Sharp kinetic trace theory

Abstract

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 $\alpha_p=1/(p+1)$, the estimate holds on every bounded $\mathrm{C}^{1,\alpha}$ domain with $\alpha\ge\alpha_p$, and it fails for every $0<\alpha<\alpha_p$ on some strictly convex bounded domain of exact regularity $\mathrm{C}^{1,\alpha}$. 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 $\omega_2$ estimate and density instead yield $\omega_p$-trace operators.