The authors establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. It is shown that the relative compactness in all variables of a bounded family of nonnegative functions \(f_\lambda(x,v)\in L^1\) satisfying some appropriate transport relation \[ v\cdot \nabla_x f_\lambda =(1-\Delta_x)^{\beta/2}(1-\Delta_v)^{\alpha/2} g_\lambda \] may be inferred solely from additional integrability and compactness with respect to \(v\).