Transfer of regularity by kinetic mollification along critical trajectories

We give a trajectory-based proof of transfer-of-regularity estimates à la Bouchut-Hörmander for kinetic equations at the weak scale of local diffusion. The proof is based on kinetic mollification along critical kinetic trajectories, whose endpoint map has precisely the kinetic scaling. Instead of using Fourier multipliers or the Kolmogorov fundamental solution, we work directly in physical space. The key step is an explicit formula for the mollification defect and corresponding critical support, size, and difference estimates for the kernels. These bounds yield the sharp, scale-invariant homogeneous estimates through maximal-function and Littlewood-Paley estimates on the kinetic group.