The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals

By Jonathan Bennett, Anthony Carbery, Michael Christ, Terence Tao. 2005

306 citations recorded by the source corpus.

Abstract.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 Carlen, Lieb, and Loss [CLL]. 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/77e0d67f182a23b933e6cfa370a890cc87b8d44c

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Citations in the corpus

Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, methodology): “Ball [1], states that given two inputs f and f , (45) BL(L,p; f)BL(L,p; f) ≤ sup x∈Rd BL(L,p; (g j )) BL(L,p; f ∗ f ), where g j (y) := fj(Ljx− y)f ′ j(y) and f ∗ f ′ := (fj ∗ f ′ j); see [1], [11] and [8].”
Monotonicity and its analytic and geometric implications (2012, background): “It is easy to see (more or less from the argument given above) that [5] does not hold on a general manifold M with Ric ≥ 0.”
On the multilinear restriction and Kakeya conjectures (2005, background): “Although our proof of Proposition 3.1 is rather less direct than the one given in [ 4 ] (which is closely related to the heat-flow approach of [11]), it does seem to lend itself much better to the perturbed situation, as we will discover in the next section.”
Heat-flow monotonicity of Strichartz norms (2008, methodology): “By [ 5 ], we have that Lemma 2.1 generalises to the statement that”
Correlation and Brascamp–Lieb Inequalities for Markov Semigroups (2009, methodology): “This is exemplified by the paper [7] where the Gaussianextremizable cases of the Euclidean Brascamp–Lieb inequality are reduced to the geometric Brascamp–Lieb inequality.”