A Sharp Stability Estimate in Tensor Tomography

where γ runs over the set of all geodesics with endpoints on ∂M . All potential fields dv given by (dv)ij = 1 2 (∇ivj +∇jvi) with v = 0 on ∂M belong to the kernel of I. The ray transform I is called s-injective if this is the only obstruction to injectivity, i.e., if If = 0 implies that f is potential. S-injectivity can only hold under certain assumptions on (M, g). A natural conjecture is that it holds on simple manifolds, see the definition below. So far it is known to be true for some classes of simple manifolds only, including generic simple manifolds, see [11, 13, 4, 16]. In the cases where s-injectivity is known, there is also a stability estimate that is not sharp. In [11], it is of conditional type with a loss of a derivative, see (2) below. In [16], the estimate is not of conditional type but there is still a loss of a derivative, see (3) below. On the other hand, if f is a function, or an 1-tensor (an 1-form), there is a sharp estimate, see [15]. The purpose of this paper is to prove a sharp estimate for the ray transform of 2-tensors. The geodesic ray transform is a linearization of the boundary distance function and plays an important role in the inverse kinematic problem (known also as boundary or lens rigidity), see e.g., [11, 15, 18, 17] and the references there. There, one wants to recover (M, g) given the distance function on ∂M × ∂M or the scattering relation σ : (x, ξ) 7→ (y, η) that maps a given x ∈ ∂M and a given incident direction ξ to the exit point y and the exit direction η of the geodesic issued from (x, ξ).