Radial source estimates in H\"older-Zygmund spaces for hyperbolic dynamics

We prove a radial source estimate in H\"older-Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov-Zworski. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns-Katok conjecture. We show in particular that in any dimension $\geq 2$, in the space of negatively-curved metrics, $C^{3+\varepsilon}$-close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou-Knieper and the second author. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journ\'e's lemma: the smooth Liv\v{s}ic Theorem of de La Llave-Marco-Moriy\'on, the smooth Liv\v{s}ic cocycle theorem of Nitic\=a-T\"or\"ok for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt and others.