Integrability of optimal mappings. (English. Russian original) Zbl 1124.28002
Math. Notes 80, No. 4, 518-531 (2006); translation from Mat. Zametki 80, No. 4, 546-560 (2006).
The author contributes to the study of the Monge-Kantorovich optimal transportation problem. In response to practical problems arising in a wide range of areas including statistical physics, expert systems and queueing theory, such studies have generated work in various fields of the mathematical sciences such as functional analysis, probability theory, statistics, linear and stochastic programming, information theory and cybernetics, as well as matrix theory.
This paper studies the integrability of optimal maps \(T\) taking a probability measure \(\mu\) to another measure \(g \cdot \mu\), and assumes that \(T\) minimizes the cost function \(c\) and that \(\mu\) satisfies some special inequality relative to \(c\) (the infimum-convolution inequality or the logarithmic \(c\)-Sobolev inequality). The results obtained are applied to the analysis of measures of the form \(\exp(-|x|^{\alpha})\). The bibliography contains 25 references mostly in English and includes many in Russian and in French, representing some of the vast and scattered literature on and the international nature of the work.
This paper studies the integrability of optimal maps \(T\) taking a probability measure \(\mu\) to another measure \(g \cdot \mu\), and assumes that \(T\) minimizes the cost function \(c\) and that \(\mu\) satisfies some special inequality relative to \(c\) (the infimum-convolution inequality or the logarithmic \(c\)-Sobolev inequality). The results obtained are applied to the analysis of measures of the form \(\exp(-|x|^{\alpha})\). The bibliography contains 25 references mostly in English and includes many in Russian and in French, representing some of the vast and scattered literature on and the international nature of the work.
Reviewer: Pao-Sheng Hsu (Columbia Falls)
MSC:
| 28A12 | Contents, measures, outer measures, capacities |
| 39B72 | Systems of functional equations and inequalities |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Keywords:
infimum-convolution inequality; logarithmic Sobolev inequality; transportation inequality; Monge-Kantorovich variational problemReferences:
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