I have found the following claim stated very clearly at least once in the published literature (see below):

> Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \mathbb{R}^{n+1}$, strictly hyperbolic with respect to the level surfaces of the first coordinate, which I will denote by $t$.  Let $u$ be a distribution in $H^s_\mathrm{loc}(\Omega)$ for some $s \in \mathbb{R}$, such that $Pu=0$. Then the restriction (i.e. pullback) of $u$ to each of the level sets $\Sigma_{\tau} := \Omega \cap \{ t = \tau \}$ is itself in $H^s_\mathrm{loc}(\Sigma_\tau)$.

In fact, a more general statement may be found in Section 2 of:

<cite authors="Bao, Gang; Symes, William W.">_Bao, Gang; Symes, William W._, [**A trace theorem for solutions of linear partial differential equations**](http://dx.doi.org/10.1002/mma.1670140803), Math. Methods Appl. Sci. 14, No.8, 553-562 (1991). [ZBL0754.35023](https://zbmath.org/?q=an:0754.35023).</cite>

There, the authors say that the statement boxed above follows from standard energy estimates, and refer to e.g. Taylor's book on pseudo-differential operators for the arguments. However, I meet the following difficulties when attempting to fill in all details:

 - Ostensibly, the "standard energy estimates" discussed in Taylor's book (and elsewhere, e.g. in Hörmander's) pertain distributional solutions which (in the case $\Omega = \mathbb{R}^{n+1}$) are in spaces $\bigcap_{j=0}^{m-1} C^j([\tau_1,\tau_2]; H^{s-j})$, where $m$ is the order of $P$. I, on the other hand, am interested in distributions (locally) in $H^s$ in both time and space.
 - In any case, as far as I am aware, the usual energy estimate controls the evolution in time of the sum of (appropriate) "spatial" Sobolev norms of (appropriate) $t$-derivatives of $u$ restricted to the level sets of $t$. Specifically, in the case $Pu=0$ the energy estimate says that the supremum of this quantity over a compact interval $[\tau_0,\tau_1]$ is bounded by a constant times its value at $\tau_0$.  To prove that the restriction of $u$ to the $t=\tau_0$ surface is (locally) in $H^s$, presumably we would like to control the $H^s$ norm of this restriction and place an upper bound on it, but this does not seem to be the scenario in the energy inequality.
 
By the way, I am not doubting that $u$ can be pulled back to a spacelike hypersurface, as a distribution. I know that this is the case because $WF(u)$ is disjoint from the normal bundle of the hypersurface whenever $Pu$ is smooth.