Guillarmou's Normal Operator for Magnetic and Thermostat Flows

TL;DR

The paper extends Guillarmou’s normal operator program from geodesic flows to magnetic and thermostat flows on closed manifolds. By constructing generalized normal operators and proving their ellipticity as order $-1$ pseudodifferential operators under Anosov-type dynamics and transversality assumptions, it derives stability estimates for magnetic X-ray transforms and, in key cases, injectivity results. A magnetic potential–solenoidal decomposition clarifies the kernel structure and enables parametrices, coercivity, and Livšic-based arguments to connect ray transforms with invariant distributions. These results generalize boundary-based normal operator theory to closed manifolds, with implications for rigidity problems in magnetic and thermostat systems. Overall, the work provides a robust microlocal framework for understanding X-ray-type transforms in richer dynamical settings beyond geodesic flows.

Abstract

Guillarmou's normal operator over a closed Anosov manifold is analogous to the classical normal operator of the geodesic X-ray transform over manifolds with boundary. In this paper, we generalize this normal operator, under some dynamical assumptions, to thermostat flows as well as to the case of the magnetic flows. In particular, we show that these generalized normal operators are elliptic pseudodifferential operators of order -1 in each case. As an application, we prove a stability estimate for the magnetic X-ray transform.

Theorems & Definitions (52)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: 2011-faure-sjostrand, Lefeuvre*Theorem 9.1.6
• Lemma 2.6: 2016-dyatlov-zworski