1 Introduction↩︎

Consider the unit disk \(\mathbb{D}_H = \{z\in \mathbb{C}, |z|\le 1\}\), with interior \(\mathbb{D}_H^\circ\) equipped with the Poincaré metric \[\label{eq:Poincare95metric} g_H = c^{-2}(z)|\mathrm{d}z|^2 ,\qquad c(z) \mathrel{\vcenter{:}}= \frac{1-|z|^2}{2}, \index{g_H}\index{c}\tag{1}\] of constant curvature \(-1\). The space of oriented geodesics through \(( \mathbb{D}_H^\circ, g_H)\) can be modeled as the open manifold \({\cal G}_{\mathrm h}= ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \mathbb{R}_a\) arising in the “horocyclic” parameterization \(\gamma_{\beta,a}(t)\) of unit-speed hyperbolic geodesics given in 4 . Our main object of study is the hyperbolic X-ray transform \[I_0^H f (\beta,a) \mathrel{\vcenter{:}}= \int_ \mathbb{R}f(\gamma_{\beta,a}(t))\;\mathrm{d}t, \qquad (\beta,a)\in {\cal G}_{\mathrm h}, \label{eq:hypXrt}\index{I_0}\tag{2}\] initially defined for \(f\in C_c^\infty( \mathbb{D}_H^\circ)\) and with values in \(C_c^\infty ({\cal G}_{\mathrm h})\).

Such a transform is a classical object of study [1]–[6] that continues to receive recent attention, see [7], [8]; in particular, it is long known that the transform above is injective over integrands with compact support or in the Schwartz class, that its range over such integrands can be described in terms of moment conditions [9], and a variety of global and semi-local inversion formulas can be derived [10], [11]. This transform is of interest due to its applications in imaging in media with variable refractive index, for its connections with the inverse conductivity problem [12]–[14], and for geophysical data processing [15].

More recently, integral-geometric questions generalizing the study of 2 have been considered on asymptotically hyperbolic geometries (geometric models generalizing \(( \mathbb{D}_H^\circ, g_H)\) to non-constant curvature, see e.g. Section 3.2.3). This includes questions of injectivity of the associated X-ray transform and its connection with rigidity questions [16], [17], stability estimates [18], and the study of associated non-abelian X-ray transforms [19].

Formulating stability statements as in [18] begs the question of finding the appropriate Sobolev scales that sharply capture the mapping properties of an X-ray transform, or further yet, finding explicit functional settings, Fréchet or Sobolev, where a “normal” operator (i.e. the X-ray transform composed with its adjoint) can be made invertible. Such issues have only been looked at very recently, starting with the works [20], [21], as they lead, among other applications, to fulfilling crucial criteria in novel toolboxes in statistical inverse problems; see also [22]. In geometries without conjugate points, where the normal operator is a pseudo-differential operator in the interior, answering these questions requires to turn one’s focus to the boundary geometry and identify the pseudo-differential calculus (factoring in boundary behavior) where the normal operator is invertible. Such constructions depend on the boundary model, whether incomplete convex, asymptotically hyperbolic, or asymptotically Euclidean/conic, etc. The first case is currently the most documented (see the recent topical review [23]), while the understanding of the other ones is still in its beginnings. This work represents a first step in this direction in the case of asymptotically hyperbolic geometries. Focusing on the Poincaré disk model as a first step is of relevance as it has the further desireable property that special normal operators (given below) can be found to be in the functional calculus of some distinguished differential operators of wedge type (see Remark 11), and this further informs which pseudo-differential algebra and Sobolev spaces to use.

One of the salient features of this article is that, on the topic of mapping properties of the hyperbolic X-ray transform, one can find invertibility settings and sharp functional settings to obtain tame and compact mapping properties, by leveraging (i) recent results established in the case of the Euclidean disk (an incomplete model), see [21], [24], [25], and (ii) the projective equivalence between the Euclidean disk and the Poincaré disk (see Proposition 30 below). Such a projective equivalence is of course not new, and has been exploited in prior work, e.g. in [9], to derive a range characterization via moment conditions, or in [26] to study partial data problems for a transform related to 2 . Here, we exploit this equivalence further, notably paying close attention to the change in boundary behavior incurred through this projective equivalence which bridges a prototype of an incomplete, convex geometry with a prototype of a complete, asymptotically hyperbolic one. On the hyperbolic side, this requires compactifying \(\mathbb{D}_H^\circ\) (in the obvious way) and \({\cal G}_{\mathrm h}\) (in a manner dictated by the projective equivalence), which then allows to discuss function spaces with non-trivial behavior at infinity, where the hyperbolic X-ray transform enjoys sharp mapping properties, compactness, and good invertibility. A summary of the main results is as follows:

\(\bullet\) In Theorem 8, we give several singular value decompositions (SVDs) for appropriately weighted versions of 2 , in particular contexts where the associated normal operator is compact, a desirable feature for several applied purposes.

\(\bullet\) Theorem 17 provides a functional framework (Sobolev spaces on \(\mathbb{D}_H\) and on the data space \({\cal G}_{\mathrm h}\) that account for boundary behavior) where one can formulate sharp and global stability estimates for the transform 2 .

\(\bullet\) Corollary 1 gives a surjectivity result for the backprojection operator (defined in 9 below); such a result is the hyperbolic analogue of [27]. The latter result, in the context of simple surfaces, is a crucial building block to several subsequent results in the analysis of X-ray and attenuated X-ray transforms, see [28].

\(\bullet\) Theorem 21 gives a generalization of the range characterization of \(I_0^H\) in terms of hyperbolic moment conditions (initially formulated in [9] for functions of Schwartz class), to functions with non-trivial boundary behavior and finite Sobolev regularity.

\(\bullet\) Proposition 14 and Theorem 15 give new functional relations with distinguished differential operators on \(\mathbb{D}_H\) and \({\cal G}_{\mathrm h}\), where the differential operators \({\cal L}_{\gamma}^H\) on \(\mathbb{D}_H\) (see 22 ) and \({\cal T}_\gamma^H\) on \({\cal G}_{\mathrm h}\) (see 23 ) when \({\cal G}_{\mathrm h}\) is suitably compactified, are of wedge type (see e.g., [29]). This is in contrast with prior work on the topic, where relations between 2 and the hyperbolic Laplacian have been found (see e.g., [7], [30]), and where prior studies of the X-ray transform on asymptotic hyperbolic spaces make systematic use of the uniformly degenerate (or 0-) calculus of Mazzeo-Melrose ([31]), the natural calculus where the hyperbolic Laplacian belongs.

\(\bullet\) For one specific weighted X-ray transform (the case \(\gamma=0\) below), the functional relations just discussed simplify, and the infinite-dimensional co-kernel of the X-ray transform can be identified with the help of boundary operators \(P_-^H\), \(C_-^H\), analogous to those defined on simple surfaces in [24], [32], [33]. This leads to the refined range characterization given in Theorem 22, some of which required a refinement (to Sobolev scales) of the range characterization of the X-ray transform [32] in the case of the Euclidean disk (see Sec.3.1.4).

The current results open the way for further studies, which the authors plan to address in future work, including the study of analogous questions for tensor fields (where the study of “tensor fields with boundary behavior”, viewed as functions on the sphere bundle \(S \mathbb{D}_H^\circ\), see 7 , requires discussing an appropriate compactification of the latter), and an in-depth study of the link between the X-ray transform on asymptotically hyperbolic manifolds and wedge operators.

We now discuss the main results in the next section and give an outline of the remainder at the end of it.

2 Main results↩︎

2.1 Preliminaries↩︎

On \(\mathbb{D}_H\) equipped with its standard smooth structure, we define the distinguished boundary defining function (‘bdf’, here and below) \[x(z) \mathrel{\vcenter{:}}= \frac{1-|z|^2}{1+|z|^2}, \quad z\in \mathbb{D}_H. \label{eq:xbdf}\index{x}\tag{3}\] With \(\omega = \arg(z)\) and \(x\) as in 3 , the measure induced by the Riemannian volume form \(\mathrm{d}V_H\) of \(( \mathbb{D}_H^\circ, g_H)\) is given by \(x^{-2} \mathrm{d}x \mathrm{d}\omega\).

2.1.1 Geodesics of \(( \mathbb{D}^\circ_H,g_H)\)↩︎

There are several models for the manifold of oriented geodesics on \(( \mathbb{D}_H^{\circ},g_H)\). Throughout the presentation, we will focus on a parameterization of the space of oriented geodesics by means of the manifold \({\cal G}_{\mathrm h}= ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \mathbb{R}_a\) by writing \[\gamma_{\beta,a}(t) \mathrel{\vcenter{:}}= e^{i\beta} \frac{(2+ia) {\mathrm x}(t) + ia}{ia {\mathrm x}(t) - 2+ia}, \quad {\mathrm x}(t)\mathrel{\vcenter{:}}= \tanh \Big( \frac{t}{2} \Big), \quad (\beta,a,t)\in ( \mathbb{R}/2\pi \mathbb{Z}) \times \mathbb{R}\times \mathbb{R}. \label{eq:hyp95fan95beam}\tag{4}\] This parameterization arises from horocycles as we explain in Section3.2.2. Orientation reversion arises via the involution \[{\cal S}_A^H\colon {\cal G}_{\mathrm h}\to {\cal G}_{\mathrm h}, \qquad (\beta,a)\mapsto (\beta+\pi + 2\tan^{-1}a, -a).\] The map \({\cal S}_A^H\) arises as the composition of the scattering map with the antipodal map, see Section5.1 for details. Some results will also be given in vertex parameterization 69 for completeness.

2.1.2 Hyperbolic X-ray transform↩︎

Recall the definition of \(I_0^H\) in Eq. 2 . If \(f\in C_c^\infty( \mathbb{D}_H)\), then the smoothness of \(I_0^H f\) comes from the smoothness of the map \((\beta,a,t)\mapsto \gamma_{\beta,a}(t)\). To see why \(I_0^H f\) has compact support, we define on \({\cal G}_{\mathrm h}\) the function \[\mu_{\mathrm h}(\beta,a) = \mu_{\mathrm h}(a)\mathrel{\vcenter{:}}= \cos(\tan^{-1}a) = (1+a^2)^{-1/2}. \label{eq:mu}\index{\mu_{\mathrm h}}\tag{5}\] Then a direct calculation shows that with \(\gamma_{\beta,a}(t)\) as in 4 , we have \[x(\gamma_{\beta,a}(t)) \le \mu_{\mathrm h}(a), \qquad t\in \mathbb{R}, \label{eq:xcontrol}\tag{6}\] see e.g. 84 . In particular, if \(f = 0\) on \(x^{-1} (0,\varepsilon)\), where \(0<\varepsilon<1\), then \(I_0^H f =0\) on \(\mu_{\mathrm h}^{-1}(0,\varepsilon)\). Because \(I_0^H f\) does not depend on orientation, we obtain the symmetry condition \[(I_0^H f) \circ {\cal S}_A^H = I_0^H f \qquad \text{on }\quad {\cal G}_{\mathrm h}.\]

2.1.3 Hyperbolic backprojection operator↩︎

The unit sphere bundle of \(( \mathbb{D}_H^{\circ}, g_H)\) is denoted \[S \mathbb{D}_H^{\circ} = \{(z,w)\in T \mathbb{D}_H^{\circ}\colon g_H (w,w) = 1\}. \label{eq:SDH}\tag{7}\] In Section 3.2.2, we define a smooth map \[\pi_\mathrm{h}\colon S \mathbb{D}_H^{\circ} \to {\cal G}_{\mathrm h}, \label{eq:F}\tag{8}\] mapping \((z,w)\in S \mathbb{D}_H^{\circ}\) to the unique oriented geodesic in \({\cal G}_{\mathrm h}\) passing through \((z,w)\). With \(\pi_\mathrm{h}\) defined in 8 , for \(g\in C^\infty({\cal G}_{\mathrm h})\), we define the hyperbolic backprojection of \(g\) by \[(I_0^H)^\sharp g (z) \mathrel{\vcenter{:}}= \int_{S_z \mathbb{D}_H^{\circ}} g(\pi_\mathrm{h}(z,w))\;\mathrm{d}S_z(w), \label{eq:backproj}\tag{9}\] where \(\mathrm{d}S_z(w)\) is the Lebesgue measure on the unit tangent circle \(S_z \mathbb{D}_H^{\circ}\).

