Abstract
We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form L^2mapsto H^{1/2}_{T}, where the H^{1/2}_{T}-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.
Highlights
Let us begin with the simplest case of the Radon transform in R2 in parallel beam geometry
The main question we address in the present paper is the existence of a stability estimate analogous to (1.1) but in a geometric setting, namely, when R2 and the lines in the plane are replaced by a Riemannian manifold and its geodesics
The geodesic X-ray transform acts on functions defined on the unit sphere bundle of a compact oriented d-dimensional Riemannian manifold (M, g) with smooth boundary ∂ M (d ≥ 2)
Summary
Let us begin with the simplest case of the Radon transform in R2 in parallel beam geometry (see [15] for more details). The geodesic X-ray transform acts on functions defined on the unit sphere bundle of a compact oriented d-dimensional Riemannian manifold (M, g) with smooth boundary ∂ M (d ≥ 2). This is when we assume in addition that the sectional curvature of M is non-positive In this case a stability estimate is available as follows: Theorem 1.2 ([21] and [23, Theorem 4.3.3]) Let (M, g) be a connected compact manifold with strictly convex boundary and non-positive sectional curvature. The space HT1(∂+ S M) is defined by restriction, and HT1/2(∂+ S M) is defined by complex interpolation between L2(∂+ S M) and HT1 (∂+ S M) With this definition we may state our main result: Theorem 1.3 Let (M, g) be a connected compact manifold with strictly convex boundary and non-positive sectional curvature.
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