Abstract
We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of all such transforms in a neighborhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absence of connection, then for any smooth enough connection multiplied by a complex parameter, the corresponding transform is injective for all values of that parameter outside a discrete set. Range characterizations are also provided.
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1 Introduction
This paper is concerned with the attenuated X-ray transform on a non-trapping surface. We shall consider attenuations determined by a \(n\times n\) matrix A of complex-valued 1-forms (a \(GL(n,\mathbb {C})\)-connection).
Consider (M, g) a compact-oriented Riemannian surface with smooth boundary. We let \(SM = \{ (\mathrm {x},v)\in TM:\ |v|_g = 1 \}\) be the unit tangent bundle with geodesic flow \(\varphi _t:SM\rightarrow SM\), defined on the domain
where \(\tau (\mathrm {x},v)\) is the first time at which the geodesic \(\gamma _{(\mathrm {x},v)}\) with initial conditions \((\mathrm {x},v)\) hits the boundary \(\partial M\). Recall that \(\varphi _{t}\) is defined as \(\varphi _{t}(\mathrm {x},v):=(\gamma _{(\mathrm {x},v)}(t),\dot{\gamma }_{(\mathrm {x},v)}(t))\), with infinitesimal generator \(X_{(\mathrm {x},v)} = \frac{\mathrm{d}\varphi _t}{\mathrm{d}t}(\mathrm {x},v)|_{t=0}\).
The manifold is said to be non-trapping if \(\tau (\mathrm {x},v)<\infty \) for all \((\mathrm {x},v)\in SM\). In this paper, we consider non-trapping surfaces where \(\partial M\) is strictly convex, meaning that the second fundamental form of \(\partial M\subset M\) is positive definite. This is already enough to imply that M is a disk (cf. [22, Proposition 2.4]). If in addition (M, g) has no conjugate points, we say that the surface is simple.
Given A, consider the matrix weight \(w:SM\rightarrow GL(n,\mathbb {C})\) that arises as a solution of the transport equation on SM:
where \(\partial _{+}(SM)\) denotes the set of \((\mathrm {x},v)\in \partial (SM)\) such that v points inside M, i.e. \(\langle \nu (\mathrm {x}),v\rangle \le 0\) where \(\nu \) is the outer unit normal at \(\partial M\). We define the attenuated X-ray transform associated with the connection A,
as
If \(f\in C^{\infty }(M,\mathbb {C}^{n})\), we shall set \(I_{A,0}(f):=I_{A}(f\circ \pi )\), where \(\pi :SM\rightarrow M\) is the footpoint projection.
A question of fundamental importance in the subject is whether \(I_{A,0}\) is injective. In [21], the authors prove that in the case of a unitary connection and a Higgs field, the corresponding ray transform on a simple surface is injective. In [23], the same authors provide a range characterization for the attenuated ray transform. The salient feature here is that the connection need not be unitary (or, equivalently, Hermitian) in the sense that \(A \ne -A^*\).
To put the X-ray transform (3) into perspective, consider first a general matrix weight \(w:SM\rightarrow GL(n,\mathbb {C})\). For each fixed \(\mathrm {x}\in M\), the quantity \(w^{-1}Xw(\mathrm {x},v)\) may be expanded in the velocities v as
Hence, the transport equation (2) tells that the X-ray transform with connection picks up precisely the term in the expansion above that gives linear dependence in velocities. If we were to pick just \(\Phi (\mathrm {x})\), we would have a Higgs field or potential and for \(n=1\) and g flat, this reduces to the usual attenuated ray transform that is so prominent in SPECT (single-photon emission computed tomography). Inversion formulas and range characterization for this very important case were obtained in [2, 17, 18]. Even for \(n=1\), if the weight w is allowed to be arbitrary, the attenuated ray transform in 2D may not be injective [4], but one may speculate that if the expansion in (4) is finite, injectivity may persist. In dimensions \(\ge 3\), the game changes and very general injectivity results have been obtained in [24].
Besides the motivation coming from medical imaging and SPECT, there is another reason for considering the problem of injectivity of \(I_{A,0}\) and it has to do with the non-linear inverse problem of recovering A from its scattering data, or non-abelian X-ray transform. It would be impossible to do justice to the literature on the topic here, but we refer to [5, 19, 21, 24] and references therein; the last two references also have a discussion about a pseudo-linearization procedure that allows to connect the linear and non-linear problems.
Our approach in order to invert such transforms explicitly is a generalization of inversion formulas derived in [26] and further analyzed in [10] for geodesic ray transforms on simple surfaces. The first author then provided generalizations of such approaches in the case of symmetric differentials on simple surfaces [12] and recently provided inversion formulas for geodesic X-ray transforms with scalar Higgs-field type attenuations [13] (that is, \(w^{-1}Xw=\Phi (\mathrm {x})\) and \(n=1\)).
In [5], it is proved that \(I_{A,0}\) is injective for an arbitrary \(GL(n,\mathbb {C})\)-connection when (M, g) is a domain in \(\mathbb {R}^{2}\); this uses a delicate theorem about existence of holomorphic integrating factors established in [6, Theorem 5]. Generic injectivity for the case of simple manifolds, including the case when both g and A are real analytic, is proved in [32]. In [25], injectivity of \(I_{A,0}\) is proved for an arbitrary \(GL(n,\mathbb {C})\)-connection whenever (M, g) is a negatively curved simple manifold. In spite of all this progress, the following question remains open:
Question
Let (M, g) be a simple surface and A a \(GL(n,\mathbb {C})\)-connection. Is \(I_{A,0}\) injective?
Note that, the question has a positive answer for \(n=1\); this follows essentially from the methods in [28], cf. Proposition 22 below. Theorem 5 below provides several new instances in which \(I_{A,0}\) is proved to be injective, and we shall also provide range characterizations. Numerical simulations illustrating the effectiveness of our approach will appear in future work. We now proceed to state our results in detail.
2 Main Results
Let (M, g) be a non-trapping Riemannian surface with strictly convex boundary, and let \(\nu \) denote the outer unit normal to \(\partial M\). The unit sphere bundle SM is a 3-dimensional compact manifold with boundary, which can be written as the union \(\partial (SM) =\partial _{+}(SM) \cup \partial _{-}(SM) \),
The standard volume forms on SM and \(\partial (SM)\) that we will use are defined by
where \(\mathrm{d}M\) (resp. \(\mathrm{d}\partial M\)) is the volume form of M (resp. \(\partial M\)), and \(\mathrm{d}S\) is the volume form of the unit circle \(S_{\mathrm {x}}\) in \(T_{\mathrm {x}}M\). For \((x,v)\in \partial _+(SM)\), let \(\mu (x,v)=-\langle \nu (x), v \rangle \) and \(L^{2}_{\mu }(\partial _{+}(SM),\mathbb {C}^{n})\) be the space of \(\mathbb {C}^{n}\)-valued functions on \(\partial _{+}(SM)\) with inner product
Suppose now that A is a \(GL(n,\mathbb {C})\)-connection. This simply means that A is an \(n\times n\) matrix whose entries are 1-forms. Its curvature is defined as the 2-form \(F_{A}:=dA+A\wedge A\) (i.e., \(F_{A}\) is a matrix of 2-forms with components \((F_A)_{ij} = dA_{ij} + \sum _{k=1}^n A_{ik}\wedge A_{kj}\)). Using the star operator \(\star \) associated with the metric, we will often consider \(\star F_{A}:M\rightarrow \mathbb {C}^{n\times n}\).
Let us define the attenuated ray transform with connection A, as follows. Let \(X := \frac{\mathrm{d}\varphi _t}{\mathrm{d}t}|_{t=0}\) be the geodesic vector field on the unit circle bundle, and let \(X_\perp := [X,V]\) where V is the so-called vertical derivative (see Sect. 3.1). For \(f\in C(SM, \mathbb {C}^n)\), we define \(I_A f := u_A^f|_{\partial _+ (SM)}\), where \(u_A^f\) denotes the unique solution u to the transport problem
In this case, \(I_A:C(SM,\mathbb {C}^n)\rightarrow C(\partial _+(SM),\mathbb {C}^n)\) extends by density into a bounded operator \(I_A:L^2(SM,\mathbb {C}^n)\rightarrow L^2_\mu (\partial _+(SM),\mathbb {C}^n)\). Upon defining \(\pi :SM\rightarrow M\) to be the canonical projection and \(A_V:= V(A)\), restrictions of \(I_A\) of interest are given by \(I_{A,0} f := I_{A} [f\circ \pi ]\) for \(f\in C^\infty (M,\mathbb {C}^n)\), extendible to \(L^2(M,\mathbb {C}^n)\) by continuity; \(I_{A,\perp } f := I_A [(X_\perp - A_V) (f\circ \pi )]\) for \(f\in C_0^\infty (M,\mathbb {C}^n)\) (i.e., a smooth function vanishing at the boundary), extendible to \(H^1_0(M,\mathbb {C}^n)\) by continuity. Note that, \(I_{A,\perp }\) can also be defined on functions which do not vanish at \(\partial M\), and the difference will be studied in much detail in Sect. 5.
Upon defining the operators
where \(\pi _0:L^2(SM)\rightarrow L^2(M)\) denotes the fiberwise average\(\pi _0 f(\mathrm {x}) := \frac{1}{2\pi } \int _{S_\mathrm {x}} f(\mathrm {x},v)\ \mathrm{d}S(v)\), we first derive the following formulas, true on any non-trapping Riemannian surface with strictly convex boundary:
Theorem 1
Let (M, g) be a non-trapping Riemannian surface with boundary. Then the following equations hold:
Formulas (5)–(6) take the form of filtered-backprojection algorithms, where the operator \(B_{A,+} H Q_{A,-}\) (defined in Sect. 3.3, see (19) and (21)) can be viewed as a filter in data space, while the operators \(I_{-A^*,0}^*, I_{-A^*,\perp }^*\), formal adjoints of \(I_{-A^*,0}:L^2(M,\mathbb {C}^n)\rightarrow L^2_\mu (\partial _+ (SM),\mathbb {C}^n)\) and \(I_{-A^*,\perp }:H^1_0(M,\mathbb {C}^n)\rightarrow L^2_\mu (\partial _+ (SM),\mathbb {C}^n)\), respectively, are sometimes referred to as backprojection operators.
Remark 2
While the transform \(I_{A,\perp }\) can be defined for smooth functions with non-zero boundary values, Eq. (6) no longer holds in this augmented space, as is illustrated on the Euclidean transform \(I_\perp \) without connection in [15, Proposition 5]. There, decomposing \(f\in H^1(M)\) into \(f = f_0 + f_\partial \) where \(f_\partial \) is the harmonic extension of the trace of f and \(f_0 \in H^1_0(M)\), it is then shown that formula (6) applied to \(I_\perp f\) recovers \(f_0 + \frac{1}{4} f_\partial \) and not \(f_0 + f_\partial \).
