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.