A sharp stability estimate for tensor tomography in non-positive curvature
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Mathematische Zeitschrift
A sharp stability estimate for tensor tomography in non-positive curvature Gabriel P. Paternain1 · Mikko Salo2 Received: 18 January 2020 / Accepted: 16 September 2020 © The Author(s) 2020
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 1/2 1/2 establish a stability estimate of the form L 2 → HT , where the HT -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.
1 Introduction To motivate our results, let us begin with the simplest case of the Radon transform in R2 in parallel beam geometry (see [15] for more details). Example If f ∈ Cc∞ (R2 ), the Radon transform of f is R f (s, v) =
∞
−∞
f (sv + tv ⊥ ) dt, s ∈ R, v ∈ S 1 ,
where v ⊥ is the rotation of v by 90◦ counterclockwise. The Fourier transform of R f in the s variable, denoted by (R f )˜( · , v), satisfies the Fourier slice theorem (R f )˜(σ, v) = (2π)1/2 fˆ(σ v), σ ∈ R, v ∈ S 1 .
B
Gabriel P. Paternain [email protected] Mikko Salo [email protected]
1
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK
2
Department of Mathematics and Statistics, University of Jyväskylä, Jyvaskyla, Finland
123
G. P. Paternain, M. Salo
Using the Plancherel theorem and polar coordinates, we obtain that ∞ 2 2 ˆ f L 2 (R2 ) = f L 2 (R2 ) = | fˆ(σ v)|2 σ dv dσ 0 S1 1 ∞ | fˆ(σ v)|2 |σ | dv dσ = 2 −∞ S 1 ∞ 1 |(R f )˜(σ, v)|2 |σ | dv dσ. = 4π −∞ S 1 In particular, this implies the stability estimate f L 2 (R2 ) ≤
1 R f H 1/2 (R×S 1 ) T (4π)1/2
(1.1)
˜ with the mixed Sobolev norm h H 1/2 (R×S 1 ) = (1 + σ 2 )1/4 h(σ, v) L 2 (R×S 1 ) . T
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. There are two features we wish 1/2 to preserve from (1.1): one is its L 2 → H 1/2 nature and the other is that the HT only incorporates “half of the derivatives” of the target space (space of geodesics). Let us first be more precise about the geometric setting. 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). Let S M denote the unit sphere bundle on M, i.e. S M := {(x, v) ∈ T M : |v|g = 1}. We define the volume form on S M by d 2d−1 (x, v) = d V d (x) ∧ d Sx (v), where d V d is the volume form on M and d Sx is the volume form on the fibre Sx M. The boundary of S M is ∂ S M := {(x, v) ∈ S M : x ∈ ∂ M}. On ∂ S M the natural volume form is d 2d−2 (x, v) = d V d−1 (x) ∧ d Sx (v), where d V d−1 i
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