An Algebraic Brascamp–Lieb Inequality

Abstract

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Higher order transversality in harmonic analysis

“…Here dµ M denotes Lebesgue measure on M , and T x M denotes the tangent space to M at x. Theorem 2.2 follows several nonlinear Brascamp-Lieb inequalities established under various additional (structural) hypotheses on the manifold M , and obtained by different methods -see [27,17,19,84,43,20] for further discussion. We also refer the reader to recent developments by Duncan [55,56] on nonlinear Brascamp-Lieb inequalities, including certain global estimates and stability results.…”

The case of equality in geometric instances of Barthe's reverse Brascamp-Lieb inequalit

“…Guo, Zhang [58]. Various versions of the Brascamp-Lieb inequality and its reverse form have been obtained by Balogh, Kristaly [6] Barthe [9], Barthe, Cordero-Erausquin [10], Barthe, Cordero-Erausquin, Ledoux, Maurey [11], Barthe, Wolff [13,14], Bennett, Bez, Flock, Lee [15], Bennett, Bez, Buschenhenke, Cowling, Flock [16], Bobkov, Colesanti, Fragalà [19], Bueno, Pivarov [26], Chen, Dafnis, Paouris [34], Courtade, Liu [36], Duncan [40], Ghilli, Salani [46], Kolesnikov, Milman [68], Livshyts [72,73], Lutwak, Yang, Zhang [77,78], Maldague [79], Marsiglietti [80], Rossi, Salani [89,90].…”