非椭圆型薛定谔极大算子推广的反例
Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator
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中文总结 AI 辅助
针对非椭圆型薛定谔极大算子的推广情形，本文构造反例并证明了一类无穷多项式符号$P$对应的正则性必要条件为$s\geq 1/2$。
中文摘要 AI 辅助
对于$P=X_1^2+\cdots + X_n^2$，记$T_t^Pf(x)$为线性薛定谔方程在时刻$t$的解。1980年，Carleson提出问题：初始数据函数$f\in H^s(\mathbb{R}^n)$需满足的最小正则性，以保证当$t\rightarrow 0$时，$T_t^Pf(x)$逐点收敛到$f(x)$。Bourgain通过构造薛定谔极大算子的反例，证明$s\geq n/(2(n+1))$是必要条件；Du和Zhang则证明$s> n/(2(n+1))$是充分条件。Rogues、Vargas和Vega研究了非椭圆型薛定谔极大算子的类似问题，其中$P = X_1^2-X_2^2 \pm X_3^2\pm \cdots \pm X_n^2$，并证明对所有$n\geq 2$，$s\geq 1/2$是必要条件，$s>1/2$是充分条件。在本文中，我们构造了非椭圆型情况推广的反例，证明了一类无穷多项式符号$P$对应的$s\geq 1/2$是必要条件。
英文摘要
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schrödinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resolved by Bourgain, who constructed counterexamples for the Schrödinger maximal operator to show that $s\geq n/(2(n+1))$ is necessary, and Du and Zhang, who proved that $s> n/(2(n+1))$ is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schrödinger maximal operator, where $P = X_1^2-X_2^2 \pm X_3^2\pm \cdots \pm X_n^2$, and proved that, for all $n\geq 2$, $s\geq 1/2$ is necessary and $s>1/2$ is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of $s\geq 1/2$ for an infinite class of polynomial symbols $P$.
发表机构
- Georg-August-Universität Göttingen（哥廷根大学）
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