论文标题
平面单变量函数的分区和相关双曲抛物面扰动的傅立叶限制定理
Partitions of flat one-variate functions and a Fourier restriction theorem for related perturbations of the hyperbolic paraboloid
论文作者
论文摘要
我们通过研究双曲线抛物面$ z = xy $的局部扰动，继续研究双曲线表面的傅立叶限制，该扰动$ z = xy $，$ z = xy+h（y），其中$ h（y）$是一种平稳的功能，在起源上是平坦的。以前已经处理过有限类型的扰动的情况，但是平面案件施加了几个新障碍。通过将$ | h'''| $分解为固定尺寸$λ的间隔，我们可以应用在上一篇文章中设计的方法，但是由于我们对$ h $的高阶衍生物的控制放宽了控制$ h $的高阶衍生物，因此我们被迫重新制作双线性方法的波数据包，以使波数据包进行缓慢衰减。另一个问题在于从双线性估计到线性估计的段落，我们需要单调性$ h''''。$。
We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which $|h'''|$ is of a fixed size $λ,$ we can apply methods devised in preceding papers, but since we loose control on higher order derivatives of $h$ we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of $h'''.$