If, in addition, (M, g) is simple, then the operators \(W_A\) and \(W_{A,\perp }\) extend as compact operators \(W_A,W_{A,\perp }:L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\), see Lemma 16 below. In particular, Eqs. (5) and (6) are Fredholm equations, invertible up to a finite-dimensional kernel. In fact, we can prove something with stronger implications:
Theorem 3
For any analytic \(C^1(M, (\Lambda ^1)^{n\times n})\)-valued family of connections \(\lambda \mapsto A_\lambda \), the corresponding \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\)-valued families of operators \(\lambda \mapsto W_{A_\lambda }\) and \(\lambda \mapsto W_{A_\lambda ,\perp }\) are analytic.
By analytic Fredholm theory (see, e.g., [27, Thm. VI.14]), Theorem 3 implies that for \(\lambda \mapsto A_\lambda \) analytic \(C^1(M, (\Lambda ^1)^{n\times n})\)-valued, if \(Id + W_{A_{\lambda _0}}^2\) is invertible for some value \(\lambda _0\), then this remains true for all complex values \(\lambda \) outside a discrete set, which from the Fredholm equations implies that \(I_{A_\lambda ,0}\) is injective for all such values (note that if \(I_{A,0}(f)=0\), then \(f\in C_0^\infty (M,\mathbb {C}^n)\), cf. Proposition 27 below). For similar purposes of generic uniqueness in inverse problems, prior uses of analytic Fredholm theory have appeared for instance in [30] in the case of the radiative transport equation, and in [31] in the case of Calderón’s inverse conductivity problem.
We then focus on obtaining estimates for the error operators \(W_A, W_{A,\perp }\), whose study starts in [10, 26] in the case without connection (call \(W\equiv W_0\) the corresponding operator). Obtaining transparent estimates is not obvious, as the constants derived in [10] are not well controlled by intrinsic geometric quantities. Such estimates have recently been obtained in [8] on surfaces with negative curvature, allowing non-trivial trapping. In an attempt to quantify simplicity and obtain more transparent error estimates, we recall that the absence of conjugate points is equivalent to the non-vanishing of the following function \(b:\mathcal {D}\rightarrow \mathbb {R}\) outside the set \(\{(\mathrm {x},v,0):\ (\mathrm {x},v)\in SM\}\), solution of
Since additionally, \(\lim _{t\rightarrow 0^+} \frac{|b(\mathrm {x},v,t)|}{t} = 1\) for every \((\mathrm {x},v)\in SM\), and since \(\mathcal {D}\) is compact, the following claim is obvious
in which case we say that (M, g) is a simple Riemannian surface with constants \(C_1, C_2\). A finer analysis of the Schwartz kernels of the error operators then allows to prove the following theorem. In the statement, for an \(n\times n\) matrix M, we denote \(\Vert M\Vert := ( {\text {tr }}(B^*B))^{\frac{1}{2}}\) its Frobenius norm.
Theorem 4
Let (M, g) be a simple Riemannian surface with constants \(C_1, C_2\) as in (7) and Gaussian curvature \(\kappa (\mathrm {x})\). Given the \(C^1\) connection A with curvature \(F_A\), let us denote \(\alpha _A := \sup _{(\mathrm {x},v)\in SM}\{ \Vert (A + A^*)/2\Vert (\mathrm {x},v)\}\) and \(\tau _\infty \) the diameter of M. There exist constants \(C,C'\) depending on \((n, C_1, C_2, \tau _\infty ,\alpha _A)\) such that
As consequences of Theorems 3 and 4, we obtain the following main conclusions.
Theorem 5
Let (M, g) be a simple surface and A a \(C^1\) connection. Then the following conclusions hold:
-
(i)
If \(\kappa \) is constant and A is flat, the operators \(W_A\) and \(W_{A,\perp }\) vanish identically and Theorem 1 implies that the transforms \(I_{A,0}, I_{A,\perp }, I_{-A^*, 0}\), and \(I_{-A^*, \perp }\) are all injective, with explicit, one-shot inversion formulas.
-
(ii)
Injectivity still holds if \((n, C_1, C_2, \tau _\infty , \alpha _A,\Vert \star F_A\Vert _\infty ,\Vert \mathrm{d}\kappa \Vert _\infty , \text {Vol }M)\) are such that the right-hand side of (8) is less than 1, with a Neumann series type inversion.
-
(iii)
If (M, g) is such that the operator \(Id + W^2\) is injective, then for every \(\lambda \in \mathbb {C}\) outside a discrete set, the transforms \(I_{\lambda A,0}, I_{\lambda A,\perp }, I_{-\lambda A^*,0}\), and \(I_{-\lambda A^*,\perp }\) are all injective.
In the statement of Theorem 5, injectivity of \(I_{-A^*,0}\) and \(I_{-A^*,\perp }\) comes from the fact that one may consider the Fredholm equations (5)–(6) corresponding to \(I_{-A^*,0}\) and \(I_{-A^*,\perp }\), and since we prove in Lemma 13 that \(W_A\) and \(W_{-A^*,\perp }\) are \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) adjoints, then invertibility of \(Id + W_A^2\) is equivalent to invertibility of \(Id + W_{-A^*,\perp }^2\). It is conjectured that \(Id+W^{2}\) is injective on any simple surface.
Finally, we provide a range characterization for the operators \(I_{A,0}\) and \(I_{A,\perp }\) whenever the operator \(I_{-A^*,0}\) is injective. In order to obtain such a characterization, we must first establish a series of results building equivalence of injectivities between transforms with different connections. This takes us to formulating a few key results. In what follows, we denote by \(I_{A,m} := I_A|_{\Omega _m}\), where \(\Omega _{m}\) is defined as \(\Omega _{m}=\text {Ker}(V-im Id)\cap C^{\infty }(SM,\mathbb {C}^{n})\) and V is the vertical vector field (see Sect. 3).
-
If \(I_{A,0}\) is injective, then so is \(I_{A+\omega \mathbb {I}_n,0}\) for any scalar one-form \(\omega \). See Proposition 22.
-
If \(I_{A,0}\) is injective, then \(I_{A,m}\) is injective for any \(m\in \mathbb {Z}\). See Proposition 23.
-
\(I_{A,0}\) is injective if and only if \(I_{A,\perp }\) is injective. In particular, all conclusions above hold if we only assume \(I_{A,\perp }\) injective instead. See Propositions 26 and 23.
For the range characterization, we extend \(I_{A,\perp }\) to all functions in \(C^\infty (M,\mathbb {C}^n)\) and not just those vanishing at the boundary. One may define the formal operators \(P_{A,\pm }:= B_{A,-} H_\pm Q_{A,+}\), where \(B_{A,-}\) and \(Q_{A,+}\) are defined in Sect. 3.3 and \(H_\pm \) denote odd and even fiberwise Hilbert transforms (see Sect. 3). The operators \(P_{A,\pm }\), defined in the smooth setting on a space denoted \({{\mathcal {S}}}_A^{\infty }(\partial _{+}(SM),\mathbb {C}^n)\) (see (20)), are boundary operators which only depend on the scattering relation and the scattering data \(C_A\), and they allow to describe the ranges of \(I_{A,0}\) and \(I_{A,\perp }\) as follows.
Theorem 6
(Range characterization of \(I_{A,0}\) and \(I_{A,\perp }\)) Suppose that (M, g) is a simple surface and \(I_{-A^*,0}\) is injective, and let \(\mathcal {I}\in C^\infty (\partial _+ (SM),\mathbb {C}^n)\). Then the following claims hold.
-
(i)
\(\mathcal {I}\) belongs to the range of \(I_{A,0}:C^\infty (M,\mathbb {C}^n)\rightarrow C^\infty (\partial _+ (SM),\mathbb {C}^n)\) if and only if there exists \(w\in {{\mathcal {S}}}_A^{\infty }(\partial _{+}(SM),\mathbb {C}^n)\) such that \(\mathcal {I}= P_{A,-}w\).
-
(ii)
\(\mathcal {I}\) belongs to the range of \(I_{A,\perp }:C^\infty (M,\mathbb {C}^n)\rightarrow C^\infty (\partial _+ (SM),\mathbb {C}^n)\) if and only if there exists \(w\in {{\mathcal {S}}}_A^{\infty }(\partial _{+}(SM),\mathbb {C}^n)\) such that \(\mathcal {I}= P_{A,+}w\).
Such range characterizations were previously established in [26] in the case without connection, in [1, 23] in the case of unitary connections and Higgs fields, and recently in the case of the attenuated transform [3]. The range characterization for \(I_0\) was recently proved by the first author to be the generalization of the classical moment conditions for compactly supported functions in the Euclidean case, see [14, Theorem 2.3].
2.1 Outline
The remainder of the article is organized as follows: We recall generalities on the geometry of the unit circle bundle, transport equations with connection, with additional remarks on the symmetries in the data space \(L^2_\mu (\partial _+ (SM),\mathbb {C}^{n})\) in Sect. 3. In Sect. 4, we prove Theorem 1 and study the error operators \(W_A,W_{A,\perp }\) in detail, including the proof of Theorem 4. In Sect. 5, injectivity of ray transforms corresponding to different connections or different harmonic levels are inter-related, and the relation between the transform \(I_A\) over one-forms and the transform \(I_{A,\perp }\) is refined. Finally, based on additional preparatory results from Sect. 5 (namely, Proposition 22), Sect. 6 presents the range characterization and the proof of Theorem 6.
3 Preliminaries
3.1 Setting and Notation
Throughout this section, we will assume that (M, g) is a non-trapping surface with strictly convex boundary. As a consequence, it is simply connected (hence orientable).
3.1.1 Geometry of the Unit Tangent Bundle
We briefly recall standard notation for the unit sphere bundle SM, see e.g. [21] for more detail. The vector field \(X\in T(SM)\) can be completed into a global framing \(\{X,X_\perp ,V\}\) of T(SM) with structure equations
The Sasaki metric on T(SM) is then the unique metric making this frame orthonormal, with volume form which we denote \(\mathrm{d}\Sigma ^3\). This measure gives rise to an inner product space \(L^2(SM,\mathbb {C}^{n})\), where the circle action on tangent fibers induces the orthogonal decomposition
Upon defining \(\Omega _k = C^\infty (SM,\mathbb {C}^{n}) \cap H_k\), a function \(u\in C^\infty (SM,\mathbb {C}^{n})\) decomposes uniquely as \(u = \sum _{k\in \mathbb {Z}} u_k\) where each \(u_k\) belongs to \(\Omega _k\). If \(u\in L^2(SM,\mathbb {C}^{n})\), then each \(u_k\) belongs to \(H_k \cong q^k L^2(M,\mathbb {C}^{n})\), where q denotes a non-vanishing element of \(C^\infty (SM,\mathbb {C}) \cap \ker (V-iId)\), whose existence is guaranteed by simple connectedness.
3.1.2 Scattering Relation
For \((\mathrm {x},v)\in SM\), let us denote
both endpoints of the geodesic passing through \((\mathrm {x},v)\). Let \(\alpha :\partial (SM)\rightarrow \partial (SM)\) the scattering relation, i.e. \(\alpha |_{\partial _\pm (SM)} = \varphi _\pm |_{\partial _\pm (SM)}\).
