A variation on Hölder–Brascamp–Lieb inequalities

Abstract

The class of Hölder–Brascamp–Lieb inequalities is a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis–Whitney inequalities. The full range of exponents was classified in a paper of Bennett, Carbery, Christ, and Tao [Math. Res. Lett. 17 (2010), no. 4, 647–666]. In a setting similar to that of Ivanisvili and Volberg [J. Lond. Mat. Sci (2) (2015), no. 3, 657–674], we introduce a notion of size for these inequalities which generalizes L p 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 maximizers to exist when the underlying linear maps match those of the convolution inequality of Young.