Propagation of singularities for equations with $C^r$ coefficients for $r>1$

We observe that, for $r>1$, $s$ in an $r$-dependent interval, $p$ a homogeneous pseudodifferential symbol of order $m$ having $C^r$ regularity in space, and $u\in H^{s+m-r}({\mathbb{R}}^n)$ such that $p(x,D)u\in H^s({\mathbb{R}}^n)$, each point in the $H^{s+m-1}$ wavefront set of $u$ lies on a maximally extended null bicharacteristic of $p$ which is contained in the $H^{s+m-1}$ wavefront set of $u$. In fact, for $r=2$ slightly less than $C^{1,1}$ regularity suffices, and here the results apply to manifolds with bounded Ricci curvature.