3.1.3 Transport Equations on the Unit Tangent Bundle
As in the Sect. 1, for \(f:SM\rightarrow \mathbb {C}^n\) and A a \(GL(n,\mathbb {C})\)-connection, we define \(u_A^f\) to be the unique solution to the transport problem
Let us denote \(U_A:SM\rightarrow GL(n,\mathbb {C})\) the unique matrix solution W to the problem
From this solution, we define the scattering data
We also define the attenuation function \(E_A:\mathcal {D}\rightarrow GL(n,\mathbb {C})\) as
unique solution of the (x, v)-dependent ODE
and in terms of which many kernels will be expressed below. For h defined on \(\partial _+ (SM)\), define \(h_{\psi , A}\) the unique solution u to the transport problem
With the definition of \(U_A\), we have, for \((\mathrm {x},v)\in SM\),
or, equivalently for the last one,
As the matrix A is not necessarily skew-Hermitian, the connections A and \(-A^*\) are distinct, though we will see below that it is helpful to consider the transforms associated to both jointly. The first important identity to notice is
since both functions coincide with the unique solution W to the transport problem
3.1.4 Decomposition of \(X+A\) and the Guillemin–Kazhdan Operators
We may decompose \(X + A = \mu _+ + \mu _-\) where \(\mu _\pm : \Omega _k \rightarrow \Omega _{k\pm 1}\) is defined by \(\mu _\pm :=\eta _{\pm }^{A}:=\eta _\pm + A_{\pm 1}\), where \(\eta _{\pm } := \frac{1}{2} (X \pm i X_\perp )\) are the Guillemin–Kazhdan operators, see [9]. Then \(\frac{1}{i} (\mu _+ - \mu _-) = X_\perp - A_V\), where \(A_V := V(A) = i(A_1-A_{-1})\). Moreover,
In this paper, we work exclusively with the case in which M is a disk, and hence, we can consider global isothermal coordinates (x, y) on M such that the metric can be written as \(\mathrm{d}s^2=\mathrm{e}^{2\lambda }(\mathrm{d}x^2+\mathrm{d}y^2)\) where \(\lambda \) is a smooth real-valued function of (x, y). This gives coordinates \((x,y,\theta )\) on SM where \(\theta \) is the angle between a unit vector v and \(\partial /\partial x\). Then \(\Omega _k\) consists of all functions \(u=h(x,y) \mathrm{e}^{ik\theta }\) where \(h \in C^{\infty }(M,\mathbb {C}^n)\). In these coordinates, a connection \(A = A_z \mathrm{d}z + A_{\bar{z}} \mathrm{d}\bar{z}\) (with \(z = x+iy\)) takes the form \(A(x,y,\theta ) = \mathrm{e}^{-\lambda } (A_z(x,y) \mathrm{e}^{i\theta } + A_{\bar{z}}(x,y) \mathrm{e}^{-i\theta })\), and we can give an explicit description of the operators \(\mu _{\pm }\) acting on \(\Omega _{k}\). For \(\mu _{-}\) we have (cf. [20, Eq. (24)])
From this expression, we may derive the following lemma which will be used later on:
Lemma 7
Given \(f\in \Omega _{k-1}\), there is \(u\in \Omega _{k}\) and \(v\in \Omega _{k-2}\) such that \(\mu _{-}u=f\) and \(\mu _{+}v=f\).
Proof
We only prove the claim for \(\mu _{-}\); the one for \(\mu _{+}\) is proved similarly. If we write \(f=g\mathrm{e}^{i(k-1)\theta }\), using (13) we see that we only need to find \(h\in C^{\infty }(M,\mathbb {C}^n)\) such that
But it is well known that there exists a smooth \(F:M\rightarrow GL(n,\mathbb {C})\) such that \(\bar{\partial }F+A_{\bar{z}}F=0\), and hence the solvability of (14) reduces immediately to the standard solvability result for the Cauchy–Riemann operator, namely, given a smooth b, there is a such that \(\bar{\partial }a=b\). The existence of F above follows right away from the fact that a holomorphic vector bundle over the disk is holomorphically trivial [7, Theorems 30.1 and 30.4], see also [6, 16] for alternative proofs. \(\square \)
3.1.5 Hilbert Transform and Commutator Formulas
An important operator for what follows is the fiberwise Hilbert transform\(H:L^2(SM,\mathbb {C}^{n})\rightarrow L^2(SM,\mathbb {C}^{n})\), diagonal on the harmonic decomposition in the fiber, and such that \(H|_{H_k} = -i\,\mathrm{sign}(k) Id|_{H_k}\), with the convention \(\,\mathrm{sign}(0) =0\).
Using the splitting \(X+A=\mu _{+}+\mu _{-}\), it is immediate to derive the commutator formulas (see [21, Lemma 2.2] for instance)
as well as the following identities, obtained by computing \([H^2,X+A]\) in two ways:
3.2 Decompositions of the Data Space
Denote \(a(\mathrm {x},v) = (\mathrm {x},-v)\) the antipodal map (or rotation by \(\pi \)). A function f is even/odd on SM if \(f\circ a= +/- f\). Also define the antipodal scattering relation to be the mapping \(\alpha _a:\partial (SM)\rightarrow \partial (SM)\)
\(\alpha _a\) is an involution and \(\alpha _a(\partial _\pm (SM))\subset \partial _\pm (SM)\). It is straightforward to see that the function \(G(\mathrm {x},v) := U_A(\mathrm {x},-v)\) solves the transport problem
so that \(G(\mathrm {x},v) = U_A(\mathrm {x},v) C_A (a(\varphi _-(\mathrm {x},v)))\) for every \((\mathrm {x},v)\in SM\). In particular, this implies the relation
Writing this for \(U_A(\mathrm {x},v)\), we obtain the identity
We will use this to characterize the symmetries of the ray transforms over even and odd integrands. In particular, the identity (17) means that
Note also the obvious identities
Lemma 8
If \(f\in C^\infty (SM)\) satisfies \(f\circ a= \pm f\), then the data \(I_A f\) satisfy
Proof
We only treat the case of f even, the odd case being similar. We write
and the identity follows since \(C_A(a(\mathrm {x},v))^{-1} = C_A(\alpha (\mathrm {x},v))\) by (17). The proof is complete. \(\square \)
This motivates a decomposition of \(C^\infty (\partial _+ (SM)) = \mathcal {V}_{A,+} \oplus \mathcal {V}_{A,-}\), where we define
This decomposition is unique and given explicitly by \(h = h_{A,+} + h_{A,-}\) with
Such symmetries, via extension as first integrals of \(X+A\), generate even and odd functions on SM:
Lemma 9
If \(h\in \mathcal {V}_{A,+}\) (\(\mathcal {V}_{A,-}\)) then the function \(h_{\psi ,A}\) is even (odd) on SM.
Proof
Suppose \(h\in \mathcal {V}_{A,+}\) (the case of \(\mathcal {V}_{A,-}\) is similar). Then, for any \((\mathrm {x},v)\in SM\),
hence the proof. \(\square \)
In general, the decomposition \(\mathcal {V}_{A,+} \oplus \mathcal {V}_{A,-}\) is not orthogonal in \(L^2_\mu (\partial _+ SM)\). In this context of non-Hermitian connections, the more natural relation is the following.
Lemma 10
For any \(GL(n,\mathbb {C})\)-connection A, the following decompositions hold, orthogonal in the \(L^2_\mu (\partial _+(SM))\) sense:
In particular, if the connection is unitary \(A = -A^*\), then \(C^\infty (\partial _+(SM)) = \mathcal {V}_{A,+} {\mathop {\oplus }\limits ^{\perp }} \mathcal {V}_{A,-}\).
Proof
It is enough to prove the first equality, as the second follows by considering the connection \(-A^*\).
The claimed decomposition is unique, because for \(h\in C^\infty (\partial _+ (SM))\), a unique decomposition \(h = h_+ + h_- \in \mathcal {V}_{A,+} \oplus \mathcal {V}_{-A^*,-}\) is given by
obtained by fulfilling the conditions of \(\mathcal {V}_{A,+}\) and \(\mathcal {V}_{-A^*,-}\) and using that \(C_{-A^*} = (C_A^*)^{-1}\) by virtue of (12).
On to proving orthogonality of the decomposition, let \(h\in \mathcal {V}_{A,+}\) and \(w\in \mathcal {V}_{-A^*, -}\). We write
We then write
Plugging this last expression into the first equality and applying Santalo’s formula, we arrive at
since \(\tau \circ \varphi _-\) is even in v and by virtue of Lemma 9; \(h_{\psi ,A}\) is even in v; and \(w_{\psi ,-A^*}\) is odd in v. The lemma is proved. \(\square \)
3.3 Boundary Operators
Extending notation from [23], we define for \(w\in C(\partial _+ (SM), \mathbb {C}^n)\)
Note that \(Q_+ w\in C(\partial (SM), \mathbb {C}^n)\). We also define
where the second equality is established in [23, Lemma 5.1]. This is due in part to the fact that
so that the operator \(Q_{A,+}:{{\mathcal {S}}}_A^\infty (\partial _+ (SM),\mathbb {C}^n)\rightarrow C^\infty (\partial _+ (SM),\mathbb {C}^n)\) makes sense. We also introduce \(B_{A,\pm }:C^\infty (\partial (SM),\mathbb {C}^n)\rightarrow C^\infty (\partial _+ (SM),\mathbb {C}^n)\),
(Note the sign difference with [23].) \(B_{A,-}\) appears naturally in the fundamental theorem of calculus along a geodesic: for \((\mathrm {x},v)\in \partial _+ (SM)\),
so that
We state without proof the following straightforward claims.
Lemma 11
-
(i)
If \(h\in \mathcal {V}_{A,+}\), then \(Q_{A,+} h\) is even in v and \(Q_{A,-} h\) is odd in v.
-
(ii)
If \(h\in \mathcal {V}_{A,-}\), then \(Q_{A,+} h\) is odd in v and \(Q_{A,-} h\) is even in v.
-
(iii)
If \(q\in C^\infty (\partial (SM),\mathbb {C}^n)\) is even in v, then \(B_{A,\pm } q \in \mathcal {V}_{A,\pm }\).
-
(iv)
If \(q\in C^\infty (\partial (SM), \mathbb {C}^n)\) is odd in v, then \(B_{A,\pm } q \in \mathcal {V}_{A,\mp }\).
4 Inversion Formulas and Control of the Error Operators
For \(f:M\rightarrow \mathbb {C}^n, f\circ \pi :SM\rightarrow \mathbb {C}^n\) is an even function of v and \((X_\perp - A_V)f\) is odd in v, and hence from Lemma 8 we have
We will see below that it is somehow natural to consider the inversion of operators \(I_{A,0}, I_{A,\perp }, I_{-A^*, 0}\), and \(I_{-A^*, \perp }\) together.
4.1 Adjoints
For \(f\in C^\infty (M,\mathbb {C}^n)\) and \(h\in C^\infty (\partial _+ (SM), \mathbb {C}^n)\), we compute (denote \(\left\langle {\cdot },{\cdot }\right\rangle \) the Hermitian product on \(\mathbb {C}^n\))
So we deduce that
A similar argument with \(f\in C^\infty _0(SM)\), and using the fact that \(X_\perp ^* = -X_\perp \) when either function in the \(L^2(SM,\mathbb {C}^n)\) inner product vanishes at the boundary, yields that
A direct use of Lemma 9 and inspection on symmetries yields the following:
Lemma 12
4.2 Fredholm Equations for \(I_{A,0}\) and \(I_{A,\perp }\)—Proof of Theorem 1
As discussed in the Sect. 1, let us define the operators
We now prove Theorem 1 before studying the operators \(W_{A}\) and \(W_{A,\perp }\) further. For the proof below, let us make the comment that, in terms of solutions of elementary transport problems of the form \(u_A^f\) and \(h_{A,\psi }\) defined in Sect. 3, the solution to the problem,
is \(u = u_A^f + (w - I_{A}f)_{\psi ,A}\), and the solution to the problem,
is \(u = u_A^f + ((C_A^{-1} h)\circ \alpha )_{\psi , A}\).
