On the range of the attenuated magnetic ray transform for connections and Higgs fields

For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,\Phi}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $\Phi$. We give a description for the range of $I_{A,\Phi}$ acting on $\mathbb C^n$-valued tensor fields.

Here ∇ is the Levi-Civita connection of g. The triple (M, g, Ω) is said to be a magnetic system. In the absence of a magnetic field, that is Ω = 0, we recover the ordinary geodesic flow and ordinary geodesics. Note that magnetic geodesics are not time reversible, unless Ω = 0. Note also that magnetic geodesics have constant speed, and hence are restricted to a specific energy level -we will consider the magnetic flow on the unit sphere bundle SM = {(x, v) ∈ T M : |v| = 1}. This is not a restriction at all from a dynamical point of view, since other energy levels may be understood by simply changing Ω to cΩ, where c ∈ R.
Magnetic flows were first considered in [2,3] and it was shown in [4,11,15,16,17,19] that they are related to dynamical systems, symplectic geometry, classical mechanics and mathematical mechanics. Inverse problems related to magnetic flows were studied in [1,5,6] Let Λ stand for the second fundamental form of ∂M and ν(x) for the inward unit normal to ∂M at x. We say that ∂M is strictly magnetic convex if for all (x, ξ) ∈ S(∂M ). Note that if we replace ξ by −ξ, we can put an absolute value in the right-hand side of (2). In particular, magnetic convexity is stronger than the Riemannian analogue. For x ∈ M , we define the magnetic exponential map at x to be the partial map exp µ x : T x M → M given by exp µ x (tξ) = π • φ t (ξ), t ≥ 0, ξ ∈ S x M. In [5] it is shown that, for every Definition 1.1. We say that a magnetic system (M, g, α) is a simple magnetic system if ∂M is strictly magnetic convex and the magnetic exponential map exp µ x : In this case M is diffeomorphic to the unit ball of Euclidean space, and therefore Ω = dα for some 1-form α on M . This definition is a generalization of the notion of a simple Riemannian manifold. The latter naturally arose in the context of the boundary rigidity problem [14]. Throughout this paper we only consider simple magnetic systems.
1.2. Attenuated magnetic ray transform for a unitary connection and Higgs field. We consider a unitary connection and a skew-Hermitian Higgs field on the trivial bundle M × C n . We define a unitary connection as a matrix-valued smooth map A : T M → u(n) which for fixed x ∈ M is linear in v ∈ T x M , and define a skew-Hermitian Higgs field as a matrix-valued smooth map Φ : M → u(n). The connection A induces a covariant derivative which acts on sections of M × C n by d A := d + A. Saying A is unitary means the following holds for the inner product of sections s 1 , Pairs of unitary connections and skew-Hermitian Higgs fields (A, Φ) are very important in the Yang-Mills-Higgs theories, since they correspond to the most popular structure groups U (n) or SU (n), see [7,8,12,13].
On the boundary of M , we consider the set of inward and outward unit vectors defined as where ν is the unit inner normal to ∂M . The magnetic geodesics entering M can be parametrized by ∂ + SM . We say that a magnetic system (M, g, Ω) is nontrapping if for any (x, v) ∈ SM the time τ + (x, v) when the magnetic geodesic γ x,v , with x = γ x,v (0), v =γ x,v (0), exits M is finite. In particular, simple magnetic systems are non-trapping [5].
Let G µ denote the generating vector field of the magnetic flow φ t . Given f ∈ C ∞ (SM, C n ), consider the following transport equation for u : Here A and Φ act on functions on SM by matrix multiplication. This equation has a unique solution u f , since on any fixed magnetic geodesic the transport equation is a linear system of ODEs with zero initial condition.
