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 $\mathbf{B}\mathbf{L}(G,\mathit{\sigma },\mathit{p})$ associated to a Brascamp–Lieb datum: locally compact groups $G$ and $G_j$, a family of homomorphisms ${\sigma }_j:G\to G_j$ 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 $\mathbf{B}\mathbf{L}(G,\mathit{\sigma },\mathit{p})$ is equal to the constant $\mathbf{B}\mathbf{L}(\mathfrak{g},,\mathit{p})$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathrm{d}{\sigma }_j$ is the differential of ${\sigma }_j$. 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 $\mathbf{B}\mathbf{L}(G,\mathit{\sigma },\mathit{p})$ in terms of $\mathit{\sigma }$ and $\mathit{p}$ and find an explicit expression for the constant, similar to those found by Bennett and Jeong in the abelian case.