The one-frequency cohomological equation, Brjuno-like functions and Khintchine-L\'evy numbers

Abstract

In the paper we consider the one-frequency cohomological equation \begin{equation*} (\partial_x + \omega \partial_y) g(x,y) = a(x,y) \end{equation*} on the 2-torus with unknown $g$ and analytic initial data $a$. We identify all the frequencies $\omega$ for which the equation has an analytic solution and express the analytic solvability condition in terms of two Brjuno-like functions, providing explicit estimates on the sup-norm of $g$. As an example we estimate the Brjuno-like functions for Diophantine and Khintchine-L\'evy numbers. We also construct an example of an arbitrarily small function $a$ for which an analytic $g$ does not exist when one of the Brjuno-like functions has infinite value.