Proof of Theorem 1
Here and below, for a function \(u(\mathrm {x},v)\) defined on SM, we write the even/odd decomposition with respect to v as \(u = u_+ + u_-\), where \(u_{\pm }(\mathrm {x},v):= (u(\mathrm {x},v)\pm u(\mathrm {x},-v))/2\).
We first prove the inversion formula for \(I_{A,0}\), i.e., Eq. (5).
Start from the equation
so that \(u_A^f|_{\partial _+ (SM)} = I_{A,0} f\). Direct application of the left equation of (16) to the transport equation gives
where the last step comes from the fact that \(\pi _0 (X_\perp - A_V) H u_{A,+}^f = 0\). It now remains to write a transport problem for \(Hu_{A,-}^f\), for which we use the formula for \([H,X+A]\):
which upon projecting onto even harmonics, yields
This equation gives us \(Hu_{A,-}^f = u_A^{W_Af} + h_{\psi ,A}\), where
Plugging this expression of \(Hu_{A,-}^f\) back into (25) yields the formula
Finally, inspection of symmetries shows that \(B_{A,-}HQ_{A,-} I_{A,0} f \in \mathcal {V}_{A,+}\), so that using Lemma 9 and the expression of \(I_{-A^*,\perp }^*\), it is annihilated by \(I_{-A^*,\perp }^*\). As a conclusion, we arrive at (5).
We now prove the inversion formula for \(I_{A,\perp }\), i.e., Eq. (6).
Start from the equation
so that \(u_A^{(X_\perp -A_V)f}|_{\partial _+ (SM)} = I_{A,\perp } f\). Note that since \(I_{A,\perp }f \in \mathcal {V}_{A,-}, Q_{A,-} I_{A,\perp }f\) is even in v on SM, and in particular we have
Applying the Hilbert transform to the transport equation above, using the commutators and projecting onto odd harmonics, we obtain
From this equation, we get that the function \((Hu_{A,+}^{(X_\perp -A_V)f} - f)\) is nothing but
where
Upon applying \(\pi _0\) (fiber average) to (28), we obtain
As in the inversion of \(I_{A,0}\), we notice that \(B_{A,-} H Q_{A,-} I_{A,\perp } f\in \mathcal {V}_{A,-}\) and as such is annihilated by \(I_{-A^*,0}^*\). As a conclusion, the reconstruction formula, in its final form, looks like (6). \(\square \)
4.3 Properties of the Error Operators
Unlike the geodesic case studied in [26], \(W_A\) and \(W_{A,\perp }\) are not always \(L^2(M)\)-adjoints. Consider the Eqs. (5) and (6) corresponding to the connection \(-A^*\). They give
Inspecting the right-hand sides suggests that, for instance, taking the adjoint equation to (5) would yield (30). A partial answer to this heuristic guess is to establish:
Lemma 13
The operators \(W_A\) and \(W_{-A^*,\perp }\) are \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) adjoints. As a consequence, so are \(W_{-A^*}\) and \(W_{A,\perp }\). In particular, if \(A = -A^*\), then \(W_A\) and \(W_{A,\perp }\) are adjoints.
Proof
It is enough to check it for \(f,g\in C^\infty _0(M)\). We compute
The crossed term is zero because \(I_{A,0}f\in \mathcal {V}_{A,+}\) while \(I_{-A^*,\perp }g \in \mathcal {V}_{-A^*, -}\) and both spaces are orthogonal by virtue of Lemma 10. The lemma is proved. \(\square \)
The next result establishes that in the case where the metric is simple, the reconstruction formulas (5), (6), (29), and (30) are in fact Fredholm equations, as the operators \(W_{A,0}\) and \(W_{A,\perp }\) are compact. In order to prove this, we need to make explicit their Schwartz kernels, which in turns requires some recalls about Jacobi fields.
4.3.1 Jacobi Fields and Simplicity
Variations of the exponential map are computed following [11]. For \(\xi \in T_{(\mathrm {x},v)} SM\) uniquely written as \(\xi = a(0) X_{(\mathrm {x},v)} + b(0) X_{\perp (\mathrm {x},v)} + c(0) V_{(\mathrm {x},v)}\), there exist scalar functions \(a(\mathrm {x},v,t),b(\mathrm {x},v,t),c(\mathrm {x},v,t)\) such that (keeping \((\mathrm {x},v)\) implicit)
where Y(t) denotes \(Y(\varphi _t(\mathrm {x},v))\) for \(Y \in \{X,X_\perp ,V\}\). Due to the structure equations, we deduce that a, b, c, defined on \(\mathcal {D}\), solve the system
Particular Jacobi fields of interest are \(\mathrm{d}\varphi _t (X_\perp )\) described by \((a_1 \equiv 0, b_1, c_1)\) with initial condition \((a_1,b_1,c_1)(0) = (0,1,0)\), and \(\mathrm{d}\varphi _t (V)\) described by \((a_2\equiv 0, b_2, c_2)\) with initial condition \((a_2,b_2,c_2)(0) = (0,0,1)\). In particular, as stated in Sect. 2, the absence of conjugate points on a surface (M, g) is equivalent to the non-vanishing of the function \(b_2\) outside \(\{t=0\}\).
4.3.2 Kernels of \(W_A\) and \(W_{A,\perp }\)
We now make explicit the kernels of the operators \(W_A\) and \(W_{A,\perp }\) defined in (24), by showing the following:
Lemma 14
The operators \(W_A, W_{A,\perp }\) take the form
with respective kernels, in exponential coordinates, given by
The proof of Lemma 14 makes use of the following property, whose proof we relegate to the Appendix:
Lemma 15
For every \((\mathrm {x},v)\in SM\),
Proof of Lemma 14
Proof of (31). Using the definition (24),
with \(E_A\) defined in (11). In this expression, the only term which differentiates f is given by \(X_\perp (f(\varphi _t(\mathrm {x},v)))\), which we rewrite as
The corresponding term can then be rewritten as
In the last right-hand side, the first term vanishes identically and the second cancels out the boundary term in (33) thanks to Lemma 15. We then arrive at an expression for \(w_A\) as
which yields (31) after applying a product rule.
Proof of (32). On to \(W_{A,\perp }\), we write
We rewrite the only term which differentiates f as
Integrating this term by parts on \(S_\mathrm {x}\) in the expression of \(W_{A,\perp } h(\mathrm {x})\), this creates a boundary term of the form
which vanishes since by assumption \(h\in C_0^\infty (M)\). For the remaining term, we obtain the expression for \(w_{A,\perp }\) as
hence (32). \(\square \)
4.3.3 The Operators \(W_A\) and \(W_{A,\perp }\) are Compact
In what follows, for a \(\mathbb {C}^{n\times n}\) matrix B, we denote \(\Vert B\Vert = ( {\text {tr }}(B^* B))^{\frac{1}{2}}\) its Frobenius norm. For an operator of the form
where we have defined \({{\mathcal {W}}} (\mathrm {x},\mathrm {y}) := \frac{1}{2\pi } \frac{w(\mathrm {x},Exp_\mathrm {x}^{-1} (\mathrm {y}))}{b_2(\mathrm {x}, Exp_\mathrm {x}^{-1} (\mathrm {y}))}\), we may obtain an estimate on the \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) norm of W by computing
whenever the right-hand side is finite, and where \(\Vert \cdot \Vert _\rho \) denotes the spectral norm on \(\mathbb {C}^{n\times n}\). Using that \(\Vert \cdot \Vert _\rho \le \Vert \cdot \Vert \) and changing variable \(\mathrm {y}= Exp_\mathrm {x}(v,t)\), we arrive at the following estimate, to be used below
which implies both continuity and compactness of W whenever the right-hand side is finite.
Lemma 16
The operators \(W_A\) and \(W_{A,\perp }\) (and by duality via Lemma 13, \(W_{-A^*}\) and \(W_{-A^*, \perp }\)) are \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) compact.
Proof
The proof mainly consists in looking at the behavior of \(w_A\) and \(w_{A,\perp }\) defined in (31) and (32) near \(t =0\). Near \(t=0\), the following expansions hold:
where we used that
In particular, \(V(E_A^{-1}(\mathrm {x},v,t))/b_2(\mathrm {x},v,t) = -A_V(\mathrm {x},v) + \mathcal {O}(t)\). Together with the fact that the functions \(\frac{Vb_2}{b_2^2}\) and \(V\left( \frac{b_1}{b_2} \right) \) vanish as \(t\rightarrow 0\) (see for instance [12, 26]), this allows to deduce that
This means in particular that the function \(\frac{k(\mathrm {x},v,t)}{b_2(\mathrm {x},v,t)}\) where \(k \in \{w_A, w_{A,\perp }\}\) is bounded near \(t=0\). Since it inherits the regularity of \(b_1, b_2, A\) outside \(t=0\) and \(b_2\) does not vanish outside \(\{t=0\}\) because (M, g) is simple, the only problem was at \(t=0\). For each operator, changing variables \(\mathrm {y}(v,t) = \gamma _{\mathrm {x},v}(t)\) with change of volume \(\mathrm{d}M_\mathrm {y}= |b_2(\mathrm {x},v,t)|\, \mathrm{d}t\ \mathrm{d}S(v)\), the Schwarz kernels of \(W_A\) and \(W_{A,\perp }\) are of the form \(\frac{k(\mathrm {x},v(\mathrm {y}),d(\mathrm {x},\mathrm {y}))}{b_2(\mathrm {x},v(\mathrm {y}),d(\mathrm {x},\mathrm {y}))}\) with \(k\in \{w_A,w_{A,\perp }\}\), and they are bounded near the diagonal and away from the diagonal. Since \(M\times M\) has finite volume, these kernels belong to \(L^2(M\times M)\), and thus the operators \(W_A, W_{A,\perp }: L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) are compact. \(\square \)
4.4 Analytic Fredholm Approach—Proof of Theorem 3
We start with some preliminary estimates for ordinary differential equations. Let us define, for \(A\in C^1(M,(\Lambda ^1)^{n\times n})\),
In addition, for a function \(B:\mathcal {D}\rightarrow \mathbb {C}^{n\times n}\), we define \(\Vert B\Vert _{F,\infty } := \sup _{(\mathrm {x},v,t)\in \mathcal {D}} \Vert B(\mathrm {x},v,t)\Vert \), making \((C^0(\mathcal {D},\mathbb {C}^{n\times n}),\Vert \cdot \Vert _{F,\infty })\) into a Banach space. Moreover, since the Frobenius norm is submultiplicative, so is \(\Vert \cdot \Vert _{F,\infty }\). We also consider the Banach space \((C^1(\mathcal {D},\mathbb {C}^{n\times n}), \Vert \cdot \Vert _{F,\infty ,1})\) with the norm
also submultiplicative. Now for a problem of the form
a priori estimates yield an estimate of the form
Moreover, for \(Y \in \{X,X_\perp ,V\}\), one may derive ODE’s for YU of the form
which upon using (36) implies an estimate of the form
where \(C'\) is independent of U, A, or F. Similarly, a problem of the form
is equivalent to a problem for \(W = U - \mathbb {I}_n\):
for which (37) applies. Combining this with the triangle inequality, and using that \(\Vert A(\varphi _t)\Vert _{F, \infty , 1}\le \Vert A\Vert _{C^1(M,(\Lambda ^1)^{n\times n})}\), we arrive at
Finally, let us note that estimates (36), (37), and (38) also hold if the connection is right-multiplied in the ODEs considered instead of left-multiplied. With these estimates in mind, we are ready to prove Theorem 3.
Proof of Theorem 3
We prove the statement for \(W_{A_\lambda }\) only, as the proof from \(W_{A_\lambda ,\perp }\) is similar. Recall that \(w_A\), the kernel of \(W_A\) up to exponential map, is given by
with \(E_A\) as defined in (11). Estimates on \(W_{A_\lambda }\) boil down to studying how \(\lambda \mapsto w_{A_\lambda }\) behaves in the \(C^0(\mathcal {D},\mathbb {C}^{n\times n})\) topology, which in turn requires to look at how \(\lambda \mapsto E_{A_\lambda }\) behaves in the \(C^1(\mathcal {D},\mathbb {C}^{n\times n})\) topology. Denote \(E_\lambda = E_{A_\lambda }\) for short. Fix \(\lambda _0\in \mathbb {C}\) and consider \(\lambda \) close to \(\lambda _0\), write by assumption
and where \(A'_{\lambda _0} \in C^1(M, (\Lambda ^1)^{n\times n})\). Define \(E'_{\lambda _0}\) the unique solution to
since \(A'_{\lambda _0}(\varphi _t) E_{\lambda _0}(t)\in C^1(\mathcal {D},\mathbb {C}^{n\times n})\), (37) gives us that \(E'_{\lambda _0}\in C^1(\mathcal {D},\mathbb {C}^{n\times n})\). Moreover, we have the relation \(E_\lambda = E_{\lambda _0} + (\lambda -\lambda _0) E'_{\lambda _0} + (\lambda -\lambda _0) F_\lambda \), where \(F_\lambda \) satisfies
Since \(\lim _{\lambda \rightarrow \lambda _0} \Vert B_\lambda (\varphi _t) E_\lambda + A'_{\lambda _0}(\varphi _t) (E_\lambda -E_{\lambda _0})\Vert _{C^1(\mathcal {D},\mathbb {C}^{n\times n})} = 0\), then estimate (37) implies that \(\lim _{\lambda \rightarrow \lambda _0} \Vert F_\lambda \Vert _{C^1(\mathcal {D},\mathbb {C}^{n\times n})} = 0\) thus \(\lambda \mapsto E_\lambda \) is an analytic \(C^1(\mathcal {D},\mathbb {C}^{n\times n})\) function. Similarly, we obtain
Thus, upon defining
we obtain that \(w_{A_{\lambda _0}}'\in C^0(\mathcal {D},\mathbb {C}^{n\times n})\) and
where the estimate on \(v_\lambda \) easily follows from the estimates on \(G_\lambda \) and \(B_\lambda \). In addition, let us analyze the behavior of \(w'_{A_{\lambda _0}}\) near \(t=0\) since the ratio \(w'_{A_{\lambda _0}}/b_2\) will be continuous, hence bounded, elsewhere. Looking at the ODE satisfied by \(E'_{\lambda _0}\), we have, near \(t=0\), the expansions
With the additional vanishing of \(V(b_1/b_2)\) as \(t\rightarrow 0\), this is enough to establish that \(\lim _{t\rightarrow 0} w'_{A_{\lambda _0}} = 0\), and from the relation (39), we also have \(\lim _{t\rightarrow 0} v_\lambda = 0\). Thus, \(w'_{A_{\lambda _0}}/b_2\) and \(v_\lambda /b_2\) are bounded on \(\mathcal {D}\), and since \(\mathcal {D}\) has finite volume, this clearly implies
which, by virtue of estimate (35), implies that the operator \(W'_{A_{\lambda _0}}\) defined in terms of the kernel \(w'_{A_{\lambda _0}}\) as in (34), is \(L^2(M,\mathbb {C}^n)\rightarrow L^2(M,\mathbb {C}^n)\) continuous. Reasoning similarly on \(V_\lambda \), we also obtain that
Theorem 3 is proved. \(\square \)
4.5 Further Error Estimates—Proof of Theorem 4
We now refine the previous result by estimating the operator norms of \(W_A\) and \(W_{A,\perp }\) explicitly. In particular, we now define two functions of interest which appeared in the proof of Lemma 16:
in terms of which the kernels \(w_A\) and \(w_{A,\perp }\) are written as
Using the fact that
we now establish the following:
Lemma 17
The functions \(K_\ell \) for \(\ell = 1,2\) satisfy the following ODEs on \(\mathcal {D}\):
Proof
We only treat \(K_2\), as the case of \(K_1\) is similar. We compute directly, keeping the variables \((\mathrm {x},v)\) implicit and writing \({\dot{E}} \equiv \frac{\mathrm{d}E}{\mathrm{d}t}\):
Note that \(b_2\) is a scalar function, so we can commute it. We also have used that \(\dot{b}_2 = -c_2\). The proof is complete. \(\square \)
The next result, whose proof we relegate to the Appendix, is an explicit bound on the quantities \(V\left( \frac{b_1}{b_2} \right) \) and \(V\left( \frac{1}{b_2} \right) \). Such quantities first appeared in [26] and arise as the kernels, up to exponential map, of the error operators W and \(W^*\) of the geodesic ray transform without connection. In what follows, we recall that for (M, g) a simple surface, we may define \(C_1(M,g) := \min _{\mathcal {D}} \frac{|b_2(\mathrm {x},v,t)|}{t} >0\) and \(C_2(M,g) := \max _{\mathcal {D}} \frac{|b_2(\mathrm {x},v,t)|}{t}>0\).
Lemma 18
Let (M, g) a simple surface with constants \(C_1, C_2\) as in (7). Then the functions \(V\left( \frac{b_1}{b_2} \right) \) and \(V\left( \frac{1}{b_2} \right) \) satisfy the following estimates:
We now prove the main result of this section, Theorem 4.
Proof of Theorem 4
In order to apply estimate (35) to \(W_A\) and \(W_{A,\perp }\), we now bound the functions \(w_A\) and \(w_{A,\perp }\) using expressions in (41). The equation satisfied by \(E = E_A\) implies
so that we may obtain the estimate
The same estimate holds for \(E^{-1}\). Integrating (42) using E as integrating factor, we deduce the following integral representations (we keep \((\mathrm {x},v)\) implicit)
where in the last equality, we have used (52) (proved in the Appendix). We now bound the Frobenius norm of the left-hand sides, using submultiplicativity of \(\Vert \cdot \Vert _F\):
Similarly, using that \(|b_2(\varphi _s,t-s)|\le C_2(t-s)\), we can arrive at the exact same bound for \(\Vert b_2K_1-b_1K_2\Vert (t)\). Given the form of \(w_A\) and \(w_{A,\perp }\) in (41), and the fact that bounds on \(V\left( \frac{b_1}{b_2} \right) \) and \(V\left( \frac{1}{b_2} \right) \) are the same and bounds on \(\Vert K_2\Vert \) and \(\Vert b_2 K_1 - b_1 K_2\Vert \) are the same, this will yield the same bound on \(w_A\) or \(w_{A,\perp }\). Therefore, let us focus on \(w_{A,\perp }\): using the previous bound together with Lemma 18
Using \((a+b)^2\le 2(a^2 + b^2)\) to bound \(\Vert w_{A,\perp }(\mathrm {x},v,t)\Vert ^2\) and using (35), we arrive at
Therefore, (8) holds with
valid both for \(W_A\) and \(W_{A,\perp }\) as explained above. Theorem 4 is proved. \(\square \)
5 Injectivity Equivalences and Implications
The purpose of this section is twofold. It first clarifies the relation between the transform \(I_{A}\) restricted to one-forms, and the transform \(I_{A,\perp }\). Second, it serves as preparation for the range characterization results stated in the next section.
5.1 On the Range Decomposition of \(I_A\)
We now prove that the range of \(I_{A}\) acting on 1-forms (i.e. acting on \(\Omega _{-1}\oplus \Omega _{1}\)) decomposes into (i) the range of \(I_{A,\perp }\) defined on \(H^1_0\) and (ii) the ranges of \(I_A\) restricted to \(\ker ^{\pm 1} \mu ^*_{\pm } := \Omega _{\pm 1} \cap \ker \mu ^*_{\pm }\) (or their \(L^2\) versions), and that the sum vanishes if and only if all three components vanish.
The first thing to observe is the following lemma which follows right away form the ellipticity of \(\mu _{\pm }\).
Lemma 19
Let (M, g) be a Riemannian surface with boundary and A a \(C^1\) connection. The following decompositions hold, orthogonal for the \(L^2(SM,\mathbb {C}^{n})\) inner product (hence unique):
-
(i)
For every \(f\in \Omega _1\), there exists \(v\in \Omega _0\) with \(v|_{\partial M} = 0\) and \(g_1\in \ker ^1 \mu _+^*\) such that \(f = \mu _+ v + g_1\).
-
(ii)
For every \(f\in \Omega _{-1}\), there exists \(v\in \Omega _0\) with \(v|_{\partial M} = 0\) and \(g_{-1}\in \ker ^{-1} \mu _-^*\) such that \(f = \mu _- v + g_{-1}\).
Lemma 19 implies that any one-form \(\omega = \omega _1 + \omega _{-1}\) decomposes uniquely as follows: write \(\omega _1 = \mu _+ a + g_1\) and \(\omega _{-1} = \mu _- b + g_{-1}\), with \(a,b \in H^1_0(M)\) and \(g_{\pm 1}\in \ker ^{\pm 1} \mu _{\pm }^*\). Upon defining \(g_p := (a+b)/2\) and \(g_s := (i(a-b)/2)\), the sum can be rewritten as
The transport equation
can then be rewritten as
where the functions u and \(u + g_p\) agree on \(\partial SM\) so that
We will say that \(I_{A}\) acting on 1-forms is solenoidal injective if whenever \(I_{A}(\omega )=0\), there is smooth \(p:M\rightarrow \mathbb {C}^{n}\) with \(p|_{\partial M}=0\) such that \(\omega =d_{A}p=dp+Ap\). Lemma 19 implies the following:
Lemma 20
For any \(C^1\) connection \(A, I_A\) is solenoidal injective on one-forms if and only if, for any \(f\in C_0^1(M), g_1\in \ker ^{1} \mu _+^*\) and \(g_{-1} \in \ker ^{-1} \mu _-^*, I_A g_{-1} + I_{A,\perp } f + I_A g_1 = 0\) implies \(f = g_1 = g_{-1} = 0\).
