  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.
