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. pass through sparse and local bounds, the latter proved by an iteration Fourier expansion, telescoping, Cauchy–Schwarz inequality. allow $$2^{m-1}<p\le \infty $$ m dimension cube, extending earlier result that required $$p=2^m$$ . The threshold $$2^{m-1}$$ is sharp in our theorems.