[proofplan] The proof uses the constant-coefficient Cauchy representation of the [wave equation](/page/Wave%20Equation) as a sum of half-wave Fourier integral operators, together with microlocal cutoffs to ignore the low-frequency region, which cannot contribute to the wave front set. Since the initial data are conormal along $Y$, their wave front sets are contained in $N^*Y \setminus 0$. The half-wave Fourier integral operator wave front mapping theorem transports these covectors along the two characteristic Hamilton-flow branches $\tau = \pm |\xi|$. Finally, elliptic regularity for $P$ excludes noncharacteristic covectors, so the only possible singularities of $u$ lie on the null bicharacteristics issuing from the characteristic lifts of $N^*Y \setminus 0$. [/proofplan]

[step:Represent the Cauchy solution by microlocal half-wave propagators]The theorem statement defines $\mathcal{S}'(\mathbb{R}^n)$, $|D_x|$, $C(t)$, and $S(t)$. Since $\mathcal{E}'(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$, the compactly supported distributions $u_0$ and $u_1$ may be acted on by these Fourier multipliers. For each fixed $t \in \mathbb{R}$, the distributional Cauchy solution is \begin{align*} u(t,\cdot) = C(t)u_0 + S(t)u_1. \end{align*} Equivalently, away from $\xi = 0$, the operators $C(t)$ and $S(t)$ are finite sums of the half-wave propagators \begin{align*} E_+(t) := e^{it|D_x|}, \qquad E_-(t) := e^{-it|D_x|} \end{align*} composed with classical pseudodifferential amplitudes of orders $0$ and $-1$. More precisely, fix a nonzero spacetime covector being tested and choose a symbol $\psi : T^*\mathbb{R}^n \to [0,1]$ that is smooth, homogeneous of degree $0$ for $|\xi| \geq 1$, and equal to $1$ on a conic neighbourhood of the corresponding nonzero spatial covector. On this conic neighbourhood, the factors $\psi(D_x)C(t)$ and $\psi(D_x)S(t)$ are, after localization in $t$ on compact intervals, spacetime Fourier integral operators associated with the two half-wave canonical graphs. The part cut off by $1-\psi$ is not asserted to be globally low-frequency; it is only microlocally irrelevant at the tested covector because its full symbol vanishes in a conic neighbourhood of that covector. If one also inserts a separate low-frequency cutoff supported near $\xi=0$, then the multipliers in that low-frequency piece are smooth in $(t,\xi)$, with $|\xi|^{-1}\sin(t|\xi|)$ extended smoothly at $\xi=0$ by the value $t$; after multiplying in $t$ by any cutoff in $C_c^\infty(\mathbb{R}_t)$, the inverse Fourier kernels are smooth in both the time and spatial variables on that compact time interval. Thus the only microlocally relevant high-frequency pieces are the two half-wave Fourier integral operators.[/step]