Definition 1.2. The attenuated magnetic ray transform of f ∈ C ∞ (SM, C n ), with attenuation determined by a unitary connection A : T M → u(n) and a skew-Hermitian Higgs field Φ : M → C n , is given by It is clear that a general function f ∈ C ∞ (SM, C n ) cannot be determined by its attenuated magnetic ray transform, since f depends on more variables than I A,Φ f . Moreover, one can easily see that the functions of the following type are always in the kernel of I A,Φ However, in applications one often needs to invert the transform I A,Φ acting on functions on SM arising from symmetric tensor fields. Further, we will consider this particular case.
Let f = f i1··· im dx i1 ⊗ · · · ⊗ dx im be a C n -valued, smooth symmetric m-tensor field on M . Then a tensor field induces a smooth function f m ∈ C ∞ (SM, C n ) by We denote by C ∞ (S m (M ), C n ) the bundle of smooth C n -valued, (covariant) symmetric m-tensor fields on M . When m = 1, we also use the notation C ∞ (Λ 1 (M ), C n ). By I m A,Φ we denote the following operator . We will frequently identify the tensor field f ∈ C ∞ (S m (M ), C n ) with the corresponding function f m ∈ C ∞ (SM, C n ).
The magnetic field and Higgs field couple tensors of degrees m and m − 1, therefore, we have to consider m-tensors and (m − 1)-tensors simultaneously. This observation implies that we have to study I m A,Φ which acts on the product space C ∞ (S m (M ), C n ) × C ∞ (S m−1 (M ), C n ) and is defined as 1.3. Range description. In this paper we give a characterization of the range of I A,Φ . More precisely, we give a description for the functions in C ∞ (∂ + SM, C n ) which are in the range of I A,Φ . For the description we use the following boundary data: the scattering relation (see Section 2.1), the scattering data of the pair (A, Φ) (see Section 2.2) and the fibrewise Hilbert transform at the boundary (see Section 2.5). Given w ∈ C ∞ (∂ + SM, C n ) we define w ♯ to be the unique solution to transport equation Moreover, we define S ∞ A,Φ (∂ + SM, C n ) to be the set of all w ∈ C ∞ (∂ + SM, C n ) such that w ♯ is smooth.
We introduce the operator B A,Φ : C(∂(SM ), C n ) → C(∂ + SM, C n ) defined by Let a be a smooth function and f = ( Clearly the operator P is completely determined by the scattering relation S and scattering data C A,Φ .
Recall a connection A induces an operator d A , acting on C n -valued differential forms on M by the formula d A α = dα + A ∧ α. By d * A we denote the dual of d A with respect to L 2 -norm on the space of forms. Then it is not difficult to check that d * A = − ⋆ d A ⋆. We use the notation H A for the space of all 1-forms η with d A η = d * A η = 0 and  * η = 0 where  : ∂M → M is the inclusion map. The elements of this space are called A-harmonic forms. Note that H A is a finite dimensional space, since the equations defining H A are an elliptic system with regular boundary condition, see [27,Section 5.11]. Since M is a disk, we have H A = 0 whenever A = 0.
We can now state our main result.
be a two-dimensional simple magnetic system, A a unitary connection and Φ a skew-Hermitian Higgs field. Then a function u ∈ A,Φ (∂ + SM, C n ) and for some η ∈ H A . Theorem 1.3 was proved by Paternain, Salo and Uhlmann [21] in the case of the geodesic flow. This was used to give a description of the range of unattenuated geodesic ray transform acting on tensors. For the characterization of the range of the ray transform on higher order tensors our main result is stated in Section 3: The crucial difficulty when one deals with a magnetic field or Higgs field is the fact that the concomitant transport equation couples different Fourier components. Even if one restricts oneself to the geodesic case, by adding a Higgs field this difficulty already presents an obstacle to the approach of [21]. The key idea that is utilized to overcome this, and which represents the major contribution of this paper, is a result on the "simultaneous" surjectivity of the adjoints of the ray transform I 0 A,Φ and I 1 A,Φ -this is made precise in Theorem 4.2. This in turn relies on the ellipticity of the operator If the latter domain is chosen, then the kernel of the ray transform has a natural obstruction, and in particular, is not identically 0.
