Clean Composition Theorem for Fourier Integral Operators

Theorem

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Let $X$, $Y$, and $Z$ be smooth manifolds with $n_Y=\dim Y$. Let $C_1 \subset (T^*X \setminus 0) \times (T^*Y \setminus 0)$ and $C_2 \subset (T^*Y \setminus 0) \times (T^*Z \setminus 0)$ be closed conic immersed Lagrangian submanifolds in the untwisted kernel convention: if $K_A$ and $K_B$ are the Schwartz kernels of the operators below, then $C_1$ and $C_2$ are the conic Lagrangians containing $\operatorname{WF}(K_A)$ and $\operatorname{WF}(K_B)$ without applying the antipodal map to the intermediate cotangent variables. Let $A \in I^{m_1}(X,Y;C_1)$ and $B \in I^{m_2}(Y,Z;C_2)$ be properly supported Fourier integral operators in the chapter's kernel-order convention. Let $\operatorname{WF}'(A) \subset (T^*X \setminus 0) \times (T^*Y \setminus 0)$ and $\operatorname{WF}'(B) \subset (T^*Y \setminus 0) \times (T^*Z \setminus 0)$ denote the operator wavefront relations associated to the Schwartz kernels $K_A$ and $K_B$ in the same untwisted kernel convention. Assume that on every microlocal patch used in the composition, $K_A$ and $K_B$ admit nondegenerate positively homogeneous phase parametrizations \begin{align*} \phi_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{R} \end{align*} and \begin{align*} \phi_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{R} \end{align*} with classical amplitudes in the symbol orders prescribed by the chapter's kernel-order convention, and that the corresponding oscillatory integrals are interpreted by the standard regularized oscillatory-integral definition. Define the negative clean fibre product \begin{align*} C_1 \times_{T^*Y}^{-} C_2 := \{(x,\xi;y,\eta;y,\eta';z,\zeta) \in C_1 \times C_2 : \eta + \eta' = 0\}. \end{align*} Equivalently, $C_1 \times_{T^*Y}^{-} C_2$ is the intersection of $C_1 \times C_2$ with \begin{align*} \{(x,\xi;y,\eta;y',\eta';z,\zeta): y=y', \ \eta+\eta'=0\} \end{align*} inside \begin{align*} (T^*X \setminus 0) \times (T^*Y \setminus 0) \times (T^*Y \setminus 0) \times (T^*Z \setminus 0). \end{align*} Assume that this intersection is clean with constant excess $e$, meaning that for every $p \in C_1 \times_{T^*Y}^{-} C_2$, \begin{align*} T_p(C_1 \times_{T^*Y}^{-} C_2) = T_p(C_1 \times C_2) \cap T_p\{y=y', \ \eta+\eta'=0\}, \end{align*} and the dimension excess over the transverse intersection value is the same integer $e$. Assume moreover that, for the combined local phase \begin{align*} \Phi: U_X \times U_Y \times U_Z \times (\mathbb{R}^{N_1} \setminus 0) \times (\mathbb{R}^{N_2} \setminus 0) &\to \mathbb{R} \end{align*} defined by \begin{align*} \Phi(x,y,z,\theta,\sigma)=\phi_1(x,y,\theta)+\phi_2(y,z,\sigma), \end{align*} the critical set of $\Phi$ in the integration variables $(y,\theta,\sigma)$ is a clean critical manifold which maps diffeomorphically onto the corresponding microlocal piece of $C_1 \times_{T^*Y}^{-} C_2$, and the normal Hessian of $\Phi$ in the variables $(y,\theta,\sigma)$ has constant rank equal to the number of integration variables minus the fibre dimension determined by the excess $e$. In the chapter's kernel-order convention, assume the clean stationary phase normalization for this situation gives precisely the order shift \begin{align*} \frac{e}{2}. \end{align*} Let \begin{align*} \pi_{XZ}: C_1 \times_{T^*Y}^{-} C_2 &\to (T^*X \setminus 0) \times (T^*Z \setminus 0) \end{align*} be the map \begin{align*} (x,\xi;y,\eta;y,-\eta;z,\zeta) &\mapsto (x,\xi;z,\zeta). \end{align*} Assume that, on the closed conic subset of $C_1 \times_{T^*Y}^{-} C_2$ [lying over](/theorems/2876) $\operatorname{WF}'(A) \times \operatorname{WF}'(B)$, the map $\pi_{XZ}$ is proper and has constant rank. Assume further that \begin{align*} C_1 \circ C_2 := \pi_{XZ}(C_1 \times_{T^*Y}^{-} C_2) \end{align*} is an admissible immersed conic Lagrangian target for the class $I^m(X,Z;C_1 \circ C_2)$ in the chapter's definition. Then the composition $AB$ is a well-defined properly supported Fourier integral operator from $Z$ to $X$, and \begin{align*} AB \in I^{m_1+m_2+e/2}(X,Z;C_1 \circ C_2) \end{align*} in the chapter's kernel-order convention. Equivalently, after translating the same operators into the standard half-density operator-order convention, the composed operator has order equal to the sum of the two input operator orders plus the clean excess contribution displayed above.

