Hypoelliptic estimates in radiative transfer

We derive the hypoelliptic estimates for a kinetic equation of the form partial derivative tf + k.del(x)f = (-Delta(d))(beta)h, for (t, x, k) is an element of R x Rd+1 x S-d where d >= 1, beta >= 0, S-d, is the unit sphere in Rd+1 and Delta(d) is the Laplace-Beltrami operator on S-d. 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 is an element of S-d, we obtain estimates for the fractional derivatives of f in the (t, x) variables. Our proof follows the method of Bouchut based on the regularization of the momentum variable and on averaging lemmas on the sphere