2026/07/27 by Lukas Niebel, Lisa Valentini
#math.AP
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≥2, 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 ⋅ n|,|v ⋅ n|p\, 1≤ p<∞. Writing αp=1/(p+1), the estimate holds on every bounded C1,α domain with α≥αp, and it fails for every 0<α<αp on some strictly convex bounded domain of exact regularity C1,α. In particular, the natural trace (p=1) has the regularity threshold C1,1/2. On bounded C1,1/2 domains in d≥2, 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 C1 domain. In the unrestricted Gaussian model, for each 1≤ p<2, we construct counterexamples on every bounded C1,1 domain in dimension d≥2, answering Question 1.8 of Albritton, Armstrong, Mourrat, and Novack (2024) negatively. For 2≤ p<∞, the Gaussian ω2 estimate and density instead yield ωp-trace operators.