  This paper is the first in a series of two articles whose aim is to extend a
recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length
spectrum from the case of compact negatively-curved Riemannian manifolds to the
case of manifolds with hyperbolic cusps. In this first paper, we deal with the
linear (or infinitesimal) version of the problem and prove that such manifolds
are spectrally rigid for compactly supported deformations. More precisely, we
prove that the X-ray transform on symmetric solenoidal 2-tensors is injective.
In order to do so, we expand the microlocal calculus developed by Bonthonneau
and Bonthonneau-Weich to be able to invert pseudodifferential operators on
Sobolev and H\"older-Zygmund spaces modulo compact remainders. This theory has
an interest on its own and is extensively used in the second paper
[arXiv:1910.02154] in order to deal with the nonlinear problem.
