Guillarmou's Normal Operator

• Guillarmou’s Normal Operator is a fundamental elliptic pseudodifferential operator that generalizes the classical X-ray normal operator to closed Anosov manifolds.
• It combines pushforward and pullback techniques with a flow-invariant operator to deliver explicit inversion formulas and stability estimates essential for tensor tomography.
• Its ellipticity on the solenoidal subspace underpins robust Sobolev mapping properties that facilitate accurate reconstructions in geometric inverse problems.

Guillarmou’s normal operator is a fundamental object in the microlocal and analytic theory of the X-ray transform, especially on closed Riemannian manifolds with hyperbolic or Anosov dynamical systems. It generalizes the classical X-ray normal operator from boundary value problems to the setting of closed manifolds, allowing for deep analysis of tensor tomography, stability estimates, and invariant distribution theory in geometric inverse problems. The normal operator arises from a combination of pushforward and pullback procedures intertwined with a flow-invariant operator on the unit tangent bundle, and in notable cases, possesses an explicit pseudodifferential structure and inversion formulas.

1. Geometric and Dynamical Context

Let $(M,g)$ denote a closed, oriented Riemannian $n$-manifold, with unit tangent bundle $SM$. The fundamental geometric flows are the geodesic flow, magnetic flow (associated to a closed $2$-form $\Omega$), and thermostat flow (generated by an external field $Y: SM \rightarrow (v^\perp)$). Each is represented as a flow $\varphi_t: SM \rightarrow SM$, whose infinitesimal generator $F$ is a smooth, first-order differential operator: $F = (v, 0)$ in the geodesic case, augmented by $Y$ in the magnetic/thermostat cases.

The Anosov condition is central: the tangent bundle of $SM$ admits a continuous, $\varphi_t$-invariant splitting $T(SM) = \mathbb{R} F \oplus E_s \oplus E_u$, corresponding to stable/unstable manifolds with uniform contraction/expansion under the flow. Technical requirements include volume preservation, mixing properties, transversality, and applicability of the smooth Livšic theorem—all of which underpin both the analytic properties of the normal operator and the well-posedness of calibration and inversion.

2. Definition and Construction of the Normal Operator

The geodesic X-ray transform maps a function or tensor field to its integrals over closed orbits (periodic geodesics): for $f \in C^\infty(SM)$ and a closed orbit $\gamma$, $I f(\gamma) = \int_0^T f(\varphi_t(x,v)) dt$. For tensors of rank $m$, one uses the canonical identification $\pi_m^* h(x,v) = h_x(v, \ldots, v)$, defining $I_m[h] = I(\pi_m^* h)$.

The adjoint $I_m^*$ corresponds to integration in fibers with respect to the Liouville measure, and the normal operator is

$$N_m = I_m^* I_m = \pi_{m*} (\Pi + 1 \otimes 1) \pi_m^*,$$

where $\Pi$ is the flow-invariant operator central to Guillarmou’s construction, and $1 \otimes 1$ projects onto constant distributions. $\Pi$ is expressed via meromorphic continuations of the flow-resolvent $R_\pm(z) = (\mp F - z)^{-1}$, taken as

$\Pi = - (R_+^{\rm hol}(0) + R_-^{\rm hol}(0)).$

In this setup, $\Pi$ is self-adjoint, flow-invariant, annihilates $F$-derivatives, and is nonnegative.

3. Analytic and Pseudodifferential Properties

Guillarmou’s normal operator $N_m$ is a classical elliptic pseudodifferential operator of order $-1$, acting on $m$-tensors, with block form relating $S^mT^*M$ and $S^{m-1}T^*M$. The principal symbol is given, in local coordinates and with respect to the (m)- and (m-1)-tensor decomposition, by

$$\sigma(N_m)(x,\xi) = \frac{2\pi}{|\xi|}\, \mathrm{diag}\left[C_{n,m}^{-1} \pi_{\ker i_\xi}|_{S^m},\; C_{n,m-1}^{-1} \pi_{\ker i_\xi}|_{S^{m-1}}\right],$$

where $i_\xi$ is contraction with $\xi$, the $C_{n,m}$ constants depend on $n$ and $m$, and $\pi_{\ker i_\xi}$ projects onto the kernel of contraction.

