ANALYSIS & PDE msp Volume 9 No. 8 2016 GABRIEL P. PATERNAIN AND HANMING ZHOU INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM ANALYSIS AND PDE Vol. 9, No. 8, 2016 dx.doi.org/10.2140/apde.2016.9.1903 msp INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM GABRIEL P. PATERNAIN AND HANMING ZHOU We establish an equivalence principle between the solenoidal injectivity of the geodesic ray transform acting on symmetric m-tensors and the existence of invariant distributions or smooth first integrals with prescribed projection over the set of solenoidal m-tensors. We work with compact simple manifolds, but several of our results apply to nontrapping manifolds with strictly convex boundary. 1. Introduction The present paper studies the geodesic ray transform of a compact simply connected Riemannian manifold with no conjugate points and strictly convex boundary. Our main objective is to establish an equivalence principle between injectivity of the ray transform acting on solenoidal symmetric m-tensors and the existence of solutions to the transport equation (associated with the geodesic vector field) with prescribed projection over the set of solenoidal m-tensors. The Radon transform in the plane is the most fundamental example of the geodesic ray transform. It packs the integrals of a function f in R2 over straight lines: Rf .s; !/D Z 1 1 f .s!C t!?/ dt; s 2 R; ! 2 S1: Here !? is the rotation of ! by 90 degrees counterclockwise. The properties of this transform are well studied [Helgason 1999] and constitute the theoretical underpinnings for many medical imaging methods such as CT and PET. Generalizations of the Radon transform are often needed. In seismic and ultrasound imaging one finds ray transforms where the measurements are given by integrals over more general families of curves, often modeled as the geodesics of a Riemannian metric. Moreover, integrals of tensor fields over geodesics are ubiquitous in rigidity questions in differential geometry and dynamics. In this paper we will relate the injectivity properties of the geodesic ray transform with a well-studied subject in classical mechanics: the existence of special first integrals of motion along geodesics. Some Riemannian metrics admit distinguished first integrals; e.g., the geodesic flow of an ellipsoid in R3 admits a nontrivial first integral which is quadratic in momenta. As recently shown in [Kruglikov and Matveev 2016], a generic metric does not admit a nontrivial first integral that is polynomial in momenta, but here we will show a complementary statement going in the opposite direction: from the injectivity of the geodesic ray transform on tensors, we will show that it is possible to construct a smooth first integral with any prescribed polynomial part. In other words, given a polynomial F of degree m in momenta MSC2010: primary 53C65; secondary 58J40. Keywords: geodesic ray transform, first integral, tensor tomography, invariant distribution. 1903 1904 GABRIEL P. PATERNAIN AND HANMING ZHOU satisfying a natural restriction condition (related with the transport equation, see Section 7), we will show that we can find a smooth function G whose dependence on momenta is of order >m such that F CG is a first integral of the geodesic flow. Generically G is nonvanishing and not polynomial in momenta. Let us now explain our results in more detail. The geodesic ray transform acts on functions defined on the unit sphere bundle of a compact oriented n-dimensional Riemannian manifold .M; g/ with boundary @M (n 2). Let SM denote the unit sphere bundle on M ; i.e., SM WD f.x; / 2 TM W kkg D 1g: We define the volume form on SM by d†2n1.x; /D jdV n.x/^ dx./j, where dV n is the volume form on M and dx./ is the volume form on the fiber SxM. The boundary of SM is given by @SM WD f.x; / 2 SM W x 2 @M g. On @SM the natural volume form is d†2n2.x; /D jdV n1.x/^ dx./j, where dV n1 is the volume form on @M. We define two subsets of @SM, @˙SM WD ˚ .x; / 2 @SM W ˙h; .x/ig  0 ; where .x/ is the outward unit normal vector on @M at x. It is easy to see that @CSM \ @SM D S.@M/: Given .x; / 2 SM, we denote by x; the unique geodesic with x;.0/D x and P x;.0/D  and let .x; / be the first time when the geodesic x; exits M. We say that .M; g/ is nontrapping if .x; / <1 for all .x; / 2 SM. Definition 1.1. The geodesic ray transform of a function f 2 C1.SM/ is the function If .x; /D Z .x;/ 0 f x;.t/; P x;.t/  dt; .x; / 2 @CSM: Note that if the manifold .M; g/ is nontrapping and has strictly convex boundary, then I WC1.SM/! C1.@CSM/, and Santaló’s formula (see Section 2) implies that I is also a bounded map L2.SM/! L2.@CSM/, where d.x; /D jh.x/; ijd†2n2.x; / and L2.@CSM/ is the space of functions on @CSM with inner product .u; v/L2.@CSM/ D Z @CSM u Nv d: Given f 2 C1.SM/, what properties of f may be determined from the knowledge of If ? Clearly a general function f on SM is not determined by its geodesic ray transform alone, since f depends on more variables than If . In applications one often encounters the transform I acting on special functions on SM that arise from symmetric tensor fields, and we will now consider this case. We denote by C1.Sm.T M// the space of smooth covariant symmetric tensor fields of rank m on M with L2 inner product .u; v/ WD Z M ui1imvi1im dV n; INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1905 where vi1im D gi1j1   gimjmvj1jm . There is a natural map `m W C1.Sm.T M//! C1.SM/ given by `m.f /.x; / WDfx.; : : : ; /. We can now define the geodesic ray transform acting on symmetric m-tensors simply by setting Im WD I ı `m. Let d D r be the symmetric inner differentiation, where r is the Levi-Civita connection associated with g, and  denotes symmetrization. It is easy to check that if v D dp for some p 2 C1.Sm1.T M// with pj@M D 0, then Imv D 0. The tensor tomography problem asks the following question: are such tensors the only obstructions for Im to be injective? If this is the case, then we say I is solenoidal injective or s-injective for short. The problem is wide open for compact nontrapping manifolds with strictly convex boundary (but see [Uhlmann and Vasy 2016; Stefanov et al. 2014]). There are more results if one assumes the stronger condition of being simple, i.e., .M; g/ is simply connected, has no conjugate points and strictly convex boundary. For simple surfaces, the tensor tomography problem has been completely solved [Paternain et al. 2013]. For simple manifolds of any dimension, solenoidal injectivity is known for I0 and I1 [Muhometov 1977; Anikonov and Romanov 1997]. For m-tensors, m 2, the tensor tomography problem is still open, but some substantial partial results were established under additional assumptions; see, e.g., [Pestov and Sharafutdinov 1988; Sharafutdinov 1994; Stefanov and Uhlmann 2005; Paternain et al. 2015a; Stefanov et al. 2014]. Let us explain a bit further the term “solenoidal injective”. Consider the Sobolev spaceHk.Sm.T M// naturally associated with the L2 inner product defined above. By [Sharafutdinov 1994; Sharafutdinov et al. 2005], there is an orthogonal decomposition of L2 symmetric tensors fields. Given v 2Hk.Sm.T M//, k  0, there exist uniquely determined vs 2Hk.Sm.T M// and p 2HkC1.Sm1.T M// such that v D vsC dp; ıvs D 0; pj@M D 0; where ı is the divergence. We call vs and dp the solenoidal part and potential part of v respectively. Moreover, we denote by Hk.Smsol.T M// and C1.Smsol.T M// the subspaces of Hk.Sm.T M// and C1.Sm.T M// respectively whose elements are solenoidal symmetric tensor fields. Solenoidal injec- tivity of Im simply means that Im is injective when restricted to C1.Smsol.T M//. Let I denote the adjoint of I using the L2 inner products defined above; that is, .Iu; '/D .u; I'/ for u 2 L2.SM/, ' 2 L2.