Free Functional Inequalities on the Circle

TL;DR

This paper explores free functional inequalities on the circle, highlighting differences from classical cases and introducing modified tools like Wasserstein distance to analyze transportation and entropy inequalities.

Contribution

It introduces new phenomena in free inequalities on the circle and develops modified Wasserstein tools for their analysis, revealing classical counterparts not previously studied.

Findings

Modified Wasserstein distance for the circle

Differences in free Poincaré inequality due to rotation invariance

Identification of classical counterparts to free inequalities

Abstract

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.