Local bounds for singular Brascamp-Lieb forms with cubical structure

TL;DR

This paper establishes sharp $L^p$ bounds for singular Brascamp-Lieb forms with cubical structure, extending previous results to a broader range of $p$ values using Fourier analysis and iterative techniques.

Contribution

It extends existing bounds for singular Brascamp-Lieb forms to a larger $p$ range, specifically for $p > 2^{m-1}$, with a novel proof approach.

Findings

Established $L^p$ bounds for $p > 2^{m-1}$

Extended previous results from $p=2^m$ to a broader range

Proved the sharpness of the $2^{m-1}$ threshold

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.