AI 中文总结
研究上半空间临界Sobolev迹不等式的定量稳定性，通过证明迹泡谱非退化性，得出Sobolev迹亏缺控制到迹泡流形梯度距离的\(\max\{2,p\}\)次幂的结论。
AI 中文摘要
设\(n\geq3\)且\(1\lt p\lt n\)。我们证明了上半空间中临界Sobolev迹不等式的定量稳定性估计。更确切地说，Sobolev迹亏缺控制到迹泡流形的梯度距离的\(\max\{2,p\}\)次幂。证明的核心部分是迹泡的谱非退化性：线性化加权Steklov问题的前两个特征子空间恰好是幅度、伸缩和平切平移模式。
英文摘要
Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.
Comments60 pages