The operator $N_m$ is elliptic on the solenoidal subspace $\ker D^*_\mu$, where $D^*_\mu$ is the magnetic divergence operator, adjoint to the symmetrized covariant derivative. This ellipticity enables construction of parametrices and, consequently, stability estimates crucial for geometric inverse problems.

4. Explicit Inversion and Attenuated Normal Operators

In negative constant curvature, Guillarmou’s normal operator admits an explicit inversion formula. For a closed surface $(M,g)$ of constant curvature $K<0$, the operator is constructed by the push-pull

$\Pi_0 = \pi_{0*}\, \Pi\, \pi_0^*,$

and similarly its attenuated versions for $\Re z > 0$,

$$\Pi_0^{(z)} = \pi_{0*} \left[\int_{\mathbb{R}} e^{-|t|z} \varphi_t^*\, dt \right] \pi_0^*.$$

For mean-zero functions, there is an explicit inversion formula:

$\Delta S_K\, \Pi_0 f = -8\pi^2 f,$

where $S_K$ integrates along geodesics with exponential attenuation depending on curvature. This formula leads to the construction of $X$-invariant distributions with prescribed pushforward and highlights representation-theoretic connections observed by Guillarmou–Monard (Richardson, 13 Jan 2025).

5. Stability Estimates and Sobolev Mappings

The ellipticity of $N_m$ yields precise a priori estimates. For solenoidal $f$,

$\|f\|_{H^s} \leq C \| N_m f \|_{H^{s+1}},$

and because $N_m = I_m^* I_m$, with $I_m^*: H^{s+1}(SM) \to H^s(M)$ bounded,

$\|f\|_{H^s(M)} \leq C \| I_m f \|_{H^{s+1}(SM)}.$

Sobolev norms thus reflect no loss of derivatives beyond the order of $N_m$. The multiplicative constant depends on flow expansion/contraction rates, geometric pinching (e.g., via curvature), regularity bounds, and the magnitude of $Y$ in the magnetic/thermostat case (Muñoz-Thon et al., 28 Dec 2025).

6. Comparison with Classical Boundary Value X-ray Theory

In traditional boundary rigidity and X-ray problems, the normal operator arises as $N = I^*I$, where $I$ integrates along geodesics between boundary points. Both $N$ and Guillarmou’s $N_m$ are elliptic pseudodifferential operators of order $-1$, central to stability and uniqueness. Key differences stem from the lack of boundary in the Anosov/closed setting, the role of Ruelle–resolvent theory (in place of Hilbert space trace methods), and the structure of smoothing remainders (cutoff/resolvent-based rather than boundary convexity-induced). The closed case handles all $m$-tensors directly, while the boundary case relies on delicately managed trace and extension operators.

Setting	Normal Operator	Order
Closed Anosov (Guillarmou’s)	$N_m = I^*_m I_m$	$-1$
Boundary value (classical)	$N = I^* I$	$-1$

7. Applications and Further Developments

Guillarmou’s normal operator provides the analytic backbone for inversion formulas, explicit construction of $X$-invariant distributions, and stability estimates—core tools in tensor tomography and spectral rigidity. Recent advances extend the theory to magnetic and thermostat flows, employing microlocal and representation-theoretic strategies (Muñoz-Thon et al., 28 Dec 2025, Richardson, 13 Jan 2025). Explicit formulas connect to Helgason’s spherical transform, while approximate inversion on variable-curvature manifolds remains a subject of ongoing research.

The regularity of $X$-invariant distributions constructed via the normal operator is limited by its pseudodifferential order $-1$, precisely matching the expected Sobolev mapping properties. The theory strongly suggests analogous results on Anosov manifolds with variable curvature, where the main analytic and microlocal framework, incorporating anisotropic Sobolev spaces and resolvent expansions, remains applicable with suitable modifications to control remainder terms.

• (Muñoz-Thon et al., 28 Dec 2025) Guillarmou's Normal Operator for Magnetic and Thermostat Flows
• (Richardson, 13 Jan 2025) An inversion formula for the X-ray normal operator over closed hyperbolic surfaces