.Now, on functions, d s = ∇ , and as d s (cid:122) differs from d s by a 0th order oper-ator, (cid:107)∇ u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L + C (cid:107) u (cid:107) L automatically. In particular, (4.4) followsif one shows (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L for u ∈ ˙ H , , or equivalently (by density) for u ∈ C ∞ c (Ω j ), which is a Poincar´e inequality.To prove this Poincar´e inequality, notice that (cid:107) e − (cid:122) /x ( x D x ) e (cid:122) /x u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L certainly, so it suffices to estimate (cid:107) u (cid:107) L in terms of the L norm of e − (cid:122) /x ( x D x ) e (cid:122) /x u = ( x D x + i (cid:122) ) u. But for any operator P , writing P R = ( P + P ∗ ) / P I = ( P − P ∗ ) / (2 i ) for thesymmetric and skew-symmetric parts, (cid:107) P u (cid:107) = (cid:107) P R u (cid:107) + (cid:107) P I u (cid:107) + (cid:104) i [ P R , P I ] u, u (cid:105) . It is convenient here to use a metric dx x + hx where h is a metric, independent of x , on the level sets of x , using some product decomposition. For then the metricdensity is x − ( n +1) | dx | | dh | , so with P = x D x + i (cid:122) , P ∗ = x D x + i ( n − x − i (cid:122) ,so P R = x D x + i n − x, P I = (cid:122) − n − x, i [ P R , P I ] = ix n − , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 29 we have (cid:107) ( x D x + i (cid:122) ) u (cid:107) L = (cid:13)(cid:13)(cid:13)(cid:16) x D x + i n − (cid:17) u (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13)(cid:16) (cid:122) − n − x (cid:17) u (cid:13)(cid:13)(cid:13) L − n − (cid:104) x u, u (cid:105) = (cid:13)(cid:13)(cid:13)(cid:16) x D x + i n − (cid:17) u (cid:13)(cid:13)(cid:13) L + (cid:68)(cid:16) ( (cid:122) − n − x ) − n − x (cid:17) u, u (cid:69) . Now, if Ω j ⊂ { x ≤ x } , as long as x > (cid:16) (cid:122) − n − x (cid:17) − n − x is positive (and thus bounded below by a positive constant) on [0 , x ], which isautomatic for sufficiently small x , or indeed for bounded x and sufficiently large (cid:122) , one obtains that (cid:107) u (cid:107) L ≤ C (cid:107) ( x D x + i (cid:122) ) u (cid:107) L , and thus in summary that (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L , as desired. This proves the lemma for functions, at least in the case of sufficientlysmall x .This actually suffices for our application, but in fact one can do better by notingthat in fact even in general this gives us the estimate (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) + C (cid:107) u (cid:107) L ( { x ≤ x ≤ x } ) for suitable small x >
0. But by the standard Poincar´e inequality, using the van-ishing at x = x , one can estimate the last term in terms of C (cid:48) (cid:107) d s (cid:122) u (cid:107) , which givesthe general conclusion for functions. Here, to place us properly in the standardPoincar´e setting, we note that with φ = e (cid:122) /x u , the last required estimate is equiva-lent to the weighted estimate (cid:107) e − (cid:122) /x φ (cid:107) L ( { x ≤ x ≤ x } ) ≤ C (cid:107) e − (cid:122) /x dφ (cid:107) L ( { x ≤ x ≤ x } ) ,and now the weights are bounded, so can be dropped completely.It remains to deal with one-forms. For this we use that (4.1) and (4.3) give that(4.6) δ s (cid:122) d s (cid:122) = 12 ∇ ∗ ∇ + 12 (cid:122) + 12 d (cid:122) δ (cid:122) + A + ˜ R, where A ∈ Diff ( X ) is independent of (cid:122) and ˜ R ∈ x Diff ( X ); this follows byrewriting ∇ ∗ (cid:122) ∇ (cid:122) using (4.3), which modifies R in (4.1) to give (4.6). Thus, in fact(4.7) (cid:107) d s (cid:122) u (cid:107) = 12 (cid:107)∇ u (cid:107) + 12 (cid:122) (cid:107) u (cid:107) + 12 (cid:107) δ s (cid:122) u (cid:107) + (cid:104) Au, u (cid:105) + (cid:104) ˜ Ru, u (cid:105) . Since A ∈ Diff ( X ), |(cid:104) Au, u (cid:105)| ≤ C (cid:107) u (cid:107) ˙ H , (cid:107) u (cid:107) L , and there is a similar estimate forthe last term. This gives an estimate, for sufficiently large (cid:122) ,(4.8) (cid:107)∇ u (cid:107) + (cid:122) (cid:107) u (cid:107) ≤ C (cid:107) d s (cid:122) u (cid:107) + C (cid:107) x / u (cid:107) , with the constant C on the right hand side depending on (cid:122) , and thus (cid:104) (1 − Cx ) u, u (cid:105) ≤ C (cid:107) d s (cid:122) u (cid:107) . Again, if x is sufficiently small, this gives (cid:107) u (cid:107) ≤ C (cid:107) d s (cid:122) u (cid:107) , and thus the invertibility, while if x is larger, this still gives (cid:107) u (cid:107) L ≤ C (cid:107) d s (cid:122) u (cid:107) L + C (cid:107) u (cid:107) L ( { x ≤ x ≤ x } ) . One can then finish the proof as above, using the standard Poincar´e inequality forone forms, see [23, Section 6, Equation (28)]. (cid:3)
A slight modification of the argument gives:
Lemma 4.4.
The operator on functions ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) , considered as a map ˙ H ,r sc → ¯ H − ,r sc is invertible for all (cid:122) > and all r ∈ R .On the other hand, there exists (cid:122) > such that for (cid:122) ≥ (cid:122) , the operator ∆ (cid:122) ,s = δ s (cid:122) d s (cid:122) on one forms is invertible as a map ˙ H ,r sc → ¯ H − ,r sc for all r ∈ R .Proof. Since the function case is completely analogous, we consider one forms tobe definite. Also note that (full) elliptic regularity would automatically give thisresult if not for ∂ int Ω j .An isomorphism estimate ∆ (cid:122) ,s : ˙ H ,r sc → ¯ H − ,r sc is equivalent to an isomorphismestimate x − r ∆ (cid:122) ,s x r : ˙ H , → ¯ H − , . But the operator on the left is ∆ (cid:122) ,s + F ,where F ∈ x Diff . Thus, x − r ∆ (cid:122) ,s x r is of the form (4.6), with only ˜ R changed.The rest of the proof then immediately goes through. (cid:3) Before proceeding with the analysis of the Dirichlet Laplacian, we first discussthe analogue of Korn’s inequality that will be useful later.
Lemma 4.5.
