The Mañé-Conze-Guivarc'h lemma for intermittent maps of the circle

We study the existence Of Solutions g to the functional inequality f <= g circle T - g + beta where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an indifferent fixed point, and beta is the maximum ergodic average of f. Using a method due to T. Bousch, we show that continuous Solutions g always exist when the Holder exponent of f is close to 1. In the converse direction, we construct explicit examples of continuous functions f with low Holder exponent for which no continuous solution g exists. We give sharp estimates oil the best possible Holder regularity of a solution g given the Holder regularity of f