2.2 Extension of \(I_0^H\) to non-compactly supported integrands↩︎

We first explain how to extend definition 2 to spaces of non-compactly supported smooth functions. To this end, it will be useful to compactify \({\cal G}_{\mathrm h}\) into \(\overline{{\cal G}_{\mathrm h}}\mathrel{\vcenter{:}}= ( \mathbb{R}/2\pi \mathbb{Z})_\beta \times [-\infty,\infty]_a\), where the factor \([-\infty,\infty]_a\) is given the smooth structure of a compact manifold with boundary using the stereographic projection map so that the function \(\mu_{\mathrm h}\) defined in 5 , extended by zero on \(\partial\overline{{\cal G}_{\mathrm h}}\), becomes a smooth bdf.

The manifold \(\overline{{\cal G}_{\mathrm h}}\) is then diffeomorphic to the inward-pointing boundary of the Euclidean disk \(( \mathbb{D}_E = \{z\in \mathbb{C}, |z|\le 1\}, |dz|^2)\), denoted \(\partial_+ S \mathbb{D}_E = ( \mathbb{R}/2\pi \mathbb{Z})_\beta \times [-\pi/2,\pi/2]_\alpha\) (with bdf \(\mu(\beta,\alpha) = \cos\alpha\)), via the horocycle-to-fan-beam map \[\Psi_\mathrm{hf}\colon \overline{{\cal G}_{\mathrm h}}\to \partial_+ S \mathbb{D}_E, \qquad (\beta,a)\mapsto (\beta,\tan^{-1} a), \label{eq:Psihf}\tag{10}\] and where, by convention, \(\tan^{-1} (\pm \infty) = \pm \frac{\pi}{2}\).

On the Euclidean disk, an important subspace of \(C^\infty(\partial_+ S \mathbb{D}_E)\) for capturing the range of the X-ray transform is \(C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E)\), defined in 46 . Accordingly, on \(\overline{{\cal G}_{\mathrm h}}\) it will be crucial to define \[C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}}) \mathrel{\vcenter{:}}= \Psi_\mathrm{hf}^* C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E), \label{eq:CalphaG}\tag{11}\] where \(\Psi_\mathrm{hf}^*\) denotes the pullback by \(\Psi_\mathrm{hf}\). The following characterization holds.

Let \(C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\) be the space of smooth functions on \(\mathbb{D}_H\) admitting Taylor expansions off of \(\mathbb{D}_H\) only made of even terms when written in terms of the special bdf \(x\) defined in 3 . Then, a first observation is the following.

2.3 Singular value decompositions↩︎

We now turn toward mapping properties of \(I_0^H\) between Hilbert spaces. Below we write \[L^2_\pm ({\cal G}_{\mathrm h}, \mu_{\mathrm h}^{-2\gamma}\mathrm{d}\beta\mathrm{d}a) \mathrel{\vcenter{:}}= L^2({\cal G}_{\mathrm h}, \mu_{\mathrm h}^{-2\gamma}\mathrm{d}\beta\mathrm{d}a) \cap \ker ( ({\cal S}_A^H)^* \mp id), \label{eq:Lpm}\tag{13}\] where \(({\cal S}_A^H)^*\) denotes the pullback by \({\cal S}_A^H\).

We now give a full singular value decomposition of the operator 14 . As it will be inherited from that of the X-ray transform on the Euclidean disk, we recall some notation. Let us define the family of functions, for \(n\in \mathbb{N}_0\) and \(k\in \mathbb{Z}\), \[\psi_{n,k}^\gamma (\beta,\alpha)= (\cos\alpha)^{2\gamma+1} e^{i(n-2k)(\beta+\alpha+\pi/2)} p_n^{\gamma} (\sin\alpha), \label{eq:psinkgamma}\index{\psi_{n,k}^\gamma} \quad (\beta,\alpha)\in \partial_+ S \mathbb{D}_E,\tag{15}\] where \(p_n^\gamma:[-1,1]\to \mathbb{R}\) is a scalar multiple of the \(n\)-th Jacobi polynomial⁴, orthogonal for the weight \((1-x^2)^{\gamma+1/2}\), normalized here for convenience so that \(\int_{-1}^1 (p_n^\gamma(x))^2 (1-x^2)^{\gamma+1/2}\mathrm{d}x = (2\pi)^{-1}\). This makes \(\{\psi_{n,k}^\gamma\}_{n\ge 0,\;k\in \mathbb{Z}}\) an orthonormal basis for the space \(L_+^2(\partial_+ S \mathbb{D}_E, \mu^{-2\gamma}\mathrm{d}\beta\mathrm{d}\alpha)\).

With \((I_0^E)^\sharp\), the backprojection operator associated with the Euclidean disk, we define \[Z_{n,k}^\gamma \mathrel{\vcenter{:}}= (I_0^E)^\sharp \mu^{-2\gamma-1} \psi_{n,k}^\gamma, \qquad n\ge 0, \qquad 0\le k\le n. \label{eq:Zernike}\index{Z_{n,k}^\gamma}\tag{16}\] Those are equal, up to normalization, to the generalized disk Zernike polynomials (\(P_{n-k,k}^\gamma\) in the convention of [34]), and they are a Hilbert basis for \(L^2( \mathbb{D}_E,d^\gamma \mathrm{d}V_E)\), where \[d(z) \mathrel{\vcenter{:}}= 1-|z|^2 \in C^\infty( \mathbb{D}_E), \label{eq:d}\tag{17}\] and \(\mathrm{d}V_E\) is the Euclidean volume form. An important map for what follows is the following: \[\Phi\colon \mathbb{D}_H\to \mathbb{D}_E, \qquad \Phi(z) \mathrel{\vcenter{:}}= \frac{2z}{1+|z|^2}. \label{eq:Phi}\index{\Phi}\index{ \mathbb{D}_E}\tag{18}\] The map \(\Phi\) is an isometry between \(( \mathbb{D}_H^{\circ},g_H)\) and \(( \mathbb{D}_E^{\circ},g_K)\), where \(g_K\) is the Beltrami-Klein hyperbolic metric on the unit disk. The latter is projectively equivalent to the Euclidean metric, in the sense that the geodesics of the two metrics agree up to parameterization. It can be seen that the map \(\Psi_\mathrm{hf}\) in 10 is induced by \(\Phi\), see Section 3.2, and that with \(x\) and \(d\) as in 3 and 17 , one has \[\label{eq:bdfs} \Phi^* d = x^2.\tag{19}\] This implies that \(\Phi^*:C^\infty( \mathbb{D}_E)\to C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\) is an isomorphism. Returning to \({\cal G}_{\mathrm h}\), the family \[\psi_{n,k}^{\gamma,H} \mathrel{\vcenter{:}}= \mu_{\mathrm h}\Psi_\mathrm{hf}^*\;\psi_{n,k}^\gamma, \qquad n\ge 0, \quad k\in \mathbb{Z}, \label{eq:psinkgammaH}\tag{20}\] is then a Hilbert basis of \(L^2_+ ({\cal G}_{\mathrm h}, \mu_{\mathrm h}^{-2\gamma}\mathrm{d}\beta\mathrm{d}a)\) defined in 13 .

2.4 Distinguished differential operators and special relations↩︎

On \(\mathbb{D}_H\), define the differential operator \[\begin{align} {\cal L}_\gamma^H &= - x^{-2\gamma-1} \partial_x \left( x^{2\gamma+1} (1-x^2) \partial_x \right) - \frac{1}{1-x^2} \partial_\omega^2 + (1+\gamma)^2 id, \nonumber \\ &=- (1-x^2) \partial_x^2 - \left( \frac{2\gamma+1}{x} - (2\gamma+3)x \right)\partial_x - \frac{1}{1-x^2} \partial_\omega^2 + (1+\gamma)^2 id. \end{align} \label{eq:L95wedge}\index{{\cal L}_\gamma^H}\tag{22}\]

One may find that the space \(C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\) described in Section 2.2 is a subspace of \(L^2( \mathbb{D}_H, x^{2\gamma+3}\mathrm{d}V_H)\) if and only if \(\gamma>-1\), that \({\cal L}_\gamma^H (C_{\mathrm{ev}}^\infty( \mathbb{D}_H))\subset C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\), and that \(({\cal L}_\gamma^H, C_{\mathrm{ev}}^\infty( \mathbb{D}_H))\)⁵ is symmetric with respect to the inner product \(L^2( \mathbb{D}_H, x^{2\gamma+3}\mathrm{d}V_H)\). More specifically, we have the following proposition.

On \({\cal G}_{\mathrm h}\), define the differential operator \(T \mathrel{\vcenter{:}}= \partial_\beta - (1+a^2) \partial_a\), and \[{\cal T}_\gamma^H = -T^2 + 2(\gamma+1) a T + (\gamma^2 - 2(\gamma+1)a^2 - 1) id, \label{eq:TgH}\index{\mathcal{T}_\gamma^H}\tag{23}\] so that \({\cal T}_\gamma^H = \mu_{\mathrm h}\circ \Psi_\mathrm{hf}^{*} \circ {\cal T}_\gamma \circ \Psi_\mathrm{hf}^{-*} \circ \mu_{\mathrm h}^{-1}\), where \({\cal T}_\gamma\) is the operator on \(\partial_+S \mathbb{D}_E\) defined in 52 . Then, if \(\gamma>-1\), one finds that \(\mu_{\mathrm h}^{2\gamma+2} C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}})\subset L^2_+ ({\cal G}_{\mathrm h}, \mu_{\mathrm h}^{-2\gamma}\mathrm{d}\beta\mathrm{d}a)\), that \({\cal T}_\gamma^H (\mu_{\mathrm h}^{2\gamma+2} C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}}))\subset \mu_{\mathrm h}^{2\gamma+2} C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}})\) (using [25]), and that the operator \(({\cal T}_\gamma^H, \mu_{\mathrm h}^{2\gamma+2} C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}}))\) is symmetric with respect to \(L^2_+ ({\cal G}_{\mathrm h}, \mu_{\mathrm h}^{-2\gamma}\mathrm{d}\beta\mathrm{d}a)\). More specifically, we have the following proposition.

