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

Abstract

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.