A sharp stability estimate for tensor tomography in non-positive curvature

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 $L^2\mapsto H_T^{1/2}$, where the $H_T^{1/2}$-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.

Revised version to appear in Mathematische Zeitschrift, 24p. The appendix of version 1 has been removed and incorporated in shorter form into the main text. The appendix still can be found in v1 and could be useful for other purposes