[guided]The first point is that the Cauchy solution is not treated as an abstract distribution; it is represented by the standard constant-coefficient wave propagators declared in the theorem statement. Since $\mathcal{E}'(\mathbb{R}^n) \subset \mathcal{S}'(\mathbb{R}^n)$, the compactly supported distributions $u_0$ and $u_1$ may be acted on by the Fourier multipliers $C(t)$ and $S(t)$. For compactly supported distributional data $u_0,u_1 \in \mathcal{E}'(\mathbb{R}^n)$, the standard Cauchy evolution gives \begin{align*} u(t,\cdot) = C(t)u_0 + S(t)u_1. \end{align*} The symbol $|D_x|^{-1}$ is singular at $\xi = 0$, so we must say why this causes no microlocal problem. Wave front sets live over nonzero covectors. Thus, to test whether a fixed nonzero covector can belong to $WF(u)$, choose a smooth conic cutoff $\psi : T^*\mathbb{R}^n \to [0,1]$ which equals $1$ in a conic neighbourhood of the corresponding spatial covector and avoids the zero section. On the support of $\psi$, the multiplier $|\xi|^{-1}$ is a classical symbol of order $-1$. Therefore, after localizing in time on compact intervals, $\psi(D_x)S(t)$ is a legitimate microlocal spacetime Fourier integral operator. Away from $\xi=0$, the trigonometric identities \begin{align*} \cos(t|\xi|) = \frac{1}{2}e^{it|\xi|} + \frac{1}{2}e^{-it|\xi|} \end{align*} and \begin{align*} |\xi|^{-1}\sin(t|\xi|) = \frac{1}{2i}|\xi|^{-1}e^{it|\xi|} - \frac{1}{2i}|\xi|^{-1}e^{-it|\xi|} \end{align*} show that the microlocal pieces of $C(t)$ and $S(t)$ are sums of half-wave propagators with pseudodifferential amplitudes. The complement of $\psi$ should be interpreted microlocally, not as a purely low-frequency operator. Its symbol vanishes in the conic neighbourhood being tested, so it cannot contribute to that tested covector. Separately, the genuinely low-frequency piece can be cut off by a compactly supported frequency cutoff near $\xi=0$; its multipliers have compact frequency support and are smooth in $(t,\xi)$, including the smooth extension of $|\xi|^{-1}\sin(t|\xi|)$ at $\xi=0$. After also multiplying by an arbitrary cutoff in $C_c^\infty(\mathbb{R}_t)$, these compact-frequency multipliers have inverse Fourier kernels that are smooth in both $t$ and $x$ on the chosen compact time interval. Applying such locally-in-time smoothing operators to compactly supported distributions produces no microlocal singularity at any nonzero spacetime covector. Thus only the two high-frequency half-wave branches remain relevant for the desired wave front inclusion.[/guided]

[step:Use conormality to localize the initial wave front set] By the meaning of conormal Cauchy data along $Y$, the initial distributions satisfy \begin{align*} WF(u_0) \cup WF(u_1) \subset N^*Y \setminus 0. \end{align*} Thus every nonzero covector of the initial data has the form $(y,\eta)$ with $y \in Y$ and $\eta \in N_y^*Y \setminus \{0\}$. [/step]

[step:Identify the canonical relations of the two half-wave branches] For the half-wave operators $E_+(t)$ and $E_-(t)$, the relevant phase functions are, microlocally away from $\eta=0$, \begin{align*} \phi_+ : \mathbb{R}_t \times \mathbb{R}_x^n \times \mathbb{R}_y^n \times (\mathbb{R}_\eta^n \setminus \{0\}) \to \mathbb{R} \end{align*} defined by \begin{align*} \phi_+(t,x,y,\eta) := (x-y)\cdot \eta + t|\eta|, \end{align*} and \begin{align*} \phi_- : \mathbb{R}_t \times \mathbb{R}_x^n \times \mathbb{R}_y^n \times (\mathbb{R}_\eta^n \setminus \{0\}) \to \mathbb{R} \end{align*} defined by \begin{align*} \phi_-(t,x,y,\eta) := (x-y)\cdot \eta - t|\eta|. \end{align*} The associated canonical relations in \begin{align*} T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \times T^*\mathbb{R}_y^n \end{align*} are the graphs obtained from \begin{align*} \partial_\eta \phi_\pm(t,x,y,\eta) = 0. \end{align*} For $\phi_+$ this critical equation is \begin{align*} x-y+t\frac{\eta}{|\eta|}=0, \end{align*} and for $\phi_-$ it is \begin{align*} x-y-t\frac{\eta}{|\eta|}=0. \end{align*} The output covectors are computed from the phase derivatives: \begin{align*} \tau = \partial_t\phi_\pm = \pm |\eta|, \qquad \xi = \partial_x\phi_\pm = \eta. \end{align*} Therefore the two branches send an input covector $(y,\eta)$, $\eta \neq 0$, to covectors satisfying \begin{align*} (t,x,\tau,\xi) = \left(t, y \mp t\frac{\eta}{|\eta|}, \pm|\eta|,\eta\right), \end{align*} with the upper sign corresponding to $\phi_+$ and the lower sign corresponding to $\phi_-$. Define the canonical graphs \begin{align*} \Lambda_\pm \subset T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0 \times T^*\mathbb{R}_y^n \setminus 0 \end{align*} by declaring that $((t,x,\tau,\xi),(y,\eta)) \in \Lambda_\pm$ exactly when $\eta \neq 0$ and \begin{align*} (t,x,\tau,\xi) = \left(t, y \mp t\frac{\eta}{|\eta|}, \pm|\eta|,\eta\right). \end{align*} Thus $\Lambda_\pm(A)$ denotes the image of a conic set $A \subset T^*\mathbb{R}^n \setminus 0$ under this graph relation. The Hamilton vector field of \begin{align*} p(t,x,\tau,\xi)=\tau^2-|\xi|^2 \end{align*} is \begin{align*} H_p = 2\tau\,\partial_t - 2\sum_{j=1}^n \xi_j\,\partial_{x_j}. \end{align*} Since $p$ is independent of $t$ and $x$, $\tau$ and $\xi$ are constant along its integral curves. The curves just described are exactly reparametrizations of the integral curves of $H_p$ through \begin{align*} (0,y,\pm|\eta|,\eta). \end{align*} They are null because \begin{align*} p(0,y,\pm|\eta|,\eta)=|\eta|^2-|\eta|^2=0. \end{align*} [/step]

