Brascamp–Lieb inequalities generalise classical results such as Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. In the context of euclidean spaces these inequalities have been intensively studied, for their own intrinsic interest and for their applications in areas such as partial differential equations and incidence geometry as well as for their connections to problems in theoretical computer science. Brascamp–Lieb inequalities involving finitely generated abelian groups have also been examined, as have those involving finite-dimensional compact abelian groups. Recently Jon Bennett and the speaker considered Brascamp–Lieb inequalities on general locally compact abelian groups.

It therefore seems timely to consider Brascamp–Lieb inequalities on nonabelian groups. It might seem that the lack of commutativity would make things more difficult. However, it also means that there are many fewer potential inequalities to consider, and in some situations, all possible inequalities are built up out of H¨older’s inequality and Young’s convolution inequality. We illustrate this by means of examples, using compact Lie groups and Heisenberg-like groups.