Sharp boundary regularity properties for hypoelliptic kinetic equations

We establish sharp boundary regularity results for solutions to kinetic Fokker-Planck equations under prescribed inflow boundary conditions, providing precise quantification of the boundary hypoelliptic regularization effect. For equations with rough coefficients, we characterize the behaviours for solutions on grazing and incoming boundaries. In particular, in the absence of influxes and sources, an explicit exponential infinite-order vanishing estimate is derived near incoming boundaries. When the coefficients are regular, we obtained the optimal Hölder regularity on grazing boundaries and general Schauder-type estimates away from them.