We prove that the sequence of averaged quantities R R m un(x, p) ρ(p)dp, is strongly precompact in L 2 loc (R d ), where ρ ∈ L 2 c (R m ), and un ∈ L 2 (R m ; L s (R d )), s ≥ 2, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If s > 2 then the coefficients can be discontinuous with respect to the space variable x ∈ R

more »

... , otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the H-measures. Content 1. Introduction 239 2. Statement of the main result 243 3. Auxiliary results 246 4. Proof of the main theorem 253 5. Ultra-parabolic equation with discontinuous coefficients 255 References 258