Sobolev regularity of solutions of the cohomological equation

TL;DR

This paper advances the understanding of the Sobolev regularity of solutions to the cohomological equation for translation flows on higher genus surfaces, establishing sharp results and their relation to Lyapunov exponents.

Contribution

It refines the theory for typical translation surfaces, providing sharp Sobolev regularity results and linking obstructions to Kontsevich-Zorich exponents.

Findings

Exact determination of the dimension of obstruction spaces in Sobolev classes.

Sharp regularity results for typical translation surfaces.

Relation between distributional obstructions and Lyapunov exponents.

Abstract

We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For a fixed arbitrary translation surface and a typical direction, our results are probably not optimal but are the best which can be achieved with the available harmonic analysis techniques we have introduced in an earlier paper.