Fourier duality in the Brascamp–Lieb inequality

Abstract

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

Higher order transversality in harmonic analysis

“…Furthermore, for torsion-free discrete groups, they showed that if the constant C is finite then it must equal 1 -a conclusion consistent with the classical Young's convolution inequality on the integers. Similar results may be established in the compact setting, and ultimately give rise to an abstract duality principle of the form (2.10) -see [28] for more general conclusions and clarification.…”