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**The Brascamp-Lieb inequalities: Finiteness, structure and extremals.**
*(English)*
Zbl 1132.26006

Summary: We consider the Brascamp-Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred Gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. Our main tool is a monotonicity formula for positive solutions to heat equations in linear and multilinear settings, which was first used in this type of setting by E. A. Carlen, E. H. Lieb, and M. Loss [J. Geom. Anal. 14, No. 3, 487–520 (2004; Zbl 1056.43002)]. In that paper, the heat flow method was used to obtain the rank-one case of Lieb’s fundamental theorem concerning exhaustion by Gaussians; we extend the technique to the higher-rank case, giving two new proofs of the general-rank case of Lieb’s theorem.

### MSC:

26D15 | Inequalities for sums, series and integrals |

15A45 | Miscellaneous inequalities involving matrices |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

35K05 | Heat equation |

44A35 | Convolution as an integral transform |

35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |