Rough Curvature Bounds and Free Functional inequalities

Abstract

We study the transport inequalities, gradient estimates entropy, and Ricci curvature. A free probability method of the Wasserstein metric on the trace-states space is considered. We give a free Brunn-Minkowski inequality, and show the Talagrand inequality for the semicircular law and energy of eigenvalues of Beta ensembles. We also show the Ricci curvature for metric measure spaces by optimal transport, and consider the mass transportation and rough curvature bounds for discrete spaces. We investigate the combinatorial dimension and certain norms in the method of harmonic analysis, and characterize the relationships between combinatorial measurements and Orlicz norms. Also Characterization of dimension free concentration in terms of transportation and Poincar'e inequalities with dimension free concentration of measure are shown, mass transportation evident of free functional inequalities and free Poincar'e inequalities are confirm.