On implicitly oscillatory quadrilinear integrals

TL;DR

This paper investigates bounds for quadrilinear integrals involving real analytic submersions, establishing conditions under which these integrals can be majorized by Sobolev norms, with proofs relying on sublevel set inequalities.

Contribution

It introduces a new majorization result for quadrilinear integrals under specific geometric conditions, extending previous understanding of oscillatory integral bounds.

Findings

Established upper bounds for quadrilinear integrals under certain conditions

Identified necessary geometric conditions for the bounds to hold

Connected the bounds to sublevel set inequalities from a companion paper

Abstract

For quadrilinear functionals $\int_B \prod_{j=1}^4 (f_j\circ\varphi_j)$, where $B\subset{\mathbb R}^2$ is a ball, $\varphi_j:B\to{\mathbb R}^1$ are real analytic submersions, and $f_j\in L^\infty({\mathbb R}^1)$ are bounded and measurable, we seek a majorization of the integral by a product of negative order Sobolev norms of the factors $f_j$. An obvious necessary condition is that any smooth solution of $\sum_j (g_j\circ\varphi_j)\equiv 0$, in any connected open set, must be constant. Assuming this condition and certain auxiliary hypotheses, we establish an upper bound of the desired type. The proof relies in part on a three term sublevel set inequality established in a companion paper.