Proof
\((\implies )\) Suppose \(I_A\) solenoidal injective and assume that \(I_A g_{-1} + I_{A,\perp } f + I_A g_1 = 0\). Then from solenoidal injectivity, this means that there exists a function h defined on M vanishing on \(\partial M\) such that
rewritten differently this means that \((\mu _+ + \mu _-)h = g_1 - i (\mu _+ - \mu _-) f + g_{-1}\), which upon projecting onto Fourier modes 1 and \(-1\), implies
Uniqueness of such decompositions implies \(h+if = h-if = g_1 = g_{-1} = 0\), so \(f = h =0\).
Let \(\omega \) be such that \(I_A = 0\). Using Lemma 19, we can write
with \(g_p, g_s\) functions on M vanishing at \(\partial M\) and \(g_{\pm 1} \in \ker ^{\pm 1} \mu _{\pm }^*\). Then
which by assumption implies \(g_s = g_{-1} = g_1 = 0\), thus \(\omega = (X+A) g_p\). Hence \(I_A\) is solenoidal injective on one-forms. \(\square \)
Given a \(\mathbb {C}^{n}\)-valued 1-form \(\omega =\omega _{-1}+\omega _{1}\), there is an alternative decomposition to (43) which uses slightly different boundary conditions. For this, one considers the elliptic operator
where \(\Lambda ^{1}(M)\) is the set of all \(\mathbb {C}^{n}\)-valued 1-forms, given by
Now let \({\mathfrak {H}}_{A}\) denote the finite-dimensional space of 1-forms h such that \(d_{A}h=d_{A}\star h=0\) and \(j^*h=0\), where \(j:\partial M\rightarrow M\) is the inclusion map. Using D it is easy to show that given \(\omega \in \Lambda ^{1}(M)\) there are \((p,f)\in C^{\infty }_{0}(M,\mathbb {C}^{n})\times C^{\infty }(M,\mathbb {C}^{n})\) and \(h\in \mathfrak {H}_{-A^{*}}\) such that
Note that \(\mathfrak {H}_{-A^{*}}\) is the ortho-complement to the range of D (compare this with [23, Lemma 6.1]). Observe also that we can express (44) as
where \(h_{\pm 1}\in \ker ^{\pm 1} \mu _{\pm }^*\), but the difference with (43) is that now we do not require f to vanish at the boundary and instead we have \(j^{*}h=0\). We will return to this alternative decomposition after proving Theorem 6.
5.2 Injectivity for Scalar Perturbations of Connections
Given a connection A on a simple surface (M, g), we first start by giving a characterization of the injectivity for \(I_{A,0}\). Recall that a function f defined on SM is so-called (fiberwise) holomorphic (resp. antiholomorphic) if \((Id + iH)f = f_0\) (resp. \((Id - iH)f = f_0\)).
Proposition 21
(Characterization of injectivity of \(I_{A,0}\)) Let A be a \(GL(n,\mathbb {C})\)-connection. Then \(I_{A,0}\) is injective if and only if the following is true: for any \(f,u \in C^\infty (SM,\mathbb {C}^{n})\) satisfying \((X+A)u = -f\) with \(u|_{\partial SM} =0\),
-
(i)
If f is holomorphic and even, then u is holomorphic, odd.
-
(ii)
If f is antiholomorphic and even, then u is antiholomorphic and odd.
Proof
\((\implies )\) Suppose \(I_{A,0}\) injective. We only prove (i), as (ii) is similar. Let u, f as in the statement with f holomorphic. Then \((Id - iH)f = f_0\). Moreover, projecting the transport equation onto odd harmonics, we obtain \((X+A)u_+ = 0\) with boundary condition \(u_+|_{\partial SM} = 0\), hence \(u_+ = 0\), thus u is odd. We then compute
which upon integrating along geodesics implies that
\(I_{A,0} [f_0 -i ((X_\perp - A_V)u)_0 ] = 0\). By assumption, this implies
\(f_0 + i((X_\perp - A_V)u)_0 = 0\). In particular,
\((X+A)(Id-iH)u = 0\) with \((Id-iH)u|_{\partial SM} = 0\), hence \((Id-iH)u = 0\), which means that u is holomorphic, hence the proof. 
Suppose (i), (ii) are satisfied. Let f be a smooth function such that \(I_{A,0} f = 0\), then there exists \(u:SM\rightarrow \mathbb {C}^n\) with \(u|_{\partial SM} = 0\) and such that \((X+A)u = -f\). f is even, both holomorphic and antiholomorphic, thus by (i) and (ii), u is odd, both holomorphic and antiholomorphic, thus \(u=0\), hence \(f = 0\). Proposition 21 is proved. \(\square \)
The next result relies on the key concept of holomorphic integrating factor for scalar connections, which we now recall. Given a one-form \(\omega \), there exists \(v:SM\rightarrow \mathbb {C}\) holomorphic, even solution of \(Xv = - \omega \). This is based on injectivity of the unattenuated transform \(I_0\), cf. [21, Theorem 4.1]. The construction goes as follows: First one may write \(\omega = Xf + X_\perp g\) for g vanishing at \(\partial M\). Then we are left looking for v such that \(X(v+f) = - X_\perp g\). One can construct \(u = (Id + iH) h_\psi \) with \(h_\psi \) even such that
By surjectivity of \(I_0^*\), one can find h such that \(I_0^* h = 2\pi (h_\psi )_0 = -2\pi ig\) and for such an h, the function \(v = -f + (Id + iH) h_\psi \) is a holomorphic, even solution of \(Xv = -\omega \). As a result, the functions \(\mathrm{e}^v\) and \(\mathrm{e}^{-v}\) are non-vanishing holomorphic, even, solutions of \(X \mathrm{e}^{\pm v} \pm \omega \mathrm{e}^{\pm v} = 0\). Using the same h, we can then construct \(w = -f - (Id - iH)h_\psi \), antiholomorphic solution of \(Xw = -\omega \) giving rise to antiholomorphic integrating factors \(\mathrm{e}^{\pm w}\) solutions of \(X \mathrm{e}^{\pm w} \pm \omega \mathrm{e}^{\pm w} = 0\).
With the use of such integrating factors, we are then able to establish the following.
Proposition 22
For any \(GL(n,\mathbb {C})\)-connection A, if \(I_{A,0}\) is injective, then for any smooth one-form \(\omega \), so is \(I_{A + \omega \mathbb {I}_n, 0}\).
Proof
Suppose \(I_{A,0}\) injective and let \(\omega \) be a one-form. We use the characterization from Proposition 21 to show that \(I_{A+\omega \mathbb {I}_n,0}\) is injective by satisfying (i), (ii). Let u, f be such that \((X+A +\omega )u = -f\) with \(u|_{\partial SM} = 0\). If f is holomorphic even, then u is odd since \((X+A+\omega )u_+ = 0\) with zero boundary condition. Let \(\mathrm{e}^v\) a holomorphic, even, integrating factor for \(\omega \), then we can recast \((X+A + \omega )u = -f\) as \((X+A) (\mathrm{e}^{-v}u) = - \mathrm{e}^{-v}f\), where \(\mathrm{e}^{-v} f\) is holomorphic, even and \(\mathrm{e}^{-v}u\) vanishes at \(\partial SM\). Then since A satisfies (i), this implies that \(\mathrm{e}^{-v}u\) is holomorphic, odd, and hence \(u = \mathrm{e}^{v}(\mathrm{e}^{-v}u)\) is holomorphic, odd. The proof of (ii) is similar. \(\square \)
Such a result allows to derive injectivity results for several restrictions of \(I_A\) to other subspaces of \(C^\infty (SM)\), as they amount to studying transforms with connections which are translated from one another by a scalar one-form. Here and below, we denote \(I_{A,k}\) the transform \(I_A\) restricted to \(\Omega _k\).
Proposition 23
Suppose \(I_{A,0}\) injective, then the following conclusions hold.
-
(i)
For any \(k\in \mathbb {Z}\), the transform \(I_{A,k}\) is injective.
-
(ii)
\(I_A\) is solenoidal injective over one-forms. In particular, \(I_{A,\perp }\) is injective.
Remark 24
In particular, both statements imply that \(I_{A,k}|_{\ker ^k \mu _+^*}\) and \(I_{A,-k}|_{\ker ^{-k} \mu _-^*}\) are both injective for every \(k=0,1,2\dots \). However, this can be proved to always hold, see Proposition 27 below.
Proof
Suppose \(I_{A,0}\) injective.
Proof of (i). Let \(f\in \Omega _k\) such that \(I_{A,k} f = 0\). Write \(f = q^k \tilde{f}\) for q a non-vanishing section of \(\Omega _1\) and \(\tilde{f}:M\rightarrow \mathbb {C}^n\). Then if u is the unique solution to
the function \(q^{-k} u\) satisfies
so that \((q^{-k} u)|_{\partial _+ SM} = I_{A+kq^{-1} Xq \mathbb {I}_n, 0} \tilde{f}\). In particular, this implies that
Since \(I_{A+kq^{-1} Xq \mathbb {I}_n, 0}\) is injective by virtue of Proposition 22, then \(\tilde{f} = 0\), hence \(f = 0\).
Proof of (ii). Suppose \(I_A (\omega _1 + \omega _{-1}) = 0\), then there exists u such that \((X+A)u = - \omega _{-1} - \omega _1\) with \(u|_{\partial SM} =0\). In particular, u is even since \(u_-\) is a first integral of \(X+A\) vanishing at \(\partial SM\). If \(q\in \Omega _1\) is non-vanishing, the equation \((X+A)u = -\omega _{-1} - \omega _1\) can be rewritten as
If \(\mathrm{e}^v\) is a holomorphic, even, solution of \(X \mathrm{e}^v - q^{-1}Xq \mathrm{e}^v = 0\), then this equation can be rewritten as
and since the right-hand side is holomorphic and even, then by injectivity of \(I_{A,0}, \mathrm{e}^{-v}qu\) is holomorphic and odd. Then \(u = q^{-1} \mathrm{e}^v (\mathrm{e}^{-v} qu)\) has harmonic content no less than \(-1\) and since u is even, \(u_{-1} = 0\) as well, so u is holomorphic. Similarly using an antiholomorphic integrating factor, one may show that u is antiholomorphic, so we conclude that \(u = u_0\) with \(u_0|_{\partial M} = (u|_{\partial SM})_0 = 0\), and the relation \((X+A)u_0 = - \omega _1 - \omega _{-1}\) implies that \(I_{A}\) is solenoidal injective over one-forms. The proof is complete. \(\square \)
Finally, the next two propositions aim at showing that \(I_{A,\perp }\) injective implies that \(I_{A,0}\) injective.
Proposition 25
(Characterization of injectivity of \(I_{A,\perp }\)) Let A be a smooth GL(n) connection. Then \(I_{A,\perp }:C_0^\infty (M)\rightarrow C^\infty (\partial _+ (SM))\) is injective if and only the following is true: for any \(f,u \in C^\infty (SM)\) satisfying \((X+A)u = -f\) with \(u|_{\partial SM} =0, f\) odd and u even,
-
(i)
If \(f_k = 0\) for all \(k< -1\) and \(f_{-1}\perp \ker ^{-1} \mu _-^*\), then u is holomorphic.