@CSM/. A simple application of Santaló’s formula yields I' D ']; where '].x; / WD ' x;..x;//; P x;..x;// (see Section 2 for details). Observe that by definition, '] is constant along orbits of the geodesic flow. If we are now interested in Im, we note that Im D `m ı I and hence we just need to compute `m. This is easy (see Section 2) and one finds Lmf WD `mf .x/i1im WD gi1j1   gimjm Z SxM f .x; /j1    jm dx./: 1906 GABRIEL P. PATERNAIN AND HANMING ZHOU The fundamental microlocal property of the geodesic ray transform is that, for simple manifolds, ImIm is a pseudodifferential operator of order 1 on a slightly larger open manifold engulfing M. Moreover, it has a suitable ellipticity property when acting on solenoidal tensors [Sharafutdinov et al. 2005]. This has been exploited to great effect to derive surjectivity of Im knowing injectivity of Im [Pestov and Uhlmann 2005; Dairbekov and Uhlmann 2010] for mD 0; 1. Since the range of Im is contained in the space of solenoidal tensors, by saying Im is surjective we mean that the range of Im equals the latter. Surjectivity of Im for tensors of order 0 and 1 has been the key for the recent success in the solution of several long standing questions in 2D [Salo and Uhlmann 2011; Pestov and Uhlmann 2005; Paternain et al. 2012; 2013; 2014; Guillarmou 2014]. However, very little is known about surjectivity for m 2 and this largely motivates the present paper. The surjectivity properties of the adjoint of the geodesic ray transform reveal themselves in the existence of solutions f to the transport equation Xf D 0 with prescribed values for Lmf in the space of solenoidal tensors. Here X is the geodesic vector field acting on distributions by duality (recall that X preserves the volume form d†2n1). A distribution f on SM is said to be invariant if it satisfies Xf D 0. As we already mentioned, in this paper we mainly study the relation among the injectivity of Im, the surjectivity of its adjoint Im on solenoidal tensor fields and the existence of some invariant distributions or smooth first integrals associated with solenoidal tensor fields. On a compact nontrapping manifold with strictly convex boundary, the geodesic ray transform Im is extendable to a bounded operator Im WHk.Sm.T M//!Hk.@CSM/ for all k  0 [Sharafutdinov 1994, Theorem 4.2.1]. Moreover, it can be easily checked that Im.H k 0 .S m.T M///Hk0 .@CSM/ and hence we can define Im by duality acting on negative Sobolev spaces to obtain a bounded operator: Im WHk.@CSM/!Hk.Sm.T M//: In other words, for ' 2 Hk.@CSM/, we have Im' is defined by .Im'; u/ D .'; Imu/ for all u 2 Hk0 .S m.T M//. Let C 1˛.@CSM/ denote the set of smooth functions ' for which '] is also smooth. Our main result is the following theorem: Theorem 1.2. Let M be a compact simple Riemannian manifold. Then the following are equivalent: (1) Im is s-injective on C1.Sm.T M//. (2) For every u 2 L2.Smsol.T M//, there exists ' 2H1.@CSM/ such that uD Im'. (3) For every u 2 L2.Smsol.T M//, there exists f 2H1.SM/ satisfying Xf D 0 and uD Lmf . (4) For every u 2 C1.Smsol.T M//, there exists ' 2 C 1˛.@CSM/ such that uD Im'. (5) For every u 2 C1.Smsol.T M//, there exists f 2 C1.SM/ with Xf D 0 such that Lmf D u. We observe that by [Sharafutdinov et al. 2005, Theorem 1.1], s-injectivity of Im on L2.Sm.T M// is equivalent to s-injectivity of Im on C1.Sm.T M//. INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1907 Let us return to the subject of special first integrals associated with the geodesic flow. By considering the vertical Laplacian  on each fiber SxM of SM, we have a natural L2 decomposition L2.SM/DL m0Hm.SM/ into vertical spherical harmonics. We set m WD Hm.SM/ \ C1.SM/. Then a function u belongs to m if and only if uDm.mCn 2/u, where nD dimM. The maps `m W C1.Sm.T M//! Œm=2M kD0 m2k and Lm W Œm=2M kD0 m2k! C1.Sm.T M// are isomorphisms. These maps give natural identification between functions in m and trace-free symmetricm-tensors (for details on this, see [Guillemin and Kazhdan 1980b; Dairbekov and Sharafutdinov 2010; Paternain et al. 2015a]). If .M; g/ is a simple manifold with Im s-injective, Theorem 1.2(5) says that given any u 2 C1.Smsol.T M// there is a first integral of the geodesic flow f such that Lmf D u. In other words, if we let F D L1m u 2 LŒm=2 kD0 m2k and G D f F, we see that F is polynomial of degree m in velocities and it can be completed by adding G to obtain a first integral. We also see that (taking the even or odd part of f if necessary) G 2Lk1mC2k . These were the functions mentioned earlier in the introduction. If G were to be zero, then there would be a first integral that is polynomial in velocities and generically these do not exist. We note that the paper [Paternain et al. 2015a] also constructs invariant distributions (they are not smooth in general) with prescribed m-th polynomial component using a different method (a Beurling transform), but it requires nonpositive curvature for it to work. As already mentioned, here we use instead the normal operator ImIm. The results in [Pestov and Uhlmann 2005; Dairbekov and Uhlmann 2010] prove that (1) implies (4) or (5) in Theorem 1.2 for mD 0; 1, so the main contribution in the theorem is to cover the case m 2 and also to provide additional invariant distributions associated with L2 solenoidal tensors. The proof of Theorem 1.2 relies on a solenoidal extension of tensor fields. For mD 0 no extension is needed and for mD 1 the situation is considerably simpler and an extension result is already available in [Kato et al. 2000]. Paradoxically the need for a solenoidal extension does not arise in the more complicated setting of Anosov manifolds since there is no boundary. In this setting, an analogous result to Theorem 1.2 (in the L2 setting) has been recently proved by C. Guillarmou [2014, Corollary 3.7] and these ideas gave rise to a full solution to the tensor tomography problem on an Anosov surface. Since in 2D the tensor tomography problem has been fully solved [Paternain et al. 2013], we derive: Corollary 1.3. Let .M; g/ be a compact simple surface. For every u 2 C1.Smsol.T M//, there exists f 2 C1.SM/ with Xf D 0 such that Lmf D u. We shall also give an alternative proof of the corollary using results from [Paternain et al. 2015b]. The alternative proof avoids the smooth solenoidal extension and sheds some light on the relationship between the transport equation and the solenoidal condition. 