On implicitly oscillatory quadrilinear integrals

Abstract

For quadrilinear functionals ${\int }_B{\prod }_{j=1}^4(f_j\circ {\phi }_j)$, where $B\subset {\mathbb{R}}^2$ is a ball, ${\phi }_j:B\to {\mathbb{R}}^1$ 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 ${\sum }_j(g_j\circ {\phi }_j)\equiv 0$, 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.