**Abstract : ** On a closed manifold endowed with a uniformly hyperbolic flow — or Anosov,
in the literature —, a certain number of dynamical/geometrical problems (structural stability,
marked length spectrum rigidity, study of transparent connections, ...) involve a class of equations called cohomological equations.
Usually, one can construct ``by hand" a Hölder continuous solution to these equations but proving smoothness is harder.
I will explain how one can relate the study of these equations to microlocal analysis. The key estimate to prove is a
radial source estimate in Hölder-Zygmund spaces (and more generally, in Besov spaces), which is a kind of propagation
of singularities in phase space. This was first used in the context of hyperbolic flows by Dyatlov-Zworski in Sobolev spaces.
However, their proof is based on a positive commutator argument and the sharp Gärding inequality and does not seem to generalize
to Hölder-Zygmund spaces.
This is an ongoing project with Yannick Guedes Bonthonneau.