The One Dimensional Free Poincar\'e Inequality

In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincar\'e is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincar\'e inequality revolves around the counting number operator for the Chebyshev polynomials of first kind with respect to the arcsine law on $[-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic potential with external fields as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities.