Free Functional Inequalities on the Circle

In this paper we deal with free functional inequalities on the circle. There are some interesting changes as opposed to the classical case. For example, the free Poincar\'e inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have a classical counterpart, which does not seem to have been investigated before.