AI 中文总结
研究正交函数的分形 Hardy-Littlewood-Sobolev 不等式，通过傅里叶分析重证 Adams 不等式，避免 Schatten 类和变分论证，可恢复\(-\Delta - \mu\)负特征值相关界及负特征值之和的 Rozenblum 界。
AI 中文摘要
我们证明了正交函数的 Lieb 型 Hardy-Littlewood-Sobolev 不等式的分形版本。一方面，这可视为正交函数的迹定理；另一方面，能恢复关于\(-\Delta - \mu\)负特征值个数的 Rozenblum-Tashchiyan 界，其中\(\mu\)是壳势。还通过 Lieb-Thirring 动力学不等式恢复了负特征值之和的 Rozenblum 界。证明直接，避免了 Schatten 类和变分论证。先通过傅里叶分析重新证明了 Adams 的分形 Hardy-Littlewood-Sobolev 不等式（针对单函数），得到所需端点估计及 Frostman 测度相互作用能的界。
英文摘要
We prove a fractal version of Lieb's Hardy-Littlewood-Sobolev inequality for orthonormal functions. On the one hand, this can be viewed as a trace theorem for orthonormal functions. On the other, it allows us to recover the Rozenblum-Tashchiyan bound for the number of negative eigenvalues of $-Δ-μ$, where $μ$ is a shell potential. We also recover Rozenblum's bound for the sum of negative eigenvalues via a Lieb-Thirring kinetic inequality. Our proof is direct, avoiding both Schatten classes and variational arguments. We first reprove Adams' fractal Hardy-Littlewood-Sobolev inequality (for single functions) via Fourier analysis. This yields the required endpoint estimate as well as a bound for the interaction energy of Frostman measures.
Comments14 pages
