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Open PDE & Analysis Seminar

Dr Thibault Lefeuvre (Université Paris-Saclay)

Microlocal regularity of solutions to cohomological equations

Friday 15th May 2020, 3pm (Paris time)

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.

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Thomas Alazard (ENS Paris-Saclay)         Nicolas Burq (Orsay)         Iván Moyano (Nice)