  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.
