In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.

在本文中，我们研究了获得与抛物面的矩形块及其摄动有关的傅立叶限制算子的Lebesgue空间不等式的问题。我们针对这些不等式中的算子范数的依赖关系对矩形的边长提出一个猜想，证明该猜想来自椭圆超曲面的限制猜想（对此做了些许重构），并证明（如果有效）该猜想是本质上是锋利的。这些问题在退化超曲面的约束不等式的研究中很自然地出现。我们通过使用我们的积极结果证明具有某些加性结构的一类超曲面的新的约束不等式来证明这种联系。