The Brascamp--Lieb inequality on compact Lie groups and its extinction on homogeneous Lie groups

We study the Brascamp–Lieb inequalities on locally compact nonabelian groups and the Brascamp–Lieb constants $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ associated to a Brascamp–Lieb datum: locally compact groups $G$ and $G_j$, a family of homomorphisms $σ_j: G \to G_j$ and Lebesgue indices $p_j$. We focus on homogeneous Lie groups and compact Lie groups. For homogeneous Lie groups $G$, we show that the constant $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ is equal to the constant $\mathbf{BL}(\mathfrak{g}, \boldsymbol{\mathrm{d}σ}, \boldsymbol{p})$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathrm{d}σ_j$ is the differential of $σ_j$. For Heisenberg-like groups $G$, we show that the only inequalities that can occur are multilinear Hölder inequalities. For compact Lie groups, we find necessary and sufficient conditions for finiteness of the constant $\mathbf{BL}(G, \boldsymbolσ, \boldsymbol{p})$ in terms of $\boldsymbolσ$ and $\boldsymbol{p}$ and find an explicit expression for the constant, similar to those found by Bennett and Jeong in the abelian case.

💡 Research Summary

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The paper investigates Brascamp–Lieb (BL) inequalities in the setting of non‑abelian locally compact groups, with a focus on homogeneous Lie groups and compact Lie groups. A BL datum consists of a locally compact group (G), a family of continuous homomorphisms (\sigma_j : G \to G_j) into locally compact groups (G_j), and a vector of exponents (p = (p_1,\dots,p_J)) with (p_j\in