1908 GABRIEL P. PATERNAIN AND HANMING ZHOU The rest of the paper is organized as follows. Section 2 contains some preliminaries. In Section 3 we establish the L2 and C1 compactly supported solenoidal extension of tensor fields. This necessitates at some point the use of the generic nonexistence of nontrivial Killing tensor fields recently proved in [Kruglikov and Matveev 2016]. Section 4 uses the well-established microlocal analysis to prove a surjectivity result for ImIm following the strategy in [Dairbekov and Uhlmann 2010]. Section 5 establishes various boundedness properties on Sobolev spaces that allow us to extend the relevant operators to negative Sobolev spaces (i.e., distributions). Section 6 bundles up everything together and proves Theorem 1.2. Section 7 gives an alternative proof of Corollary 1.3 and clarifies the connection between solenoidal tensors and the transport equation. 2. Preliminaries In this section we provide details about the regularity properties of the operators introduced in the previous section. First we describe the basic notation we will use frequently in the rest of the paper. Given a compact Riemannian manifold M with boundary, we define C1c .Mint/ WD ff 2 C1.M/ W suppf Mintg; Hkc .Mint/ WD ff 2Hk.M/ W suppf Mintg for k 2 Z: Then for any s > 0, s 2 Z, we say H s0 .M/ is the completion of C1c .Mint/ under the H s norm. Now let M be a compact manifold. Given f 2 C1.SM/ and u 2 C1.Sm.T M//, we have .`mu; f /D Z SM uj1jm.x/j1    jmf .x; / d†2n1 D Z M uj1jm.x/ Z SxM f .x; /j1    jm dx./ dV n.x/: This means that Lm D `m W C1.SM/! C1.Sm.T M// is given by Lmf .x/i1im D gi1j1   gimjm Z SxM f .x; /j1    jm dx./: Since the metric tensor g is smooth, for the sake of simplicity, we identify Lmf with its dual, Lmf .x/ j1jm D Z SxM f .x; /j1    jm dx./: On the other hand, it is easy to see that the map `m can be extend to the bounded operator `m WHk.Sm.T M//!Hk.SM/ for any integer k  0. In particular `m.Hk0 .Sm.T M///Hk0 .SM/. Therefore we can define Lm WHk.SM/!Hk.Sm.T M// (1) in the sense of distributions and it is bounded. INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1909 Next, if M is compact nontrapping with strictly convex boundary, we study the properties of I and its adjoint I. Recall a useful integral identity called Santaló’s formula. Lemma 2.1 [Sharafutdinov 1999, Lemma 3.3.2]. Let M be a compact nontrapping Riemannian manifold with strictly convex boundary. For every function f 2 C.SM/, the equalityZ SM f .x; / d†2n1.x; /D Z @CSM d.x; / Z .x;/ 0 f x;.t/; P x;.t/  dt holds. Notice that the definition of compact dissipative Riemannian manifold (CDRM) in [Sharafutdinov 1999] is equivalent to compact nontrapping manifolds with strictly convex boundary. Now let ' 2 C 1˛.@CSM/ and f 2 C1.SM/. By Santaló’s formula, .If; '/D Z @CSM '.x; / d Z .x;/ 0 f x;.t/; P x;.t/  dt D Z @CSM d Z .x;/ 0 '] x;.t/; P x;.t/  f x;.t/; P x;.t/  dt D Z SM ']f d†2n1: Thus I' D '] with I W C 1˛.@CSM/! C1.SM/ bounded. By the proof of [Sharafutdinov 1994, Theorem 4.2.1], one can extend I to a bounded operator I WHk.SM/!Hk.@CSM/ and I.Hk0 .SM//Hk0 .@CSM/ for any integer k 0 (notice that I.C1c ..SM/int//C1c ..@CSM/int/). Thus we can define the bounded operator I WHk.@CSM/!Hk.SM/ (2) in the sense of distributions. Given u2Hk0 .Sm.T M// and ' 2Hk.@CSM/, we have Im' is defined in the sense of distributions: .Im'; u/ WD .I'; `mu/D .'; I ı `mu/D .'; Imu/: Lemma 2.2. Given a compact nontrapping Riemannian manifold M with strictly convex boundary, Im D Lm ı I WHk.@CSM//!Hk.Sm.T M// is a bounded operator. 1910 GABRIEL P. PATERNAIN AND HANMING ZHOU To conclude this section, we briefly discuss X, the generating vector field of the geodesic flow on the unit sphere bundle SM, acting on distributions. Since X is a differential operator on SM, it is obvious that X WHkC1.SM/!Hk.SM/; k  0: For f 2 Hk.SM/ and h 2 HkC10 .SM/ (so Xh 2 Hk0 .SM/), we define Xf 2 Hk1.SM/ in the sense of distributions (notice that the volume form d†2n1 is invariant under the geodesic flow): .Xf; h/ WD .f;Xh/: 3. Solenoidal extensions In the paper [Kato et al. 2000], the authors proved the existence of compactly supported solenoidal extensions of solenoidal 1-forms to some larger manifold in both L2 and smooth cases. Proposition 3.1. Let  be a bounded simply connected domain, with smooth boundary, contained in some Riemannian manifold M. Let U be an open neighborhood of  with @U smooth. Then there exists a bounded map E W L2sol.T /! L2U;sol.T M/ such that Ej D Id. Moreover, E.C1sol .T // C1U;sol.T M/. Here L2U;sol.T M/ and C1U;sol.T M/ denote the subspaces of L2sol.T M/ and C1sol .T M/ respec- tively consisting of elements supported in U. Our goal is to extend this result to symmetric tensor fields of higher rank. However, for tensor fields of higher rank, new ideas are required and the argument is more involved. L2 solenoidal extensions. We first prove the extension in the L2 category by solving a suitable elliptic system. Proposition 3.2. Let  be a bounded simply connected domain, with smooth boundary, contained in some Riemannian manifold .M; g/. Let U be an open neighborhood of  with @U smooth. Then given m  2, K  2 and  > 0, there exist a Riemannian metric Qg and a bounded map E W L2.Smsol.T g//! L2.SmU;sol.T  QgM// such that k QggkCK < , Qgj D g and Ej D Id. Proof. Suppose u 2L2.Smsol.T g//, i.e., ıuD 0 in the sense of distributions. By the Green’s formula for symmetric tensor fields (see [Sharafutdinov 1994]) one can define the boundary contraction of u with the outward unit normal vector  on @ in the sense of distributions; i.e., for v 2H 1.Sm1.T g// we have .u; dv/ D .ju; v/@: (3) Since the trace operator T W H 1.Sm1.T g// ! H 1=2.Sm1.@T g//, T v D vj@, is surjective, ju 2H1=2.Sm1.@T g// is well-defined, and in local coordinates .ju/i1i2im1 D ui1i2im1j j: By (3), for v 2H 1.Sm1.T g// with dv D 0 (Killing tensor fields on ), we have .ju; v/@ D 0. INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1911 It is known that generic (in the CK-topology for K  2) metrics admit only trivial integrals polynomial in momenta [Kruglikov and Matveev 2016]; i.e., for a generic metric h, the only Killing tensor fields are of the form chk, where c 2 R and hk D .h˝   ˝„ ƒ‚ … k h/ is the symmetric tensor product of k copies of h. Thus given any  > 0 and K  2, there is a smooth metric Qg with k Qg gkCK <  and Qgj D g so that .U n; Qg/ (thus .U; Qg/) does not have nontrivial Killing tensor fields. Define f D ju on @; 0 on @U: Let D WD U n and consider the following boundary value problem for systems of second-order partial differential equations: 8ˆ<ˆ : ıdw D 0 in D; jdw D f 2H1=2.Sm1.@T QgD//; w 2H 1.Sm1.T QgD//: (4) Here  is the outward unit normal vector on @D for D; notice j@ D. We claim that the system (4) is a regular elliptic system (also called coercive in some texts). Assume that the claim is true for the moment and let us continue the proof. Next, we study the solutions of the homogeneous problem. Let ıdv D 0 and jdvj@D D 0 for some v 2H 1.Sm1.T QgD//; by ellipticity, v is smooth. Applying Green’s formula, one hasZ D hdv; dvi dV n.x/D Z D hıdv; vi dV n.x/C Z @D hjdv; vi dV n.x/D 0; i.e., dv  0. So the solution set of the homogeneous problem is KD ˚v 2 C1.Sm1.T QgD// W dv  0 ; the set of Killing tensor fields of rank m 1 on D. Now by [McLean 2000, Theorem 4.11], (4) is solvable in H 1.Sm1.T QgD// for the given boundary condition f if and only if .v; f /@D D 0 for all v 2 K. Note that .D; Qg/ does not have nontrivial Killing tensor fields. If m is even, the only Killing .m1/-tensor field is vD 0; then .v; f /@D D .0; f /@D D 0. If m is odd, the Killing .m1/-tensor fields in D are of the form v D c Qg.m1/=2jD . Thus we can extend v to v D c Qg.m1/=2jU, which is also a Killing tensor field in . By the definition of f , .v; f /@D D.v; ju/@ D.v; ıu/ .dv; u/ D 0; since ıuD 0, dv D 0 in . Thus the system (4) is solvable. Let w 2H 1.Sm1.T QgD// be a solution of (4) (the set of all solutions is wCK) and define EuD 8<: u in ; dw in D; 0 in MnU: 1912 GABRIEL P. PATERNAIN AND HANMING ZHOU It is easy to see that Eu 2 L2.Sm.T QgM// and supp Eu U. In particular, for v 2H 1.Sm1.T QgM//, .