Regularity of the moments of the solution of a Transport Equation

Let u = u(x, v) satisfy the Transport Equation u+v·∂ xu=f, x∈RN, v∈RNwhere f belongs to some space of type Lp(dx ⊗ dμ(v)) (where μ is a positive bounded measure on RN). We study the resulting regularity of the moment ∝ u(x, v) dμ(v) (in terms of fractional Sobolev spaces, for example). Counter-examples are given in order to test the optimality of our results.