Suppose Ω j is a domain in X as above. For (cid:122) > and r ∈ R , (cid:107) u (cid:107) ¯ H ,r sc (Ω j ) ≤ C ( (cid:107) x − r d s (cid:122) u (cid:107) L (Ω j ) + (cid:107) u (cid:107) x − r L (Ω j ) ) . for one-forms u ∈ ¯ H ,r sc (Ω j ) .Proof. First note that if one lets ˜ u = x − r u , then (cid:107) u (cid:107) ¯ H ,r sc (Ω j ) is equivalent to (cid:107) ˜ u (cid:107) ¯ H , (Ω j ) , and (cid:107) x − r d s (cid:122) u (cid:107) L (Ω j ) + (cid:107) u (cid:107) x − r L (Ω j ) is equivalent to (cid:107) d s (cid:122) ˜ u (cid:107) L (Ω j ) + (cid:107) ˜ u (cid:107) L (Ω j ) since the commutator term through d s (cid:122) can be absorbed into a sufficientlylarge multiple of (cid:107) u (cid:107) x − r L (Ω j ) = (cid:107) ˜ u (cid:107) L (Ω j ) . Thus, one is reduced to proving thecase r = 0.Let ˜Ω j be a domain in X with C ∞ boundary, transversal to ∂X , containing Ω j .We claim that there is a continuous extension map E : ¯ H , (Ω j ) → ˙ H , ( ˜Ω j ) suchthat(4.9) (cid:107) d s (cid:122) Eu (cid:107) L (˜Ω j ) + (cid:107) Eu (cid:107) L (˜Ω j ) ≤ C ( (cid:107) d s (cid:122) u (cid:107) L (Ω j ) + (cid:107) u (cid:107) L (Ω j ) ) , u ∈ ¯ H , (Ω j ) , i.e. Eu is also continuous when on both sides the gradient is replaced by the symmet-ric gradient in the definition of an H -type space. Once this is proved, the lemmacan be shown in the following manner. By (4.1) of Lemma 4.1 any v ∈ ˙ H , ( ˜Ω j ),in particular v = Eu , satisfies, for any (cid:15) > (cid:107)∇ v (cid:107) L (˜Ω j ) + (cid:107) v (cid:107) L (˜Ω j ) ≤ (cid:107) d s (cid:122) v (cid:107) L (˜Ω j ) + (cid:107) v (cid:107) L (˜Ω j ) + C (cid:107) v (cid:107) L (˜Ω j ) (cid:107) v (cid:107) ¯ H , (˜Ω j ) ≤ (cid:107) d s (cid:122) v (cid:107) L (˜Ω j ) + C (cid:48) (cid:107) v (cid:107) L (˜Ω j ) + (cid:15) (cid:107) v (cid:107) H , (˜Ω j ) and now for (cid:15) > v = Eu , noting that E is an extension map so (cid:107)∇ u (cid:107) L (Ω j ) + (cid:107) u (cid:107) L (Ω j ) ≤ (cid:107)∇ Eu (cid:107) L (˜Ω j ) + (cid:107) Eu (cid:107) L (˜Ω j ) , we deduce, using (4.9) in the last step, that (cid:107) u (cid:107) ¯ H , (Ω j ) ≤ C ( (cid:107) d s (cid:122) Eu (cid:107) L (˜Ω j ) + (cid:107) Eu (cid:107) L (˜Ω j ) ) ≤ C (cid:48) ( (cid:107) d s (cid:122) u (cid:107) L (Ω j ) + (cid:107) u (cid:107) L (Ω j ) ) , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 31 completing the proof of the lemma.Thus, it remains to construct E . By a partition of unity, this can be reducedto a local extension, local on X . Since ∂ Ω j is transversal to ∂X , near points on ∂X ∩ ∂ Ω j one can arrange that locally (in a model in which a neighborhood of p is identified with an open set in R n ) ∂ Ω j is the hypersurface x n = 0, Ω j is x n > ∂X near points on ∂ Ω j .Since H s,r sc ( X ), ˙ H s,r sc ( ˜Ω j ), ¯ H s,r sc (Ω j ), are locally, and also for compactly supportedelements in the chart, are preserved by local diffeomorphisms of X to R n in thesense that X is replaced by R n , Ω j by R n + (by virtue of these spaces are well definedon manifolds with boundary, without additional information on metrics, etc., up toequivalence of norms), it suffices to prove that there is a local extension map E that has the desired properties.Let Φ k ( x (cid:48) , x n ) = ( x (cid:48) , − kx n ) for x n <
0, and consider a variation of the standardconstruction of an H ( R n + ) extension map on one-forms as follows. (Note that theusual extension map is given by trivialization of a bundle, in this case using dx j asa local basis of sections, and extending the coefficients using the extension map onfunctions.) Let E given by( E (cid:88) j u j dx j )( x (cid:48) , x n ) = (cid:88) k =1 c k Φ ∗ k ( (cid:88) u j dx j ) , x n < , and ( E (cid:88) j u j dx j )( x (cid:48) , x n ) = (cid:88) u j dx j , x n ≥ , with c k chosen so that E : C ( R n + ) → C ( R n ). We can achieve this mappingproperty as follows. We have, with ∂ j acting as derivatives on the components, orequivalently but invariantly as Lie derivatives in this case,Φ ∗ k u j dx j = u j ( x (cid:48) , − kx n ) dx j , j (cid:54) = n, Φ ∗ k u n dx n = − ku n ( x (cid:48) , − kx n ) dx n ,∂ i Φ ∗ k u j dx j = ( ∂ i u j )( x (cid:48) , − kx n ) dx j , i, j (cid:54) = n,∂ i Φ ∗ k u n dx n = − k ( ∂ i u n )( x (cid:48) , − kx n ) dx n , i (cid:54) = n,∂ n Φ ∗ k u j dx j = − k ( ∂ n u j )( x (cid:48) , − kx n ) dx j , j (cid:54) = n,∂ n Φ ∗ k u n dx n = k ( ∂ n u j )( x (cid:48) , − kx n ) dx n , so the requirements for matching the derivatives at x n = 0, which gives the C property, are, for j (cid:54) = n , c + c + c = 1 , − c − c − c = 1 , while for j = n − c − c − c = 1 ,c + 4 c + 9 c = 1 , which gives a 3-by-3 system − − −
31 4 9 c c c = . The matrix on the right is a Vandermonde matrix, and is thus invertible, so onecan find c k with the desired properties. With this, E : C c ( R n + ) → C c ( R n ) hasthe property that (cid:107) E u (cid:107) H ( R n ) ≤ C (cid:107) u (cid:107) H ( R n + ) , since each term in the definition of E has derivatives ∂ i satisfying (cid:107) ∂ i Φ ∗ k u (cid:107) L ( R n ) ≤ C (cid:107) ∂ i u (cid:107) L ( R n + ) , and since E u ∈ C c ( R n ) assures that the distributional derivative satisfies ∂ i E u ∈ L ( R n ), whosesquare norm can be calculated as the sum of the squared norms over R n + = { x n > } and R n − = { x n < } . Correspondingly, E extends continuously, in a uniquemanner, to a map H ( R n + ) → H ( R n ).Before proceeding we note that with this choice of coefficients, E defined as theanalogous map on functions, is actually the standard H extension map. However,on one-forms the same choice, defined in terms of pull-backs, i.e. natural operations,as above, rather than trivializing the form bundle, does not extend continuouslyto H . On the other hand, if one trivializes the bundle and uses the H extensionmap, one does not have the desired property (4.9) for symmetric differentials, aproperty that we check below with our choice of extension map.Notice that, with Φ ∗ k acting on 2-tensors as usual, for all i, j , dx i ⊗ ( ∂ i Φ ∗ k u j dx j ) + ( ∂ j Φ ∗ k u i dx i ) ⊗ dx j = Φ ∗ k (( ∂ i u j + ∂ j u i ) dx i ⊗ dx j ) , as follows from a direct calculation, or indeed from the naturality of the symmetricgradient d s = d sg for a translation invariant Riemannian metric g : the two sidesare the ij component of 2 d s Φ ∗ k , resp. 2Φ ∗ k d s , as for such a metric the symmetricgradient is actually independent of the choice of the metric (in this class). Since,summed over i, j , the left hand side is the symmetric gradient of Φ ∗ k (cid:80) u j dx j in x n <
0, while the right hand side is the pull-back of the symmetric gradient from x n >
0, this shows that (cid:107) d s Φ ∗ k u (cid:107) L ( R n − ) ≤ C (cid:107) d s u (cid:107) L ( R n + ) . This proves that one has (cid:107) d s E u (cid:107) L ( R n ) ≤ C (cid:107) d s u (cid:107) L ( R n + ) . Now, using a partition of unity { ρ k } to localize on Ω j , as mentioned above, thisgives a global extension map from H (Ω j ): (cid:80) ψ k E ,k ρ k , where ψ k is identically 1near supp ρ k . While d s depends on the choice of a metric, the dependence is viathe 0th order term, i.e. one has d sg u = d sg u + Ru for an appropriate 0th order R .Using the Euclidean metric in the local model, this shows that (cid:107) d sg ψ k E ,k ρ k u (cid:107) L ( R n ) ≤ C ( (cid:107) d sg ρ k u (cid:107) L ( R n + ) + (cid:107) ρ k u (cid:107) L ( R n + ) ) . Since d s (cid:122) differs from d s by a 0th order term, one can absorb this in the L norm(using also the continuity of the extension map from L to L ): (cid:107) d sg, (cid:122) ψ k E ,k ρ k u (cid:107) L ( R n ) ≤ C ( (cid:107) d sg , (cid:122) ρ k u (cid:107) L ( R n + ) + (cid:107) ρ k u (cid:107) L ( R n + ) ) . Summing over k proves (4.9), and thus the lemma. (cid:3) We now return to the analysis of the Dirichlet Laplacian.
Corollary 4.6.
Let φ ∈ C ∞ c (Ω j \ ∂ int Ω j ) . Then on functions, for (cid:122) > , k ∈ R ,the operator φ ∆ − (cid:122) ,s φ : ¯ H − ,k sc → ˙ H ,k sc is in Ψ − , ( X ) . There is (cid:122) > such thatthe analogous conclusion holds for one forms for (cid:122) ≥ (cid:122) . NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 33
Proof.