1.4. Structure of the paper. The structure of the paper is as follows. In Section 2 we recall some facts and definitions from [1,23] that will be used in our paper. The proof of Theorem 3.2 is given in Section 3. In Section 4 we discuss the surjectivity properties of the adjoint of the attenuated ray transform. Finally, in Section 5 we give the proof of Theorem 1.3.
The scattering relation S : From the above comments on τ + , we conclude that the scattering relation S is a smooth map. For a given w ∈ C ∞ (∂ + SM, C n ), the transport equation

2.2.
Scattering data of a unitary connection and skew-Hermitian Higgs field. Let U A,Φ : SM → U (n) be the unique solution of the transport equation has the same scattering data. It was proved in [1] that (A, Φ) can be determined by the scattering data C A,Φ up to such a gauge equivalence. Now, for a given w ∈ C ∞ (∂ + SM, C n ) consider the unique solution w ♯ : SM → C n to the transport equation Using the scattering relation S and the scattering data C A,Φ , we introduce the operator Lemma 4.2] in terms of the operator Q A,Φ as follows: Using the fundamental solution U A,Φ , we can also give an integral expression for the ray transform. Recall A,Φ f . Integrating from 0 to τ + (x, v) for (x, v) ∈ ∂ + SM we obtain the following expression

2.3.
Geometry and Fourier analysis on SM . Since M is assumed to be oriented there is a circle action on the fibres of SM with infinitesimal generator V called the vertical vector field. Let X denote the generator of the geodesic flow of g. We complete X, V to a global frame of T (SM ) by defining the vector field X ⊥ := [V, X], where [·, ·] is the Lie bracket for vector fields. It is easy to see that the generator of magnetic flow φ t can be expressed in terms of the global frame (X, X ⊥ , V ) in the following form G µ = X + λV, where λ is the unique function satisfying Ω = λd Vol g with d Vol g being the area form of M .
For any two functions u, v : SM → C n define an inner product: where dΣ 3 is the Liouville measure of g on SM . The space L 2 (SM, C n ) decomposes orthogonally as a direct sum where H k is the eigenspace of −iV corresponding to the eigenvalue k. Any function u ∈ C ∞ (SM, C n ) has a Fourier series expansion 2.4. Some elliptic operators of Guillemin and Kazhdan. Now we introduce the following first order elliptic operators due to Guillemin and Kazhdan [9] η + , η − : By the commutation relations [−iV, η + ] = η + and [−iV, η − ] = −η − we see that We will use these operators in the last two sections.
2.5. Fibrewise Hilbert transform. An important tool in our approach is the fibrewise Hilbert transform H : C ∞ (SM, C) → C ∞ (SM, C), which we define in terms of Fourier coefficients as where we use the convention sgn(0) = 0. Moreover, H(u) = k H(u k ). Note that The following commutator formula, which was derived by Pestov and Uhlmann in [24] and generalized in [1,23], will play an important role.

Range characterizations for higher order tensors
Let κ denote the canonical line bundle of M , whose complex structure is that induced by its metric g. For k ∈ N we denote by Γ(M, κ ⊗k ) the set of k-th tensor power of canonical line bundle. It was explained in [20, Section 2] the set Γ(M, κ ⊗k ) can be identified with Ω k . Roughly speaking, for a given ξ ∈ Γ(M, κ ⊗k ) we obtain a corresponding function on Ω k via the one-to-one map SM ∋ (x, v) → ξ x (v ⊗k ).