Combining ?? and ?? , we see that the operator \((I^H_0 x^{2+2\gamma})^* I^H_0 x^{2+2\gamma}\) commutes with \({\cal L}_\gamma^H\), and by symmetry arguments, also with the \(L^2( \mathbb{D}_H, x^{2\gamma+3}\mathrm{d}V_H)\)-graph closure of \((-\partial_\omega^2, C_{\mathrm{ev}}^\infty( \mathbb{D}_H))\), denoted \(-\partial_\omega^2\) below.

Given the spectrum of each of the operators \((I^H_0 x^{2+2\gamma})^* I^H_0 x^{2+2\gamma}\), \({\cal L}_\gamma^H\), and \(-\partial_\omega^2\), we arrive at the following result.

2.5 Sobolev mapping properties↩︎

For \(s \ge 0\), define the space⁶ \[H_w^{s,\gamma} ( \mathbb{D}_H) \mathrel{\vcenter{:}}= {\cal D}\left(({\cal L}_{\gamma}^H)^{s/2}\right), \label{eq:Sobw}\index{H_w^{s,\gamma} ( \mathbb{D}_H)}\tag{26}\] which can be obtained as the completion of \(C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\) for the norm \[\|u\|_{H_w^{s,\gamma}}^2 = \sum_{n,k} (n+1+\gamma)^{2s} |u_{n,k}|^2, \qquad u = \sum_{n,k} u_{n,k} \widehat{\Phi^* Z_{n,k}^\gamma}. \label{eq:Hsnorm}\tag{27}\]

Theorem 17 implies the following important result.

2.6 Range characterization and moment conditions.↩︎

As a corollary of Theorem 8 and Theorem 17[item:isomorphismH2], we immediately state without proof.

The above spectral characterization can also be understood in terms of moment conditions, most appropriately formulated in vertex coordinates, which are described in Section 3.2.1. Given a function \(u\) on \(\overline{{\cal G}_{\mathrm h}}\), we denote by \(u^{ {\mathrm v}}\) the function \(u\) pulled back to vertex coordinates. By virtue of Lemma 2 below, this means \[u^ {\mathrm v}(\omega,s) \mathrel{\vcenter{:}}= u\left(\omega + \frac{3\pi}{2} + 2\tan^{-1}s, \frac{-2s}{1-s^2}\right), \qquad (\omega,s)\in \mathbb{R}/2\pi \mathbb{Z} \times (-1,1).\]

Theorem 21 is the extension to integrands with non-trivial boundary behavior of the characterization appearing in [9] in the case of compactly supported and Schwartz class integrands.

2.7 The case \(\gamma=0\). Boundary operators and range characterizations.↩︎

More can be said about Corollary 2 in the case \(\gamma = 0\): First, the spectral decay can be expressed in terms of natural anisotropic Sobolev scales; second, the infinite-dimensional co-kernel can be understood in terms of boundary operators, akin to those defined in [32], [33] in the context of simple Riemannian surfaces.

Sobolev spaces on \(\overline{{\cal G}_{\mathrm h}}\). Upon defining \(C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}})\mathrel{\vcenter{:}}= \Psi_\mathrm{hf}^* C_\alpha^\infty(\partial_+ S \mathbb{D}_E)\), where the space \(C_\alpha^\infty(\partial_+ S \mathbb{D}_E)\) is defined in 45 , the spaces \(\mu_{\mathrm h}^2 C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}})\) and \(\mu_{\mathrm h}C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}})\) are two distinct core domains of \(L^2 (\overline{{\cal G}_{\mathrm h}}, \mathrm{d}\beta\mathrm{d}a)\)-essential-self-adjointness for \({\cal T}_0^H\). Accordingly, we denote by \({\cal T}_{0,D}^H\) (resp. \({\cal T}_{0,N}^H\)) the graph closure of \(({\cal T}_0, \mu_{\mathrm h}^2 C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}}))\) (resp., of \(({\cal T}_0, \mu_{\mathrm h}C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}}))\)), thought of as a Dirichlet (resp. Neumann) realization of \({\cal T}_0^H\) based on the Taylor expansions of elements in \(\mu_{\mathrm h}^2 C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}})\) (resp. \(\mu_{\mathrm h}C_{\alpha}^\infty(\overline{{\cal G}_{\mathrm h}})\)). The involution \({\cal S}_A^H\) induces the even/odd decompositions \(L^2 = L^2_+ \stackrel{\perp}{\oplus} L^2_-\) and \(C_{\alpha}^\infty= C_{\alpha,+}^\infty\oplus C_{\alpha,-}^\infty\), with projection maps \((id \pm ({\cal S}_A^H)^*)/2\). Moreover, since \(\mu_{\mathrm h}= \mu_{\mathrm h}\circ{\cal S}_A^H\), we also have \(\mu_{\mathrm h}C_{\alpha}^\infty= \mu_{\mathrm h}C_{\alpha,+}^\infty\oplus \mu_{\mathrm h}C_{\alpha,-}^\infty\). Then, smoothness scales that respect the symmetries and the boundary behavior of \(I_0^H x^2\) will arise by defining, for \(s\ge 0\), \[\begin{align} &H_{T,D,\pm}^s (\overline{{\cal G}_{\mathrm h}}) \mathrel{\vcenter{:}}= ({\cal T}_{0,D}^H)^{-s/2} \left( L^2_\pm (\overline{{\cal G}_{\mathrm h}}, \mathrm{d}\beta\mathrm{d}a) \right), \\ &H_{T,N,\pm}^s (\overline{{\cal G}_{\mathrm h}}) \mathrel{\vcenter{:}}= ({\cal T}_{0,N}^H)^{-s/2} \left( L^2_\pm (\overline{{\cal G}_{\mathrm h}}, \mathrm{d}\beta\mathrm{d}a) \right). \label{eq:SobTH} \end{align}\tag{37}\] Then, observing that \({\cal T}_0^H\) commutes with the projections \((id \pm ({\cal S}_A^H)^*)/2\), one can obtain the decompositions \({\cal D}(({\cal T}_{0,\bullet}^H)^{s/2}) = H_{T,\bullet,+}^s (\overline{{\cal G}_{\mathrm h}}) \oplus H_{T,\bullet,-}^s (\overline{{\cal G}_{\mathrm h}})\) for \(\bullet\in \{D,N\}\).

In Section 5, we define two “boundary” operators \[\label{eq:CPCinf} C_-^H \colon \mu_{\mathrm h}^2 C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}}) \to \mu_{\mathrm h}^2 C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}}), \qquad P_-^H \colon \mu_{\mathrm h}C_{\alpha,-}^\infty(\overline{{\cal G}_{\mathrm h}}) \to \mu_{\mathrm h}^2 C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}})\tag{38}\] which, for every \(s\ge 0\), extend to bounded operators \[C_-^H \colon H_{T,D,+}^s (\overline{{\cal G}_{\mathrm h}}) \to H_{T,D,+}^s (\overline{{\cal G}_{\mathrm h}}), \qquad P_-^H \colon H_{T,N,-}^s (\overline{{\cal G}_{\mathrm h}}) \to H_{T,D,+}^s(\overline{{\cal G}_{\mathrm h}}), \label{eq:CP}\tag{39}\] and help characterize the range of \(I_0^H x^2\) as follows.

Aside from condition [gammazeroHit3alt] discussed below Theorem 21, Theorem 22 gives two more extensions of known results: Condition [gammazeroHit2alt] is the analogue to Pestov-Uhlmann’s range characterization of the X-ray transform on simple surfaces [32], extended to the case of the hyperbolic space and to non-smooth classes, see also Theorem 27 below; condition [gammazeroHit1alt] is the generalization to hyperbolic space of [33].

Outline. The remainder of the article is organized as follows.

We first discuss in Section 3 all building blocks needed for the main results. In Section 3.1, we discuss analogous results in the Euclidean disk, including a new Sobolev-space range characterization of the X-ray transform in Theorem 27. In Section 3.2, we discuss three parameterizations of hyperbolic geodesics (vertex, horocyclic, and cosphere bundle) and equivalences between them. We prove a Santaló-type formula (Proposition 29) for the hyperbolic disk in Section 3.3. Finally, we explicitly describe in Section 3.4 the projective equivalence between Euclidean and Poincaré disks, and the intertwining implications at the level of the X-ray transforms and backprojection operators (notably making using of Santaló’s formula).

With these building blocks being set, Section 4 provides brief proofs of the main results, Proposition 1 through Theorem 21.

Finally, Section 5 discusses the case \(\gamma=0\) and the proof of Theorem 22 (given in Section5.4), discussing along the way a hyperbolic generalization of the incoming/outgoing boundaries of the unit tangent bundle of a simple surface (Section5.1), boundary operators \(C_{\pm}^H, P_{\pm}^H\) helping characterize the range of \(I_0^H x^2\) (Section5.2), and an ambient description of the space \(C_{\alpha,+}^\infty(\overline{{\cal G}_{\mathrm h}})\) obtained through an appropriate gluing of the incoming and outgoing boundaries, usually kept disconnected in prior literature (see, e.g., [16]), in Section5.3.

3 Preliminaries↩︎

3.1 Known results on the Euclidean disk↩︎

The Euclidean disk \(\mathbb{D}_E\) is equipped with its usual smooth structure, so that the function \(d\) in 17 is a smooth bdf. We parameterize geodesics through \(\mathbb{D}_E\) via fan-beam coordinates (inward-pointing boundary): Given \(e^{i\beta}\in \partial \mathbb{D}\) and an inward-pointing vector of direction \(e^{i(\beta+\alpha+\pi)}\) for some \(\alpha\in [-\pi/2,\pi/2]\), a unit-speed parameterization of the Euclidean geodesic passing through \((e^{i\beta}, e^{i(\beta+\pi+\alpha)})\) is given by \[\begin{align} \begin{aligned} \gamma_{\beta,\alpha}^E (u) &= e^{i\beta} + (u+\cos \alpha) e^{i(\beta+\pi+\alpha)} \\ &= e^{i(\beta+\alpha+\pi)} (u + i\sin\alpha), \quad u\in [-\cos\alpha, \cos\alpha], \end{aligned} \label{eq:Egeo}\index{ \gamma_{\beta,\alpha}^E} \end{align}\tag{40}\] with endpoints \(\gamma_{\beta,\alpha}^E (-\cos\alpha) = e^{i\beta}\) and \(\gamma_{\beta,\alpha}^E (\cos\alpha) = e^{i (\beta+\pi+2\alpha)}\), attaining its vertex⁷ when \(u=0\). Then, the Euclidean X-ray transform is defined by \[I_0^E f(\beta,\alpha) = \int_{-\cos\alpha}^{\cos\alpha} f(e^{i(\beta+\alpha+\pi)} (u + i\sin\alpha))\;du, \qquad (\beta,\alpha)\in \partial_+ S \mathbb{D}_E. \label{eq:EI0}\tag{41}\]