Discussion

No discussion available for this theorem.

Proof

[proofplan] The proof is local on $X \times Y \times Z$ and microlocal near the wavefront-relevant part of the clean fibre product. We write the two kernels as oscillatory integrals whose phases parametrize $C_1$ and $C_2$, multiply the kernels, and integrate over the intermediate variable $y \in Y$. The stationary equations for the combined phase impose precisely the sign condition $\eta+\eta'=0$, so the clean intersection hypothesis gives a clean critical manifold of constant excess $e$. Clean stationary phase then produces a new oscillatory integral associated with the projected image $C_1 \circ C_2$, and the chapter's kernel-order normalization gives the order \begin{align*} m_1+m_2+\frac{e}{2}. \end{align*} [/proofplan] [step:Localize the kernels near the wavefront-relevant clean fibre product] Because $A$ and $B$ are properly supported, their composition is defined by the standard kernel composition criterion for properly supported operators (citing a result not yet in the wiki: Kernel composition criterion for properly supported operators). Thus the Schwartz kernel $K_{AB}$ is locally obtained from $K_A$ and $K_B$ by pairing over the intermediate manifold $Y$. Fix coordinate charts $(U_X,\kappa_X)$ on $X$, $(U_Y,\kappa_Y)$ on $Y$, and $(U_Z,\kappa_Z)$ on $Z$. In these charts, let \begin{align*} \phi_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{R} \end{align*} and \begin{align*} \phi_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{R} \end{align*} be nondegenerate positively homogeneous phase functions parametrizing $C_1$ and $C_2$, respectively, on the chosen microlocal patches. Let \begin{align*} a_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{C} \end{align*} and \begin{align*} a_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{C} \end{align*} be classical amplitudes in the precise symbol orders prescribed by the chapter's kernel-order convention for $I^{m_1}(X,Y;C_1)$ and $I^{m_2}(Y,Z;C_2)$. We write $\mathcal{L}^{N_1}$, $\mathcal{L}^{N_2}$, and $\mathcal{L}^{\dim Y}$ for [Lebesgue measure](/page/Lebesgue%20Measure) in the chosen coordinate variables. After inserting microlocal cutoffs supported in these coordinate patches, the kernels have the local form, interpreted as regularized oscillatory integrals, \begin{align*} K_A(x,y)=\int_{\mathbb{R}^{N_1}} e^{i\phi_1(x,y,\theta)} a_1(x,y,\theta) \, d\mathcal{L}^{N_1}(\theta) \end{align*} and \begin{align*} K_B(y,z)=\int_{\mathbb{R}^{N_2}} e^{i\phi_2(y,z,\sigma)} a_2(y,z,\sigma) \, d\mathcal{L}^{N_2}(\sigma), \end{align*} where $x \in U_X$, $y \in U_Y$, $z \in U_Z$, $\theta \in \mathbb{R}^{N_1} \setminus 0$, and $\sigma \in \mathbb{R}^{N_2} \setminus 0$. [guided] We first reduce the assertion to the local oscillatory-integral calculation. Proper support is the hypothesis that prevents the intermediate integration over $Y$ from escaping to infinity: for compact sets in the outer variables, only a compact set of intermediate points $y \in Y$ contributes. Therefore the standard kernel composition criterion for properly supported operators (citing a result not yet in the wiki: Kernel composition criterion for properly supported operators) allows us to compute the composed kernel locally by multiplying the two kernels and integrating over $Y$. Choose coordinate charts $(U_X,\kappa_X)$, $(U_Y,\kappa_Y)$, and $(U_Z,\kappa_Z)$. On a conic microlocal patch of $C_1$, choose a nondegenerate positively homogeneous phase function \begin{align*} \phi_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{R}. \end{align*} This means that the critical set of $\phi_1$ in the auxiliary variable $\theta \in \mathbb{R}^{N_1} \setminus 0$ parametrizes the relevant piece of $C_1$ by \begin{align*} (x,y,\theta) \mapsto (x,\partial_x\phi_1(x,y,\theta);y,\partial_y\phi_1(x,y,\theta)). \end{align*} Likewise, on a conic microlocal patch of $C_2$, choose \begin{align*} \phi_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{R} \end{align*} so that the corresponding parametrization is \begin{align*} (y,z,\sigma) \mapsto (y,\partial_y\phi_2(y,z,\sigma);z,\partial_z\phi_2(y,z,\sigma)). \end{align*} The amplitudes \begin{align*} a_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{C} \end{align*} and \begin{align*} a_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{C} \end{align*} belong to the symbol classes corresponding to the orders $m_1$ and $m_2$ in the chapter's kernel-order convention. Thus, after