ıEu; v/M D.Eu; dv/M D.dw; dv/D .u; dv/ D.jdw; v/@D .ju; v/@ D.ju; v/@ .ju; v/@ D 0: Thus Eu is solenoidal in the sense of distributions, and Eu 2 L2.SmU;sol.T QgM//. Moreover, by [McLean 2000, Theorem 4.11], we have the stability estimate kEuk2 L2.M/ D kuk2L2./Ckdwk2L2.D/  kuk2L2./CCkjuk2H1=2.@/  C 0kuk2L2./; i.e., E is bounded.  The only thing left to prove is the claim about ellipticity. Lemma 3.3. The system (4) above is a regular elliptic system. Proof. It is well known that ıd is a self-adjoint elliptic operator; see, for example, [Sharafutdinov 1994]. We just need to show that the Neumann boundary value problem satisfies the Lopatinskii condition. To check the Lopatinskii condition, we follow a similar procedure to that in the proof of [Sharafutdinov 1994, Theorem 3.3.2]. We choose local coordinates .x1; x2; : : : ; xn1, xnD t  0/ in a neighborhood W of x0 D .x0; 0/ 2 @D in D so that @D\W D ft D 0g and gij .x0/D ıij . Define d0 D pd and ı0 D pı, the principal symbols of d and ı respectively. Then we need to show that the boundary value problem for systems of ordinary differential equations( ı0.x 0; 0;  0;Dt /d0.x0; 0;  0;Dt /w.t/D 0; j @ @t d0.x 0; 0;  0;Dt /w.t/jtD0 D f0 has a unique solution in NC for all  0 2 Rn1nf0g and f0 2 Sm1.Rn/, symmetric .m1/-tensors on Rn. HereDt Did=dt , and for the sake of simplicity, we drop the space variables .x0; 0/ from the symbols so NC WD ˚ w 2 Sm1.Rn/jfx0gŒ0;1/ W ı0. 0;Dt /d0. 0;Dt /w D 0 and w decays rapidly together with all derivatives as t !C1 : Since the equation det ı0. 0; /d0. 0; / D 0 has real coefficients with no real root for  0 ¤ 0, it is not difficult to see that dimNCD dimSm1.Rn/. Thus it is sufficient to show that the homogeneous problem( ı0. 0;Dt /d0. 0;Dt /w.t/D 0; j @ @t d0. 0;Dt /w.t/jtD0 D 0 (5) has only the zero solution in NC. By a similar computation to that in the proof of [Sharafutdinov 1994, Theorem 3.3.2], we have the following Green’s formula. Let v.t/ 2 C1Œ0;1/! Sm.Rn/ and w.t/ 2 C1Œ0;1/! Sm1.Rn/ INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1913 such that both of them decay rapidly together with all derivatives as t !C1. If j @ @t v.0/D 0 (notice that different from [Sharafutdinov 1994], here we use the Neumann boundary condition at t D 0) thenZ 1 0 hı0. 0;Dt /v; wi dt D Z 1 0 hv; d0. 0;Dt /wi dt: (6) Now if w.t/ 2NC is a solution to (5), let v.t/D d0. 0;Dt /w.t/. By (6) we obtain d0. 0;Dt /w.t/D 0: Notice that .d0./w/i1im D im mX kD1 ikwi1 yik im ; where the y over ik means this index is omitted. Let im D n and  D . 0;Dt /. We obtain the system of first-order ordinary differential equations .d0. 0;Dt /w/ni1im1 D im  .`C 1/Dtwi1im1 C X ik¤n ikwni1 yik im1  D 0; where ` D `.i1; : : : ; im1/ is the number of occurrences of the index n in .i1; : : : ; im1/. Since limt!C1w.t/D 0, by induction on `, the only solution to the above first-order homogeneous system is w  0, and this shows that (4) satisfies the Lopatinskii condition.  Smooth solenoidal extensions. In this subsection we achieve C1 solenoidal extensions for tensors of arbitrary rank. Observe that the approach we use is quite different from the one of [Kato et al. 2000]. Proposition 3.4. Let  be a bounded connected domain, with smooth boundary, contained in some Riemannian manifold .M; g/. Let U be an open neighborhood of  with @U smooth. Then given m 2, K  2 and  > 0, there exist a Riemannian metric Qg and a bounded map E WHk.Smsol.T g//! L2.SmU;sol.T  QgM// for some integer k  2 such that k QggkCK < , Qgj D g, Ej D Id and E.C1.Smsol.T g/// C1.SmU;sol.T QgM//: To prove the proposition, we start with the following lemma on the existence of solenoidal extensions that might not be compactly supported. Lemma 3.5. Let  be a bounded connected domain, with smooth boundary, contained in some Riemann- ian manifold .M; g/. There exists an open neighborhood U of  such that every u 2 C1.Smsol.T // can be extended to Qu 2 C1.Smsol.T U// with Quj D u. Proof. Let u 2 C1.Smsol.T //, i.e., ıuD 0, in local coordinates uD uj1jmdxj1 ˝   ˝ dxjm and .ıu/i1im1 D gjkrjuki1im1 D 0; (7) where rjuki1im1 D @juki1im1 €`jku`i1im1 m1X sD1 €`j isu`ki1 yis im1 : (8) 1914 GABRIEL P. PATERNAIN AND HANMING ZHOU Pick x0 2 @. We follow the idea of the proof in [Stefanov and Uhlmann 2005, Lemma 4.1] and choose semigeodesic coordinates .x1; : : : xn1; xn/D .x0; xn/ near x0 with @D fxnD 0g and @nD  the unit outward (with respect to ) vector normal to @; thus gkn D ıkn ; €nkn D €knn D 0 for all k D 1; 2; : : : ; n: We extend the components uj1jm , js < n for all 1  s  m, smoothly to U (note that U n is determined by the semigeodesic neighborhood of @), and denote the extensions by vj1jm . We will construct the other components in fxn > 0g by induction on the number of appearances of n in j1 : : : jm. By equations (7) and (8), if i1; : : : ; im1 < n, @nvni1im1 m1X sD1 X ` 0 and K  2, there is a smooth metric Qg with k QggkCK < and QgjV Dg so that .D; Qg/ does not have nontrivial Killing tensor fields. Now if m is even, the only Killing .m1/-tensor field on .D; Qg/ is v D 0. Then .v; f /D D .0; f /D D 0: If m is odd, Killing .m1/-tensor fields on .D; Qg/ are of the form v D c Qg.m1/=2jD . Thus we can extend v to v D c Qg.m1/=2jU, which is also a Killing tensor field in . By Green’s formula, .v; f /D D .v; ıw/D D.dv;w/DC .v; jw/@D D.v; ju/@ D.v; ıu/ .dv; u/ D 0: since ıuD 0 and dv D 0 in . Here D is the unit outward normal vector on @D and .jw/i1i2im1 D wi1i2im1jj: Now by [Delay 2012, Theorem 1.3], there exist uD 2C1.Sm.T M// with suppuD U n such that ıuD Df . It is not difficult to check that the symmetric differentiation d satisfies the kernel restriction condition (KRC) and the asymptotic Poincaré inequality (API) of [Delay 2012]. We define EuDwCuD . Then ıEuD ıwC ıuD D f f D 0; i.e., Eu 2 C1.SmU;sol.T QgM//. Moreover, Euj D u. The argument above gives a construction for compactly supported smooth solenoidal extensions. One can further check that the extension can be constructed in a stable way. In view of the ODEs (9), the solution is controlled by the initial value and the nonhomogeneous term on the right side under Sobolev norms; see, e.g., [Han 2011]. By induction on the number of appearances of n and repeatedly differentiating (9), we have that kuV kH1.V n/  C  kjukHk1 .@/C X is0 independent of u. Since C1.Smsol.T // is dense inHk.Smsol.T // under theHk norm, we can extend E to a bounded map fromHk toL2 with the same properties, which completes the proof.  Remark 3.6. We expect that the L2 norm of Eu can be bounded by the L2 norm of uj through sharper estimates, similar to the result under the L2 setting in the previous subsection. However, the Hk space is enough for carrying out the argument under the smooth setting in the next section; see Lemma 4.3. 