3.1.1 The space \(C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E)\)↩︎

Recall that the scattering relation is the map \[\mathcal{S}^E:\partial_\pm S \mathbb{D}_E\to \partial_\mp S \mathbb{D}_E,\quad (z,w)\mapsto \varphi_{\tau(z,\pm w)}(z,w),\] where \(\tau(z,w)\) denotes the exit time for the orbit of the geodesic flow with initial data \((z,w)\) and \(\partial_- S \mathbb{D}_E=( \mathbb{R}/2\pi \mathbb{Z})_{\beta}\times [\pi/2,3\pi/2]_\alpha\) is the outward pointing boundary expressed in fan-beam coordinates, in which \((z,w) = (e^{i\beta}, e^{i(\beta+\pi+\alpha)})\). The scattering relation is expressed in fan beam coordinates as \[\quad {\cal S}^E (\beta,\alpha) = (\beta+\pi+2\alpha, \pi-\alpha). \label{eq:EscatRel}\tag{42}\] Such a relation allows to extend a function on \(\partial_+ S \mathbb{D}_E\) to \(\partial S \mathbb{D}_E\) by evenness or oddness with respect to \({\cal S}^E\) through the operators \[A_\pm u (z,w) \mathrel{\vcenter{:}}= \left\{ \begin{array}{rl} u(z,w), &\qquad (z,w)\in \partial_+ S \mathbb{D}_E \\ \pm u({\cal S}^E(z,w)), &\qquad (z,w)\in \partial_- S \mathbb{D}_E. \end{array} \right. \label{eq:ApmE}\tag{43}\] As mentioned in [33], \(A_\pm\) is bounded when viewed as an operator \(L^2( \partial_+ S \mathbb{D}_E,\mathrm{d}\beta\mathrm{d}\alpha)\to L^2( \partial S \mathbb{D}_E,\mathrm{d}\beta\mathrm{d}\alpha)\) and as an operator \(L^2( \partial_+ S \mathbb{D}_E,\mu \mathrm{d}\beta\mathrm{d}\alpha)\to L^2( \partial S \mathbb{D}_E,|\mu|\mathrm{d}\beta\mathrm{d}\alpha)\), whose adjoint in either functional setting is given by \[\label{eq:A95star} A_\pm ^*u (z,w)=u(z,w)\pm u(\mathcal{S}^E(z,w)), \quad (z,w)\in \partial _+ S \mathbb{D}_E.\tag{44}\] If \(u\in C^\infty(\partial_+ S \mathbb{D}_E)\), then \(A_{\pm} u\) is smooth away from \(\partial_0 S \mathbb{D}_E \mathrel{\vcenter{:}}=\partial_- S \mathbb{D}_E\cap \partial_+ S \mathbb{D}_E\), though making it smooth on \(\partial S \mathbb{D}_E\) requires constraining its Taylor expansion off of \(\partial_0 S \mathbb{D}_E\). Related to this issue, an important subspace of \(C^\infty (\partial_+ S \mathbb{D}_E)\) is defined (generally on a simple Riemannian surface) in [27] as \[C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E) \mathrel{\vcenter{:}}= \{ u\in C^\infty(\partial_+ S \mathbb{D}_E), \;A_+ u\in C^\infty(\partial_+ S \mathbb{D}_E) \}. \label{eq:CalphaE}\tag{45}\] In particular, functions in \(C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)\) can also be thought of as restrictions of functions that are smooth on \(\partial S \mathbb{D}_E\) and invariant under \({\cal S}^E\). We now explain how this space helps capture the mapping properties of \(I_0^E\). Since \(I_0^E\) generally does not depend on geodesic orientation, we encode this further symmetry by defining \[C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E) \mathrel{\vcenter{:}}= \{ u\in C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E),\;u\circ {\cal S}_A^E = u \}, \label{eq:CalphappE}\tag{46}\] where \({\cal S}_A^E\colon (\beta,\alpha)\mapsto (\beta+\pi+2\alpha, -\alpha)\) denotes the composition of the scattering relation and the antipodal map \[A_E:S \mathbb{D}_E\to S \mathbb{D}_E, \qquad (z,w)\mapsto (z,-w), \label{eq:AE}\tag{47}\] and encodes the orientation-reversing map of geodesics.

We now claim that for any \(\gamma>-1\), \[I_0^E d^{\gamma} (C^\infty( \mathbb{D}_E)) \subset \mu^{2\gamma+1} C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E). \label{eq:mappingI0E}\tag{48}\] Indeed, notice that with \(\gamma_{\beta,\alpha}^E (u)\) defined in 40 , we have \(d(\gamma_{\beta,\alpha}^E(u)) = \cos^2\alpha - u^2\). Then for \(f\in C^\infty( \mathbb{D}_E)\), changing variable \(u = t(\cos \alpha)\) for \(t\in [-1,1]\) in the integral 41 , we obtain \[I_0^E d^\gamma f (\beta,\alpha) = \mu^{2\gamma+1} \int_{-1}^1 (1-t^2)^\gamma f\left( e^{i (\beta+\pi+\alpha)} (t \cos \alpha + i \sin \alpha)\right)\;dt.\] This last integral is seen to be well-defined and smooth on \(\partial S \mathbb{D}_E\), invariant under both \({\cal S}^E\) and \({\cal S}_A^E\). This exactly means that its restriction to \(\partial_+ S \mathbb{D}_E\) belongs to \(C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E)\).

3.1.2 Singular value decomposition↩︎

We follow notation from [25], replacing \(I_0\) there by \(I_0^E\) here. For any \(\gamma>-1\), defining \[L_+^2 (\partial_+ S \mathbb{D}_E, \mu^{-2\gamma}\mathrm{d}\beta\mathrm{d}\alpha) \mathrel{\vcenter{:}}= L^2 (\partial_+ S \mathbb{D}_E, \mu^{-2\gamma}\mathrm{d}\beta\mathrm{d}\alpha) \cap \ker (({\cal S}_A^E)^*-id),\] the operator \[I_0^E d^\gamma \colon L^2 ( \mathbb{D}_E, d^\gamma \mathrm{d}V_E) \longrightarrow L_+^2 (\partial_+ S \mathbb{D}_E, \mu^{-2\gamma}\mathrm{d}\beta\mathrm{d}\alpha) \label{eq:I0dgamma}\tag{49}\] is bounded, with Hilbert space adjoint \((I_0^E)^\sharp \mu^{-2\gamma-1}\), where \((I_0^E)^\sharp\) denotes the Euclidean backprojection operator.

3.1.3 Intertwining differential operators, functional relations, and \(C^\infty( \mathbb{D}_E)\)-mapping properties↩︎

From [25], we have, for any \(\gamma>-1\), \[\begin{align} I_0^E d^\gamma \circ {\cal L}_\gamma &=& {\cal T}_\gamma \circ I_0^E d^\gamma \qquad \text{on} \quad C^\infty( \mathbb{D}_E), \\ {\cal L}_\gamma \circ (I_0^E)^\sharp \mu^{-2\gamma-1} &=& (I_0^E)^\sharp \mu^{-2\gamma-1} \circ {\cal T}_\gamma \qquad \text{ on } \quad C^\infty( (\partial_+ S \mathbb{D}_E)^{\circ}), \end{align}\] where \[\begin{align} {\cal L}_\gamma &\mathrel{\vcenter{:}}= -(1-\rho^2)^{-\gamma} \rho^{-1}\partial_\rho\left((1-\rho^2)^{\gamma+1} \rho\partial_\rho\right)-\rho^{-2}\partial _\omega^2+(1+\gamma^2)id,\tag{51} \\ {\cal T}_\gamma &\mathrel{\vcenter{:}}= -\mu^{2\gamma} T \mu^{-2\gamma} T + \gamma^2 id, \qquad T\mathrel{\vcenter{:}}= \partial_\beta-\partial_\alpha.\tag{52} \end{align}\] The Zernike polynomials \(\{Z_{n,k}^\gamma\}_{n,k}\) make up the full eigenvalue decomposition of the \(L^2_\gamma\)-essentially-self-adjoint operator \(({\cal L}_\gamma, C^\infty( \mathbb{D}_E))\), whose graph closure we denote by \({\cal L}_{\gamma,C^\infty}\), as \({\cal L}_\gamma Z_{n,k}^\gamma = (n+1+\gamma)^2 Z_{n,k}^\gamma\) for all \(n,k\). One may then define the scale of Hilbert spaces \[\begin{align} \widetilde{H}^{s,\gamma}( \mathbb{D}_E) &\mathrel{\vcenter{:}}= {\cal D}({\cal L}_{\gamma,C^\infty}^{s/2})\\ &= \left\{u = \sum_{n\ge 0} \sum_{k=0}^n u_{n,k} \widehat{Z_{n,k}^\gamma}, \qquad \sum_{n,k} (n+1+\gamma)^{2s} |u_{n,k}|^2 <\infty\right\}. \label{eq:Hsgamma}\index{\widetilde{H}^{s,\gamma}( \mathbb{D})} \end{align}\tag{53}\] Combined with Theorem 24, we obtain a description of the range of \(I_0^E d^\gamma\) when defined on this special Sobolev scale: \[I_0^E d^\gamma \left( \widetilde{H}^{s,\gamma}( \mathbb{D}_E) \right) = \left\{ \sum_{n\ge 0} \sum_{k=0}^n u_{n,k}\;\sigma_{n,k}^\gamma \psi_{n,k}^\gamma, \qquad \sum_{n,k} (n+1+\gamma)^{2s} |u_{n,k}|^2 <\infty\right\}. \label{eq:rangeSobolev}\tag{54}\]

Further, large-\(n\) estimates on \(\sigma_{n,k}^\gamma\) in [25] show that for \(s\ge 0\), \[\widetilde{H}^{s+\max (1,1+\gamma), \gamma}( \mathbb{D}_E) \subset (I_0^E)^\sharp \mu^{-2\gamma-1} I_0^E d^\gamma(\widetilde{H}^{s,\gamma}( \mathbb{D}_E))\subset \widetilde{H}^{s+\min (1,1+\gamma), \gamma}( \mathbb{D}_E). \label{eq:SobolevMapping}\tag{55}\] In particular, the normal operator \((I_0^E)^\sharp \mu^{-2\gamma-1} I_0^E d^\gamma\) is a tame map with tame inverse of \(C^\infty( \mathbb{D}_E)\), when the latter is equipped with the graded family of semi-norms \(\{\|\cdot\|_{\widetilde{H}^{p,\gamma}}\}_{p\in \mathbb{N}_0}\).

3.1.4 The special case \(\gamma=0\)↩︎

In this case, the structure of the singular values simplifies, as \(\sigma_{n,k}^0 = \frac{\sqrt{4\pi}}{\sqrt{n+1}}\) does not depend on \(k\), allowing to refine some range descriptions. Below, we recall the results from [21], but also give a Sobolev refinement, in Theorem 27 below, of [32]. This requires defining new Hilbert scales on \(\partial_+ S \mathbb{D}_E\), which generalize those previously defined in [21].

