标题： 波算子对于有限秩扰动的拉普拉斯算子的$L^p$有界性 显示英文标题

标题： The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations

摘要： 本文研究了有限秩扰动下的拉普拉斯算子的波算子的$L^p$有界性\begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d. \end{equation*}。对于维度$d\ge 3$，我们证明波算子$W_\pm(H,H_0)$在$L^p$上对于整个范围$1\le p\le \infty$是有界的。 这扩展了Nier和第三作者的工作\cite{NS}，通过解决之前未探索的端点情况下的有界性问题$p=1$和$p=\infty$。在低维情况下$d = 1, 2$，我们首次建立了波算子的$L^p$-有界性。此外，我们揭示了端点情况$p = 1$中的一个有趣的二分法：\begin{itemize} \item 如果对于每个$1\le i\le N$$\int_{\mathbb{R}^d} \varphi_i(x) \, x̣ = 0$都成立，那么波算子在$L^p(\mathbb{R}^d)$上对于所有$1 \leq p \leq \infty$都是有界的。 \item 如果存在至少一个$i$($1\le i\le N$) 使得$\int_{\mathbb{R}^d}\varphi_i(x)x̣\ne0$，则波算子对于$1 < p < \infty$保持有界，并满足弱型$(1,1)$估计，但不能在$L^1(\mathbb{R}^d)$上有界。 \end{itemize} 显示英文摘要

摘要： This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d. \end{equation*} For dimensions $d\ge 3$, we prove that the wave operators $W_\pm(H,H_0)$ are bounded on $L^p$ for the full range $1\le p\le \infty$. This extends the work of Nier and the third author \cite{NS} by resolving the previously unexplored question of boundedness at the endpoint cases $p=1$ and $p=\infty$. In lower dimensions $d = 1, 2$, we establish the $L^p$-boundedness of the wave operators for the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case $p = 1$: \begin{itemize} \item If $\int_{\mathbb{R}^d} \varphi_i(x) \, \d x = 0$ holds for every $1\le i\le N$, then the wave operators are bounded on $L^p(\mathbb{R}^d)$ for all $1 \leq p \leq \infty$. \item If there exists at least one $i$ ($1\le i\le N$) such that $\int_{\mathbb{R}^d}\varphi_i(x)\d x\ne0$, then the wave operators remain bounded for $1 < p < \infty$ and satisfy weak type $(1,1)$ estimates, but fail to be bounded on $L^1(\mathbb{R}^d)$. \end{itemize}