论文标题
在隐式振荡的四联积分上
On implicitly oscillatory quadrilinear integrals
论文作者
论文摘要
对于quadriLinear函数$ \ int_b \ prod_ {j = 1}^4（f_j \circcφ_j）$，其中$ b \子集r}^1）$是有界和可测量的，我们通过负顺序的sobolev norms $ f_j $ sobolev norms of sobolev norms of the积分的数量。一个明显的必要条件是，$ \ sum_j（g_j \circcφ_j）的任何平滑解决方案\ equiv 0 $，在任何连接的打开集中，都必须是恒定的。假设这种情况和某些辅助假设，我们建立了所需类型的上限。证明部分依赖于同伴论文中确定的三个任期内的不平等现象不平等。
For quadrilinear functionals $\int_B \prod_{j=1}^4 (f_j\circφ_j)$, where $B\subset{\mathbb R}^2$ is a ball, $φ_j:B\to{\mathbb R}^1$ 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 $\sum_j (g_j\circφ_j)\equiv 0$, 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.