-
(ii)
If \(f_k = 0\) for all \(k> 1\) and \(f_1 \perp \ker ^1 \mu _+^*\), then u is antiholomorphic.
Proof
\((\implies )\) Suppose \(I_{A,\perp }\) injective. We only prove (i) as (ii) is similar. Let u, f as in the statement with \(f_k = 0\) for all \(k< -1\) and \(f_{-1} \perp \ker ^{-1} \mu _-^*\). In particular, from Lemma 19, we can write \(f_{-1} = \mu _- v_0\) with \(v_0|_{\partial M} = 0\). Then \((Id - iH)f = 2f_{-1} = 2\mu _- v_0\). The function \((Id - iH) u\) solves
With \(2\mu _- v_0 = (X+A) v_0 - i (X_\perp -A_V) v_0\), the equation above becomes
Upon integrating along geodesics, we get
which by injectivity of \(I_{A,\perp }\) implies \(u_0 + v_0 = 0\). Then the transport equation above becomes
which implies \((Id - iH) u = -v_0\), thus u is holomorphic.
Suppose (i), (ii) satisfied. Let \(h\in C_0^\infty (M)\) such that \(I_{A,\perp } h = 0\), then there exists \(u:SM\rightarrow \mathbb {C}^n\) with \(u|_{\partial SM} = 0\) and such that \((X+A)u = -(X_\perp - A_V) h\). Then u is even since \((X+A) u_- = 0\) with \(u_-|_{\partial SM} = 0\). Then \(f = (X_\perp - A_V) h = i (\mu _- h- \mu _+ h)\) satisfies requirements for both (i) and (ii), so that u is both holomorphic and antiholomorphic. Then \(u= u_0\) with \(u_0|_{\partial M} = 0\). Then the relation \((X+A)u_0 = - (X_\perp - A_V) h\) implies \(\mu _+ (iu_0 + h) = 0\) and \(\mu _- (iu_0 - h) = 0\). Since \((iu_0\pm h)|_{ \partial M} = 0\), this implies \(iu_0 \pm h = 0\), hence \(u_0 = h = 0\), and \(I_{A,\perp }\) is injective. Proposition 25 is proved. \(\square \)
Proposition 26
Let A be a \(GL(n,\mathbb {C})\) connection. If \(I_{A,\perp }\) is injective, then so is \(I_{A,0}\).
Proof
Suppose \(I_{A,\perp }\) injective so that it satisfies (i) and (ii) in Proposition 25. Let f such that \(I_{A,0}f = 0\). Then there exists u odd such that \((X+A)u = -f\) with \(u|_{\partial SM} =0\). With q a non-vanishing section of \(\Omega _1\) and \(\omega := -q^{-1} Xq\), this implies
Let \(\mathrm{e}^w\) a holomorphic, even function such that \(X \mathrm{e}^w + \omega \mathrm{e}^w = 0\), then the equation above can be rewritten as
where \(qf\mathrm{e}^{-w}\) is odd and \(qu\mathrm{e}^{-w}\) is even. Moreover, \(qf\mathrm{e}^{-w}\) is holomorphic, thus satisfies the requirement for (i), hence \(qu\mathrm{e}^{-w}\) is holomorphic, hence \(u = u_{-1} + u_1 + u_3 \dots \). Using a similar argument with (ii), we can then cancel all \(u_k\)’s for \(k\ge 2\). Thus \(u = u_{-1} + u_1\). Projecting the equation \((X+A)u = -f\) onto \(\Omega _2\) and \(\Omega _{-2}\) gives \(\mu _+ u_1 = \mu _- u_{-1} = 0\), and since \(u_1|_{\partial SM} = u_{-1} |_{\partial SM} =0\), this implies \(u_1 = u_{-1} = 0\), hence \(f = 0\). \(\square \)
We conclude by proving the following result which has independent interest.
Proposition 27
Suppose there is \(u\in \Omega _{k}\) such that \(I_{A,k}(u)=0\). Then u has vanishing jet at \(\partial M\). In particular, \(I_{A,k}\) is injective when restricted to \(\text {Ker}\,\mu _{\pm }\).
Proof
The main observation is that \(N=I_{A,k}^{*} I_{A,k}\) is an elliptic classical \(\Psi DO\) of order \(-1\) in the interior of any simple manifold engulfing M, see [23, Sect. 5] and references therein. Hence consider a slightly larger simple manifold \(M_{1}\) containing M and extend u by zero to \(M_{1}\) (A is extended in any smooth way). Thus \(Nu=0\) in the interior of \(M_{1}\) and by elliptic regularity we deduce that u is smooth in \(M_{1}\). Since u vanishes outside M, this clearly implies that u has zero jet at the boundary of M.
Suppose in addition \(\mu _{-}(u)=0\). If we write \(u=h\mathrm{e}^{ik\theta }\) then using (13) we see that \(\bar{\partial }(h\mathrm{e}^{k\lambda })+A_{\bar{z}}h\mathrm{e}^{k\lambda }=0\). Using the existence of \(F:M\rightarrow GL(n,\mathbb {C})\) such that \(\bar{\partial } F+A_{\bar{z}}F=0\) as in Lemma 7 we see that \(\bar{\partial }(F^{-1}h\mathrm{e}^{k\lambda })=0\). Since h vanishes on \(\partial M\), this is enough to conclude that \(u=0\). A similar argument applies to elements in the kernel of \(\mu _{+}\) (or their adjoints). \(\square \)
6 Range Characterization
We start with a standard surjectivity result.
Theorem 28
Suppose \(I_{A,0}\) is injective. Given \(f\in C^{\infty }(M,\mathbb {C}^n)\) there exists \(h\in {{\mathcal {S}}}_{-A^{*}}^{\infty }(\partial _{+}(SM),\mathbb {C}^n)\) such that \(I_{A,0}^*(h)=f\).
The proof of this result is now well understood and we omit it. It follows from injectivity of \(I_{A,0}\) and the fact that \(I^*_{A,0} I_{A,0}\) is an elliptic classical \(\Psi DO\) of order \(-1\) in the interior of any simple manifold engulfing M, see [23, Sect. 5] and references therein.
From the expression \(I_{A,0}^* h = 2\pi (h_{-A^*,\psi })_0\), upon setting \(u = 2\pi \ h_{-A^*,\psi }\in C^\infty (SM,\mathbb {C}^n)\), Theorem 28 is equivalent to stating that for every \(f\in C^{\infty }(M,\mathbb {C}^n)\), there exists \(u\in C^\infty (SM,\mathbb {C}^n)\) satisfying \((X-A^*)u = 0\) and \(u_0 = f\).
The next result is less standard and it is based on the solvability result given by Lemma 7 and follows the strategy of the proof of [23, Theorem 5.5].
Theorem 29
Suppose \(I_{A,0}\) is injective. Given \(f\in C^{\infty }(M,\mathbb {C}^n)\) there exists \(h\in {{\mathcal {S}}}_{-A^{*}}^{\infty }(\partial _{+}(SM),\mathbb {C}^n)\) such that \(I_{A,\perp }^*(h)=f\).
Proof
Consider the purely imaginary 1-form
where \(q\in \Omega _1\) is nowhere vanishing (e.g. in global isothermal coordinates \(q=\mathrm{e}^{i\theta }\)). Observe that if \(u:SM\rightarrow \mathbb {C}^n\) is any smooth function then
where \(m\in \mathbb {Z}\). First we show the following result which is interesting in its own right:
Lemma 30
Suppose \(I_{A,0}\) injective. Given any \(f\in \Omega _m\), there exists \(w\in C^{\infty }(SM,\mathbb {C}^n)\) such that
-
1.
\((X-A^*)w=0\),
-
2.
\(w_m=f\).
Proof
Since \(I_{A,0}\) is injective, by Proposition 22, \(I_{A-ma \mathbb {I}_n,0}\) is injective (with a defined in (45)), thus by Theorem 28, there is \(u\in C^{\infty }(SM,\mathbb {C}^n)\) such that \(0 = (X-A^* +m \bar{a} \mathbb {I}_n)u= (X-A^* -m a \mathbb {I}_n)u\) and \(u_0=q^{-m}f\). If we let \(w:=q^{m}u\), then clearly \(w_m=f\) and by (46) we also have \((X-A^*)w=0\). \(\square \)
As before, consider the operators \(\mu _{\pm }=\eta _{\pm }^{A}=\eta _{\pm }+A_{\pm 1}\). Clearly,
We need the following solvability result which is a direct consequence of Lemma 7.
Lemma 31
Given \(f\in C^{\infty }(M,\mathbb {C}^n)\) there are \(w_1\in \Omega _1\) and \(w_{-1}\in \Omega _{-1}\) such that
Proof
Obviously the claim is equivalent to showing that there exists \(w_1\in \Omega _1\) such that \(\eta _{-}^{-A^{*}}(w_{1})=-f/4\pi i\) and \(w_{-1}\in \Omega _{-1}\) such that \(\eta _{+}^{-A^{*}}(w_{1})=f/4\pi i\). This follows directly from Lemma 7. \(\square \)
We are now in good shape to complete the proof of Theorem 29. Given \(f\in C^{\infty }(M,\mathbb {C}^n)\), we consider the functions \(w_{\pm 1}\in \Omega _{\pm 1}\) given by Lemma 31. By Lemma 30 we can find odd functions \(p,q\in C^{\infty }(SM,\mathbb {C}^n)\) solving the transport equation \((X-A^*)p=(X-A^*)q=0\) and with \(p_{-1}=w_{-1}\) and \(q_{1}=w_{1}\). Then the smooth function
satisfies \((X-A^*)w=0\) thanks to Eq. (47). Upon defining \(h = w|_{\partial _{+}SM}\) so that \(w = h_{\psi ,-A^*}\), we then obtain that h satisfies
as desired. \(\square \)
Finally, with the surjectivity Theorems 28 and 29, we are now ready to prove Theorem 6. As explained in the Sect. 1, define \(P_A:{{\mathcal {S}}}_A^{\infty }(\partial _{+}(SM),\mathbb {C}^n) \rightarrow C^{\infty }(M,\mathbb {C}^n)\), as follows:
The operator \(P_A\) is a boundary operator which only depends on the scattering relation and the scattering data \(C_A\). Upon splitting the Hilbert transform H into its projections onto even and odd harmonics (call them \(H_+\) and \(H_-\)), we obtain the splitting \(P_{A} = P_{A,+} + P_{A,-}\), where we have defined \(P_{A,\pm }:= B_{A,-} H_\pm Q_{A,+}\).