microlocal cutoffs, the two kernels are represented as \begin{align*} K_A(x,y)=\int_{\mathbb{R}^{N_1}} e^{i\phi_1(x,y,\theta)} a_1(x,y,\theta) \, d\mathcal{L}^{N_1}(\theta) \end{align*} and \begin{align*} K_B(y,z)=\int_{\mathbb{R}^{N_2}} e^{i\phi_2(y,z,\sigma)} a_2(y,z,\sigma) \, d\mathcal{L}^{N_2}(\sigma). \end{align*} This localization is harmless because the desired conclusion is microlocal, and the statement assumes properness and constant rank on the closed conic subset [lying over](/theorems/2876) $\operatorname{WF}'(A)\times\operatorname{WF}'(B)$. [/guided] [/step] [step:Write the composed kernel with the combined phase] Define the combined phase \begin{align*} \Phi: U_X \times U_Y \times U_Z \times (\mathbb{R}^{N_1} \setminus 0) \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{R} \end{align*} by \begin{align*} \Phi(x,y,z,\theta,\sigma)=\phi_1(x,y,\theta)+\phi_2(y,z,\sigma). \end{align*} Define the product amplitude \begin{align*} a: U_X \times U_Y \times U_Z \times (\mathbb{R}^{N_1} \setminus 0) \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{C} \end{align*} by \begin{align*} a(x,y,z,\theta,\sigma)=a_1(x,y,\theta)a_2(y,z,\sigma). \end{align*} The local expression for the composed kernel is then the regularized oscillatory integral \begin{align*} K_{AB}(x,z)=\operatorname{Os}\!\int_{U_Y}\int_{\mathbb{R}^{N_1}}\int_{\mathbb{R}^{N_2}} e^{i\Phi(x,y,z,\theta,\sigma)} a(x,y,z,\theta,\sigma) \, d\mathcal{L}^{N_2}(\sigma) \, d\mathcal{L}^{N_1}(\theta) \, d\mathcal{L}^{\dim Y}(y), \end{align*} where $\operatorname{Os}\!\int$ denotes the standard limit obtained by inserting a compactly supported cutoff in $(\theta,\sigma)$ which tends to $1$ together with all derivatives on compact conic sets. The support cutoffs and proper support ensure that the $y$-integration is over a compact subset of the coordinate patch for fixed compact subsets of $U_X \times U_Z$, while the regularized oscillatory-integral definition justifies the product and pushforward at the distributional level. [/step] [step:Identify the stationary equations with the negative fibre product] The variables integrated out in the composed kernel are $y$, $\theta$, and $\sigma$. Hence the relevant critical equations are \begin{align*} \partial_\theta\phi_1(x,y,\theta)=0, \end{align*} \begin{align*} \partial_\sigma\phi_2(y,z,\sigma)=0, \end{align*} and \begin{align*} \partial_y\phi_1(x,y,\theta)+\partial_y\phi_2(y,z,\sigma)=0. \end{align*} On the critical set of $\partial_\theta\phi_1$ and $\partial_\sigma\phi_2$, the phase parametrizations give \begin{align*} \xi=\partial_x\phi_1(x,y,\theta), \end{align*} \begin{align*} \eta=\partial_y\phi_1(x,y,\theta), \end{align*} \begin{align*} \eta'=\partial_y\phi_2(y,z,\sigma), \end{align*} and \begin{align*} \zeta=\partial_z\phi_2(y,z,\sigma). \end{align*} Therefore the remaining stationarity equation in $y$ is exactly $\eta+\eta'=0$. Thus the critical set of $\Phi$ in the integration variables maps to \begin{align*} C_1 \times_{T^*Y}^{-} C_2. \end{align*} Conversely, every point of the fibre product lying in the chosen parametrizing patches is represented by such a critical point of $\Phi$. The clean-intersection hypothesis therefore says precisely that this critical set is a clean critical manifold of constant excess $e$. [/step] [step:Apply clean stationary phase along the clean critical manifold] We now apply the clean [stationary phase theorem](/theorems/8198) to the phase $\Phi$ in the integration variables $(y,\theta,\sigma)$. The theorem requires a clean critical manifold and constant normal Hessian rank. These are exactly the phase-level hypotheses of this theorem: the critical set maps diffeomorphically onto the corresponding microlocal piece of $C_1 \times_{T^*Y}^{-} C_2$, and the Hessian of $\Phi$ transverse to that critical manifold has constant rank. Therefore clean stationary phase applies to the regularized oscillatory integral for $K_{AB}$. Its conclusion is that, modulo a smooth kernel, $K_{AB}$ is an oscillatory integral whose phase parametrizes the image of the critical manifold under the map retaining the exterior covectors \begin{align*} (x,y,z,\theta,\sigma) \mapsto (x,\partial_x\phi_1(x,y,\theta);z,\partial_z\phi_2(y,z,\sigma)). \end{align*} Under the identification made in the previous step, this map is exactly $\pi_{XZ}$. The product amplitude $a=a_1a_2$ has the combined symbol order corresponding to $m_1+m_2$ in the chapter's kernel-order convention. By the explicit normalization hypothesis in