[proofplan] The proof first uses the microlocal definition of singular support: a distribution is smooth at a base point exactly when no nonzero covector over that point lies in its wave front set. Therefore the singular support is the base projection of the wave front set. The assumed inclusion $\operatorname{WF}(K_U)\subset C$ immediately gives the abstract containment. The two examples then identify the base projection of the wave canonical relation: straight-line Hamilton flow for the Euclidean symbol $|\xi|$ gives the light cone, while Riemannian geodesic flow for $|\xi|_g$ gives the distance sphere before the injectivity radius. [/proofplan]

[step:Project the wave front set to obtain the abstract singular support inclusion]Define the base manifold \begin{align*} X := I \times Y \times Y. \end{align*} Let $\pi_X: T^*X \to X$ denote the cotangent bundle projection. For every distribution $u \in \mathcal{D}'(X)$, the defining relation between wave front set and singular support is \begin{align*} \operatorname{sing\,supp}(u) = \pi_X(\operatorname{WF}(u)). \end{align*} Here this standard microlocal identity says precisely that $u$ is smooth near a point $q \in X$ if and only if no nonzero covector in $T_q^*X$ belongs to $\operatorname{WF}(u)$; this is the wave-front characterization of singular support (citing a result not yet in the wiki: singular support is the base projection of the wave front set). Applying this identity to $u = K_U \in \mathcal{D}'(X)$ gives \begin{align*} \operatorname{sing\,supp}(K_U) = \pi_X(\operatorname{WF}(K_U)). \end{align*} By the hypothesis $\operatorname{WF}(K_U) \subset C$, applying the map $\pi_X$ to both sides gives \begin{align*} \pi_X(\operatorname{WF}(K_U)) \subset \pi_X(C). \end{align*} Since $X = I \times Y \times Y$, this is exactly \begin{align*} \operatorname{sing\,supp}(K_U) \subset \pi_{I \times Y \times Y}(C). \end{align*}[/step]

[guided]Set \begin{align*} X := I \times Y \times Y. \end{align*} The kernel $K_U$ is a distribution on $X$, so its wave front set is a closed conic subset of $T^*X \setminus 0$, and its singular support is a subset of the base space $X$. The key point is the standard microlocal equivalence between smoothness and absence of wave-front directions. For a distribution $u \in \mathcal{D}'(X)$ and a point $q \in X$, the statement "$u$ is smooth in a neighbourhood of $q$" is equivalent to the statement that no nonzero covector $\zeta \in T_q^*X \setminus \{0\}$ lies in $\operatorname{WF}(u)$. Equivalently, the points where $u$ fails to be smooth are exactly the base points of covectors in $\operatorname{WF}(u)$. Thus \begin{align*} \operatorname{sing\,supp}(u) = \pi_X(\operatorname{WF}(u)), \end{align*} where $\pi_X: T^*X \to X$ is the base projection. This is the standard wave-front characterization of singular support (citing a result not yet in the wiki: singular support is the base projection of the wave front set). Now apply this identity to the particular distribution $u = K_U$. We obtain \begin{align*} \operatorname{sing\,supp}(K_U) = \pi_X(\operatorname{WF}(K_U)). \end{align*} The theorem assumes that the wave front set of the kernel is contained in the canonical relation: \begin{align*} \operatorname{WF}(K_U) \subset C. \end{align*} Since $\pi_X$ is a set map, it preserves inclusions of subsets in the sense that $A \subset B$ implies $\pi_X(A) \subset \pi_X(B)$. Therefore \begin{align*} \pi_X(\operatorname{WF}(K_U)) \subset \pi_X(C). \end{align*} Combining the last two displayed relations gives \begin{align*} \operatorname{sing\,supp}(K_U) \subset \pi_X(C). \end{align*} Finally, because $X$ was defined to be $I \times Y \times Y$, the map $\pi_X$ is the same map denoted in the statement by $\pi_{I \times Y \times Y}$. This proves the abstract containment.[/guided]

