Adjoint Brascamp-Lieb inequalities

The Brascamp-Lieb inequalities are a generalization of the H & ouml;lder, 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 Lp$L<^>p$ 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 Lp$L<^>p$ inequalities for various tomographic transforms. We conclude with some open questions.