the theorem statement, clean stationary phase along a clean critical manifold of excess $e$ shifts the resulting kernel order by \begin{align*} \frac{e}{2}. \end{align*} Hence the resulting kernel has order \begin{align*} m_1+m_2+\frac{e}{2}. \end{align*} This is the same rule as in the standard half-density operator-order convention after applying the chapter's stated translation between kernel order and operator order. [guided] The stationary phase step has two separate hypotheses, and both must be checked at the level of the actual phase variables. First, the critical set of $\Phi$ with respect to $(y,\theta,\sigma)$ must be a smooth clean critical manifold. The theorem statement makes this explicit: for each microlocal phase representation, this critical set maps diffeomorphically onto the corresponding piece of \begin{align*} C_1 \times_{T^*Y}^{-} C_2. \end{align*} The equations defining that critical set are $\partial_\theta\phi_1=0$, $\partial_\sigma\phi_2=0$, and $\partial_y\phi_1+\partial_y\phi_2=0$. The first two equations place the two points on $C_1$ and $C_2$, and the last equation is precisely the negative matching condition $\eta+\eta'=0$. Second, clean stationary phase requires constant rank of the Hessian normal to the critical manifold. This is also now a hypothesis of the theorem statement: the normal Hessian of $\Phi$ in the variables $(y,\theta,\sigma)$ has constant rank equal to the number of integration variables minus the fibre dimension determined by the excess $e$. Therefore the clean stationary phase theorem applies to the regularized oscillatory integral defining $K_{AB}$. The theorem then replaces the integration in the normal directions by a new amplitude and leaves the clean critical directions as parameters. The exterior covectors of the resulting oscillatory integral are obtained from the derivatives of the combined phase in the non-integrated variables: \begin{align*} (x,y,z,\theta,\sigma) \mapsto (x,\partial_x\phi_1(x,y,\theta);z,\partial_z\phi_2(y,z,\sigma)). \end{align*} Because the critical set has already been identified with the negative fibre product, this map is exactly \begin{align*} \pi_{XZ}: C_1 \times_{T^*Y}^{-} C_2 \to (T^*X \setminus 0) \times (T^*Z \setminus 0). \end{align*} Thus the new oscillatory integral is associated with the projected canonical relation. Finally we track the order. The product amplitude $a=a_1a_2$ has the combined symbol order corresponding to $m_1+m_2$ in the chapter's kernel-order convention, because symbol orders add under multiplication. The theorem statement explicitly records the chapter normalization used by this composition theorem: clean stationary phase for a clean critical manifold with excess $e$ contributes the order shift \begin{align*} \frac{e}{2}. \end{align*} Therefore the composed kernel has order \begin{align*} m_1+m_2+\frac{e}{2}. \end{align*} This gives the same order formula after translating to the standard half-density operator-order convention. [/guided] [/step] [step:Use properness and constant rank to identify the composed Lagrangian target] The theorem assumes that $\pi_{XZ}$ is proper and has constant rank on the closed conic subset of $C_1 \times_{T^*Y}^{-} C_2$ [lying over](/theorems/2944) $\operatorname{WF}'(A)\times\operatorname{WF}'(B)$. Properness ensures that the pushforward of the local oscillatory representation has no escaping contribution at infinity in the fibre-product variables. Constant rank ensures that the projected image is locally an immersed conic submanifold on the wavefront-relevant set. By the additional hypothesis in the statement, this image \begin{align*} C_1 \circ C_2=\pi_{XZ}(C_1 \times_{T^*Y}^{-} C_2) \end{align*} is an admissible immersed conic Lagrangian target for the class $I^m(X,Z;C_1\circ C_2)$. The local oscillatory integrals produced above are therefore precisely local representatives of an element of \begin{align*} I^{m_1+m_2+e/2}(X,Z;C_1\circ C_2). \end{align*} The microlocal pieces patch by the standard equivalence of phase parametrizations in the definition of Fourier integral distributions. Hence \begin{align*} AB \in I^{m_1+m_2+e/2}(X,Z;C_1\circ C_2). \end{align*} This proves the asserted composition statement in the chapter's kernel-order convention, and the final equivalence with the half-density operator-order convention follows from the same normalization translation used in the chapter's definition of the input classes. [/step]

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