This follows from the usual parametrix identity. Namely, by Lemma 4.1,∆ (cid:122) ,s has a parametrix B ∈ Ψ − , ( X ) so that B ∆ (cid:122) ,s = Id + F L , ∆ (cid:122) ,s B = Id + F R , with F L , F R ∈ Ψ −∞ , −∞ sc ( X ). Let ψ ∈ C ∞ c (Ω j \ ∂ int Ω j ) be identically 1 on supp φ .Thus, ψ = B ∆ (cid:122) ,s ψ − F L ψ = Bψ ∆ (cid:122) ,s + B [∆ (cid:122) ,s , ψ ] − F L ψ and ψ = ψ ∆ (cid:122) ,s B − ψF R = ∆ (cid:122) ,s ψB + [ ψ, ∆ (cid:122) ,s ] B − ψF R . Then ψ ∆ − (cid:122) ,s ψ = ψ ∆ − (cid:122) ,s ∆ (cid:122) ,s ψB + ψ ∆ − (cid:122) ,s ([ ψ, ∆ (cid:122) ,s ] B − ψF R )= ψ B + Bψ ∆ (cid:122) ,s ∆ − (cid:122) ,s ([ ψ, ∆ (cid:122) ,s ] B − ψF R )+ ( B [∆ (cid:122) ,s , ψ ] − F L ψ )∆ − (cid:122) ,s ([ ψ, ∆ (cid:122) ,s ] B − ψF R )= ψ B + Bψ ([ ψ, ∆ (cid:122) ,s ] B − ψF R )+ ( B [∆ (cid:122) ,s , ψ ] − F L ψ )∆ − (cid:122) ,s ([ ψ, ∆ (cid:122) ,s ] B − ψF R ) . Multiplying from both the left and the right by φ gives φ ∆ − (cid:122) ,s φ = φBφ + φBψ ([ ψ, ∆ (cid:122) ,s ] B − ψF R ) φ + φ ( B [∆ (cid:122) ,s , ψ ] − F L ψ )∆ − (cid:122) ,s ([ ψ, ∆ (cid:122) ,s ] B − ψF R ) φ. Now, the first two terms on the right hand side are in Ψ − , , resp. Ψ −∞ , −∞ sc , inthe latter case using the disjointness of supp dψ and φ for [ ψ, ∆ (cid:122) ,s ] Bφ , resp. that F L ∈ Ψ −∞ , −∞ sc for ψF R φ . For this reason, ([ ψ, ∆ (cid:122) ,s ] B − ψF R ) φ and φ ( B [∆ (cid:122) ,s , ψ ] − F L ψ ) are smoothing, in the sense that they map H s,r sc ( X ) to H s (cid:48) ,r (cid:48) sc ( X ) for any s (cid:48) , r (cid:48) , s, r , and they also have support so that they map into functions supported inΩ j \ ∂ int Ω j , and they also can be applied to functions on Ω j . As ∆ − (cid:122) ,s is continuous¯ H − ,k sc (Ω j ) → ˙ H ,k sc (Ω j ), this shows that the last term is continuous from H s,r sc ( X )to H s (cid:48) ,r (cid:48) sc ( X ) for any s (cid:48) , r (cid:48) , s, r , which means that it has a Schwartz (rapidly decayingwith all derivatives) Schwartz kernel, i.e. it is in Ψ −∞ , −∞ sc ( X ). This completes theproof. (cid:3) Corollary 4.7.
Let φ ∈ C ∞ c (Ω j \ ∂ int Ω j ) , χ ∈ C ∞ (Ω j ) with disjoint support andwith χ constant near ∂ int Ω j . Let (cid:122) , (cid:122) as in Corollary 4.6. Then the operator χ ∆ − (cid:122) ,s φ : ¯ H − ,k sc (Ω j ) → ˙ H ,k sc (Ω j ) in fact maps H s,r sc ( X ) → ˙ H ,k sc (Ω j ) for all s, r, k .Similarly, φ ∆ − (cid:122) ,s χ : ¯ H − ,k sc (Ω j ) → ˙ H ,k sc (Ω j ) in fact maps ¯ H − ,k sc (Ω j ) → H s,r sc ( X ) for all s, r, k .Proof. Since the second statement follows by duality, it suffices to prove the first.As χφ = 0, we can write χ ∆ − (cid:122) ,s φ = [ χ, ∆ − (cid:122) ,s ] φ = ∆ − (cid:122) ,s [∆ (cid:122) ,s , χ ]∆ − (cid:122) ,s φ. By Corollary 4.6, [∆ (cid:122) ,s , χ ]∆ − (cid:122) ,s φ ∈ Ψ −∞ , ∞ sc ( X ) since it is in Ψ − , ( X ) (this usessupp dχ disjoint from ∂ int Ω j ) but dχ and φ have disjoint supports. Thus, it maps H s,r sc ( X ) → H − ,k sc ( X ), and thus, in view of supp dχ , to ¯ H − ,k sc (Ω j ), giving theconclusion. (cid:3) Corollary 4.8.
Let φ ∈ C ∞ c (Ω j \ ∂ int Ω j ) , χ ∈ C ∞ (Ω j ) with disjoint support andwith χ constant near ∂ int Ω j . Let (cid:122) , (cid:122) as in Corollary 4.6.Then φ S (cid:122) , Ω j φ ∈ Ψ , ( X ) , while χ S (cid:122) , Ω j φ : H s,r sc ( X ) → x k L (Ω j ) and φ S (cid:122) , Ω j χ : x k L (Ω j ) → H s,r sc ( X ) for all s, r, k .Proof. This is immediate from S (cid:122) , Ω j = Id − d s (cid:122) ∆ − (cid:122) ,s, Ω j δ s (cid:122) and the above results con-cerning ∆ − (cid:122) ,s, Ω j , using that d s (cid:122) and δ s (cid:122) are differential operators, and thus preservesupports. (cid:3) We also need the Poisson operator associated to ∂ int Ω j . First note that if H isa (codimension 1) hypersurface in Ω j which intersects ∂ Ω j away from ∂ int Ω j , anddoes so transversally, then the restriction map γ H : ˙ C ∞ (Ω j ) → ˙ C ∞ ( H ) , with the dots denoting infinite order vanishing at ∂ Ω j , resp. ∂H , as usual, in factmaps, for s > / γ H : H s,r sc (Ω j ) → H s − / ,r sc ( H )continuously. This can be easily seen since the restriction map is local, and locallyin Ω j , one can map a neighborhood of p ∈ ∂H to a neighborhood of a point p (cid:48) ∈ ∂ R n − in R n by a diffeomorphism so that H is mapped to R n − , and thus bythe diffeomorphism invariance of the spaces under discussion, the standard R n resultwith the usual Sobolev spaces H s ( R n ) = H s, ( R n ), using that weights commutewith the restriction, gives (4.10). The same argument also shows that there is acontinuous extension map(4.11) e H : H s − / ,r sc ( H ) → H s,r sc (Ω j ) , γ H e H = Id , since the analogous result on R n is standard, and one can localize by multiplyingby cutoffs without destroying the desired properties.Considering Ω j inside a larger domain Ω (cid:48) , with ∂ int Ω j satisfying the assumptionsfor H , we have a continuous extension map ¯ H s,r sc (Ω j ) → ¯ H s,r sc (Ω (cid:48) ) by local reductionto R n . Correspondingly, we also obtain restriction and extension maps γ ∂ int Ω j : ¯ H s,r sc (Ω j ) → H s − / ,r sc ( ∂ int Ω j ) , e ∂ int Ω j : H s − / ,r sc ( ∂ int Ω j ) → ¯ H s,r sc (Ω j ) . With this background we have:
Lemma 4.9.