Sobolev scales on \(\partial_+ S \mathbb{D}_E\). From the mapping property (see [25]) \[T(C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)) \subset \mu C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E), \qquad T (\mu C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)) \subset C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E), \label{eq:Tmaps}\tag{56}\] the operator \({\cal T}_0 \mathrel{\vcenter{:}}= -T^2\) can be equipped with domain \(\mu C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)\) or\(C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)\), and \({\cal T}_0\) is \(L^2(\partial_+ S \mathbb{D}_E,\mathrm{d}\beta\mathrm{d}\alpha)\)-symmetric, and, in fact, essentially self-adjoint with respect to either of them. Indeed, the case of \(\mu C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)\) is covered in [25], and the case of \(C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E)\) can be proved using similar arguments. From the looks of their Taylor expansions off of \(\partial_0 S \mathbb{D}_E\), these domains can be thought of as Dirichlet and Neumann core domains, respectively. Thus, we denote by \({\cal T}_{0,D}, {\cal T}_{0,N}\) the \(L^2(\partial_+ S \mathbb{D}_E,\mathrm{d}\beta\mathrm{d}\alpha)\)-graph closure of \(({\cal T}_0, \mu C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E))\) and \(({\cal T}_0, C_{\alpha}^\infty(\partial_+ S \mathbb{D}_E))\), respectively. The involution \({\cal S}_A^E\) induces the even/odd decompositions \(L^2 = L^2_+ \stackrel{\perp}{\oplus} L^2_-\) and \(C_{\alpha}^\infty= C_{\alpha,+}^\infty\oplus C_{\alpha,-}^\infty\), with projection maps \((id \pm ({\cal S}_A^E)^*)/2\). Since \(\mu \circ {\cal S}_A^E = \mu\), we also have \(\mu C_{\alpha}^\infty= \mu C_{\alpha,+}^\infty\oplus \mu C_{\alpha,-}^\infty\). This allows us to define the following anisotropic Dirichlet and Neumann Sobolev scales for \(s\ge 0\): \[H_{T,\bullet,\pm}^s (\partial_+ S \mathbb{D}_E) \mathrel{\vcenter{:}}= {\cal T}_{0,\bullet}^{-s/2} (L^2_\pm (\partial_+ S \mathbb{D}_E,\mathrm{d}\beta\mathrm{d}\alpha)), \qquad \bullet \in \{D,N\}, \label{eq:SobT}\tag{57}\] so that \({\cal D}({\cal T}_{0,\bullet}^{s/2}) = H_{T,\bullet,+}^s (\partial_+ S \mathbb{D}_E) \oplus H_{T,\bullet,-}^s (\partial_+ S \mathbb{D}_E)\) for \(\bullet\in \{D,N\}\). The spaces \(H_{T,D,+}^s\) coincide with the spaces \(H_{T,+}^s(\partial_+ SM)\) originally introduced in [21].

Let us define, for \(n\ge 0\) and \(k\in \mathbb{Z}\), the \(L^2(\partial_+ S \mathbb{D}_E, \mathrm{d}\beta\mathrm{d}\alpha)\)-normalized functions \[\begin{align} \psi_{n,k} \mathrel{\vcenter{:}}= \frac{(-1)^n}{2\pi} e^{i(n-2k)(\beta+\alpha)} (e^{i(n+1)\alpha} + (-1)^n e^{-i(n+1)\alpha}), \\[0.2cm] \phi_{n,k} \mathrel{\vcenter{:}}= \frac{(-1)^n}{2\pi} e^{i(n-2k)(\beta+\alpha)} (e^{i(n+1)\alpha} - (-1)^n e^{-i(n+1)\alpha}). \end{align} \label{eq:phipsiE}\tag{58}\] Up to scalar multiplication, the functions \(\psi_{n,k}\) coincide with \(\psi_{n,k}^0\) defined in 15 , and \(\psi_{n,k}\) (resp. \(\phi_{n,k}\)) corresponds to \(u'_{p,q}\) (resp. \(v'_{p,q}\)) defined in [24] upon reindexing \((p,q)\mapsto (n = -p+2q, k = q-p)\). These families are Hilbert bases of \(L^2_+(\partial S \mathbb{D}_E, \mathrm{d}\beta\mathrm{d}\alpha)\) and \(L^2_-(\partial S \mathbb{D}_E, \mathrm{d}\beta\mathrm{d}\alpha)\), respectively, and from [24] we also have the characterizations \[\begin{align} \mu C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E) = \Bigg\{ \sum_{n\ge 0}\sum_{k\in \mathbb{Z}} u_{n,k}\;\psi_{n,k} \Bigg\}, \quad C_{\alpha,-}^\infty(\partial_+ S \mathbb{D}_E) = \Bigg\{ \sum_{n\ge 0}\sum_{k\in \mathbb{Z}} u_{n,k}\;\phi_{n,k} \Bigg\}, \label{eq:spans} \end{align}\tag{59}\] where the expansions have rapid decay, in the sense that for all \(a,b\in \mathbb{N}\) one has \(\sup_{n,k}\{ |u_{n,k}| (n+1)^{2a} (n-2k)^{2b} \}<\infty\). Noticing that \(\psi_{n,k}, \phi_{n,k}\) are eigenvectors of \(-T^2\), each with corresponding eigenvalue \((n+1)^2\), we deduce the following characterizations: \[\begin{align} \begin{aligned} H_{T,D,+}^s (\partial_+ S \mathbb{D}_E) &= \Bigg\{ \sum_{n\ge 0}\sum_{k\in \mathbb{Z}} u_{n,k}\;\psi_{n,k}, \quad \sum_{n,k} (n+1)^{2s} |u_{n,k}|^2 <\infty \Bigg\}, \\[0.2cm] H_{T,N,-}^s (\partial_+ S \mathbb{D}_E) &= \Bigg\{ \sum_{n\ge 0}\sum_{k\in \mathbb{Z}} v_{n,k}\;\phi_{n,k}, \quad \sum_{n,k} (n+1)^{2s} |v_{n,k}|^2 <\infty \Bigg\}. \end{aligned} \label{eq:HTcharac} \end{align}\tag{60}\]

Boundary operators and range characterizations. Out of the scattering relation and fiberwise Hilbert transform, one can construct a ‘boundary’ operator \[\begin{align} C_- \colon \mu C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E) \to \mu C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E), \end{align}\] that directly captures the infinite-dimensional co-kernel of \(I_0^E\). Specifically, \[C_- \mathrel{\vcenter{:}}= \frac{1}{2} A_-^* H_- A_-, \label{eq:Cminus}\tag{61}\] where \(A_-, A_-^*\) are defined in 43 and 44 and \(H\) is the odd part of the fiberwise Hilbert transform on \(\partial S \mathbb{D}_E\), whose integral expression is given by 98 . Explicitly, one finds that \[\label{eq:Cminus95spectral} C_-\psi_{n,k} = i (1_{k<0} - 1_{k>n}) \psi_{n,k}, \qquad n\ge 0,\;k\in \mathbb{Z};\tag{62}\] in particular, \(C_- \colon H_{T,D,+}^s(\partial_+ S \mathbb{D}_E)\to H_{T,D,+}^s(\partial_+ S \mathbb{D}_E)\) is bounded for every \(s\ge 0\).

We now discuss a Sobolev scale generalization of the Pestov-Uhlmann range characterization [32] in terms of the boundary operator \[P_- = A_-^* H_- A_+, \label{eq:Pminus}\tag{64}\] defined analogously to 61 . In this case, we have that \(P_- (C_{\alpha,-}^\infty(\partial_+ S \mathbb{D}_E)) \subset \mu C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E)\), \(P_-\) extends by continuity as an operator \(P_-\colon L^2_-(\partial_+ S \mathbb{D}_E)\to L^2_+ (\partial_+ S \mathbb{D}_E)\) and, reformulating [24] in the current notation, \(P_-\) has singular value decomposition \[P_- \phi_{n,k} = -2i\;1_{0\le k\le n}\;\psi_{n,k}, \qquad n\ge 0,\quad k\in \mathbb{Z}, \label{eq:SVDP}\tag{65}\] In particular, it becomes immediate that \(P_- \colon H_{T,N,-}^{s+1/2}(\partial_+ S \mathbb{D}_E)\to H_{T,D,+}^{s+1/2} (\partial_+ S \mathbb{D}_E)\) is bounded for any \(s\ge 0\), and its range coincides with \(I_0^E (\widetilde{H}^{s,0} ( \mathbb{D}_E))\), which we state here without proof.

This is a Sobolev generalization of [32] and the slight refinement [24], which provides the smooth counterpart \[\label{eq:PU95Smooth} I_0^E (C^\infty( \mathbb{D}_E)) = P_- \left( C_{\alpha,-}^\infty(\partial_+ S \mathbb{D}_E) \right).\tag{66}\]

3.2 Models of hyperbolic geodesics↩︎

Recall that the unit sphere bundle \(S \mathbb{D}_H^\circ\) in \(( \mathbb{D}_{H}^\circ,g_H)\) is defined in 7 . In what follows, we will sometimes use the identification \[\label{eq:identification95sphere95bundle} (z,w)=\big(z,{2\undefined(c\;e^{i\theta}\partial_{{z}})}\big)\in S \mathbb{D}_H^{\circ} \quad \leftrightarrow \quad (z,\theta)\in \mathbb{D}_H^{\circ}\times ( \mathbb{R}/2\pi \mathbb{Z}). \index{\theta}\tag{67}\] A distinguished \(g_H\)-unit speed geodesic passing through the \(g_H\)-unit vector \((z,\theta) = (0,0)\) is given by the function \({\mathrm x}(t)\) defined in 4 for which \(( {\mathrm x}(0),\dot{\mathrm x}(0)) = (0,1/2)\). Any other geodesic on \(\mathbb{D}_H\) can be obtained as the image of \({\mathrm x}(t)\) under an element of \(\mathrm{PSU}(1,1)\): Specifically, the unit-speed geodesic passing through \((z_1, \theta)\) at \(t=0\) is given by \[\gamma_{z_1,\theta}(t) \mathrel{\vcenter{:}}= T_{z_1,\theta} ( {\mathrm x}(t)), \qquad T_{z_1,\theta} (z) \mathrel{\vcenter{:}}= \frac{e^{i\theta}z + z_1}{1+e^{i\theta} \overline{z_1} z}. \label{eq:geodesic}\tag{68}\]

We now discuss certain models of all oriented geodesics on \(( \mathbb{D}_H,g_H)\).

3.2.1 Vertex parameterization↩︎

Using 68 , define, for \((\omega,s)\in ( \mathbb{R}/2\pi \mathbb{Z})\times (-1,1)\), \[\gamma^{{\mathrm v}}_{\omega,s}(t) \mathrel{\vcenter{:}}= T_{s e^{i\omega}, \omega+\pi/2} ( {\mathrm x}(t)) = e^{i\omega} \frac{s+i {\mathrm x}(t)}{1+is {\mathrm x}(t)}, \qquad t\in \mathbb{R}, \label{eq:Hgeo}\tag{69}\] with \({\mathrm x}(t)\) defined in 4 . This geodesic is parameterized so as to attain its vertex (point closest to the origin) at \(t=0\), equal to \(s e^{i\omega}\), with direction \(\omega+\pi/2\). In this way the set \(( \mathbb{R}/2\pi \mathbb{Z})_{\omega}\times (-1,1)_s\) gives all oriented geodesics, with orientation-reversing involution \((\omega,s)\mapsto (\omega+\pi, -s)\). Note the important identity used below, with \(x\) as in 3 : \[x(\gamma^{{\mathrm v}}_{\omega,s}(t)) = \frac{1-s^2}{1+s^2} \frac{1- {\mathrm x}(t)^2}{1+ {\mathrm x}(t)^2} = \frac{1-s^2}{1+s^2} \frac{1}{\cosh(t)}. \label{eq:xvertex}\tag{70}\]

3.2.2 Horocyclic parameterization↩︎

We now discuss the derivation of the parameterization 4 , whose analogy with Euclidean fan-beam coordinates will become clear in Proposition 30.

