Mass Transportation Proofs of Free Functional Inequalities, and Free Poincare Inequalities

TL;DR

This paper provides direct mass transportation proofs for various free probability inequalities, including transportation, Log-Sobolev, and Brunn-Minkowski inequalities, with sharp constants and spectral interpretations.

Contribution

It introduces direct mass transportation proofs for free probability inequalities, avoiding random matrix approximation, and extends results to measures on \\R_{+} with applications to Marcenko-Pastur distribution.

Findings

Established sharp constants for free inequalities.

Connected free Poincaré inequalities to spectral properties of Jacobi operators.

Extended inequalities to measures on \\R_{+} with specific examples.

Abstract

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincar\'e inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on $\R_{+}$ with the reference example of the Marcenko-Pastur distribution.