The geodesic X-ray transform has wide ranging applications in X-ray computerized tomography, the boundary rigidity problem associated with the ‘travel-time metric for seismic waves’, ultrasound transmission tomography, ‘optical tomography with a variable index of refraction’, anisotropic Calderón’s problem, etc. This paper is devoted to the study of ‘microlocal properties of the geodesic X-ray transform \(\mathcal X\)’ defined on a Riemannian manifold \(\left(M,g \right)\) with non-simple boundary ‘allowing the presence of conjugate points’. Prior to this paper, as far as known to this reviewer, the mathematical literature concerning the geodesic X-ray transform was mostly centered around simple manifolds like, [V. Guillemin and S. Sternberg, Am. J. Math. 101, 915–955 (1979; Zbl 0446.58019); G. Paternain et al., Chin. Ann. Math., Ser. B 35, 399–428 (2014; Zbl 1303.92053)], etc. Some of the recent studies highlighting the results of geodesic X-ray transforms for non-simple manifolds can be found in [B. Frigyik et al., J. Geom. Anal. 18, No. 1, 89–108 (2008; Zbl 1148.53055); F. Monard et al., Comm. Math. Phys. 337, 1491–1513, (2015; Zbl 1319.53086)], etc.
In [Guillemin and Sternberg, loc. cit.] it is shown “that the normal operator \({\mathcal N} = {{\mathcal X}^{t}} \circ \mathcal X\) is an elliptic pseudodifferential operator of order \(-1\) when \(\left(M,g \right)\) is a simple manifold”, \({\mathcal X}^{t}\) being the transpose of \({\mathcal X}\). But for non-simple Riemannian manifolds \(\left(M,g \right)\) the normal operator \( {\mathcal N}\) is not a pseudodifferential operator. P. Stefanov and G. Uhlmann [Anal. and PDE 5, No. 2, 219–260, (2012; Zbl 1271.53070)] have earlier shown “that in the case of fold caustics, an appropriately localized version of the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator (FIO)”, a result which is similar to the main result of Theorem 4 established in the present paper by the authors, yet here they “lessen the restriction to fold caustics”. Monard et al. [loc. cit.] did away with the restriction to fold caustics for the two-dimensional case. In this paper, the method adopted by the authors is similar to the one used by the second author of this paper in his earlier co-authored paper [Zbl 1319.53086] but here the geometry of conjugate points is analyzed more deeply to arrive at “a more general conclusion”.
Assumption 1. \(\left(M,g \right)\) is an \(n\)-dimensional compact, non-trapping Riemannian manifold with smooth strictly convex boundary and with \(n \geq 2\).
Under this assumption, the authors state and prove the following most important result of this paper:
Theorem 4. Suppose that \({C_S} = \emptyset\). Then the sets \({C_{{A_k}}} = {{\mathcal C}_k}\left( {{J_{R,k}}} \right) \subset {T^*}\left( {{M^{\text{int} }} \times {M^{\text{int} }}} \right)\) are either empty or are local canonical relations. On the level of operators, if \(\phi \in {C^\infty }\left( {SM} \right)\) is greater than or equal to zero everywhere and \({{\mathcal N}_\phi }\) is defined by \({{\mathcal N}_\phi } = {\pi _*} \circ {\phi ^m} \circ {F^*} \circ {F_*} \circ {\phi ^m} \circ {\pi ^*}\) then we have a decomposition \[ {N_\phi } = \Upsilon + \sum\limits_{k = 1}^{n - 1} {\left( {\sum\limits_{m = 1}^{{M_k}} {{A_{k,m}}} } \right)},\tag{1} \] where \(\Upsilon \) is a pseudodifferential operator of order \(-1\), and for each \(k\) either
\[ {A_{k,m}} \in {{\mathcal I}^{ - \left( {n - k + 1} \right)/2}}\left( {{M^{\text{int} }} \times {M^{\text{int} }},C{'_{{A_{k,m}}}};\Omega _{{M^{\text{int} }} \times {M^{\text{int} }}}^{1/2}} \right), \]
where \({C_{{A_{k,m}}}} \subset {C_{{A_k}}}\) for each \(m\), or \({M_k} = 1\) and \({A_{k,1}} = 0\) if \({C_{{A_k}}} = \phi. \) Furthermore, \(\Upsilon \) is elliptic at every point \(\eta \in {T^*}{M^{\text{int} }}\) such that there exists a \(v \in S{M^{\text{int} }}\) with \(\eta \left( v \right) = 0\) and \(\phi \left( v \right) \neq 0.\)
Next, supposing that the following assumption holds:
Assumption 2. Assume that the dimension \(n\) is at least three, that all conjugate pairs in \(S{\widetilde M^{\text{int} }} \times S{\widetilde M^{\text{int} }}\) are of order \(1\), and that \({C_{{A_1}}}\) (as mentioned in Corollary 1 of the paper) is a local canonical graph.
The authors address the problem of inverting \({\mathcal N}_{\phi}\), by using ‘the decomposition in Theorem 4 to obtain stability estimates for inversion of \(\mathcal X\)’, proving the following:
Theorem 5. If Assumption 2 is satisfied, \(\phi \in {C^\infty }\left( {SM} \right)\) is greater than or equal to zero everywhere and for every \(\eta \in {T^*}{M^{\text{int} }}\) there exists a \(v \in S{M^{\text{int} }}\) with \(\eta \left( v \right) = 0\) and \(\phi \left( v \right) \neq 0,\) then the kernel of \({{\mathcal X}_\phi }\) acting on \({L^2}\left( {\Omega _M^{1/2}} \right)\) is at most finite-dimensional and is contained in \(C_c^\infty \left( {\Omega _{{M^{\text{int} }}}^{1/2}} \right).\) Furthermore, if \({\mathcal F} \subset {L^2}\left( {\Omega _M^{1/2}} \right)\) is a closed subspace complementary to the kernel of \({{\mathcal X}_\phi }\) then \[ {\left\| {{{\mathcal X}_\phi }\left[ f \right]} \right\|_{{L^2}\left( {\Omega _{\partial \_SM}^{1/2}} \right)}} \sim {\left\| f \right\|_{{H^{ - 1/2}}\left( {\Omega _M^{1/2}} \right)}}\tag{2} \] for all \(f \in {\mathcal F}.\)
In the opinion of the reviewer these remarkable results should find some interesting practical applications in X-ray tomography in the coming days, further, he sees eye to eye with the authors’ conclusion that “while we have gone some way towards completing the microlocal analysis of the geodesic X-ray transform for nontrapping manifolds, a number of questions remain” which, the reviewer hopes that, will be the burning questions of the upcoming research in this field.