The group \(G=\mathrm{SL}(2, \mathbb{R})\) acts on the upper half plane model of hyperbolic space \[\mathbb{H}^2=\{z=x_ \mathbb{H}+iy_ \mathbb{H}\in \mathbb{C}:y_ \mathbb{H}>0\},\quad g_{ \mathbb{H}}=\frac{\mathrm{d}x_ \mathbb{H}^2+\mathrm{d}y_ \mathbb{H}^2}{y_ \mathbb{H}^2}\] by isometries via Möbius transformations, that is, \(\left[\begin{smallmatrix} a & b\\ c & d \end{smallmatrix}\right]\cdot z=\frac{az+b}{cz+d}\). An Iwasawa decomposition of \(G\) reads \(G = KNA\), where \(A,N,K\leq G\) are the subgroups \[\begin{align} A&=&\left\{A(t)=\left[\begin{smallmatrix} e^{-t/2} & 0\\ 0 & e^{t/2} \end{smallmatrix}\right]:t\in \mathbb{R}\right\}, \quad N=\left\{N(a)=\left[\begin{smallmatrix} 1 & a\\ 0 & 1 \end{smallmatrix}\right]:a\in \mathbb{R}\right\}, \\ K&=&\left\{K(\beta)=\left[\begin{smallmatrix} \cos(\beta/2) & \sin(\beta/2)\\ -\sin(\beta/2) & \cos(\beta/2) \end{smallmatrix}\right]:\beta\in \mathbb{R}\right\}. \end{align}\] Conjugating by the Cayley transform \(C(z)=\frac{z-i}{z+i}\), which is an isometry \(C:(\mathbb{H}^2,g_{\mathbb{H}})\to ( \mathbb{D}_H^\circ,g_H)\), we obtain an action of \(\mathrm{SL}(2, \mathbb{R})\) on \(\mathbb{D}_H^\circ\) by isometries. For \(z\in \mathbb{D}_H^\circ\), we then have \(K(\beta)\cdot z=e^{i\beta}z\), \(A(t)\cdot z=\frac{z- {\mathrm x}(t)}{1-z {\mathrm x}(t)}\) (with \({\mathrm x}(t)\) defined in 4 ), and \(N(a)\cdot z=\frac{z(2+ia)-ia}{ia z+(2-ia)}\). Further, \(A(t)\cdot 0\) is a geodesic, and one finds that the expression 4 of \(\gamma_{\beta,a}(t)\) is nothing but \[\gamma_{\beta,a}(t) = K(\beta)N(a)A(t)\cdot 0.\] For future reference, \(\gamma_{\beta,a}\) has endpoints \[\label{eq:endpoints} \gamma_{\beta,a}(-\infty)=e^{i\beta}\quad \text{ and }\quad\gamma_{\beta,a}(+\infty)= -e^{i\beta} \frac{1+ia}{1-ia} = e^{i(\beta+\pi + 2\tan^{-1}a)}.\tag{71}\] The parameterization 4 is called horocyclic in that, when \(\beta,t\) are frozen, the curve \(a\mapsto K(\beta)N(a)A(t)\cdot 0\) parameterizes a horocycle with ideal point \(e^{i\beta}\), generally with non-unit speed.

The map \(\pi_\mathrm{h}\). We now give the expression of the map \(\pi_\mathrm{h}\) defined in 8 , used to define the backprojection operator 9 .

The equivalence between vertex and horocyclic parameterizations is as follows.

3.2.3 Cosphere bundle parameterization↩︎

Another parameterization of the space of oriented geodesics on \(\mathbb{D}_H^\circ\) appears in recent studies of the X-ray transform on asymptotically hyperbolic manifolds [16] and is obtained by studying geodesic flows on the cosphere bundle. The purpose of this section is to show that this parameterization recovers, up to the sign, the horocycle parameterization 4 in the case of the hyperbolic disk. See Lemma 3 below.

To recall the context, let \(M\) be a \(n+1\)-dimensional compact manifold with boundary whose interior \(M^\circ\) is equipped with an asymptotically hyperbolic metric in the sense that there exist local coordinates \((\tilde{x},y)\) near \(\partial M\), where \(\tilde{x}\) is a suitable bdf such that the metric takes the form \[g = \frac{\mathrm{d}\tilde{x}^2 + \sum_{ij}h_{ij}(\tilde{x})\,\mathrm{d}y^i\,\mathrm{d}y^j}{\tilde{x}^2};\] a bdf \(\tilde{x}\) with this property is called a geodesic boundary defining function. In this context, the cosphere bundle \(S^* M^\circ\) can be viewed as a subset of the \(b\)-cotangent bundle \(^bT^*M\) (see [39]) restricted to \(M^\circ\). That is, \[S^* M^\circ = \left\{\left(\overline{\xi_0}\frac{\mathrm{d}\tilde{x}}{\tilde{x}} + \sum_i{\eta_i\,\mathrm{d}y^i}\right)\Big|_p\,:\,p\in M^\circ,\quad \overline{\xi_0}^2 + \tilde{x}^2\sum_{ij}h^{ij}\eta_i\eta_j = 1\right\}.\] Then \(S^*M^\circ\) extends smoothly as a submanifold \(\overline{S^*M^\circ}\) of \(^bT^* M\) to the boundary, with \(\overline{S^*M^\circ}\) at the boundary given by \[\partial\overline{S^*M^\circ} = \overline{S^*M^\circ}|_{\partial M} = \left\{\left(\overline{\xi_0}\frac{\mathrm{d}\tilde{x}}{\tilde{x}} + \sum_i{\eta_i\,\mathrm{d}y^i}\right)\Big|_p\,:\,p\in \partial M,\quad \overline{\xi_0}^2 = 1\right\}.\] We have \(\partial\overline{S^*M^\circ}= \partial_+S^*M^\circ \cup \partial_-S^*M^\circ\), where \[\partial_{\pm} S^*M^\circ \mathrel{\vcenter{:}}= \left\{\left(\pm\frac{\mathrm{d}\tilde{x}}{\tilde{x}} + \sum_i{\eta_i\,\mathrm{d}y^i}\right)\Big|_p\,:\,p\in\partial M,\quad \sum_i{\eta_i\,\mathrm{d}y^i}\Big|_p\in T_p^* \partial M \right\}.\] Then, the Hamiltonian vector field \(X\) corresponding to geodesic flow can be written as \(\tilde{x}\overline{X}\) where \(\overline{X}\) is smooth and transverse to \(\partial\overline{S^*M^\circ}\). Since the orbits of \(X\) and \(\overline{X}\) agree up to reparameterization on \(S^*M^\circ\), we can identify geodesics with their incoming/outgoing b-covectors in \(\partial_{\pm} S^*M^\circ\). Concretely, given a unit-speed geodesic \(\gamma(t)\), we consider the limits \[\lim_{t\to\mp\infty} g\left(\dot{\gamma}(t)\right)\] where we interpret \(g\) acting on a vector as converting it to a covector; such limits exist in \(^bT^*M\) and belong to \(\partial_{\pm} S^* M^\circ\).

In the case of \(( \mathbb{D}_H^\circ,g_H)\), a family of geodesic bdfs, indexed by a constant \(C>0\), is given by \[\tilde{x}(z) = C\frac{1-|z|}{1+|z|}, \quad z\in \mathbb{D}_H^\circ,\] in which case \(g_H = \frac{1}{\tilde{x}^2} \left( \mathrm{d}\tilde{x}^2 + \left(\frac{C^2-\tilde{x}^2}{2C}\right)^2\,\mathrm{d}\omega^2 \right)\). Then, writing \(\gamma(t) = (\tilde{x}(t),\omega(t))\), we have \[g_H(\dot{\gamma}(t)) = \frac{\dot{\tilde{x}}(t)}{\tilde{x}(t)} \frac{\mathrm{d}\tilde{x}}{\tilde{x}} + \left( \frac{C^2-\tilde{x}^2(t)}{2C} \right)^2 \frac{\dot{\omega}(t)}{\tilde{x}^2(t)} \mathrm{d}\omega.\]

Note that the measure \(\mathrm{d}\beta \mathrm{d}a\) on \({\cal G}_{\mathrm h}\) coincides with that induced by the symplectic form \(\mathrm{d}\beta\wedge \mathrm{d}\eta\) on \(T^* \mathbb{S}^1_{\beta,\eta}\).

3.3 Santaló’s formula↩︎

Any smooth vector field on \(S \mathbb{D}_H^\circ\) (and, more generally, on the sphere bundle of any oriented Riemannian surface) can be written as a linear combination over \(C^\infty( \mathbb{D}_H^\circ)\) of three globally defined vector fields \(X\), \(X_\perp\), and \(V\) (see [28]), acting on any \(f\in C^\infty (S \mathbb{D}_H^\circ)\) by \[\begin{align} Xf(z,w)&=\frac{\mathrm{d}}{\mathrm{d}t}f(\phi_t(z,w))\big|_{t=0},\\ X_\perp f{(z,w)}&=\frac{\mathrm{d}}{\mathrm{d}t}f(\psi_t(z,w))\big|_{t=0},\quad Vf{(z,w)}=\frac{\mathrm{d}}{\mathrm{d}t}f(r_t(z,w))\big|_{t=0}. \end{align}\] Here, \(\phi_t\) denotes the geodesic flow, \(\psi_t(z,w)\) is the curve in \(S \mathbb{D}_H^\circ\) arising as the parallel transport of \((z,w)\) along the geodesic with initial velocity \((z,-w^\perp)\) (where \((\cdot)^\perp\) denotes fiberwise counterclockwise rotation by angle \(\pi/2\)), and \(r_t(z,w)\) is the counterclockwise rotation of \((z,w)\) by angle \(t\). In terms of coordinates \((z,\theta)\), see 67 , they take the form (see [28]) \[\label{SM95Vector95fields} \begin{align} X&= c \Big(2\undefined(e^{i\theta}\partial_z)+2\undefined(e^{i\theta}\partial_z \log c)\partial _\theta\Big),\\ X_\perp&= c \left(2\undefined(e^{i\theta}\partial_z)-2\undefined(e^{i\theta}\partial_z \log c)\partial_\theta\right),\qquad V=\partial_\theta, \end{align}\tag{77}\] with \(c\) as in 1 . In coordinates \((z,\theta)\), the Liouville form on \(\mathbb{D}_H^{\circ}\times ( \mathbb{R}/2\pi \mathbb{Z})\) is given by \[\label{eq:Liouville} \mathrm{d}\Sigma^3(z,\theta) = \mathrm{d}V_H \wedge \mathrm{d}\theta = c^{-2}(z) \frac{\mathrm{d}\bar{z}\wedge dz}{2i}\wedge \mathrm{d}\theta;\tag{78}\] note in particular that \[\mathrm{d}\Sigma^3 (X,-X_\perp, V) = 1. \label{eq:ONframe}\tag{79}\]

The horocyclic parameterization 4 induces a smooth bijection \[\label{eq:geod95param} {\cal G}_{\mathrm h}\times \mathbb{R}\ni (\beta,a,t)\overset{F}{\longmapsto} (\gamma_{\beta,a} (t), \dot{\gamma}_{\beta,a}(t))\in S \mathbb{D}_H^\circ;\tag{80}\] it will be shown below that it is actually a diffeomorphism, and we will compute the pullback of the Liouville measure 78 .

