Mass transportation proofs of free functional inequalities, and free Poincaré inequalities |
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Authors: | Michel Ledoux |
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Institution: | a Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse, France b Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, United States c IMAR 21, Calea Grivitei Street 010702-Bucharest, Sector 1, Romania |
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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 Poincaré 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. |
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Keywords: | Functional inequalities Mass transport Spectral gap Random matrices |
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