[step:Map the conormal initial wave front set by the half-wave calculus]Apply the Fourier integral operator wave front mapping theorem to each microlocal half-wave summand: if $A$ is a Fourier integral operator with canonical relation $C_A$, then \begin{align*} WF(Af) \subset C_A \circ WF(f) \end{align*} for every distribution $f$ for which the canonical relation composition is proper on the microlocal support being tested. Here $C_A \circ WF(f)$ means the set of output covectors $\rho$ for which there exists an input covector $\nu \in WF(f)$ with $(\rho,\nu) \in C_A$. In the present case $A$ is a spacetime solution operator from $\mathbb{R}_y^n$ to $\mathbb{R}_t \times \mathbb{R}_x^n$, the canonical relation is one of the graphs $\Lambda_+$ or $\Lambda_-$ computed above, and the input distributions $u_0,u_1$ have compact support in $\mathbb{R}^n_y$. Fix a compact spacetime set $K_{t,x} \subset \mathbb{R}_t \times \mathbb{R}_x^n$, and choose a cutoff $\chi \in C_c^\infty(\mathbb{R}_t \times \mathbb{R}_x^n)$ with $\chi = 1$ on a neighbourhood of $K_{t,x}$. Let $K_y \subset \mathbb{R}_y^n$ be a compact set containing $\operatorname{supp} u_0 \cup \operatorname{supp} u_1$. The localized operator $\chi A$ is properly supported on the base variables relevant to these data, because its base projection is contained in $\operatorname{supp}\chi \times K_y$. Its canonical relation is the restriction of $C_A$ over $\operatorname{supp}\chi$, and the fiber relation remains a closed conic graph over nonzero covectors. More concretely, on $\eta \neq 0$ the formulas $(y,\eta) \mapsto (t, y \mp t\eta/|\eta|, \pm |\eta|, \eta)$ determine the output covector uniquely and retain $\eta$ as the spatial covector. Therefore, after intersecting both cotangent factors with any closed conic set whose image in cosphere variables is compact and whose base lies in $\operatorname{supp}\chi \times K_y$, the graph projection is proper: escaping fiber directions on one side are exactly escaping fiber directions on the other side. Hence the projections of $C_A \cap (T^*\operatorname{supp}\chi \times WF(u_j))$ to the two factors are proper on the compact-conic microlocal support being tested, so the mapping theorem applies to $\chi A$. Since $\chi=1$ near $K_{t,x}$, this gives the same wave front inclusion over $K_{t,x}$ as for $A$ itself. Applying this theorem to $E_+(t)$ and $E_-(t)$, and then using \begin{align*} WF(u_0) \cup WF(u_1) \subset N^*Y \setminus 0, \end{align*} gives \begin{align*} WF(C(t)u_0 + S(t)u_1) \subset \Lambda_+(N^*Y \setminus 0) \cup \Lambda_-(N^*Y \setminus 0), \end{align*} where $\Lambda_+$ and $\Lambda_-$ denote the two half-wave canonical graphs identified above. Cancellations between the two summands can only remove wave front directions, not create new ones, because \begin{align*} WF(v+w) \subset WF(v) \cup WF(w) \end{align*} for distributions $v,w$.[/step]

[guided]We now use the microlocal mapping theorem for Fourier integral operators. The theorem says the following: if $A$ is a Fourier integral operator with canonical relation $C_A$, then the wave front set of $Af$ is contained in the image of $WF(f)$ under $C_A$: \begin{align*} WF(Af) \subset C_A \circ WF(f). \end{align*} We verify the hypotheses needed here. The operators under consideration are the microlocal pieces of $E_+(t)$ and $E_-(t)$, obtained after cutting away $\xi=0$. These are Fourier integral operators from distributions on $\mathbb{R}_y^n$ to distributions on $\mathbb{R}_t \times \mathbb{R}_x^n$, with canonical relations $\Lambda_+$ and $\Lambda_-$ computed in the previous step. These relations are the closed conic graphs \begin{align*} \Lambda_\pm \subset T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0 \times T^*\mathbb{R}_y^n \setminus 0 \end{align*} that send $(y,\eta)$ to \begin{align*} \left(t, y \mp t\frac{\eta}{|\eta|}, \pm|\eta|,\eta\right). \end{align*} Thus applying the relation to a covector of the initial data gives a single propagated covector on each branch. Because the initial data $u_0$ and $u_1$ are compactly supported in $\mathbb{R}^n_y$, choose a compact set $K_y \subset \mathbb{R}^n_y$ containing $\operatorname{supp} u_0 \cup \operatorname{supp} u_1$. If $K_{t,x} \subset \mathbb{R}_t \times \mathbb{R}_x^n$ is compact, choose $\chi \in C_c^\infty(\mathbb{R}_t \times \mathbb{R}_x^n)$ with $\chi=1$ near $K_{t,x}$. The localized spacetime operator $\chi A$ has base support contained in $\operatorname{supp}\chi \times K_y$ when applied to $u_0$ or $u_1$, so it is properly supported for the microlocal region under consideration. Since the canonical relation is a closed conic graph over the nonzero covectors under consideration, its projections are proper after this localization. The graph has no fiber degeneracy away from $\eta=0$: the input covector $(y,\eta)$ determines the output covector $(t, y \mp t\eta/|\eta|, \pm|\eta|,\eta)$, and the output spatial covector is exactly $\eta$. Thus the canonical relation composition required by the mapping theorem is defined in the microlocal region being tested, and the resulting inclusion for $\chi A$ gives the same inclusion over $K_{t,x}$ because $\chi=1$ there. The conormality assumption supplies the input set: \begin{align*} WF(u_0) \cup WF(u_1) \subset N^*Y \setminus 0. \end{align*} Therefore every input covector that can contribute to the wave front set of the solution has the form $(y,\eta)$ with $y \in Y$ and $\eta \in N_y^*Y \setminus \{0\}$. Applying the two canonical graphs gives only the covectors \begin{align*} (t,x,\tau,\xi) = \left(t, y \mp t\frac{\eta}{|\eta|}, \pm|\eta|,\eta\right). \end{align*} These are precisely the two characteristic branches through the lifted conormal covectors \begin{align*} (0,y,\pm|\eta|,\eta). \end{align*} Finally, the solution is a sum of the $u_0$-term and the $u_1$-term. Wave front sets are subadditive under addition: \begin{align*} WF(v+w) \subset WF(v) \cup WF(w). \end{align*} Thus possible cancellations between the two Cauchy contributions can only shrink the wave front set. They cannot create a covector outside the union of the two propagated conormal sets.[/guided]

[step:Exclude noncharacteristic covectors by elliptic regularity]Let $(t_0,x_0,\tau_0,\xi_0) \in T^*(\mathbb{R}_t \times \mathbb{R}_x^n)\setminus 0$ satisfy \begin{align*} p(t_0,x_0,\tau_0,\xi_0) \neq 0. \end{align*} Then $P$ is elliptic at $(t_0,x_0,\tau_0,\xi_0)$, because its principal symbol is nonzero there. By [microlocal elliptic regularity for pseudodifferential operators](/theorems/8163), \begin{align*} (t_0,x_0,\tau_0,\xi_0) \notin WF(u) \end{align*} whenever \begin{align*} (t_0,x_0,\tau_0,\xi_0) \notin WF(Pu). \end{align*} Since $Pu=0$ and the zero distribution has empty wave front set, we have $WF(Pu)=\varnothing$. Hence every covector in $WF(u)$ must lie in the characteristic set \begin{align*} p^{-1}(0) = \{(t,x,\tau,\xi) : \tau^2 = |\xi|^2\}. \end{align*}[/step]

