Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty_t H^3_x$ and $\partial_t g \in L^\infty H^2_x$, then energy estimates enable to propagate sobolev regularity of the initial data set till $H^3\times H^2$, is this sharp in the following sense : can you find a metric with this regularity and an initial data set $u_0, u_1$ in say $H^4 \times H^3$ s.t. the corresponding solution of the linear wave equation does not lie even locally in $L^\infty_t H^4_x$ but only in $L^\infty_t H^3_x$ ? Seems to me true but i fail to construct a counterexample. (I would like to apologize for not understanding how to reedit from another computer and a unregistered account, sorry).
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> question reedited 

