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We prove a sharp $L^2\to H^{1/2}$ stability estimate for the geodesic X-raytransform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannianmanifold with a suitable chosen $H^{1/2}$ norm. We show that such an estimateholds for a family of such $H^{1/2}$ norms, not topologically equivalent, butequivalent on the range of the transform. The reason for this is that thegeodesic X-ray transform has a microlocally structured range.

A sharp stability estimate for the geodesic ray transform