We immediately obtain a Santaló type formula.

3.4 Projective equivalence and intertwiners↩︎

Recall the parameterization of Euclidean geodesics through \(\mathbb{D}_E\) given by 40 . It is well-known that the Euclidean disk and the Poincaré disk are projectively equivalent, explicitly so via the map \(\Phi\) defined in 18 .

3.4.1 Intertwining of X-ray transforms↩︎

3.4.2 Intertwining of backprojection operators↩︎

4 Proofs of results in sections 2.2-2.5↩︎

5 Data space structure and boundary operators - case \(\gamma=0\)↩︎

In this section, we study the structure of the data space for the X-ray transform on \(( \mathbb{D}_H^\circ,g_H)\) and construct boundary operators analogous to those on the Euclidean disk. We conclude it with the proof of Theorem 22.

5.1 Incoming and outgoing boundary for the hyperbolic sphere bundle; the scattering map.↩︎

Recall that the space of geodesics on \(\mathbb{D}_H^\circ\) is identified with \({\cal G}_{\mathrm h}= ( \mathbb{R}/2\pi \mathbb{Z})_\beta \times \mathbb{R}_a\) via the horocyclic parameterization 4 . We set \[\Gamma_\pm \mathrel{\vcenter{:}}= {\cal G}_{\mathrm h}\times\{\pm 1\}, \qquad \Gamma \mathrel{\vcenter{:}}= \Gamma_+\cup \Gamma_-.\] The spaces \(\Gamma_{+/-}\) are analogues of the interior of the incoming/outgoing boundaries \((\partial_{+/-} SM)^{\circ}\) defined for a simple Riemannian surface \(M\), and the map \((\beta,a)\mapsto (\beta,a,1)\) identifies \({\cal G}_{\mathrm h}\) with the incoming part \(\Gamma_+\). Note that by Lemma 3 and the discussion preceding it, there is already an identification of \({\cal G}_{\mathrm h}\) with \(\partial_+ S^* \mathbb{D}_H^\circ\) which induces a natural diffeomorphism \(\Gamma_\pm \to \partial_\pm S^* \mathbb{D}_H^\circ\), \((\beta,a,\pm 1)\mapsto (e^{i\beta}, \pm\frac{\mathrm{d}\tilde{x}}{ \tilde{x}}-a \mathrm{d}\omega)\).

Additionally, by virtue of Lemma 3, the hyperbolic scattering relation, to be understood as the map \({\cal S}^H \colon \Gamma_\pm \to \Gamma_{\mp}\) such that \({\cal S}^H (g(\dot{\gamma}(\pm\infty))) = g(\dot{\gamma}(\mp\infty))\) for any unit-speed geodesic \(\gamma(t)\) on \(\mathbb{D}_H^\circ\) is expressed as \[\mathcal{S}^H :\Gamma_\pm \to \Gamma_\mp, \qquad (\beta, a,\pm 1)\mapsto \big(\beta+\pi \pm 2\tan^{-1} a, a, \mp 1\big). \label{eq:SH}\tag{92}\] Finally, the antipodal involution \(A_H\) on the fibers of \(\Gamma\) takes the form \[\label{eq:hyp95antip95invol} A_H:\Gamma_\pm\to \Gamma_{\mp}, \qquad (\beta,a,\pm 1)\mapsto (\beta,-a,\mp 1),\tag{93}\] so that \({\cal S}_A^H = {\cal S}^H \circ A_H = A_H \circ {\cal S}^H\).

In terms of the identification described above, the map \(\Psi_\mathrm{hf}:{\cal G}_{\mathrm h}\to (\partial_+S \mathbb{D}_E)^\circ\) in 10 can be viewed as a map \(\Gamma_+\to (\partial_+S \mathbb{D}_E)^\circ\). We extend it to \(\Gamma\) by setting \(\Psi_\mathrm{hf}(\zeta) \mathrel{\vcenter{:}}= A_E\circ \Psi_\mathrm{hf}\circ A_H(\zeta)\) for \(\zeta\in \Gamma_-\), with \(A_E\) the Euclidean antipodal map defined in 47 . In this way, we obtain a diffeomorphism \(\Psi_\mathrm{hf}:\Gamma\to \partial S \mathbb{D}_E\setminus \partial_0 S \mathbb{D}_E\) \[\Psi_\mathrm{hf}(\beta,a,\lambda)=\begin{cases} (\beta ,\tan^{-1}(a)),\quad& (\beta,a,\lambda)\in \Gamma_+,\\ (\beta ,\pi-\tan^{-1}(a)),\quad &(\beta,a,\lambda)\in \Gamma_-, \end{cases} \label{eq:PsihfGamma}\tag{94}\] see also Figure 1. It is not hard to check the following intertwining properties: \[\label{eq:intertwining95HE} \Psi_\mathrm{hf}\circ\mathcal{S}^H= \mathcal{S}^E\circ \Psi_\mathrm{hf},\qquad \Psi_\mathrm{hf}\circ\mathcal{S}^H_A= \mathcal{S}^E_A\circ \Psi_\mathrm{hf}.\tag{95}\] Below, \(\mu_{\mathrm h}\) is extended to \(\Gamma\) so that it becomes odd with respect to \(A_H\), i.e., \(\mu_{\mathrm h}(\beta,a,\pm 1)=\pm(1+a^2)^{-1/2}\). This also implies that \(\mu_{\mathrm h}\circ {\cal S}^H=-\mu_{\mathrm h}\).

5.2 The boundary operators \(A_\pm^H\), \(C_\pm^H\), and \({P}_-^H\)↩︎

Analogously to 43 , for a measurable function on \(\Gamma_+\), we let \[A_\pm^H u(\beta,a,\lambda) \mathrel{\vcenter{:}}= \begin{cases} u(\beta,a,\lambda),\quad &(\beta,a,\lambda)\in \Gamma_+\\ \pm u(\mathcal{S}^H (\beta,a,\lambda)),\quad &(\beta,a,\lambda)\in \Gamma_-, \end{cases}\] where \({\cal S}^H\) is defined in 92 . Then, \(A_\pm^H u\) is a measurable function on \(\Gamma\). The \(L^2(\Gamma_+,\mu_{\mathrm h}^\sigma \mathrm{d}\beta\mathrm{d}a)\to L^2(\Gamma,|\mu_{\mathrm h}|^\sigma \mathrm{d}\beta\mathrm{d}a)\) adjoint for any \(\sigma \in \mathbb{R}\) is \[({A}_\pm^H)^*u(\beta, a, 1)=u(\beta, a, 1)\pm (\mathcal{S}^H)^* u(\beta, a, 1).\] The fiberwise odd Hilbert transform on \(\Gamma\) is defined for \(u\in C_c^\infty (\Gamma)\) by \[\mathcal{H}_- u(\beta,a,\pm 1) \mathrel{\vcenter{:}}= \pm \frac{1}{\pi }\mathrm{p.v.}\int_{-\infty}^\infty \frac{u_-(\beta,a',\pm 1)}{a-a'}\mathrm{d}a', \qquad u_- \mathrel{\vcenter{:}}= (u- A_H^*u)/2.\] It extends to a bounded operator on \(L^2(\Gamma,\mathrm{d}\beta\mathrm{d}a)\) with the property \(A_H^*\mathcal{H}_-=-\mathcal{H}_-\). Using it, we define the following operators on \(\Gamma_+\): \[\label{eq:CPH} {C}_-^H \mathrel{\vcenter{:}}= \frac{1}{2} (A_-^H)^*{\cal H}_{-}A_-^H,\qquad {P}_-^H \mathrel{\vcenter{:}}= (A_-^H)^*{\cal H}_{-}A_+^H.\tag{96}\] In the statement below, we denote by \(H_-\) the fiberwise Hilbert transform on the fibers of \(\partial S \mathbb{D}_E\), restricted to fiberwise odd integrands.

We can now prove the following, which, together with the mapping properties for \(C_-\), \(P_-\) mentioned in Section3.1.4 and Lemma 7 below, imply the mapping properties in 38 and 39 .

By Proposition 34 and Eqs. 62 and 65 , the following is immediate:.

5.3 The compactification \(\overline{\Gamma}\) and an ambient characterization of \(C_{\alpha,+}^\infty (\overline{{\cal G}_{\mathrm h}})\)↩︎

In this section, we construct a compactification of \(\Gamma\) (\(\overline{\Gamma}\), defined in 101 ), and upon equipping \(\overline{\Gamma}\) with a smooth structure that makes it diffeomorphic to \(\partial S \mathbb{D}_E\) (see Lemma 6), we show in Proposition 36 that \(C_{\alpha,+}^\infty (\overline{{\cal G}_{\mathrm h}})\) admits an “ambient” characterization akin to the statement \[C_{\alpha,+}^\infty(\partial_+ S \mathbb{D}_E) = \{u\in C^\infty(\partial_+ S \mathbb{D}_E)\colon A_+ u \in C^\infty(\partial S \mathbb{D}_E),\; u\circ {\cal S}_A^E = u\},\] which holds in the Euclidean case.

