Adjoint Brascamp–Lieb inequalities

Abstract The Brascamp–Lieb inequalities are a generalization of the Hölder, Loomis–Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper, we introduce an “adjoint” version of these inequalities, which can be viewed as an version of the entropic Brascamp–Lieb inequalities of Carlen and Cordero–Erausquin. As applications, we reprove a log‐convexity property of the Gowers uniformity norms, and establish some reverse inequalities for various tomographic transforms. We conclude with some open questions.