4. Surjectivity of the normal operator ImIm Since M is simple we can consider an extension zM of M which is open ( zM D zM int) and whose compact closure is also simple. It is well known that the normal operator N D ImIm is a pseudodifferential operator of order 1 on zM; see, for example, [Sharafutdinov 1994; Stefanov and Uhlmann 2004; 2008; Sharafutdinov et al. 2005]. Below is a lemma that, roughly speaking, gives a right parametrix for N on the space of solenoidal tensor fields. The proof is similar to [Sharafutdinov et al. 2005, Theorem 3.1]. Lemma 4.1. Let S be a parametrix for the operator ıd . There exists a pseudodifferential operator Q of order 1 on the bundle of symmetric m-tensor fields Sm.T  zM/ such that E DNQC dSıCK; (10) where E is the identity operator and K is a smoothing operator. Proof. Let ./ be the principal symbol of the pseudodifferential operator N and Sm .T  x zM/D fu 2 Sm.T x zM/ W juD 0g; where j Dip.ı/ W Sm.T x zM/! Sm1.T x zM/. By [Sharafutdinov 1994, Theorem 2.12.1], ./ W Sm .T x zM/! Sm .T x zM/ is an isomorphism for  ¤ 0. Thus there exists p./ such that ./p./D Id on Sm  .T x zM/. Namely, we can find some pseudodifferential operator P of order 1 such that on Sm  .T x zM/, NP DE B for some operator B of order 1. Now multiplying both sides by the “solenoidal projection” E dSı, which is of order 0, one has NP.E dSı/DE dSıR (11) defined on Sm.T  zM/. INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1917 Then we multiply both sides of (11) by ı to get ıR D R0 with R0 some smoothing operator. Let C DP1kD0Rk, which is a pseudodifferential operator of order 0 and a parametrix forER. Write (11) as NP.E dSı/C dSı DE R; and multiply both sides by C to get NP.E dSı/C C dSıC dSı 1X kD1 Rk D .E R/C DECR00; with R00 a smoothing operator. Since ıR is smoothing, dSı P1 kD1Rk is smoothing too. We arrive at the equation NP.E dSı/C C dSıCK DE; where K is a smoothing operator. Denote P.E dSı/C by Q (note that one can make Q properly supported). Then we get (10), which finishes the proof.  Let U be a small open neighborhood of M in zM. Denote the restriction operator from zM to M by rM. Then the following holds: Lemma 4.2. Suppose M is a compact simple Riemannian manifold, and assume Im is s-injective on C1.Sm.T M//. Then the operator rMN WH1c .Sm.T  zM//! L2.Smsol.T M// is surjective. Note that elements in H1c .Sm.T  zM// are defined in the sense of distributions, which are compactly supported in zM. Proof. We adopt the approach of [Dairbekov and Uhlmann 2010] for showing the surjectivity of N on 1-forms. By Lemma 4.1, NQuD uCKu for all u 2 L2c.Smsol.T  zM// with K a smoothing operator on zM. Since the simplicity is stable under small C 2-perturbations of the metric g, by Proposition 3.2, we perturb the metric of zMnM a little bit (still denoted by g) so that under the new metric zM is still simple and there exists a bounded operator E W L2.Smsol.T M//! L2.SmU;sol.T  zM// such that on L2.Smsol.T M//, rMNQE DEC rMKE : Since K is a smoothing operator, rMKE is compact on L2.Smsol.T M//, which implies that EC rMKE has closed range and finite codimension. Thus we have rMNQE W L2.Smsol.T M//! L2.Smsol.T M// has closed range and finite codimension. By the inclusion relation rMNQE.L2.Smsol.T M/// rMN.H1c .Sm.T  zM/// L2.Smsol.T M//; 1918 GABRIEL P. PATERNAIN AND HANMING ZHOU the intermediate space rMN.H1c .Sm.T  zM/// is also closed in L2.Smsol.T M//. Thus it suffices to show that the adjoint .rMN/ is injective, which will imply the surjectivity of rMN. For L2 symmetric m-tensor fields, we have the decomposition L2.Sm.T M//D L2.Smsol.T M//˚L2.SmP .T M//; (12) where L2.SmP .T M// is the potential part. Thus the dual operator of rMN is .rMN/  W L2.Smsol.T M//! .H1c .Sm.T  zM///: For u 2 L2.Smsol.T M// and v 2 H1c .Sm.T  zM//, if we denote by E0u the extension of u to zM by zero (note that generally E0u is not solenoidal on zM ), we have ..rMN/ u; v/D .u; rMNv/D .E0u;Nv/D .NE0u; v/; i.e., .rMN/ DNE0. Therefore given u 2 L2.Smsol.T M//, if NE0uD 0, then 0D .NE0u; E0u/D kImE0uk2L2.@CS zM/ D) ImE0uD 0: Since E0uD 0 outside M and zM is simple, this implies ImuD 0: By [Sharafutdinov et al. 2005, Theorem 1.1], u is smooth and ıu D 0. The s-injectivity assumption implies uD 0. This completes the proof of the lemma.  Next we prove the lemma in the smooth setting: Lemma 4.3. Suppose M is a compact simple Riemannian manifold, and assume Im is s-injective on C1.Sm.T M//. Then the operator rMN W C1c .Sm.T  zM//! C1.Smsol.T M// is surjective. Proof. By Lemma 4.1, NQuD uCKu for all u 2 C1c .Smsol.T  zM// with K a smoothing operator on zM. Since the simplicity is stable under small C 2-perturbations of the metric g, by Proposition 3.4, we perturb the metric of zMnM a little bit (still denoted by g) so that under the new metric zM is still simple and there exists a bounded operator E WHk.Smsol.T M//! L2.SmU;sol.T  zM// for some integer k  2 with E.C1.Smsol.T M/// C1.SmU;sol.T  zM// such that on Hk.Smsol.T M//, rMNQE DEC rMKE : Now the argument of [Dairbekov and Uhlmann 2010, Lemma 2.2] can be applied to tensors of any order to finish the proof.  INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1919 Remark 4.4. One can actually prove Lemmas 4.2 and 4.3 just by applying Lemma 3.5. Given a smooth solenoidal tensor u onM, by Lemma 3.5 we first extend it to a smooth solenoidal tensor Qu on an arbitrarily small open neighborhood U; then we extend Qu smoothly to zM with compact support, denoted by Eu. Note that generally Eu is not solenoidal. Since the Schwartz kernel of the parametrix S of ıd is smooth away from the diagonal  zM zM , we can choose S to make the support of its Schwartz kernel sufficiently close to  zM zM so that dS ıEuD 0 in an open neighborhood of M. This implies that rM dS ıEuD 0, i.e., rMNQEuD uC rMKEu. It also works for L2 solenoidal tensors. On the other hand, the original proof of [Dairbekov and Uhlmann 2010, Lemma 2.2] uses the existence of compactly supported solenoidal extensions of solenoidal 1-forms one more time at the very end to show that the adjoint .rMN/ is injective. However, one can also avoid this. Notice that given a 1-form f in the kernel of .rMN/, by [Dairbekov and Uhlmann 2010, equation (2.33)], f D dp for some distribution p on zM with sing suppp  @M and pj @ zM D 0. Moreover, since suppf M, we have dp D 0 outside M. As p is smooth outside M and pD 0 on @ zM, strict convexity of @M implies p 0 in zMnM. Now given a smooth solenoidal 1-form u on M, by Lemma 3.5 let Eu be the smooth compactly supported extension of u to zM which is solenoidal in a small open neighborhood (¤ zM/ of M. Since the supports of ıEu and p are disjoint, we have .f; Eu/D .dp; Eu/D .p; ıEu/D 0; which implies that f D 0, i.e., .rMN/ has trivial kernel. The argument works for tensors of arbitrary rank. At this point, we see that one can prove the surjectivity of rMN just using Lemma 3.5, without the need of knowing the generic absence of nontrivial Killing tensors [Kruglikov and Matveev 2016]. However, a perturbation of the metric seems still necessary so far for the proof of the existence of compactly supported solenoidal extensions, and Propositions 3.2 and 3.4 may find their applications in other areas. 