Local bounds for singular Brascamp-Lieb forms with cubical structure

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.