Local bounds for singular Brascamp–Lieb forms with cubical structure

Abstract

We prove a range of \(L^p\) bounds for singular Brascamp–Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy–Schwarz inequality. We allow \(2^{m-1}<p\le \infty \) with m the dimension of the cube, extending an earlier result that required \(p=2^m\). The threshold \(2^{m-1}\) is sharp in our theorems.