5. Analysis of the adjoint Im Before proving the main result, we need to extend the definition of the geodesic ray transform Im so that it acts on negative Sobolev spaces. To this end, we will study the regularity property of the adjoint of the geodesic ray transform, Im. As discussed in the Introduction, given M a compact nontrapping manifold with strictly convex boundary, the operator Im W C 1˛.@CSM/ ! C1.Sm.T M// is the product of two operators, i.e., Im D Lm ı I. We instead study the regularity properties of I and Lm. We start with the latter. Lemma 5.1. Given a compact Riemannian manifold M (with or without boundary), the operator Lm WHk.SM/!Hk.Sm.T M// is bounded for every integer k  0. Proof. Our purpose is to show that there exists a constant C > 0 such that for any w 2Hk.SM/, the following holds: kLmf kHk  Ckf kHk : (13) 1920 GABRIEL P. PATERNAIN AND HANMING ZHOU Since M is compact, by a partition of unit, it suffices to show the above inequality in local charts. Let U be a domain in SM with local coordinate system .z1; : : : ; z2n1/. We assume suppf  U. Let V be a domain in M with local coordinate system .x1; : : : ; xn/, and be a smooth function with support in V . We will show k Lmf kHk.Sm.T V //  Ckf kHk.U /: By the definition of the Hk norm of tensors, we only need to show the above inequality is true for each component of the tensor. We start with f 2 C1.SM/ with support in U ; then Lmf is also smooth. Let J D .j1    jm/ and J WD j1    jm . Then D˛x  .x/Lmf .x/ J  DD˛x  .x/ Z SxM f .x; /J dx./  DD˛x  .x/ Z Sn1 f .x; .x; //J .x; /P.x; / d./  D X ˛1C˛2C˛3D˛ D˛1x .x/ Z Sn1 D˛2x f .x; .x; // D˛3x  J .x; /P.x; /  d./ D X ˛1C˛2C˛3D˛ D˛1x .x/ Z SxM D˛2x f .x; / D˛3x  JP.x; .x; //  P 0.x; / dx./: (14) Here P and P 0 are corresponding Jacobians. For j˛j  k, according to (14), kD˛x Œ .x/Lmf .x/J k2L2.V /  X ˇ˛ Cˇ;˛ Z V Z SxM jDˇx f .x; /j2 dx./ dx  X j jj˛j C ;˛ Z U jD z f .z/j2 dz  Ckf k2Hk.U /: Thus the estimate (13) is proved when w 2 C1.SM/. For f 2Hk.SM/, since C1.SM/ is dense in Hk.SM/, by an approximation argument, it is easy to show that Lmf 2Hk.Sm.T M// and the estimate (13) holds too. This proves the lemma.  Now we turn to the analysis of the operator I, which basically is an invariant extension, along the geodesic flow, of functions on @CSM to functions on SM. It is well known that given ' 2 C1.@CSM/, '] D I.'/ is not necessarily in C1.SM/. The following subspace of C1.@CSM/ has already been considered in the Introduction: C 1˛.@CSM/ WD ˚ ' 2 C1.@CSM/ W '] 2 C1.SM/ : In particular, by [Pestov and Uhlmann 2005, Lemma 1.1], if M is compact nontrapping with strictly convex boundary, C 1˛.@CSM/D ˚ ' 2 C1.@CSM/ W A' 2 C1.@SM/ INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1921 where A'.x; /D  '.x; /; .x; / 2 @CSM; '. x;..x;//; P x;..x;///; .x; / 2 @SM: Since A' is smooth in both .@CSM/int and .@SM/int, the singularities can only come from S.@M/. We introduce the space Hk˛ .@CSM/, k  0, to be the completion of C 1˛.@CSM/ under the Hk norm. Obviously H 0˛ .@CSM/ D L2.@CSM/. It is easy to show that C1c ..@CSM/int/  C 1˛.@CSM/ (this is from the fact that @CSM is compact and the boundary @M is strictly convex), which implies that Hk0 .@CSM/Hk˛ .@CSM/. Lemma 5.2. Given a compact nontrapping manifold M with strictly convex boundary, the operator I WHk˛ .@CSM/!Hk.SM/ is bounded for any integer k  0. Proof. The idea is similar to the proof of Lemma 5.1. First we consider the case ' 2 C 1˛.@CSM/; thus '] 2 C1.SM/. Let U be a domain in @CSM with local coordinate systems .y1; : : : ; y2n2/. We assume supp'  U. Let V be a domain in SM with local coordinate systems .z1; : : : ; z2n1/, and be a smooth function with support in V . Since M is compact, it suffices to show k ']kHk.V /  Ck'kHk.U /: Since D˛z Œ .z/' ].z/D X ˇC D˛ D z .z/ Dˇz '].z/; we obtain that for j˛j  k, D˛z Œ .z/'].z/ 2L2.V / X ˇ˛ Cˇ;˛ Z V jDˇz '].z/j2 dz: Now let D D f.y; t/ W y 2 @CSM; 0  t  .y/g be a closed domain in @CSM R. Define the map ‰ WD! SM by z D‰.y; t/D y.t/; P y.t/. By [Sharafutdinov 1994, Lemma 4.2.2],Z V jDˇz '].z/j2 dz  X j jCsDjˇ j Cˇ;;s Z U Z .y/ 0 ˇˇ DyD s t ' ].z.y; t// ˇˇ2 ˇˇ˝ .y/; .x.y// ˛ˇˇ dt dy D X j jDjˇ j Cˇ; Z U Z .y/ 0 jDy '].y; t/j2 dt d.y/ .since DstDy '] DDyDst ']/ D X j jDjˇ j Cˇ; Z U .y/jDy '.y/j2 d.y/  X j jDjˇ j C 0ˇ; Z U jDy '.y/j2 d.y/ Ck'k2Hk.U /: Therefore, k']kHk.SM/  Ck'kHk.@CSM/ for ' 2 C 1˛.@CSM/. 1922 GABRIEL P. PATERNAIN AND HANMING ZHOU If ' 2Hk˛ .@CSM/, since C 1˛.@CSM/ is dense in Hk˛ .@CSM/, by an approximation argument, it is easy to show that '] 2Hk.SM/ and the operator I is bounded, which proves the lemma.  Combining the two lemmas above, we obtain the desired regularity property of Im. Proposition 5.3. Given a compact nontrapping Riemannian manifold M with strictly convex boundary, the adjoint operator of the geodesic ray transform on symmetric m-tensors Im D Lm ı I WHk˛ .@CSM/!Hk.Sm.T M// is bounded for any integer k  0. Now we can extend the definition of the geodesic ray transform so that it acts on .Hk.Sm.T M/// (the dual space is with respect to the L2 inner product) for integers k  1. Let u 2 .Hk.Sm.T M/// and ' 2Hk˛ .@CSM/. We define Imu in the sense of distributions: .Imu; '/ WD .u; Im'/: (15) By Proposition 5.3, the right-hand side of (15) is well-defined. We derive the following corollary: Corollary 5.4. Given M, a compact nontrapping manifold with strictly convex boundary, the operator Im W .Hk.Sm.T M///! .Hk˛ .@CSM// defined by (15) is bounded. Here the dual space .Hk˛ .@CSM// is also with respect to the L2 inner product. Note Hk0 .@CSM/ Hk˛ .@CSM/; thus .Hk˛ .@CSM// Hk.@CSM/. On the other hand, since C1.Sm.T M// is dense in Hk.Sm.T M// under the Hk-norm, it is clear that Hkc .Sm.T M int// .Hk.Sm.T M///; we will use the weaker map in the next section: Im WHkc .Sm.T M int//!Hk.@CSM/: (16) 6. Proof of Theorem 1.2 Now we are in a position to prove our main theorem. We start by showing that (1), (2) and (3) are equivalent. Proof. .1/ ) .2/: Since M is simple, given u 2 L2.Smsol.T M//, by Lemma 4.2, there exists v 2 H1c .Sm.T  zM// such that rM ImImv D u. Then (16) implies the existence of some Q' D Imv 2 H1.@CS zM/ such that uD rM Im Q'. For w 2H 10 .Sm.T M//, we define the distribution ' acting on Im.H 1 0 .S m.T M/// by .'; Imw/ WD . Q'; Im Qw/D .Im Q'; Qw/; where Qw 2H 10 .Sm.T  zM// is the extension of w which is zero outside M. We claim that there exists C > 0 such that j.'; Imw/j  CkImwkH1 INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1923 for all w 2H 10 .Sm.T M//. Assuming the claim, note that Imw 2H 10 .@CSM/ and by the Hahn–Banach theorem, ' can be extended to a bounded linear functional on H 10 .@CSM/, still denoted by ', i.e., ' 2H1.@CSM/. By the definition of ', j.'; Imw/j D j. Q'; Im Qw/j  CkIm QwkH1 : Therefore to prove the claim, it suffices to show that kIm QwkH1.@CS zM/  CkImwkH1.@CSM/ (17) for some C > 0. Assume at this point that inequality (17) holds and let us continue with the proof. Now '2H1.@CSM/ is well-defined. Letw 2H 10 .Sm.T M//, and let Qw be the extension ofw into zM which is zero outsideM, so Qw 2H 10 .Sm.T  zM//. Then .rM I  m Q';w/D .Im Q'; Qw/D . Q'; Im Qw/D .'; Imw/D .Im';w/: Thus uD rM Im Q' D Im'. (The choice of ' is not unique.) .2/) .3/: Given u 2L2.Smsol.T M//, by the assumption, there is ' 2H1.