Let (cid:122) , (cid:122) as in Corollary 4.6, and let k ∈ R .For ψ ∈ H / ,k sc ( ∂ int Ω j ) there is a unique u ∈ ¯ H ,k sc (Ω j ) such that ∆ (cid:122) ,s u = 0 , γ ∂ int Ω j u = ψ .This defines the Poisson operator B Ω j : H / ,k sc ( ∂ int Ω j ) → ¯ H ,k sc (Ω j ) solving ∆ (cid:122) ,s B Ω j = 0 , γ ∂ int Ω j B Ω j = Id , which has the property that, for s > / , and for φ ∈ C ∞ (Ω j ) supported away from ∂ int Ω j , φB Ω j : H s − / ,r sc ( ∂ int Ω j ) → H s,r sc (Ω j ) .Proof. The uniqueness follows from the unique solvability of the Dirichlet problemwith vanishing boundary conditions, as we already discussed, while the existenceby taking u = e ∂ int Ω j ψ − ∆ − (cid:122) ,s ∆ (cid:122) ,s e ∂ int Ω j ψ , where ∆ − (cid:122) ,s is, as before, the inverse ofthe operator with vanishing Dirichlet boundary conditions. The mapping propertyalso follows from this explicit description, the mapping properties of e ∂ int Ω j as well NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 35 as Corollary 4.7, since one can arrange that e ∂ int Ω j maps to distributions supportedaway from supp φ . (cid:3) Let Ω be a larger neighborhood of Ω; all of our constructions take place in Ω .Let ˜Ω j = Ω j \ ∂ int Ω j (so the artificial boundary is included, but not the interiorone). Let G be a parametrix for A (cid:122) in Ω ; it is thus a scattering ps.d.o. withSchwartz kernel compactly supported in ˜Ω × ˜Ω . Then GA (cid:122) = I + E , whereWF (cid:48) sc ( E ) is disjoint from a neighborhood Ω (compactly contained in Ω ) of theoriginal region Ω, and E = − Id near ∂ int Ω . Now one has G ( N (cid:122) + d s (cid:122) M δ s (cid:122) ) = I + E, as operators acting on an appropriate function space on Ω . We now apply S (cid:122) , Ω from both sides. Then N (cid:122) S (cid:122) , Ω = N (cid:122) , since N (cid:122) P (cid:122) , Ω = N (cid:122) d s (cid:122) Q (cid:122) , Ω = 0 , in view of the vanishing boundary condition Q (cid:122) , Ω imposes. On the other hand, δ s (cid:122) S (cid:122) , Ω = δ s (cid:122) − δ s (cid:122) d s (cid:122) Q (cid:122) , Ω = 0so S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + S (cid:122) , Ω E S (cid:122) , Ω . In order to think of this as giving operators on Ω , let e be the extension map fromΩ to Ω , extending functions (vector fields) as 0, and r be the restriction map.(Note that e correspondingly maps into a relatively low regularity space, such as L , even if one starts with high regularity data.) Then, with the understandingthat N (cid:122) = N (cid:122) e , r S (cid:122) , Ω GN (cid:122) = r S (cid:122) , Ω e + K , K = r S (cid:122) , Ω E S (cid:122) , Ω e . We have:
Lemma 4.10.
Let (cid:122) , (cid:122) as in Corollary 4.6.The operator K = r S (cid:122) , Ω E S (cid:122) , Ω e is a smoothing operator in the sense thatit maps x k L (Ω ) to ¯ H s,r sc (Ω ) for every s, r, k . Further, for ψ ∈ C ∞ (Ω ) withsupport in Ω , ψK ψ ∈ Ψ −∞ , −∞ sc ( X ) .Further, for any s, r, k , given (cid:15) > there exists δ > such that if e δ is the exten-sion map (by ) from Ω δ = { x ≤ δ } ∩ Ω to Ω , then (cid:107) K e δ (cid:107) L ( x k L (Ω δ ) , ¯ H s,r sc (Ω )) <(cid:15) .Proof. This follows from Corollary 4.8. Indeed, with χ ≡ ∂ int Ω but with E = − Id on supp χ , and with φ ∈ C ∞ (Ω ) vanishing near supp χ , supp φ ∩ WF (cid:48) sc ( E ) = ∅ , φ ≡ , and with T defined by the first equality, T = φ S (cid:122) , Ω E S (cid:122) , Ω φ = φ S (cid:122) , Ω χEχ S (cid:122) , Ω φ + φ S (cid:122) , Ω (1 − χ ) Eχ S (cid:122) , Ω φ + φ S (cid:122) , Ω χE (1 − χ ) S (cid:122) , Ω φ + φ S (cid:122) , Ω (1 − χ ) E (1 − χ ) S (cid:122) , Ω φ. Now, E (1 − χ ) S (cid:122) , Ω φ, φ S (cid:122) , Ω (1 − χ ) E ∈ Ψ −∞ , −∞ sc ( X ) since they are in Ψ , ( X ) andWF (cid:48) sc ( E ) ∩ supp φ = ∅ , so they are smoothing. In combination with Corollary 4.8this gives that T : H s (cid:48) ,r (cid:48) sc ( X ) → H s,r sc ( X ) continuously for all s, r, s (cid:48) , r (cid:48) , so composing with the extension and restriction maps, noting r φ = r , φe = e , proves thefirst part of the lemma.To see the smallness claim, note that K e δ = r T e δ = r ( T x − )( xe δ ) xe δ : x k L (Ω δ ) → x k L (Ω ) has norm ≤ sup Ω δ x ≤ δ , while T x − : H ,k sc ( X ) → H s,r sc ( X ) is bounded, with bound independent of δ , and the same is true for r : H s,r sc ( X ) → ¯ H s,r sc (Ω ), completing the proof. (cid:3) Now, S (cid:122) , Ω − r S (cid:122) , Ω e = − d s (cid:122) Q (cid:122) , Ω + r d s (cid:122) Q (cid:122) , Ω e = − d s (cid:122) Q (cid:122) , Ω + d s (cid:122) r Q (cid:122) , Ω e = − d s (cid:122) ( Q (cid:122) , Ω − r Q (cid:122) , Ω e )and with γ ∂ int Ω denoting the restriction operator to ∂ int Ω as above, γ ∂ int Ω ( Q (cid:122) , Ω − r Q (cid:122) , Ω e ) = − γ ∂ int Ω Q (cid:122) , Ω e , so r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + d s (cid:122) ( Q (cid:122) , Ω − r Q (cid:122) , Ω e ) + K . Thus, with B Ω being the Poisson operator for ∆ (cid:122) ,s on Ω as above, r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + d s (cid:122) ( Q (cid:122) , Ω − r Q (cid:122) , Ω e + B Ω γ ∂ int Ω Q (cid:122) , Ω e ) − d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω e + K , so S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω − S (cid:122) , Ω d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω e + S (cid:122) , Ω K . Now we consider applying this to vector fields in Ω = Ω , writing e j for theextension map to Ω j . Composing from the right, S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω e − S (cid:122) , Ω d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω e + S (cid:122) , Ω K e . Now:
Lemma 4.11.
Let (cid:122) , (cid:122) as in Corollary 4.6.The operator K (cid:48) = S (cid:122) , Ω d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω e is smoothing in the sense thatfor φ ∈ C ∞ c (Ω \ ∂ int Ω ) , φ S (cid:122) , Ω d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω e : L (Ω) → H s,r sc ( X ) for all s, r , and indeed φ S (cid:122) , Ω d s (cid:122) B Ω γ ∂ int Ω Q (cid:122) , Ω φ ∈ Ψ −∞ , −∞ sc ( X ) .Further, for any s, r, k , given (cid:15) > there exists δ > such that if Ω ⊂ Ω δ = { x ≤ δ } ∩ Ω , then (cid:107) K (cid:48) (cid:107) L ( x k L (Ω) , ¯ H s,r sc (Ω )) < (cid:15). Proof.
