Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics

We prove a radial source estimate in Hölder–Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov–Zworski [DZ16]. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns–Katok [BK85] conjecture. We show in particular that in any dimension ≥ 2 , in the space of negatively-curved metrics, C 3 + ε -close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou–Knieper and the second author [GKL22, GL19]. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journé’s lemma: the smooth Livšic Theorem of de La Llave–Marco–Moriyón [LMM86], the smooth Livšic cocycle theorem of Niticā–Török [NT98] for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt [Has92] and others.