AI 中文总结
该论文在多个速度模型中建立尖锐动力学迹估计与反例，确定迹权重边界正则性阈值，如自然迹正则性阈值为\(\mathrm{C}^{1,1/2}\)，还给出不同情况下密度产生迹算子及公式的条件，以及保范速度平移对勒贝格迹估计的影响等。
AI 中文摘要
我们在多个速度模型中建立了尖锐的动力学迹估计和反例。对于半空间位置域，在速度支持无共同界的情况下，我们证明了勒贝格和标准高斯速度测度的自然迹估计。密度在相应动能空间上产生自然迹算子和格林公式。对于\(d\geq2\)中的有界空间域，在有界支持欧几里得速度模型和球速度模型中，我们确定了迹权重\(\min\{|v\cdot n|,|v\cdot n|^p\}\)（\(1\leq p<\infty\)）的尖锐边界正则性阈值。特别地，自然迹（\(p = 1\)）的正则性阈值为\(\mathrm{C}^{1,1/2}\)。在\(d\geq2\)的有界\(\mathrm{C}^{1,1/2}\)域中，密度在有界支持欧几里得速度模型中，对于勒贝格或标准高斯测度以及球速度模型，产生自然迹算子和格林公式。保范速度平移排除了每个有界\(\mathrm{C}^1\)域上无限制的勒贝格迹估计。在无限制高斯模型中，对于每个\(1\leq p<2\)，我们在\(d\geq2\)的每个有界\(\mathrm{C}^{1,1}\)域上构造反例。对于\(2\leq p<\infty\)，高斯\(\omega_2\)估计和密度反而产生\(\omega_p\) - 迹算子。
英文摘要
We establish sharp kinetic trace estimates and counterexamples across several velocity models. For half-space position domains, without any common bound on velocity support, we prove the natural trace estimate for both Lebesgue and standard Gaussian velocity measures. Density yields natural trace operators and Green's formula on the corresponding kinetic energy spaces. For bounded spatial domains in $d\ge2$, in the bounded-support Euclidean velocity model and in the spherical velocity model, we identify the sharp boundary regularity threshold for the trace weights $\min\{|v \cdot n|,|v \cdot n|^p\}$, $1\le p<\infty$. Writing $α_p=1/(p+1)$, the estimate holds on every bounded $\mathrm{C}^{1,α}$ domain with $α\geα_p$, and it fails for every $0<α<α_p$ on some strictly convex bounded domain of exact regularity $\mathrm{C}^{1,α}$. In particular, the natural trace ($p=1$) has the regularity threshold $\mathrm{C}^{1,1/2}$. On bounded $\mathrm{C}^{1,1/2}$ domains in $d\ge2$, density yields natural trace operators and Green's formula in the bounded-support Euclidean velocity model with either Lebesgue or standard Gaussian measure, and in the spherical velocity model. Norm-preserving velocity translation rules out the unrestricted Lebesgue trace estimate on every bounded $\mathrm{C}^1$ domain. In the unrestricted Gaussian model, for each $1\le p<2$, we construct counterexamples on every bounded $\mathrm{C}^{1,1}$ domain in dimension $d\ge2$, answering Question 1.8 of Albritton, Armstrong, Mourrat, and Novack (2024) negatively. For $2\le p<\infty$, the Gaussian $ω_2$ estimate and density instead yield $ω_p$-trace operators.