By Corollary 4.6, using that δ s (cid:122) is a differential operator, ψQ (cid:122) , Ω φ ∈ Ψ −∞ , −∞ sc ( X )whenever ψ, φ ∈ C ∞ (Ω ) have disjoint supports, also disjoint from ∂ int Ω since thisoperator is in Ψ − , ( X ) directly from the corollary, and then the disjointness ofsupports gives the conclusion. Taking such ψ, φ , as one may, with φ ≡ ψ ≡ ∂ int Ω , we see that γ ∂ int Ω Q (cid:122) , Ω e : x k L (Ω) → H s,r sc ( ∂ int Ω )for all s, r, k , i.e. mapping to ˙ C ∞ ( ∂ int Ω ). The first part then follows from B Ω NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 37 mapping this to ¯ H ,r sc (Ω ) for all r , with the additional property that ˜ φB Ω mapsto H s,r sc (Ω) for all s, r if ˜ φ has properties like φ , and then Corollary 4.8 completesthe argument.For the smallness, we just need to proceed as in Lemma 4.10, writing γ ∂ int Ω Q (cid:122) , Ω e = γ ∂ int Ω ( ψQ (cid:122) , Ω φx − )( xe ) , where now ψQ (cid:122) , Ω φx − ∈ Ψ −∞ , −∞ sc ( X ), thus bounded between all weighted Sobolevspaces, with norm independent of δ , while xe : x k L (Ω δ ) → x k L (Ω ) has norm ≤ δ . (cid:3) Thus,(4.12) S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω e + K , with K smoothing and small if Ω ⊂ { x ≤ δ } , with δ suitably small. This is exactlyEquation (5.7) of [24], and from this point on we can follow the argument of the global work of Stefanov and Uhlmann [24, Section 5], with the addition of having a small rather than just compact error, giving invertibility.Restricting to Ω from the left, the key remaining step is to compute S (cid:122) , Ω − r S (cid:122) , Ω e in terms of the already existing information. As above, S (cid:122) , Ω − r S (cid:122) , Ω e = − d s (cid:122) ( Q (cid:122) , Ω − r Q (cid:122) , Ω e ) , but now we compute u = ( Q (cid:122) , Ω − r Q (cid:122) , Ω e ) f using that it is the solution of theDirichlet problem ∆ (cid:122) ,s u = 0, γ ∂ int Ω u = − γ ∂ int Ω Q (cid:122) , Ω e f , so(4.13) u = − B Ω γ ∂ int Ω Q (cid:122) , Ω e f, and using that one can compute γ ∂ int Ω Q (cid:122) , Ω e f from d s (cid:122) Q (cid:122) , Ω e f . Concretely,we have the following lemma on functions: Lemma 4.12.
Let ˙ H , (Ω \ Ω) denote the restriction of elements of ˙ H , (Ω ) to Ω \ Ω (thus, these need not vanish at ∂ int Ω ), and let ρ Ω \ Ω be a defining functionof ∂ int Ω as a boundary of Ω \ Ω , i.e. it is positive in the latter set. Suppose that ∂ x ρ Ω \ Ω > at ∂ int Ω ; note that this is independent of the choice of ρ Ω \ Ω satisfyingthe previous criteria (so this is a statement on x being increasing as one leaves Ω at ∂ int Ω ). Then on functions, for (cid:122) > , k ∈ R , the map d s (cid:122) : ˙ H ,k sc (Ω \ Ω) → x k L (Ω \ Ω) is injective, with a continuous left inverse P Ω \ Ω : x k L (Ω \ Ω) → ˙ H ,k sc (Ω \ Ω) .Proof. Consider k = 0 first.The norm of d s (cid:122) u is certainly equivalent to that of ∇ u in L (Ω \ Ω) modulo the L (Ω \ Ω) norm of u , so one only needs to prove a local Poincar´e inequality(4.14) (cid:107) u (cid:107) L (Ω \ Ω) ≤ C (cid:107) d s (cid:122) u (cid:107) L (Ω \ Ω) to conclude that (cid:107) u (cid:107) ˙ H , (Ω \ Ω) ≤ C (cid:107) d s (cid:122) u (cid:107) L (Ω \ Ω) , which proves the lemma in this case, since it proves that d s (cid:122) , between these spaces,has closed range and is injective, so it is an isomorphism between ˙ H , (Ω \ Ω)and its range, and then its inverse in this sense can be extended continuously to L (Ω \ Ω).But (4.14) can be proved similarly to Lemma 4.2, by showing that(4.15) (cid:107) u (cid:107) L (Ω \ Ω) ≤ C (cid:107) ( x D x + i (cid:122) ) u (cid:107) L (Ω \ Ω) . Here we want to use P = x D x + i (cid:122) and (cid:107) P u (cid:107) again; we need to be careful at ∂ int Ω since u does not vanish there. Thus, there is an integration by parts boundaryterm, which we express in terms of the characteristic function χ Ω \ Ω : (cid:107) P u (cid:107) L (Ω \ Ω) = (cid:104) χ Ω \ Ω P u, P u (cid:105) L (Ω ) = (cid:104) P ∗ χ Ω \ Ω P u, u (cid:105) L (Ω ) = (cid:104) P ∗ P u, u (cid:105) L (Ω \ Ω) + (cid:104) [ P ∗ , χ Ω \ Ω ] P u, u (cid:105) L (Ω ) . Similarly, (cid:107) P R u (cid:107) L (Ω \ Ω) = (cid:104) P ∗ R P R u, u (cid:105) L (Ω \ Ω) + (cid:104) [ P ∗ R , χ Ω \ Ω ] P R u, u (cid:105) L (Ω ) . On the other hand, with P I being 0th order, the commutator term vanishes for it.Correspondingly, (cid:107) P u (cid:107) L (Ω \ Ω) = (cid:104) P ∗ P u, u (cid:105) L (Ω \ Ω) + (cid:104) [ P ∗ , χ Ω \ Ω ] P u, u (cid:105) L (Ω ) = (cid:104) P ∗ R P R u, u (cid:105) L (Ω \ Ω) + (cid:104) P ∗ I P I u, u (cid:105) L (Ω \ Ω) + (cid:104) i [ P R , P I ] u, u (cid:105) L (Ω \ Ω) + (cid:104) [ P ∗ , χ Ω \ Ω ] P u, u (cid:105) L (Ω ) = (cid:107) P R u (cid:107) L (Ω \ Ω) + (cid:107) P I u (cid:107) L (Ω \ Ω) + (cid:104) i [ P R , P I ] u, u (cid:105) L (Ω \ Ω) + (cid:104) [ P ∗ , χ Ω \ Ω ] P u, u (cid:105) L (Ω ) − (cid:104) [ P ∗ R , χ Ω \ Ω ] P R u, u (cid:105) L (Ω ) . Now, as P − P R is 0th order, [ P ∗ , χ Ω \ Ω ] = [ P ∗ R , χ Ω \ Ω ], so the last two terms onthe right hand side give(4.16) (cid:104) [ P ∗ , χ Ω \ Ω ] iP I u, u (cid:105) L (Ω ) = (cid:104) x ∂ x χ Ω \ Ω ( (cid:122) − n − x ) u, u (cid:105) L (Ω ) , which is non-negative, at least if x is sufficiently small (or (cid:122) large) on ∂ int Ω since χ Ω \ Ω = χ (0 , ∞ ) ◦ ρ Ω \ Ω . Correspondingly, this term can be dropped, and oneobtains (4.15) at least if x is small on Ω just as in the proof of Lemma 4.2. Thecase of x not necessarily small on Ω (though small on Ω) follows exactly as inLemma 4.2 using the standard Poincar´e inequality, and even the case where x isnot small on Ω can be handled similarly since one now has an extra term at ∂ int Ω,away from x = 0, which one can control using the standard Poincar´e inequality.This gives (cid:107) u (cid:107) ˙ H , (Ω \ Ω) ≤ C (cid:107) d s (cid:122) u (cid:107) L (Ω \ Ω) , showing the claimed injectivity. Further, this gives a continuous inverse from therange of d s (cid:122) , which is closed in L (Ω \ Ω); one can use an orthogonal projection tothis space to define the left inverse P Ω \ Ω , completing the proof when k = 0.For general k , one can proceed as in Lemma 4.4, conjugating d s (cid:122) by x k , whichchanges it by x times a smooth one form; this changes x D x + i (cid:122) by an elementof xC ∞ ( X ), with the only effect of modifying the x n − term in (4.16), which doesnot affect the proof. (cid:3) We now turn to one forms.
Lemma 4.13.
Let ˙ H , (Ω \ Ω) be as in Lemma 4.12, but with values in one-forms, and let ρ Ω \ Ω be a defining function of ∂ int Ω as a boundary of Ω \ Ω , i.e.it is positive in the latter set. Suppose that ∂ x ρ Ω \ Ω > at ∂ int Ω ; note that thisis independent of the choice of ρ Ω \ Ω satisfying the previous criteria (so this is astatement on x being increasing as one leaves Ω at ∂ int Ω ). Then for r ≤ − ( n − / ,on one-forms the map d s (cid:122) : ˙ H ,r sc (Ω \ Ω) → H ,r sc (Ω \ Ω) NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 39 is injective, with a continuous left inverse P Ω \ Ω : H ,r sc (Ω \ Ω) → ˙ H ,r − (Ω \ Ω) .Remark . Unfortunately the argument given above for functions would give anunfavorable boundary term, so instead we proceed proving the local Poincar´e in-equality directly and using our generalized Korn’s inequality, Lemma 4.5, to avoida loss of derivatives. However, our method still produces a loss of weight, essen-tially because as presented the estimate would be natural for standard tensors, notscattering tensors, hence the presence of the loss − Proof.