[guided]We now rule out covectors that are not characteristic for the wave operator. Let \begin{align*} (t_0,x_0,\tau_0,\xi_0) \in T^*(\mathbb{R}_t \times \mathbb{R}_x^n) \setminus 0 \end{align*} satisfy \begin{align*} p(t_0,x_0,\tau_0,\xi_0) \neq 0. \end{align*} The principal symbol of $P$ is $p(t,x,\tau,\xi)=\tau^2-|\xi|^2$, so this nonvanishing condition says exactly that $P$ is elliptic at the covector $(t_0,x_0,\tau_0,\xi_0)$. Microlocal elliptic regularity says that ellipticity transfers regularity from $Pu$ back to $u$: at an elliptic covector, if that covector is not in $WF(Pu)$, then it is not in $WF(u)$. Here $Pu=0$ by the equation in the theorem statement. The zero distribution has empty wave front set, so \begin{align*} WF(Pu)=WF(0)=\varnothing. \end{align*} Therefore \begin{align*} (t_0,x_0,\tau_0,\xi_0) \notin WF(u). \end{align*} Since the same argument applies to every covector where $p \neq 0$, every covector in $WF(u)$ must lie in \begin{align*} p^{-1}(0) = \{(t,x,\tau,\xi) : \tau^2 = |\xi|^2\}. \end{align*} This is the characteristic set of the wave operator.[/guided]

[step:Conclude the propagation inclusion]Combining the half-wave wave front mapping inclusion with the noncharacteristic exclusion gives \begin{align*} WF(u) \subset \Lambda_+(N^*Y \setminus 0) \cup \Lambda_-(N^*Y \setminus 0). \end{align*} By the Hamilton-flow identification above, this right-hand side is contained in the union of the null bicharacteristics of $p$ issuing from \begin{align*} (0,y,\tau,\xi), \qquad y \in Y, \qquad (y,\xi) \in N^*Y \setminus 0, \qquad \tau=\pm|\xi|. \end{align*} Therefore $WF(u)$ is contained in the asserted union of propagated conormal characteristic lifts. This proves the theorem.[/step]

[guided]The previous steps have produced two restrictions on possible singularities. First, the half-wave mapping argument showed that singularities coming from $u_0$ or $u_1$ can only be transported by the two canonical graphs $\Lambda_+$ and $\Lambda_-$. Thus \begin{align*} WF(u) \subset \Lambda_+(N^*Y \setminus 0) \cup \Lambda_-(N^*Y \setminus 0). \end{align*} Second, the elliptic regularity step rules out every noncharacteristic covector, so the remaining covectors lie on $p^{-1}(0)$. The canonical-relation computation identified these two propagated sets with the Hamilton-flow branches through the lifted conormal covectors. More explicitly, every possible initial covector has the form $(y,\eta)$ with $y \in Y$ and $(y,\eta) \in N^*Y \setminus 0$, and the two half-wave branches lift it to \begin{align*} (0,y,\tau,\xi), \qquad \xi=\eta, \qquad \tau=\pm |\eta|. \end{align*} The Hamilton-flow computation showed that the propagated covectors are precisely reparametrizations of the integral curves of $H_p$ issuing from these points. Therefore \begin{align*} WF(u) \subset \bigcup_{y \in Y}\bigcup_{(y,\eta) \in N^*Y \setminus 0}\bigcup_{\sigma \in \{-1,1\}} \gamma_{y,\eta,\sigma}(\mathbb{R}), \end{align*} where $\gamma_{y,\eta,\sigma}$ is the null bicharacteristic of $p$ with initial point $(0,y,\sigma|\eta|,\eta)$. This is exactly the asserted propagation inclusion.[/guided]