Proof of Theorem 6
For w defined on \(\partial _+ (SM)\), recall that \(Q_{A,+} w = w_{\psi ,A}|_{\partial SM}\) and that \(B_{A,-}(u|_{\partial (SM)}) = I_A( -(X+A)u )\). Using these considerations and the commutator formulas, we are able to derive
Similarly for \(P_{A,-}\),
Since it is assumed that \(I_{-A^*,0}\) is injective, by virtue of Theorems 28 and 29, the operators \(I_{-A^*,0}^*,I_{-A^*,\perp }^*:{{\mathcal {S}}}_A^{\infty }(\partial _{+}(SM),\mathbb {C}^n) \rightarrow C^{\infty }(M,\mathbb {C}^n)\) are surjective. Combining this surjectivity with the two factorizations above, claims (i) and (ii) follow. \(\square \)
Remark 32
Examining the proof above, we then see that
In addition, on the direct sum
since \(I^*_{-A^*,0}\) vanishes on \(\mathcal {V}_{A,-}\) and \(I^*_{-A^*,\perp }\) vanishes on \(\mathcal {V}_{A,+}\), one realizes that \(P_{A,\pm }\) coincides with the restriction \(P_A|_{\mathcal {V}_{A,\pm }}\). This is also true since, following previous observations, if \(h\in \mathcal {V}_{A,+}\), then \(Q_{A,+} h\) is even and if \(h\in \mathcal {V}_{A,-}\), then \(Q_{A,+}h\) is odd, which justifies the corresponding splitting of the Hilbert transform into odd and even parts in the previous definitions.
6.1 Comparison with the Range Characterization in [23]
We conclude this section by making a comparison between Theorem 6 and the range characterization of \(I_{A}\) acting on 1-forms in [23, Theorem 1.3] when A is skew-hermitian. The first thing to observe is that due to our sign conventions \(P_{A,+}\) is precisely \(-P_{+}\) in [23], so the main difference is the presence of \(I_{A}(\mathfrak {H}_{A})\), where \(\mathfrak {H}_{A}\) was introduced in Sect. 5.1. The reason why \(\mathfrak {H}_{A}\) does not appear in Theorem 6 is that we are only considering the range of \(I_{A,\perp }\). In fact for a general \(GL(n,\mathbb {C})\)-connection A, we have
Lemma 33
Assume \(I_{A}\) is solenoidal injective on 1-forms. Then
Proof
The fact that the range splits follows directly from the decomposition (44) and the definitions. The sum is direct because of the following observation: if \(h\in \mathfrak {H}_{-A^{*}}\) is such that \(I_{A}(h)\in \text {range}\;I_{A,\perp }\) then \(h=0\). Indeed, in this case there is \(f\in C^{\infty }(M,\mathbb {C}^{n})\) such that \(I_{A}(\star d_{A}f+h)=0\). Since \(I_{A}\) is solenoidal injective, we have that there is \(p\in C^{\infty }(M,\mathbb {C}^{n})\) with \(p|_{\partial M}=0\) such that \(d_{A}p=\star d_{A}f+h\). This implies right away that \(h=0\). \(\square \)
We conclude with an example showing that \(\mathfrak {H}_{A}\) could be non-trivial. We note that \(\mathfrak {H}_{A}\) transforms isomorphically under gauge equivalences and it is trivial for \(A=0\) (hence it is zero for any flat connection). For the example, suppose M is the unit disk with the standard metric. Consider the following map \(F:\partial M={{\mathbb {S}}}^{1}\rightarrow SU(2)\) given by
Since SU(2) is simply connected F can be extended to a smooth map \(F:M\rightarrow SU(2)\). Define the \(GL(2,\mathbb {C})\)-connection \(A:=-(\bar{\partial } F)F^{-1}\mathrm{d}\bar{z}=A_{\bar{z}}\mathrm{d}\bar{z}\). Thus,
We claim that there is a non-zero 1-form h such that \(d_{A}h=d_{A}\star h=0\) and \(j^* h=0\). Indeed, let \(h:=h_{\bar{z}}\mathrm{d}\bar{z}+h_{z}\mathrm{d}z\) where
Using [23, Lemma 6.2] (which holds for all A, not just unitary ones) we see that \(\star d_{A}h=2i(\mu _{-}(h_{1})-\mu _{+}(h_{-1}))\) and thus \(d_{A}h=0\) and \(d_{A}\star h=0\) are equivalent to \(\mu _{-}(h_{1})=\mu _{+}(h_{-1})=0\). These equations hold because of (50) and \(A_{z}=0\). Finally the boundary condition \(j^*h=0\) holds because using (49)
It is now natural to ask: is there a way to characterize the finite-dimensional subspace \(I_{A}(\mathfrak {H}_{-A^{*}})\) in terms of boundary data?
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Acknowledgements
The authors thank the referees for their constructive and useful comments. FM was partially funded by NSF Grant DMS-1712790 and GPP by EPSRC Grant EP/M023842/1.
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Appendices
Appendix A: On the Simplicity Constants \(C_1, C_2\)
Recalling that we introduced constants \(C_{1,2}(M,g)\) in (7) for simple surfaces, let us mention that these constants cannot be made universal. Indeed, consider a centered disk of radius \(1-\varepsilon \) in \(\mathbb {R}^2\) with a constant curvature metric \(+1\) or \(-1\). For every \(\varepsilon >0\), these surfaces are simple, though in the first case, \(C_1\) tends to zero as \(\varepsilon \rightarrow 0\) and in the second case, \(C_2\) grows unboundedly as \(\varepsilon \rightarrow 0\). It is unclear how to sharply characterize simplicity via these constants in terms of intrinsic quantities of the surface. A sufficient way to achieve this can be done as follows: For any \(s\in \mathbb {R}\), let us denote \(s^{\pm } = \max (\pm s, 0)\), and for a non-trapping surface (M, g), let us define
The constant \(k^+(M,g)\) appears in [29, pp119-120], chosen to be a dimensionless quantity (in that it does not vary under multiplication of g by a positive number) and the condition \(k^+(M,g)<1\) is a sufficient (but not necessary) condition for the absence of conjugate points. We have the following bounds on the function b in terms of \(k^{\pm }\).
Lemma 34
Let (M, g) a non-trapping Riemannian surface such that \(k^+(M,g)<1\). Then the function \(b_2\) satisfies
Proof
Denote for brevity \(b(\mathrm {x},v,t) := -b_2(\mathrm {x},v,t)\), the unique solution to
We have for every \(\mathrm {x}\in M\), the estimate \(-\kappa ^-(\mathrm {x})\le \kappa (\mathrm {x}) \le \kappa ^+(\mathrm {x})\). Therefore, by Sturm comparison, the pointwise estimate holds
where \(\underline{b}\) and \(\overline{b}\) solve the following problems:
\(\underline{b}\) has a simple zero at \(t=0\) so we can consider the smooth function \(\frac{\underline{b}(\mathrm {x},v,t)}{t}\), and since \(k^+(M,g)<1, \underline{b}\) remains positive for \(t>0\). Direct integration of the previous ODE gives us that
On to the upper bound, we integrate the ODE satisfied by \(\overline{b}\) in a similar fashion, and obtain
so that, by Grönwall’s inequality, we obtain \(\frac{\overline{b}(t)}{t} \le \exp \left( \int _0^t u \kappa ^-(\varphi _u)\ \mathrm{d}u \right) \). Thus we can bound uniformly
The proof is complete. \(\square \)
We now prove Lemma 18, used in the proof of Theorem 4.
Proof of Lemma 18
Let us define \(\Phi (\mathrm {x},v,t) = \left[ {\begin{matrix} b_1 &{} b_2 \\ c_1 &{} c_2 \end{matrix}} \right] (\mathrm {x},v,t)\). This function satisfies
where we have defined \(\mathbb {K}:= \left[ {\begin{matrix} 0 &{} 1 \\ -\kappa &{} 0 \end{matrix}} \right] \). By Wronskian constancy, since \(\mathbb {K}\) is traceless then \(\det \Phi (\mathrm {x},v,t) \equiv 1\). The function \(V\Phi \) solves the ODE
where we have \(- V(\mathbb {K}\circ \varphi _t) = b_2(\mathrm {x},v,t)\kappa _\perp (\varphi _t(\mathrm {x},v)) \left[ {\begin{matrix} 0 &{} 0 \\ 1 &{} 0 \end{matrix}} \right] \) with \(\kappa _\perp = X_\perp \kappa \). Using \(\Phi \) itself as an integrating factor of the latter equation, we can arrive at the integral formula (keeping \((\mathrm {x},v)\) implicit)
We compute \(\Phi ^{-1} (s) \left[ {\begin{matrix} 0 &{} 0 \\ 1 &{} 0 \end{matrix}} \right] \Phi (s) = \left[ {\begin{matrix} -b_2 b_1 &{} -b_2^2 \\ b_1^2 &{} b_1b_2 \end{matrix}} \right] (s)\), and picking particular entries of \(V\Phi \), we deduce the expressions
In particular, we get
Notice further the following cocycle property:
true since, as functions of t both sides satisfy the ODE \(\frac{\mathrm{d}}{\mathrm{d}t} U + \mathbb {K}(\varphi _{t}) U = 0\) with matching condition at \(t = s\). This equality can be recasted as
In particular, looking at the \({}_{(1,2)}\) entry in this matrix equality, we obtain the relation
In particular, when (M, g) is simple with constants \(C_1, C_2\) as in (7), we can deduce the following estimate
We then obtain the exact same estimate for \(V(b_1/b_2)\):
The lemma is proved. \(\square \)
As a corollary of Lemma 18, we can revisit the case without connection here. Indeed, the function \(w(\mathrm {x},v,t) = - V\left( \frac{b_1}{b_2} \right) (t)\) is the kernel of the error operator in the case without connection, so we record the operator estimate here. We recall that
where we have defined \({{\mathcal {W}}} (\mathrm {x},\mathrm {y}) := \frac{1}{2\pi } \frac{w(\mathrm {x},Exp_\mathrm {x}^{-1} (\mathrm {y}))}{b_2(\mathrm {x}, Exp_\mathrm {x}^{-1} (\mathrm {y}))}\). Then we write
As a result, we obtain the bound:
Appendix B: Proof of Lemma 15
Proof of Lemma 15
Let us define the function
We will show that G vanishes identically by proving that \(G|_{\partial _- (SM)} =0 \) and \(XG = 0\) on SM. Recall that \(\tau \) satisfies
Using the structure equations, this implies
From (53), we deduce \(V\tau |_{\partial _- (SM)} = 0\) and for \((\mathrm {x},v)\in \partial _- (SM), b_2(\mathrm {x},v,\tau ) = b_2(\mathrm {x},v,0) = 0\), therefore \(G|_{\partial _- (SM)} = 0\). On to showing \(XG = 0\), we recall the definition \(\Phi (\mathrm {x},v,t) = \left[ {\begin{matrix} b_1 &{} b_2 \\ c_1 &{} c_2 \end{matrix}} \right] \) and set \(\Psi (\mathrm {x},v) := \Phi (\mathrm {x},v,\tau (\mathrm {x},v))\). Using (51), we have the property
where \(\tau (\mathrm {x},v)-t = \tau (\varphi _t(\mathrm {x},v))\). Differentiating \(\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\), we then obtain
which upon looking at the entries \({}_{(1,1)}\) and \({}_{(1,2)}\) gives
In short, we obtain the matrix equation
out of which \(XG =0\) is just Liouville’s formula. \(\square \)
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Monard, F., Paternain, G.P. The Geodesic X-ray Transform with a \(GL(n,\mathbb {C})\)-Connection. J Geom Anal 30, 2515–2557 (2020). https://doi.org/10.1007/s12220-017-9931-z
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DOI: https://doi.org/10.1007/s12220-017-9931-z