As in the work of the first two authors, [23, Section 6], we prove the Poincar´einequality using the identity, see [18, Chapter 3.3],(4.17) (cid:88) i [ v ( γ ( s ))] i ˙ γ i ( s ) = (cid:90) s (cid:88) ij [ d s v ( γ ( t ))] ij ˙ γ i ( t ) ˙ γ j ( t ) dt, where γ is a unit speed geodesic of the original metric g (thus not of a scatter-ing metric) with γ (0) ∈ ∂ int Ω (so v ( γ (0)) vanishes) and γ ( τ ) ∈ ∂ int Ω ∪ ∂X , with γ | (0 ,τ ) in Ω \ Ω. Identity (4.17) is just an application of the Fundamental The-orem of Calculus with the s -derivative of the l.h.s. computed using the rules ofcovariant differentiation. In this formula we use [ d s v ( γ ( t ))] ij for the componentsin the symmetric 2-cotensors corresponding to the standard cotangent bundle, andsimilarly for [ v ( γ ( s ))] i . Notice that this formula gives an explicit left inverse for d s (cid:122) , as discussed below.Here we choose γ such that x ◦ γ is strictly monotone in the sense that − ∂∂t ( x ◦ γ )is bounded below (and above) by a positive constant, thus ( x ◦ γ ) ∂∂t ( x − ◦ γ )has the same property. Note that one can construct a smooth family of suchgeodesics emanating from ∂ int Ω , parameterized by ∂ int Ω, in a manner that, with dω a smooth measure on ∂ int Ω , dω dt is equivalent to the volume form dg , i.e. alsoto dx dy . . . dy n − . Thus, for any k ≥
0, using x ( γ ( s )) ≤ x ( γ ( t )) along the geodesicsegment, t ∈ [0 , s ], | e − (cid:122) /x ( γ ( s )) x ( γ ( s )) k (cid:88) i [ v ( γ ( s ))] i ˙ γ i ( s ) | = (cid:12)(cid:12)(cid:12) (cid:90) s (cid:88) ij e − (cid:122) /x ( γ ( t )) x ( γ ( t )) k +1 [ d s v ( γ ( t ))] ij ˙ γ i ( t ) ˙ γ j ( t ) × e − (cid:122) (1 /x ( γ ( s )) − /x ( γ ( t ))) x ( γ ( t )) − dt (cid:12)(cid:12)(cid:12) ≤ n (cid:16) (cid:90) τ (cid:88) ij e − (cid:122) /x ( γ ( t )) x ( γ ( t )) k +2 | [ d s v ( γ ( t ))] ij ˙ γ i ( t ) ˙ γ j ( t ) | dt (cid:17) × (cid:16) (cid:90) s e − (cid:122) (1 /x ( γ ( s )) − /x ( γ ( t ))) x ( γ ( t )) − dt (cid:17) . Thus, | e − (cid:122) /x ( γ ( s )) x ( γ ( s )) k (cid:88) i [ v ( γ ( s ))] i ˙ γ i ( s ) | ≤ C (cid:48) (cid:16) (cid:90) τ (cid:88) ij e − (cid:122) /x ( γ ( t )) x ( γ ( t )) k +2 | [ d s v ( γ ( t ))] ij ˙ γ i ( t ) ˙ γ j ( t ) | dt (cid:17) × (cid:16) (cid:90) s e − (cid:122) (1 /x ( γ ( s )) − /x ( γ ( t ))) (cid:0) − ∂∂t ( x − ( γ ( t ))) (cid:1) dt (cid:17) ≤ C (cid:48) (cid:16) (cid:90) τ e − (cid:122) /x ( γ ( t )) x ( γ ( t )) k +2 | d s v ( γ ( t )) | (cid:96) dt (cid:17)(cid:16) (cid:90) x − ( γ ( s )) r e − (cid:122) (1 /x ( γ ( s )) − r ) dr (cid:17) for suitable r >
0, where we wrote r = x − , and we used the lower bound for( x ◦ γ ) ∂∂t ( x − ◦ γ ) in the second factor, and that γ is unit speed in the first factor,with (cid:96) being the norm as a symmetric map on T p X . The second factor on theright hand side is bounded by (2 (cid:122) ) − , so can be dropped. Now, as τ dx + ζ dy =( x τ ) dxx +( xζ ) dyx , so e.g. the dx component of d s v is x − times the dx x componentin the scattering basis, we have(4.18) | d s v ( γ ( t )) | (cid:96) ≤ Cx ( γ ( t )) − | d s v ( γ ( t )) | (cid:96) , so the right hand side is bounded from above by C (cid:48)(cid:48) (cid:122) − (cid:90) τ e − (cid:122) /x ( γ ( t )) x ( γ ( t )) k − | d s v ( γ ( t )) | (cid:96) dt Integrating in the spatial variable, γ (0) ∈ ∂ int Ω , and using that the second factoris (2 (cid:122) ) − , gives (cid:107) e − (cid:122) /x x k v ( γ (cid:48) ) (cid:107) L (Ω \ Ω) ≤ C (cid:122) − (cid:107) x k − e − (cid:122) /x d s v (cid:107) L (Ω \ Ω;Sym T ∗ X ) . Using different families of geodesics with tangent vectors covering
T X over Ω \ Ω, (cid:107) e − (cid:122) /x x k v (cid:107) L (Ω \ Ω; T ∗ X ) ≤ C (cid:122) − (cid:107) x k − e − (cid:122) /x d s v (cid:107) L (Ω \ Ω;Sym T ∗ X ) . Now, similarly to (4.18), but going the opposite direction, (cid:107) v ( p ) (cid:107) (cid:96) ≤ x ( p ) (cid:107) v ( p ) (cid:107) (cid:96) , so (cid:107) e − (cid:122) /x x k − v (cid:107) L (Ω \ Ω; sc T ∗ X ) ≤ C (cid:122) − (cid:107) x k − e − (cid:122) /x d s v (cid:107) L (Ω \ Ω;Sym T ∗ X ) . Changing the volume form as well yields (cid:107) e − (cid:122) /x x k − n +1) / v (cid:107) L (Ω \ Ω; sc T ∗ X ) ≤ C (cid:122) − (cid:107) x k − n +1) / e − (cid:122) /x d s v (cid:107) L (Ω \ Ω;Sym T ∗ X ) . With u = e − (cid:122) /x v , this gives, for u ∈ C ∞ (Ω \ Ω), vanishing at ∂ int Ω , of com-pact support,(4.19) (cid:107) u (cid:107) H ,r − (Ω \ Ω) ≤ C (cid:122) − (cid:107) d s (cid:122) u (cid:107) H ,r sc (Ω \ Ω) ,r ≤ − ( n − /
2, which then gives the same conclusion, by density and continuityconsiderations for u ∈ ˙ H ,r sc (Ω \ Ω), the desired Poincar´e estimate.To obtain the H estimate, we use Lemma 4.5, which gives, even for u ∈ ¯ H ,r − (Ω \ Ω), (cid:107) u (cid:107) H ,r − (Ω \ Ω) ≤ C ( (cid:107) d s (cid:122) u (cid:107) H ,r − (Ω \ Ω) + (cid:107) u (cid:107) H ,r − (Ω \ Ω) ) , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 41 which combined with (4.19) proves (cid:107) u (cid:107) ˙ H ,r − (Ω \ Ω) ≤ C (cid:107) d s (cid:122) u (cid:107) H ,r sc (Ω \ Ω) , u ∈ ˙ H ,r sc (Ω \ Ω) , where recall that our notation is that membership of ˙ H ,r sc (Ω \ Ω) only impliesvanishing at ∂ int Ω , not at ∂ int Ω.Taking into account the above considerations, namely choosing several familiesof geodesics to span the tangent space, and working with v = e (cid:122) /x u , the formula(4.17) then also gives an explicit formula for the left inverse. (cid:3) Recall now (4.13): u = − B Ω γ ∂ int Ω Q (cid:122) , Ω e f. Using Lemmas 4.12-4.13, we conclude that u = − B Ω γ ∂ int Ω P Ω \ Ω d s (cid:122) Q (cid:122) , Ω e f, and as e f vanishes on Ω \ Ω, S (cid:122) , Ω e f | Ω \ Ω = − d s (cid:122) Q (cid:122) , Ω e f | Ω \ Ω , so u = B Ω γ ∂ int Ω P Ω \ Ω S (cid:122) , Ω e f, and thus S (cid:122) , Ω − r S (cid:122) , Ω e = − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω S (cid:122) , Ω e . Using (4.12) this gives r S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω S (cid:122) , Ω e + r K . Using (4.12) again to express S (cid:122) , Ω e on the right hand side, we get r S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ( S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) − K ) + r K , which gives ( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω + ( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) K . We now add P (cid:122) , Ω to both sides, and use that the smallness of K when Ω is smallenough gives that Id +( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) K is invertible. Here we need tobe careful in the 2-tensor case: while K is smoothing, including in the sense ofproducing additional decay, so there is no problem with applying P Ω \ Ω regardlessof the weighted space we are considering, the result will have only a weightedestimate in H ,r − , r ≤ − ( n − /
