A Variation on Hölder-Brascamp-Lieb Inequalities

Abstract

The Hölder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.

arXiv commentNovember update to clarify an argument in Section 3

References in corpus (2)

• Hessian of Bellman functions and uniqueness of Brascamp--Lieb inequality
• Parallelepipeds obtaining HBL lower bounds