Since M is a disk, there is a nonvanishing ξ ∈ Γ(M, κ). Define a function h : SM → S 1 by setting h(x, v) := ξ x (v)/|ξ x (v)|, and hence h ∈ Ω 1 . For the description of the range of I m A,Φ we use the following unitary connection A h := −h −1 Xh Id and skew-Hermitian Higgs field Φ λ := −iλ Id. Then it is easy to see that A h and Φ λ satisfy −h −1 G µ h Id = A h + Φ λ . We start with characterizing the range of attenuated magnetic ray transform I A,Φ restricted to Ω m−1 ⊕ Ω m ⊕ Ω m+1 : We describe the range of I ± m,A,Φ and then use it in the description of the range of The range of the left hand side of (5) was described in Theorem 1.3. Thus we directly conclude the following result.
Recall that, according to [22,Section 2], there is a one-to-one correspondence between C ∞ (S m (M ), C n ) and a subspace of the set of functions on SM of the form Then we can write F = m k=−m F k for suitable F k ∈ Ω k . Therefore, we have Using this and Theorem 3.1, we obtain our second main result of the current paper.
4. Surjectivity properties of (I 1 A,Φ ) * Let dΣ 2 be the volume form on ∂(SM ). In the space of C n -valued functions on ∂ + SM define the inner product Denote the corresponding Hilbert space by L 2 µ (∂ + SM, C n ). As in [21], using the integral representation for I A,Φ and Santaló formula [5,Lemma A.8], one can show that I A,Φ can be extended to a bounded operator I A,Φ : In this section we give an expression for the adjoint of I 1 . Let f be a smooth C n -valued function on M , ω be a smooth C n -valued 1-form on M and h ∈ L 2 µ (∂ + SM, C n ), then using the Santaló formula and the integral representation for I 1 A,Φ , we have Let d Vol g (x) be a measure on M and dσ x be a measure on S x M . Then Therefore, if A is unitary and Φ is skew-Hermitian, we have Moreover, from [21, Remark 5.2] and (6) we see that the function on SM which corresponds to (I 1 A,Φ ) * (h), and is denoted again by (I 1 A,Φ ) * (h), has the following form 4.2. Normal operators. Let I 0,1 A,Φ denote the restriction of I A,Φ to Ω 0 ⊕ Ω 1 . Consider the corresponding normal operator defined as Lemma 4.1. Let (M, g, α) be a two-dimensional simple magnetic system, A a unitary connection and Φ a skew-Hermitian Higgs field. Then N 0,1 A,Φ is elliptic pseudodifferential operator in M int of order −1.
Proof. First, we consider the normal operator corresponding to I 1 A,Φ : . From the integral representation for I A,Φ and expression (6), we have Repeating the same arguments as in [5,Section 4.2], the use of [5,Lemma B.1] gives that Let β = β x 1 dx 1 + β x 2 dx 2 be a smooth 1-form on M and consider be the corresponding function on SM : Compute the terms of the Fourier expansion β = β −1 + β 1 : Therefore, we can conclude that β ∈ Ω 1 if and only if Now, consider the normal operator N A,Φ corresponding to the restricted ray transform I 0,1 A,Φ . Let f ∈ Ω 0 and ω ∈ Ω 1 . As in the case of N A,Φ , we have where, thanks to (9), we have Using [5,Lemma B.1] we see that the principal symbols of the above operators can be written as linear combinations of the principal symbols of N A,Φ , hence using the expressions for the latter from [5, Proposition 7.2] we can conclude the following For a fixed x, we can find a chart such that in this local coordinates g ij (x) = δ ij (the latter holds only at this fixed x). Then from above one can conclude that This completes the proof.

4.3.
Surjectivity of adjoints. The aim of this section is to prove the following analogue of [21,Theorem 5.4] and [21,Theorem 5.5].
