  Assume that $(u_n)$ is a sequence of solutions to heterogeneous equations
with rough coefficients and fractional derivatives, weakly converging to zero
in ${\rm L}^p(\R^{d+m})$, with $p>1$.
  We prove that the sequence of averaged quantities $(\int \rho(\my)
u_n(\mx,\my) d\my)$ is strongly precompact in $\Ljl\Rd$ for any $\rho\in
\Cc{\R^m}$, provided that restrictive non-degeneracy conditions are satisfied.
These are fulfilled for elliptic, parabolic, fractional convection-diffusion
equations, as well as for parabolic equations with a fractional time
derivative. The main tool that we are using is an adapted version of
H-distributions. As a consequence of the introduced methods, we obtain an
optimal velocity averaging result in the $\LL p$, $p\geq 2$, framework under
the standard non-degeneracy conditions, as well as a connection between the
H-measures and the H-distributions.