Consider the compactification \(\overline{ \mathbb{R}}_a=[-\infty,\infty]_a\) of the real line with the smooth structure described in Section 2.2 so that the map \(\tan^{-1}: \mathbb{R}_a\to (-\pi/2,\pi/2)\) extends to a smooth diffeomorphism \(\overline{ \mathbb{R}}_a\to [-\pi/2,\pi/2]\), still denoted by \(\tan^{-1}\). Let \(\overline{\Gamma}_{\pm} \mathrel{\vcenter{:}}= ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \overline{ \mathbb{R}}_a\times \{\pm 1\}\), and set \[\overline{\Gamma} \mathrel{\vcenter{:}}= \overline{\Gamma}_{+}\sqcup \overline{\Gamma}_-/_\sim \label{eq:Gammabar}\tag{101}\] where \(\sim\) is the smallest equivalence relation on \(\overline{\Gamma}_{-}\sqcup \overline{\Gamma}_{+}\) such that \((\beta,\pm \infty, 1)\sim (\beta,\pm \infty, -1)\). We equip \(\overline{\Gamma}\) with the quotient topology, and we now construct a smooth structure on \(\overline{\Gamma}\) making it diffeomorphic to \(\partial S \mathbb{D}_E\). Consider the following subsets of \(\overline{\Gamma}\): \[\begin{align} U_1 & \mathrel{\vcenter{:}}= \big(( \mathbb{R}/2\pi \mathbb{Z})_\beta\times (-\infty,\infty]\times \{1\}\big)/_\sim\cup\big( ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times (-\infty,\infty]\times \{-1\}\big)/_\sim,\\ U_2 & \mathrel{\vcenter{:}}= \big(( \mathbb{R}/2\pi \mathbb{Z})_\beta\times [-\infty,\infty)\times \{1\}\big)/_\sim\cup\big( ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times [-\infty,\infty)\times \{-1\}\big)/_\sim. \end{align}\] We obtain-well defined homeomorphisms \(\varphi_j:U_j\to \varphi_j(U_j)\subset ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \mathbb{R}\) by setting \[\begin{align} \varphi_1( [\beta, a,\lambda])&=\begin{cases} (\beta,\tan^{-1}({a})), & \lambda=1\\ (\beta,\pi -\tan^{-1}({a})), & \lambda=-1 \end{cases},\\ \varphi_2( [\beta, a,\lambda])&=\begin{cases} (\beta,\tan^{-1}({a})), & \lambda=1\\ (\beta,-\pi -\tan^{-1}({a})), & \lambda=-1 \end{cases}. \end{align}\] For \((\beta,\alpha)\in \varphi_2(U_1\cap U_2)=( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \big((-\pi/2,\pi/2)\cup (-3\pi/2,-\pi/2)\big)\), they satisfy \[\varphi_1\circ \varphi^{-1}_2(\beta,\alpha)= \begin{cases} (\beta,\alpha),& (\beta,\alpha)\in ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times (-\pi/2,\pi/2),\\ (\beta,\alpha+2\pi),& (\beta,\alpha)\in ( \mathbb{R}/2\pi \mathbb{Z})_\beta\times (-3\pi/2,-\pi/2), \end{cases}\] which is a diffeomorphism onto \[\varphi_1(U_1\cap U_2)=( \mathbb{R}/2\pi \mathbb{Z})_\beta\times \big((-\pi/2,\pi/2)\cup (\pi/2,3\pi/2)\big).\] Since \(\overline{\Gamma}\subset U_1\cup U_2\), we obtain a smooth structure on \(\overline{\Gamma}\) upon composing the first coordinate of \(\varphi_j\) with charts for \(( \mathbb{R}/2\pi \mathbb{Z})_\beta\) and extending the resulting charts for \(\overline{\Gamma}\) to a maximal atlas.

The scattering map extends in a natural way to a map \[\overline{\mathcal{S}}^H:\overline{\Gamma}\to \overline{\Gamma} ,\quad \overline{\mathcal{S}}^H([(\beta,{a},\lambda) ])=\begin{cases} [{\mathcal{S}}^H(\beta,a,\lambda)],\quad & a\in \mathbb{R},\\ [(\beta,\pm\infty,\lambda)], \quad & {a}=\pm \infty, \end{cases}\] and the intertwining property 95 extends to all of \(\partial S \mathbb{D}_E\): \[\label{eq:ext95scat} \bar{\Psi}_{\mathrm{hf}}\circ \overline{\mathcal{S}}^H= \mathcal{S}^E\circ\bar{\Psi}_{\mathrm{hf}}.\tag{102}\] Since \({\cal S}^E:\partial S \mathbb{D}_E\to \partial S \mathbb{D}_E\) is a smooth diffeomorphism fixing \(\partial_0 S \mathbb{D}\) and \(\overline{\mathcal{S}}^H=\bar{\Psi}_{\mathrm{hf}}^{-1}\circ \mathcal{S}^E\circ \bar{\Psi}_{\mathrm{hf}}\), Lemma 6 immediately implies the following.

Now, the following alternative characterization of the space \(C_{\alpha,+}^\infty (\overline{{\cal G}_{\mathrm h}})\) is immediate by 11 .

5.4 Proof of theorem 22↩︎

Before proving Theorem 22, we note the following lemma.

Now, we have the following proof.

6 Hyperbolic moment conditions↩︎

In this section, we discuss the hyperbolic moment conditions as defined in [9], and show in particular that they are equivalent to 34 in the present setting, when choosing \(p_m(x)= (-x)^m\). There, the authors state the hyperbolic moment conditions for rapidly decaying functions on the space \({\cal G}_{n,k}\) of totally geodesic \(k\)-dimensional planes in the \(n\)-dimensional Poincaré ball. They use the parameterization \({\cal G}_{n,k}=\{(\xi,v,r)\in \Xi_{n,k}\times \mathbb{S}^{n-1}\times [0,\infty):v\in \xi\}\), where \[\Xi_{n,k}=\big\{ \mathbb{S}^{n-1}\cap \Pi:\;\; \Pi \text{ a } n-k\text{-dimensional plane through the origin in } \mathbb{R}^n \big\}.\label{eq:Xink}\tag{103}\] Each triple in \({\cal G}_{n,k}\) corresponds to the \(k\)-geodesic plane orthogonal to the plane determined by \(\xi\), through the point \(rv\), where \(r\) denotes geodesic distance from the origin (see [9]). Then, the hyperbolic moment conditions for a rapidly decaying function \(\phi\) on \({\cal G}_{n,k}\) are stated as follows: For every \(\xi\in \Xi_{n,k}\) and \(v\in \xi\), \[\label{eq:MoCoBeTa} \int_{v\in \xi}\langle v ,v'\rangle_E^{m} \left[\int_0^{\infty}\frac{\tanh^{m+n-k-1}(r)}{\cosh(r)}\phi(\xi,v,r)\mathrm{d}r\right]\mathrm{d}S( v)=P_m(v').\tag{104}\] Here, \(\mathrm{d}S(v)\) is the measure on \(\xi\cong \mathbb{S}^{n-k-1}\) induced by the Lebesgue measure on \(\Pi\) in 103 , and \(P_m\) is a homogeneous polynomial of degree \(m\).

In our case, \(n=2\), \(k=1\), and each \(\xi= \mathbb{S}^1\cap L\), where \(L\) is a line through the origin, consists of two points. So, let \(\xi=\{\pm v_0\}\in \Xi_{2,1}\), where \(v_0\in \mathbb{S}^1\), and rewrite 104 as \[\label{eq:MoCoBeTa2} \sum_{v=\pm v_0}\langle v ,v'\rangle_E^{m} \int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v,r)\mathrm{d}r=P_m(v'),\qquad v'=\pm v_0.\tag{105}\] If 105 holds for \(v'\in \xi\), it also holds for \(-v'\) (by homogeneity of \(P_m\)). So, let \(v'=v_0\). Then, 105 is equivalent the statement that for all \(v_0\in \mathbb{S}^1\), \[\label{eq:MoCoBeTa3} \begin{align} P_m(v_0)=&\sum_{v=\pm v_0}\langle v ,v_0\rangle_E^{m} \int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v,r)\mathrm{d}r\\ =&\int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v_0,r)\mathrm{d}r+(-1)^m\int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,-v_0,r)\mathrm{d}r. \end{align}\tag{106}\] If we view \(\phi\) as a function on the double cover \({\cal G}_{2,1}'=\{(\xi,v,r)\in \Xi_{2,1}\times \mathbb{S}^{1}\times (-\infty,\infty):v\in \xi\}\) of the space of (unoriented) geodesics, we see that \(\phi(\xi,v_0,r)=\phi(\xi,-v_0,-r)\). Therefore, \[\label{eq:MoCoBeTa4} \begin{align} P_m(v_0)\overset{\phantom{\rho=-r}}{=}&\int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v_0,r)\mathrm{d}r+(-1)^m\int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v_0,-r)\mathrm{d}r\\ \overset{\rho=-r}{=}&\int_0^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v_0,r)\mathrm{d}r+(-1)^m\int_{-\infty}^{0}\frac{(-1)^m\tanh^{m}(\rho)}{\cosh(\rho)}\phi(\xi,v_0,\rho)\mathrm{d}\rho\\ \overset{\phantom{\rho=-r}}{=}&\int_{-\infty}^{\infty}\frac{\tanh^{m}(r)}{\cosh(r)}\phi(\xi,v_0,r)\mathrm{d}r. \end{align}\tag{107}\] With \({\mathrm x}(t)\) defined in 4 , since \({\mathrm x}(t)\) is a unit-speed geodesic, the geodesic distance from \(0\) to \({\mathrm x}(t)\) is \(t\). By rotation-invariance, this means that the Euclidean distance \(s\) from the origin appearing in our vertex parameterization is related to the geodesic distance \(r\) by \(s = \tanh(r/2)\), and this implies \(\tanh(r)=\frac{2s}{1+s^2}\) and \(\cosh(r)=\frac{1+s^2}{1-s^2}\). Therefore, changing variable \(r\to s\), we arrive at \[\begin{align} P_m(v_0)=&\int_{-\infty}^{\infty}\cosh(r)\tanh^{m}(r)\phi(\xi,v_0,r)\mathrm{sech}^2(r)\mathrm{d}r\\ =&\int_{-1}^{1}\frac{1+s^2}{1-s^2}\left(\frac{2s}{1+s^2}\right)^{m}\phi\big(\xi,v_0, \tanh^{-1} \Big(\frac{2s}{1+s^2}\Big)\big)\left(2\frac{1-s^2}{(1+s^2)^2}\right)\mathrm{d}s\\ =&\int_{-1}^{1} \left(\frac{2s}{1+s^2}\right)^{m}\phi^{\mathrm{v}}(\arg(v_0),s)\;\frac{2\mathrm{d}s}{1+s^2}, \end{align}\] and this recovers 34 (using the correspondence of \(\gamma^{\mathrm{v}}_{\omega,s}\), viewed as an unoriented geodesic, with \(\big(\{\pm e^{i\omega}\},e^{i\omega},\tanh^{-1}(\frac{2s}{1+s^2})\big)\in {\cal G}_{2,1}'\)).

Acknowledgments↩︎

The authors thank the anonymous referees, whose comments and suggestions improved the presentation of the article.

References↩︎

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1. Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany;
   email: nikolaos.eptaminitakis@math.uni-hannover.de↩︎

2. Department of Mathematics, University of California, Santa Cruz CA 95064; email:fmonard@ucsc.edu↩︎

3. Department of Mathematics, Northwestern University, Evanston IL 60208;
   email: yuzhou.zou@northwestern.edu↩︎

4. Commonly denoted \(P^{(\gamma+1/2,\gamma+1/2)}_n\) in the literature. Up to a scalar, this is also the ultraspherical/Gegenbauer polynomial \(C^{(\gamma+1)}_n\).↩︎

5. Here, \(({\cal L}_\gamma^H, C_{\mathrm{ev}}^\infty( \mathbb{D}_H))\) denotes the operator \({\cal L}_\gamma^H\) with domain \(C_{\mathrm{ev}}^\infty( \mathbb{D}_H)\).↩︎

6. The \(w\) subscript is in reference to the fact that \({\cal L}_\gamma^H\) is a wedge operator.↩︎

7. The point closest to the origin.↩︎