Hypoelliptic estimates in radiative transfer

We derive the hypoelliptic estimates for a kinetic equation of the form ∂tf + k · ∇xf = (−∆d)h, for (t, x, k) ∈ R× R × S, where d ≥ 1, β > 0, Sd is the unit sphere in Rd+1 and ∆d is the Laplace-Beltrami operator on Sd. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable k ∈ Sd, we obtain estimates for the fractional derivatives of f in the (t, x) variables. Our proof follows the method of [9] based on the regularization of the momentum variable and on averaging lemmas on the sphere.