薛定谔极大算子的高次推广的反例
Counterexamples for high-degree generalizations of the Schrödinger maximal operator
摘要：
在1980年Carleson提出了一个问题的最小正则性的初始数据函数在Sobolev空间$H^s(\mathbb{R}^n)$，这意味着逐点收敛的解决方案的线性薛定谔方程。经过许多作者的努力，最近Bourgain解决了这个问题（直到终点），他的薛定谔极大算子的反例构造证明了正则性的必要条件，Du和Zhang证明了一个充分条件。Carleson问题的类似物对于许多其他色散偏微分方程仍然是开放的。我们开发了一个灵活的新方法来处理这样的问题，并证明了，对于任何整数$k\geq 2$，如果一个度$k$推广的薛定谔极大算子是有界的$H^s(\mathbb{R}^n)$ $L^1(B_n(0,1))$，那么$s \geq \frac{1}{4} + \frac{n-1}{4((k-1)n+1)}.$在尺寸$n \geq 2$，对于每一个度$k \geq 3$，这是第一个结果，超过了一个长期存在的障碍在$1/4$。我们的方法是数论的，特别是适用于魏尔界，一个后果的真理黎曼假设在有限领域。
英文摘要：
In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Analogues of Carleson's question remain open for many other dispersive PDE's. We develop a flexible new method to approach such problems, and prove that for any integer $k\geq 2$, if a degree $k$ generalization of the Schrödinger maximal operator is bounded from $H^s(\mathbb{R}^n)$ to $L^1(B_n(0,1))$, then $s \geq \frac{1}{4} + \frac{n-1}{4((k-1)n+1)}.$ In dimensions $n \geq 2$, for every degree $k \geq 3$, this is the first result that exceeds a long-standing barrier at $1/4$. Our methods are number-theoretic, and in particular apply the Weil bound, a consequence of the truth of the Riemann Hypothesis over finite fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