[step:Identify the Euclidean projection by straight-line Hamilton flow] Assume $Y=\mathbb{R}^n$ and $U(t)=\cos(t\sqrt{-\Delta})$. Let $p: T^*\mathbb{R}^n \setminus 0 \to \mathbb{R}$ be the principal half-wave Hamiltonian \begin{align*} p(x,\xi) := |\xi|. \end{align*} The Hamiltonian flow of $p$ is the map $\Phi_t: T^*\mathbb{R}^n \setminus 0 \to T^*\mathbb{R}^n \setminus 0$ given by \begin{align*} \Phi_t(y,\eta) = \left(y + t\frac{\eta}{|\eta|}, \eta\right). \end{align*} For the two half-waves contributing to $\cos(t\sqrt{-\Delta})$, the projected canonical relation therefore consists, away from $t=0$, of triples $(t,x,y)$ for which there is a nonzero covector $\eta \in T_y^*\mathbb{R}^n \setminus \{0\}$ and a sign $\sigma \in \{-1,1\}$ such that \begin{align*} x = y + \sigma t\frac{\eta}{|\eta|}. \end{align*} Taking Euclidean norms gives \begin{align*} |x-y| = |t|. \end{align*} At $t=0$, the flow is the identity, so the projected relation is the diagonal $\{(0,x,x):x \in \mathbb{R}^n\}$. Hence the abstract containment from the previous step gives \begin{align*} \operatorname{sing\,supp}(K_U) \subset \{(0,x,x):x \in \mathbb{R}^n\} \cup \{(t,x,y):t \ne 0,\ |x-y|=|t|\}. \end{align*} [/step]

[step:Identify the Riemannian projection by geodesic flow before the injectivity radius] Assume that $(Y,g)$ is a Riemannian manifold, that $\Omega \subset Y$ is geodesically convex, and that $0<T$ is smaller than the relevant injectivity radius on $\Omega$. Let $I=(-T,T)$, and consider the localized kernel on $I \times \Omega \times \Omega$ of $U(t)=\cos(t\sqrt{\Delta_g})$, with $\Delta_g$ nonnegative. Let $p_g: T^*Y \setminus 0 \to \mathbb{R}$ be the principal half-wave Hamiltonian \begin{align*} p_g(y,\eta) := |\eta|_g. \end{align*} Let $\pi_Y: T^*Y \to Y$ denote the cotangent bundle base projection. The Hamiltonian flow of $p_g$ is the lifted unit-speed geodesic flow: if $\gamma_{z,\eta}: (-T,T) \to Y$ is the geodesic with initial point $z$ and initial covector direction determined by $\eta \in T_z^*Y \setminus \{0\}$ through the metric isomorphism $T_z^*Y \to T_zY$, then the base projection of the flow satisfies \begin{align*} \pi_Y(\Phi_t^g(z,\eta)) = \gamma_{z,\eta}(t) \end{align*} after normalizing the initial tangent vector to have $g$-length $1$. Therefore, for $0<|t|<T$, the projected wave relation consists of pairs $(y,z) \in \Omega \times \Omega$ joined by a unit-speed geodesic segment of length $|t|$. Since $\Omega$ is geodesically convex and $T$ is below the injectivity radius on $\Omega$, such a geodesic segment is distance minimizing and unique in the relevant local branch. Hence this condition is equivalent to \begin{align*} d_g(y,z)=|t|. \end{align*} At $t=0$, the geodesic flow is the identity on the base, so the projected relation contributes the diagonal $\{(0,y,y):y \in \Omega\}$. Applying the abstract containment to this projected canonical relation gives \begin{align*} \operatorname{sing\,supp}(K_U) \subset \{(0,y,y):y \in \Omega\} \cup \{(t,y,z):0<|t|<T,\ y,z \in \Omega,\ d_g(y,z)=|t|\}. \end{align*} This is the asserted Riemannian containment. [/step]