@CSM/ such that uD Im'. Since Im D Lm ı I, we define f D I'; then f 2 H1.SM/ and u D Lmf . Furthermore, given h 2H 20 .SM/, .Xf; h/D .f;Xh/D .I';Xh/D .';I.Xh//D 0; i.e., Xf D 0. .3/) .1/: Assume Imu D 0 for some u 2 C1.Smsol.T M//. Then it is well known that there exists h 2 C1.SM/ with hj@SM D 0 such that XhD`mu: Moreover, by [Sharafutdinov 2002, Lemma 2.3] there exists p 2C1.Sm1.T M// with pj@M D 0 such that uj@M D dpj@M . When mD 0, this just means uj@M D 0. Calculations in local coordinates show that X.`m1p/D `mdp. Thus we obtain X.hC `m1p/D`m.u dp/; with .hC `m1p/j@SM D 0. Under the projection  W SM !M, the pullback of the unit normal vector  to @M is the unit normal vector  to @SM, and in local coordinates X D  i @ @xi € ijkj k @ @ i ; where € i jk are the Christoffel symbols. By taking the boundary normal coordinates .x0; xn/ near x 2 @M (so .x/ D .x; / D @=@xn), together with the fact that .hC `m1p/j@SM D 0, we obtain that for .x; / 2 @SM , 0D`m.u dp/.x; /DX.hC `m1p/.x; /D n@xn.hC `m1p/.x; /: 1924 GABRIEL P. PATERNAIN AND HANMING ZHOU The first equality comes from the fact udpj@M D 0. Thus @.hC `m1p/.x; /D 0 for all  … Sx@M. But since h and p are smooth, and the measure of Sx@M is zero on SxM, we get @.hC`m1p/.x; /D0 for all  2 SxM, so hC `m1p 2H 20 .SM/. On the other hand, there exists f 2H1.SM/ with Xf D 0 such that uD Lmf . It follows that 0D .Xf; hC `m1p/D .f;X.hC `m1p//D .f; `m.u dp//D .Lmf; u dp/D kuk2; where the last equality comes from the fact that u is orthogonal to dp. Thus uD 0, which implies the s-injectivity.  Remark 6.1. By carrying out an argument similar to the one of [Stefanov and Uhlmann 2005, Lemma 4.1], one can actually show that there exists p 2 C1.Sm1.T M// with pj@M D 0 such that @kuj@M D @kdpj@M for all integers k  0. When mD 0, this means the boundary jet of u is zero, i.e., @kuj@M D 0 for all k  0. Note that [Stefanov and Uhlmann 2005] only considers the case that u is a symmetric 2-tensor field, but the proof works for tensors of any rank. On the other hand, given @kuj@M D @kdpj@M , one should be able to prove that hC `m1p 2HkC20 .SM/ for all k  0, i.e., hC `m1p also has zero boundary jet. However, for our purposes k D 0 is enough. The thing left to prove is the inequality (17). Actually the Hk norms of Imw and Im Qw are equivalent for arbitrary k  0, provided that w is in Hk0 .Sm.T M//. A simple calculation shows that kIm Qwk2L2 D . Qw;ImIm Qw/D.w;rM ImIm Qw/D.w;ImImw/DkImwk2L2 . We assume @ zM and @M are sufficiently close. Lemma 6.2. Let M be a compact nontrapping manifold with strictly convex boundary. Given w 2 Hk0 .S m.T M//, k  1, let Qw 2Hk0 .Sm.T  zM// be the extension of w to zM by zero. Then there exists C > 1 such that 1 C kImwkHk.@CSM/  kIm QwkHk.@CS zM/  CkImwkHk.@CSM/: (18) Proof. We only need to show (17), which is half of (18). Since @M and @ zM are close, we can assume the closure of zM is still compact nontrapping with strictly convex boundary. Given a geodesic x; on M determined by .x; / 2 @CSM, we can uniquely extend it to a geodesic y; on zM determined by .y; / 2 @CS zM. It is not difficult to see that the map T W @CSM ! @CS zM; with T .x; /D .y; /; is a diffeomorphism from @CSM onto its image T .@CSM/. On the other hand, by the definition of Qw, Imw.x; /D Im Qw.T .x; //D Im Qw.y; / and Im Qw.y; /D 0 for .y; / 2 @CS zMnT .@CSM/. Since @CSM and @CS zM are compact, similar to the proofs of Lemmas 5.1 and 5.2, we will work in local charts. Let U be a domain in @CS zM with local coordinates . Qz1; : : : ; Qz2n2/ and ' be a smooth function on @CS zM with supp'  U. In the mean time, there is a domain V in @CSM with local coordinates .z1; : : : ; z2n2/ such that T 1.U \T .@CSM// V , and is a smooth function on @CSM with T 1.U \ T .@CSM//  supp  V and  1 on T 1.U \ T .@CSM//. We first consider the case w 2 C1c .Sm.T M/int/ and show that there exists C > 0 such that k'  Im QwkHk.U /  Ck  ImwkHk.V /: INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1925 Notice that for j˛j  k, D˛Qz Œ'  Im QwD X ˇC D˛ D Qz ' DˇQz Im Qw: Thus D˛Qz Œ'  Im Qw 2L2.U / X ˇ˛ Cˇ;˛ Z U jDˇQz Im Qwj2 d Qz D X ˇ˛ Cˇ;˛ Z U\T.@CSM/ jDˇQz Im Qw. Qz/j2 d Qz  X j jj˛j C;˛ Z T1.U\T.@CSM// ˇˇ Dz Im Qw.T .z// ˇˇ2 J dz  C 0 X j jj˛j Z T1.U\T.@CSM// ˇˇ Dz .  Imw/.z/ ˇˇ2 dz  C 0 X j jj˛j Z V ˇˇ Dz .  Imw/.z/ ˇˇ2 dz  Ck  Imwk2Hk.V /; where J is the Jacobian related to the diffeomorphism T. Therefore kIm QwkHk.@CS zM/  CkImwkHk.@CSM/ for w 2 C1c .Sm.T M/int/. Now for w 2 Hk0 .Sm.T M//, there is a sequence wk 2 C1c .Sm.T M/int/, k D 1; 2; : : : , which converges to w in the Hk norm. Then it is not difficult to see that the sequence Qwk 2 C1c .Sm.T  zM// converges to Qw 2Hk0 .Sm.T  zM//. By the boundedness of the operator Im, we know Imwk and Im Qwk converge to Imw and Im Qw respectively in the Hk norm. This implies that above estimates are valid for any w 2Hk0 .Sm.T M//.  The following proposition that holds on compact nontrapping manifolds with strictly convex boundary shows that items (4) and (5) in Theorem 1.2 are equivalent and any of them implies item (1). Proposition 6.3. Let M be a compact nontrapping Riemannian manifold with strictly convex boundary and let u 2 C1.Smsol.T M//. The following are equivalent: (i) There exists ' 2 C 1˛.@CSM/ such that uD Im'. (ii) There exists f 2 C1.SM/ satisfying Xf D 0 and uD Lmf . Either of these two conditions implies s-injectivity of Im. Proof. .i/ ) .ii/: By the assumption, there is ' 2 C 1˛.@CSM/ such that u D Im' D Lm ı I'. Define f D I' D '] 2 C1.SM/ (since ' 2 C 1˛.@CSM/); then uD Lmf . Moreover, it is clear that Xf DX'] D 0 by definition. .ii/) .i/: If there exists f 2 C1.SM/ with Xf D 0, this implies that f D I.f j@CSM /. We define ' D f j@CSM 2C1.@CSM/. However, since ']D f 2C1.SM/, we know ' actually sits in the space C 1˛.@CSM/. By the assumption, uD Lmf D Lm ı I' D Im'. 1926 GABRIEL P. PATERNAIN AND HANMING ZHOU The argument that shows that any of these conditions imply s-injectivity of Im is even easier than the proof that (3) implies (1) in Theorem 1.2 since we do not have to worry about paring Xf with an element in H 20 .SM/. Assuming (ii), integration by parts yields right away that 0D .Xf; h/D .f;Xh/D .f; `m.u//D .Lmf; u/D kuk2:  Finally we show that in Theorem 1.2, item (1) implies item (4): Since M is simple, given u 2 C1.Smsol.T M//, by Lemma 4.3, there exists v 2 C1c .Sm.T  zM// such that rM ImImv D u. Then it is a standard argument that if we define ' D I.Imv/j@CSM , then Im' D u. Moreover, since I.Imv/ is smooth in the interior of S zM, we have ' 2 C 1˛.@CSM/. The proof of Theorem 1.2 is now complete. 