Averages over Spheres for Kinetic Transport Equations with Velocity Derivatives in the Right-Hand Side

We prove estimates in hyperbolic Sobolev spaces $H^{s,\delta}(R^{1+d})$, $d\geq 3$, for velocity averages over spheres of solutions to the kinetic transport equation $\partial_{t} f + v \cdot \nabla_{x} f = \Omega^{i,j}_{v} g $, where $\Omega^{i,j}_{v} g$ are tangential velocity derivatives of $g$. Our results extend to all dimensions earlier results of Bournaveas and Perthame in dimension two [J. Math. Pures Appl., 9 (2001), pp. 517-534]. We construct counterexamples to test the optimality of our results.