Theorem 4.2. Let (M, g, α) be a two-dimensional simple magnetic system, A a unitary connection and Φ a skew-Hermitian Higgs field. Given For the proof we need the following: Proof. First, we deal with the case m = 0. The statement is equivalent to the surjectivity of (I 0,1 A,Φ ) * . The proof follows the same ideas as [25,Theorem 1.4]. We only mention the main tools: The first main ingredient is Lemma 4.1, which says that N 0,1 A,Φ is an elliptic pseudo-differential operator of order −1. This ensures closed range of a suitable extension. Second key result is [1, Theorem 1.2], which in our case says that if f ∈ Ω 0 and w ∈ Ω 1 with I 0,1 A,Φ (w + f ) = 0, then w ≡ 0, f ≡ 0. This gives the desired surjectivity for (I 0,1 A,Φ ) * . Now, we prove the statement for any m ∈ Z. Fix a non-vanishing h ∈ Ω 1 . As before, consider the unitary connection A h = −h −1 Xh Id and skew-Hermitian Note that for any a ∈ C ∞ (SM, C n ) the following holds By what we have already proved above for the case m = 0, there exists a ∈ C ∞ (SM, C n ) such that and a 0 = h −m f , a 1 = h −m w. Set u := h m a, then clearly u m = f and u m+1 = w. By equality (10), we have (G µ + A + Φ)u = 0, which finishes the proof.
Proof of Theorem 4.2. Let ω be a smooth C n -valued 1-form on M and let f ∈ C ∞ (M, C n ). By Proposition 4.3, there exist u, u ′ ∈ C ∞ (SM, C n ) such that It is clear that w 0 = f and w −1 + w 1 = ω −1 + ω 1 = ω. Introduce the operators µ ± = η ± + A ±1 . Then X + A = µ − + µ + . It is easy to check that In particular, we have µ + w −1 + µ − w 1 + Φw 0 = 0. By [21,Lemma 6.2], this is equivalent to our assumption d * A ω = 2Φf . Solutions of the transport equation are unique once boundary data is specified, therefore, w ∂+SM satisfies the requirement of our theorem.

Proof of Theorem 1.3
Let w ♯ be any smooth solution of the transport equation G µ w ♯ + (A + Φ)w ♯ = 0. Applying (4) to w ♯ we get According to [21, Lemma 6.2] the following holds Collecting everything together and using (7) we have A,Φ to the above equality we obtain − 2πP A,Φ = I 1 A,Φ ⋆ d A (I 1 A,Φ ) * .
We also need the following result whose proof is postponed until after the proof of Theorem 1.3.
Lemma 5.1. Let (M, g) be a Riemannian disk, A a unitary connection and Φ a skew-Hermitian Higgs field.
(a) Let α be a smooth C n -valued 1-form. Then there exist functions a, p ∈ C ∞ (M, C n ) and η ∈ H A such that p ∂M = 0 and d A p + ⋆d A a + η = α. (b) Given f, a ∈ C ∞ (M, C n ) there is a smooth C n -valued 1-form β with ⋆d A β = f and d * A β = Φa. Proof of Theorem 1.3. Suppose that u = P A,Φ w + I 1 A,Φ η for η ∈ H A , then (11) shows that u belongs to the range of I 1 A,Φ . Conversely, if u belongs to the range of I 1 A,Φ , then u = I 1 A,Φ [ω, f ] for some smooth C n -valued function f and 1-form ω on M . By item (a) of Lemma 5.1 we can find a, p ∈ C ∞ (M, C n ) with p ∂M = 0 and η ∈ H A such that d A p + ⋆d A a + η = ω.
A . This operator acts on C n -valued graded forms and maps k-forms to k-forms. The following result directly implies item (b) of Lemma 5.1.
Lemma 5.2. Given f, a ∈ C ∞ (M, C n ) there is a smooth C n -valued 1-form β with d * A β = f and d A β = Φa d Vol g . Proof. The proof is essentially identical to the proof of [21,Lemma 6.6]. Look for β of the form β = d A u 0 + d * A u 2 where u 0 , u 2 are smooth forms. Then we need u 0 , u 2 to satisfy Writing u = u 0 + u 2 , these equations are equivalent to for some operator R of order 0. Then [21, Lemma 6.5] implies the existence of a smooth solution u, hence the existence of the desired β.