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This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without directly invoking energy considerations. But I think it is interesting in its own right.

Suppose one has a linear, second-order partial differential operator $P$ on $\mathbb{R}^{n+1}_{(t,x)}$, strictly hyperbolic with respect to the level sets of $t$. Let $\Sigma := \{t=0\}$ and $u_0, u_1 \in \mathscr{E}'(\Sigma)$. Then there is a distributional solution $u$ of $Pu=0$ such that $u$ and $\partial_t u$ (respectively) have $u_0$ and $u_1$ as their restrictions to $\Sigma$. Now let $\xi$ be a non-zero covector at $p \in \Sigma$ such that $(p, \xi) \in WF^s(u_0) \cup WF^{s-1}(u_1)$ for some $s \in \mathbb{R}$ (as in the sister question, $WF^s$ denotes the $H^s$ wave front set of a distribution).

By strict hyperbolicity, there exist precisely two non-zero $\mathbb{R}^{n+1}$-covectors at $p$, characteristic for $P$, which give $\xi$ when pulled back by the embedding $\Sigma \hookrightarrow \mathbb{R}^{n+1}$. Call these two covectors $\eta_+$ and $\eta_-$; in co-ordinates, we may write $\eta_\pm = (\tau_\pm,\xi)$. Does it follow that either $(p, \eta_+)$ or $(p,\eta_-)$ must belong to $WF^s(u)$?

Of course, for the $C^\infty$ wave front set this would be completely standard. It is also known (I think) that $H^s$ microlocal smoothness propagates: if $u_0$ is microlocally in $H^s$ at $(p,\xi)$ and $u_1$ is microlocally in $H^{s-1}$ at $(p,\xi)$ then the resulting solution $u$ is microlocally in $H^s$ at $(p,\eta_\pm)$.

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