Local functional inequalities in one-dimensional free probability

In this note we introduce and prove local and potential independent transportation, Log-Sobolev and HWI inequalities in one-dimensional free probability on compact intervals which are sharp. We recover using this approach a free transportation inequality on the whole real line which was put forward recently by Maïda and Maurel-Segala (2012) [10], . Our method is based on the operator theoretic approach developed by Ledoux and Popescu [7] to deal with the free Poincaré inequality.