Averaging regularity results for PDEs under transversality assumptions

Let u = u(x,y) and f = f(x,y) be two functions related by a PDE P(x,y,D_x,D_y)u = f; the regularity of the y‐average ∫ u(x,·)dy as a function of x is investigated knowing that of u and f. Our method consists in reducing P to a microlocal normal form under a natural transversality assumption. The 2‐microlocal regularity of u is also determined knowing that of f. These results are then applied to a homogenization problem. This article generalizes the results of [16] and [12] on velocity averaging and homogenization for kinetic equations.