7. Alternative proof of Corollary 1.3 Before giving the alternative proof, we will explain how the solenoidal condition of a tensor manifests itself at the level of the transport equation. It seems that this basic relation has not appeared before in the literature, although we believe it was known to experts. As we already pointed out in the Introduction, by considering the vertical Laplacian  on each fiber SxM of SM , we have a natural L2 decomposition L2.SM/DLm0Hm.SM/ into vertical spherical harmonics. We set m WD Hm.SM/ \ C1.SM/. Then a function u belongs to m if and only if uDm.mCn 2/u, where nD dimM. The maps `m W C1.Sm.T M//! Œm=2M kD0 m2k and Lm W Œm=2M kD0 m2k! C1.Sm.T M// are isomorphisms. These maps give natural identification between functions in m and trace-free symmetricm-tensors (for details on this, see [Guillemin and Kazhdan 1980b; Dairbekov and Sharafutdinov 2010; Paternain et al. 2015a]). The geodesic vector field X maps m to m1˚mC1 and hence we can split it as X DXCCX, where X˙ Wm!m˙1 and XC DX. Note that X`m1 D `md: Given f 2LŒm=2 kD0 m2k , in general Xf 2 LŒ.mC1/=2 kD0 mC12k . The next simple lemma charac- terizes the solenoidal condition in terms of Xf . Lemma 7.1. Xf 2mC1 if and only if Lmf is a solenoidal tensor. Proof. Note that Lmf is solenoidal if and only if .Lmf; dh/D 0 for any h 2 C1.Sm1.T M// with hj@M D 0. But .Lmf; dh/D .f; `mdh/D .f;X`m1h/D.Xf; `m1h/ INVARIANT DISTRIBUTIONS AND THE GEODESIC RAY TRANSFORM 1927 and the last term is zero if and only if .Xf /m2k1 D 0 for 0  k  Œ.m 1/=2 since `m1h 2LŒ.m1/=2 kD0 m12k .  Another way to look at the condition Xf 2mC1 is that the following equations should hold: Xfm2kCXCfm2k2 D 0 for 0 k  Œ.m 1/=2: Lemma 7.2. The following are equivalent: (1) Given a nonnegative integer m and am 2m with Xam D 0, there exists w 2 C1.SM/ such that Xw D 0 and wm D am. (2) Given a nonnegative integer m and f D PmkD0 fk such that Xf 2 m ˚ mC1, there exists w 2 C1.SM/ such that Xw D 0 and PmkD0wk D f . Proof. The fact that (2) implies (1) is quite obvious from the fact that am 2m with Xam D 0 implies Xam DXCam 2mC1. To prove that (1) implies (2) we proceed by induction on m. The case mD 0 follows right away since Xf0 21 and Xf0 D 0. Suppose the claim holds for m and let f DPmC1kD0 fk be given with Xf 2mC1˚mC2. This is equivalent to saying that X Pm kD0 fk  2m˚mC1 and XfmC1CXCfm1 D 0. By the induction hypothesis, there exists w 2C1.SM/ such that XwD 0 and wk D fk for all k m. The equation Xw D 0 in degree m is XwmC1CXCfm1 D 0 and thus X.fmC1wmC1/D 0: Using item (1) in the lemma, there exists w0 DP1mC1w0k 2 C1.SM/ such that Xw0 D 0 and w0mC1 D fmC1wmC1. Then X.wCw0/D 0 and PmC1kD0 .wCw0/k D f as desired.  Finally we show: Proposition 7.3. The following are equivalent: (1) Given a nonnegative integer m and u 2 C1.Smsol.T M//, there exists f 2 C1.SM/ with Xf D 0 such that Lmf D u. (2) Given a nonnegative integer m and am 2m with Xam D 0, there exists w 2 C1.SM/ such that Xw D 0 and wm D am. Proof. Assume (1) holds. Given am 2 m with Xam D 0, we see using Lemma 7.1 that Lmam is a solenoidal tensor. Hence there is f such that Xf D 0 and fm D L1m Lmf D am (note that Lmfk D 0 for k > m). Thus (2) holds. Conversely if (2) holds, then item (2) in Lemma 7.2 holds. Thus there exists f 2 C1.SM/ such that Xf D 0 and PŒm=2 kD0 fm2k D L1m u and (1) holds.  1928 GABRIEL P. PATERNAIN AND HANMING ZHOU Proof of Corollary 1.3. On account of Proposition 7.3, it suffices to show that given am 2 m with Xam D 0, there exists w 2 C1.SM/ such that Xw D 0 and wm D am. What makes this possible in dimension two is [Paternain et al. 2015b, Lemma 5.6], whose content we now explain. If .M; g/ is an oriented Riemannian surface, there is a global orthonormal frame fX;X?; V g of SM equipped with the Sasaki metric, where X is the geodesic vector field, V is the vertical vector field and X? D ŒX; V . We define the Guillemin–Kazhdan operators [1980a] ˙ D 12.X ˙ iX?/: If x D .x1; x2/ are oriented isothermal coordinates near some point of M, we obtain local coordinates .x; / on SM , where  is the angle between  and @=@x1. In these coordinates V D @=@ and C and  are @- and N@-type operators; see [Paternain et al. 2015a, Appendix B]. For any m 2 Z we define ƒm D fu 2 C1.SM/ W V uD imug: In the .x; /-coordinates elements of ƒm look locally like h.x/eim. Spherical harmonics may be further decomposed as 0 Dƒ0; m Dƒm˚ƒm for m 1: Any u 2 C1.SM/ has a decomposition uDP1mD1 um, where um 2ƒm. The geodesic vector field decomposes as X D CC ; where ˙ Wƒm!ƒm˙1. If m 1, the action of X˙ on m is given by X˙.emC em/D ˙emC em; ej 2ƒj ; and for mD 0, we have XCj0 D CC  and Xj0 D 0. With these preliminaries out of the way, [Paternain et al. 2015b, Lemma 5.6] says that given f 2ƒm, there is a smooth w 2 C1.SM/ with Xw D 0 and wm D f . For m D 0, this gives the desired result right away. Given am 2 m with Xam D 0 and m  1, we write am D em C em with ej 2 ƒj . Then emCCemD 0. Consider now smooth p; q with XpDXqD 0 and pmD em and qmD em. Then w D mX 1 qkC 1X m pk satisfies Xw D 0 and wm D am.  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PATERNAIN: g.p.paternain@dpmms.cam.ac.uk Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom HANMING ZHOU: hz318@dpmms.cam.ac.uk Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom mathematical sciences publishers msp Analysis & PDE msp.org/apde EDITORS EDITOR-IN-CHIEF Patrick Gérard patrick.gerard@math.u-psud.fr Université Paris Sud XI Orsay, France BOARD OF EDITORS Nicolas Burq Université Paris-Sud 11, France nicolas.burq@math.u-psud.fr Massimiliano Berti Scuola Intern. Sup. di Studi Avanzati, Italy berti@sissa.it Sun-Yung Alice Chang Princeton University, USA chang@math.princeton.edu Michael Christ University of California, Berkeley, USA mchrist@math.berkeley.edu Charles Fefferman Princeton University, USA cf@math.princeton.edu Ursula Hamenstaedt Universität Bonn, Germany ursula@math.uni-bonn.de Vaughan Jones U.C. Berkeley & Vanderbilt University vaughan.f.jones@vanderbilt.edu Vadim Kaloshin University of Maryland, USA vadim.kaloshin@gmail.com Herbert Koch Universität Bonn, Germany koch@math.uni-bonn.de Izabella Laba University of British Columbia, Canada ilaba@math.ubc.ca Gilles Lebeau Université de Nice Sophia Antipolis, France lebeau@unice.fr Richard B. Melrose Massachussets Inst. of Tech., USA rbm@math.mit.edu Frank Merle Université de Cergy-Pontoise, France Frank.Merle@u-cergy.fr William Minicozzi II Johns Hopkins University, USA minicozz@math.jhu.edu Clément Mouhot Cambridge University, UK c.mouhot@dpmms.cam.ac.uk Werner Müller Universität Bonn, Germany mueller@math.uni-bonn.de Gilles Pisier Texas A&M University, and Paris 6 pisier@math.tamu.edu Tristan Rivière ETH, Switzerland riviere@math.ethz.ch Igor Rodnianski Princeton University, USA irod@math.princeton.edu Wilhelm Schlag University of Chicago, USA schlag@math.uchicago.edu Sylvia Serfaty New York University, USA serfaty@cims.nyu.edu Yum-Tong Siu Harvard University, USA siu@math.harvard.edu Terence Tao University of California, Los Angeles, USA tao@math.ucla.edu Michael E. Taylor Univ. of North Carolina, Chapel Hill, USA met@math.unc.edu Gunther Uhlmann University of Washington, USA gunther@math.washington.edu András Vasy Stanford University, USA andras@math.stanford.edu Dan Virgil Voiculescu University of California, Berkeley, USA dvv@math.berkeley.edu Steven Zelditch Northwestern University, USA zelditch@math.northwestern.edu Maciej Zworski University of California, Berkeley, USA zworski@math.berkeley.edu PRODUCTION production@msp.org Silvio Levy, Scientific Editor See inside back cover or msp.org/apde for submission instructions. The subscription price for 2016 is US $235/year for the electronic version, and $430/year (+$55, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to MSP. 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