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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The one dimensional free Poincaré inequality


Authors: Michel Ledoux and Ionel Popescu
Journal: Trans. Amer. Math. Soc. 365 (2013), 4811-4849
MSC (2010): Primary 46L54; Secondary 60B20, 60E15, 33D45
Published electronically: April 24, 2013
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Abstract: In this paper we discuss the natural candidate for the one dimensional free Poincaré 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 Poincaré is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincaré inequality revolves around the counting number operator for the Chebyshev polynomials of the 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 energy with external fields discovered in Analyticity of the planar limit of a matrix model by S. Garoufalidis and the second author as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities.


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Additional Information

Michel Ledoux
Affiliation: Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse, France – and – Institut Universitaire de France
Email: ledoux@math.univ-toulouse.fr

Ionel Popescu
Affiliation: Department of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta Georgia 30332 – and – Institute of Mathematics of Romanian Academy, 21 Calea Grivitei Street, 010702-Bucharest, Sector 1, Romania
Email: ipopescu@math.gatech.edu, ionel.popescu@imar.ro

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05830-8
PII: S 0002-9947(2013)05830-8
Received by editor(s): May 10, 2011
Received by editor(s) in revised form: October 21, 2011
Published electronically: April 24, 2013
Additional Notes: The second author was partially supported by Marie Curie Action grant no. 249200.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.