The Brascamp-Lieb inequalities: finiteness, structure, and extremals

Details The Brascamp-Lieb inequalities: finiteness, structure, and extremals The Brascamp-Lieb inequalities: finiteness, structure, and extremals

2005-05-04

arxiv.org/abs/math/0505065v5

Mathematics

Metric Geometry

This is the final version, incorporating referee's comments

Scientific paper

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. We obtain two new proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our techniques are partly based upon monotonicity formulas for positive solutions to heat equations in linear and multilinear settings.

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