论文标题
Hörmander型振荡整体操作员的二分法
A dichotomy for Hörmander-type oscillatory integral operators
论文作者
论文摘要
在本文中，我们首先概括了波尔加因的工作，并陈述了Hörmander-type振荡整体操作员的曲率条件，我们称之为波尔加因的状况。对于傅立叶限制问题和Bochner-Riesz问题，相位函数非常满足。我们猜想，对于满足Bourgain状况的Hörmander-type振荡整体操作员，他们满足与傅立叶限制猜想相同的$ l^p $界限。为了支持我们的猜想，我们表明，每当波尔加因的状况失败时，$ l^{\ infty} \ to l^q $有限始终失败，对于某些$ q = q（n）> \ frac {2n} {n-1} {n-1} $，扩展了Bourgain的三维结果。另一方面，如果布尔加因的状况持续下去，那么我们证明了Hörmander-type振动性积分运算符的$ l^p $范围，用于$ p $，这扩展了Hickman和Zahl给出的高度傅立叶限制构想的当前最著名的范围。这在傅立叶限制问题和Bochner-Riesz问题上给出了新的进展。
In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness always fails for some $q= q(n) > \frac{2n}{n-1}$, extending Bourgain's three-dimensional result. On the other hand, if Bourgain's condition holds, then we prove $L^p$ bounds for Hörmander-type oscillatory integral operators for a range of $p$ that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl. This gives new progress on the Fourier restriction problem and the Bochner-Riesz problem.
