发表机构
Institute of Applied Physics and Computational Mathematics; University of Science and Technology Beijing; Central China Normal University（应用物理与计算数学研究所; 北京科技大学; 华中师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究明确二维薛定谔算子波算子的端点映射性质，发现s波共振可改善其端点行为，与三维及以上维度情况相反，还给出零本征值下端点有界性的充要条件。
AI 中文摘要
我们针对具有实值衰减势场V的二维薛定谔算子H=-Δ+V的波算子W±(H,-Δ)，建立了精确的端点映射性质。结合已知的非端点L^p理论，我们的结果完整分类了二维波算子的L^p映射性质，并揭示了在端点p=1和p=∞处，与通常阈值范式的意外反转。当0是H的正则点时，波算子在L^1(R²)和L^∞(R²)上不具有有界性，但满足自然的替代卡尔德隆-齐格蒙德型估计：L^1(R²)→L^{1,∞}(R²)，ℋ^1(R²)→L^1(R²)，L^∞(R²)→BMO(R²)。当0是第一类阈值奇点——即s波共振且无其他阈值阻碍时，波算子在两个端点空间L^1(R²)和L^∞(R²)上均具有有界性。因此，在二维中，s波共振可改善波算子的端点行为，这与n≥3维的情况形成鲜明对比，后者中唯一的正则情况是有利的。我们还确定了H的其余零能谱构型下的端点行为：p波共振会阻碍波算子的L^1和L^∞有界性，而在零本征值情况下，我们得到了端点有界性的充要条件，该条件用s波、p波共振的存在性以及零能本征函数满足的显式二阶调和矩抵消来表述。
英文摘要
We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-Δ)$ of two-dimensional Schrödinger operators $H=-Δ+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón--Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.
Comments65 pages. More references