2, corresponding to Lemma 4.13, so the inversionhas to be done in a sufficiently negatively weighted space, namely H ,r sc (Ω), with r ≤ − ( n − /
2. Thus,(Id +( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) K ) − ◦ (cid:16) ( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) + P (cid:122) , Ω (cid:17) = Id , and so multiplying from S (cid:122) , Ω from the right yields(4.20) (Id +( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) K ) − ◦ ( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) S (cid:122) , Ω r S (cid:122) , Ω GN (cid:122) = S (cid:122) , Ω . Now recall that N (cid:122) = e − (cid:122) /x LIe (cid:122) /x , and that for f ∈ e (cid:122) /x L (Ω), P (cid:122) , Ω e − (cid:122) /x f = 0amounts to e (cid:122) /x δ s e − (cid:122) /x ( e − (cid:122) /x f ) = 0, i.e. δ s ( e − (cid:122) /x f ) = 0. This in particular gives an inversion formula for the geodesic X-ray transform on e (cid:122) /x -solenoidalone-forms and symmetric 2-tensors.In order to state the stability estimate it is convenient to consider ( x, y, λ, ω ) ∈ SX to actually lie in sc SX via the identification (multiplying the tangent vectorby x ) ( x, y, λ ∂ x + ω ∂ y ) (cid:55)→ ( x, y, ( λ/x )( x ∂ x ) + ω ( x∂ y ))Here sc SX = ( sc T X \ o ) / R + is the sphere bundle in sc T X , and in the relevantopen set the fiber over a fixed point ( x, y ) can be identified with vectors of the form˜ λ ( x ∂ x ) + ˜ ω ( x∂ y ), ˜ ω ∈ S n − , ˜ λ ∈ R . Then the region | λ/x | < M in SX correspondsto the region | ˜ λ | < M ; this is now an open subset of sc SX . Note that in particularthat the ‘blow-down map’ ( x, y, ˜ λ, ˜ ω ) (cid:55)→ ( x, y, x ˜ λ, ˜ ω ) is smooth, and the compositemap ( x, y, ˜ λ, ˜ ω, t ) (cid:55)→ γ x,y,x ˜ λ, ˜ ω ( t ) has surjective differential. In particular, with U = {| ˜ λ | < M } , the scattering Sobolev spaces are just restrictions to a domain with smooth bound-ary. Note that U lies within the set of Ω-local geodesics; we choose M so thatsupp χ ⊂ M .This discussion, in particular (4.20), proves our main local result, for which wereintroduce the subscript c for the size of the region Ω c : Theorem 4.15.
For one forms, let (cid:122) > ; for symmetric 2-tensors let (cid:122) > be the maximum of the two constants, denoted there by (cid:122) , in Proposition 3.3 andCorollary 4.6.For Ω = Ω c , c > small, the geodesic X-ray transform on e (cid:122) /x -solenoidal one-forms and symmetric 2-tensors f ∈ e (cid:122) /x L (Ω) , i.e. ones satisfying δ s ( e − (cid:122) /x f ) =0 , is injective, with a stability estimate and a reconstruction formula f = e (cid:122) /x (Id +( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) K ) − ( r − d s (cid:122) B Ω γ ∂ int Ω P Ω \ Ω ) ◦ S (cid:122) , Ω r S (cid:122) , Ω Ge − (cid:122) /x LIf.
Here stability is in the sense that for s ≥ there exist R, R (cid:48) such that for any (suf-ficiently negative in the case of 2-tensors) r the e (cid:122) /x H s − ,r sc norm of f on Ω is con-trolled by the e (cid:122) /x H s,r + R sc norm of If on U , provided f is a priori in e (cid:122) /x H s,r + R (cid:48) sc .In addition, replacing Ω c = { ˜ x > − c } ∩ M by Ω τ,c = { τ > ˜ x > − c + τ } ∩ M , c can be taken uniform in τ for τ in a compact set on which the strict concavityassumption on level sets of ˜ x holds.Remark . Notice that the proof below gives in particular, by composing L and I , LI : e (cid:122) /x H s,r sc ( X ) → e (cid:122) /x H s,r − − s sc ( X ), s ≥
0, even though Proposition 3.1implies the mapping property LI : e (cid:122) /x H s,r sc ( X ) → e (cid:122) /x H s +1 ,r sc ( X ) (with values inscattering one-forms or 2-tensors). The loss in the derivatives by one order and ofthe decay by order ≥ L, I below.
Proof.
Given (4.20), we just need to show that for s ≥ R , R suchthat for k ∈ R , L is bounded e (cid:122) /x H s,k + R sc ( U ) → e (cid:122) /x H s,k sc ( X ) , while I is bounded e (cid:122) /x H s,k + R sc ( X ) → e (cid:122) /x H s,k sc ( U ) , NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 43 with the function spaces on X with values in either one forms or 2-tensors. To seethese boundedness statements, one proceeds as in [28, Section 3], prior to Proposi-tion 3.3, though we change our point of view slightly, as we are using the ‘blown-upspace’ sc SX rather than SX for the geodesic parameterization.Concretely, L can be written as the composition of a multiplication operator M ,by xχ (˜ λ ), resp. x χ (˜ λ ), for the one-form, resp. 2-tensor, case, times x − times asc-one-form or x − times a sc-2-tensor factor, with a − x in the def-inition of L being absorbed into the ˜ λ integral, and a push-forward in which the ˜ λ, ˜ ω variables are integrated out. The pushforward maps L ( U ) = x − (2 n − / L ( U )to L ( X ) = x − ( n +1) / L ( X ) ( L spaces without subscripts being relative to smoothnon-degenerate densities) with the weights arising from the scattering volume formsbeing x − n , resp. x − n − , times a smooth volume form. Further, it commutes withmultiplication by functions of x , so it maps e (cid:122) /x H ,k sc ( X ) to e (cid:122) /x H ,k +( n − / ( X ),and (local) lifts of scattering vector fields x D x , xD y j are still scattering vectorfields so it also maps e (cid:122) /x H s,k sc ( U ) to e (cid:122) /x H s,k +( n − / ( X ) for s ≥ s ≥
0. Also, taking into account the smoothness of χ (˜ λ ),we see that multiplication by x p χ (˜ λ ) maps e (cid:122) /x H s,k sc ( U ) → e (cid:122) /x H s,k + p sc ( U ) for all s ≥
0, so in the one form case L : e (cid:122) /x H s,k sc ( U ) → e (cid:122) /x H s,k +( n − / ( X ) , while in the 2-tensor case L : e (cid:122) /x H s,k sc ( U ) → e (cid:122) /x H s,k +1+( n − / ( X ) . On the other hand, I can be written as a pull-back to the subset U × R of sc SX × R from X , after contraction with γ (cid:48) x,y,x ˜ λ, ˜ ω ( t ), via the map γ : ( x, y, ˜ λ, ˜ ω, t ) (cid:55)→ γ x,y,x ˜ λ, ˜ ω ( t ), which has surjective differential, followed by integration over (a uni-formly controlled compact subset of) the R factor. The integration (push-forward)maps e (cid:122) /x H s,k sc ( U × R ) → e (cid:122) /x H s,k +1 / ( U ), where the 1 / γ (cid:48) x,y,x ˜ λ, ˜ ω ( t ) is x − times a scattering tangent vector, asdiscussed in Proposition 3.1. Thus, the boundedness of the pull-back as a map xL ( X ; sc T ∗ X ) → L ( U × R ) , i.e. x − ( n − / L ( X ) → x − (2 n +1) / L ( U × R ) , in the one-form case, resp. x L ( X ; Sym T ∗ X ) → L ( U × R ) , i.e. x − ( n − / L ( X ) → x − (2 n +1) / L ( U × R ) , in the 2-tensor case, follows from the surjectivity of the differential of γ . (Concretelyhere this means that as for fixed ˜ λ, ˜ ω, t , ( x, y ) (cid:55)→ γ x,y, ˜ λ, ˜ ω ( t ) = ( x (cid:48) , y (cid:48) ) is a diffeomor-phism, one can rewrite the integral expressing the squared L -norm of the pull-backin terms of the squared L -norm of the original function using Fubini’s theorem.)Further, the x coordinate along γ x,y,λ,ω , denoted by x (cid:48) in Proposition 3.1, satisfies x (cid:48) ≥ x − CM x (as | λ/x | ≤ M on U ) due to [28, Equation (3.1)], which meansthat e − (cid:122) /x x − k e (cid:122) /x (cid:48) ( x (cid:48) ) k is bounded on the curves as − (cid:122) /x + (cid:122) /x (cid:48) − k log( x/x (cid:48) )is bounded above (with the boundedness for x (cid:48) ≤ x , holding thanks to the lowerbound for x (cid:48) , being the important point; for x (cid:48) ≥ x , − (cid:122) /x − k log x being monotonefor small x can be used). Thus, the mapping property e (cid:122) /x H ,k sc ( X ) → e (cid:122) /x H ,k − n/ − ( U × R ) , resp. e (cid:122) /x H ,k sc ( X ) → e (cid:122) /x H ,k − n/ − ( U × R ) , follows by the same argument as the L boundedness. Finally, by the chain rule,using just the smoothness of γ , we obtain that any derivative of the pull-backwith respect to the standard vector fields V ∈ V ( U ) can be expressed in terms oflinear combinations with smooth coefficients of standard derivatives (with respectto V (cid:48) ∈ V ( X )) of the original function, so in particular for P ∈ Diff s ( U × R )and one-forms, P f is controlled in e (cid:122) /x H ,k − n/ − ( U × R ) in terms of derivativesof order ≤ s of f in e (cid:122) /x H ,k sc ( X ), with a similar statement for 2-tensors. Now,(with the above notation) x (cid:48) ≥ cx for some c > x/x (cid:48) is bounded), so that x factors of derivatives like x ∂ x , x∂ y j , x∂ ˜ λ , x∂ ˜ ω j being applied to the pull-backcan be turned into factors of x (cid:48) , so we see that if P ∈ Diff s sc ( X ), then P f iscontrolled in e (cid:122) /x H ,k − n/ − ( U × R ) in terms of derivatives of order ≤ s of f withrespect to the vector fields x (cid:48) ∂ x (cid:48) x (cid:48) ∂ y (cid:48) in e (cid:122) /x H ,k sc ( X ). Note here the presenceof x (cid:48) ∂ x (cid:48) rather than ( x (cid:48) ) ∂ x (cid:48) , due to the fact that when one writes the pull-backas f ( X x,y,x ˜ λ, ˜ ω ( t ) , Y x,y,x ˜ λ, ˜ ω ( t )), a derivative like x∂ y hitting it is controllable by( x (cid:48) ∂ x (cid:48) f )( ∂ y X ) and ( x (cid:48) ∂ y (cid:48) f )( ∂ y X ), with the first of these lacking an extra factor of x (cid:48) . This means that we need to have an extra decay by order s to get a boundedmap between the scattering spaces (since x (cid:48) ∂ x (cid:48) = ( x (cid:48) ) − (( x (cid:48) ) ∂ x (cid:48) )), so for s ≥ e (cid:122) /x H s,k sc ( X ) → e (cid:122) /x H s,k − s − n/ − ( U × R ) , resp. e (cid:122) /x H s,k sc ( X ) → e (cid:122) /x H s,k − s − n/ − ( U × R ) , follows, and then interpolation gives this for all s ≥
0. Thus, in the one-form case I : e (cid:122) /x H s,k sc ( X ) → e (cid:122) /x H s,k − s − n/ − / ( U × R ) , in the 2-tensor case I : e (cid:122) /x H s,k sc ( X ) → e (cid:122) /x H s,k − s − n/ − / ( U × R ) , completing the proof. (cid:3) If f ∈ x r e (cid:122) /x L (Ω) then the map f → If factors through S (cid:122) , Ω e − (cid:122) /x f = e − (cid:122) /x f − P (cid:122) , Ω e − (cid:122) /x f since Ie (cid:122) /x P (cid:122) , Ω e − (cid:122) /x f = Id s e (cid:122) /x ∆ − (cid:122) ,s, Ω e (cid:122) /x δ s e − (cid:122) /x f = 0 . By Theorem 4.15, e (cid:122) /x S (cid:122) , Ω e − (cid:122) /x f (cid:55)→ Ie (cid:122) /x S (cid:122) , Ω e − (cid:122) /x f is injective, with a stabil-ity estimate. Since e (cid:122) /x P (cid:122) , Ω e − (cid:122) /x f = d s e (cid:122) /x ∆ − (cid:122) ,s, Ω e (cid:122) /x δ s e − (cid:122) /x f, this means that we have recovered f up to a potential term, i.e. in a gauge-freemanner we have: Corollary 4.17.
Let (cid:122) > . With Ω = Ω c as in Theorem 4.15, r sufficientlynegative, c > small, if f ∈ e (cid:122) /x x r L (Ω) is a one-form then f = u + d s v , where v ∈ e (cid:122) /x x r ˙ H , (Ω) , while u ∈ e (cid:122) /x x r L (Ω) can be stably determined from If .Again, replacing Ω c = { ˜ x > − c } ∩ M by Ω τ,c = { τ > ˜ x > − c + τ } ∩ M , c can betaken uniform in τ for τ in a compact set on which the strict concavity assumptionon level sets of ˜ x holds. NVERTING THE LOCAL GEODESIC X-RAY TRANSFORM ON TENSORS 45
Corollary 4.18.
Let (cid:122) , (cid:122) be as in Theorem 4.15. With Ω = Ω c as in Theo-rem 4.15, r sufficiently negative, c > small, if f ∈ x r e (cid:122) /x L (Ω) is a symmetric2-tensor then f = u + d s v , where v ∈ e (cid:122) /x ˙ H ,r − (Ω) , while u ∈ e (cid:122) /x x r − L (Ω) can be stably determined from If .Again, replacing Ω c = { ˜ x > − c } ∩ M by Ω τ,c = { τ > ˜ x > − c + τ } ∩ M , c can betaken uniform in τ for τ in a compact set on which the strict concavity assumptionon level sets of ˜ x holds. This theorem has an easy global consequence. To state this, assume that ˜ x isa globally defined function with level sets Σ t which are strictly concave from thesuper-level set for t ∈ ( − T, x ≤ M . Thenwe have: Theorem 4.19.
Suppose M is compact. The geodesic X-ray transform is injec-tive and stable modulo potentials on the restriction of one-forms and symmetric2-tensors f to ˜ x − (( − T, in the following sense. For all τ > − T there is v ∈ ˙ H (˜ x − (( τ, such that f − d s v ∈ L (˜ x − (( τ, can be stably recov-ered from If . Here for stability we assume that s ≥ , f is in an H s -space, thenorm on If is an H s -norm, while the norm for v is an H s − -norm.Proof. For the sake of contradiction, suppose there is no v as stated on ˜ x − (( τ , > τ > − T , If = 0, and let τ = inf { t ≤ ∃ v t ∈ ˙ H ( { ˜ x > t } ) s.t. f = d s v t on { ˜ x > t }} ≥ τ . Thus, for any τ (cid:48) > τ , such as τ (cid:48) < τ + c/ c as in the uniform part of Corollar-ies 4.17-4.18 on the levels [ τ, v ∈ ˙ H ( { ˜ x > τ (cid:48) } ) such that f = d s v on { ˜ x > τ (cid:48) } . Choosing φ ∈ C ∞ ( M ) identically 1 near ˜ x ≥ τ + 2 c/
3, supported in˜ x > τ + c/ f − d s ( φv ) is supported in ˜ x ≤ τ + 2 c/
3. But then by the uniformstatement of Corollaries 4.17-4.18, there exists v (cid:48) ∈ ˙ H ( { τ − c/ < ˜ x ≤ τ + 2 c/ } )such that f − d s ( φv ) = d s v (cid:48) in τ − c/ < ˜ x < τ + 2 c/
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Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395,U.S.A.
E-mail address : [email protected] Department of Mathematics, University of Washington, Seattle, WA 98195-4350,U.S.A.
E-mail address : [email protected] Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.
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