Theorems Clean Composition Theorem for Fourier Integral Operators Attributions

Attributions & Verification

Track contributions and verify content correctness

Statement

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.

paragraph admin

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

paragraph admin

\begin{align*} \phi_1: U_X \times U_Y \times (\mathbb{R}^{N_1} \setminus 0) \to \mathbb{R} \end{align*}

latex_env admin

and

paragraph admin

\begin{align*} \phi_2: U_Y \times U_Z \times (\mathbb{R}^{N_2} \setminus 0) \to \mathbb{R} \end{align*}

latex_env admin

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.

paragraph admin

Define the negative clean fibre product

paragraph admin

\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*}

latex_env admin

Equivalently, $C_1 \times_{T^*Y}^{-} C_2$ is the intersection of $C_1 \times C_2$ with

paragraph admin

\begin{align*} \{(x,\xi;y,\eta;y',\eta';z,\zeta): y=y', \ \eta+\eta'=0\} \end{align*}

latex_env admin

inside

paragraph admin

\begin{align*} (T^*X \setminus 0) \times (T^*Y \setminus 0) \times (T^*Y \setminus 0) \times (T^*Z \setminus 0). \end{align*}

latex_env admin

Assume that this intersection is clean with constant excess $e$, meaning that for every $p \in C_1 \times_{T^*Y}^{-} C_2$,

paragraph admin

\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*}

latex_env admin

and the dimension excess over the transverse intersection value is the same integer $e$.

paragraph admin

Assume moreover that, for the combined local phase

paragraph admin

\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*}

latex_env admin

defined by

paragraph admin

\begin{align*} \Phi(x,y,z,\theta,\sigma)=\phi_1(x,y,\theta)+\phi_2(y,z,\sigma), \end{align*}

latex_env admin

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

paragraph admin

\begin{align*} \frac{e}{2}. \end{align*}

latex_env admin

Let

paragraph admin

\begin{align*} \pi_{XZ}: C_1 \times_{T^*Y}^{-} C_2 &\to (T^*X \setminus 0) \times (T^*Z \setminus 0) \end{align*}

latex_env admin

be the map

paragraph admin

\begin{align*} (x,\xi;y,\eta;y,-\eta;z,\zeta) &\mapsto (x,\xi;z,\zeta). \end{align*}

latex_env admin

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

paragraph admin

\begin{align*} C_1 \circ C_2 := \pi_{XZ}(C_1 \times_{T^*Y}^{-} C_2) \end{align*}

latex_env admin

is an admissible immersed conic Lagrangian target for the class $I^m(X,Z;C_1 \circ C_2)$ in the chapter's definition.

paragraph admin

Then the composition $AB$ is a well-defined properly supported Fourier integral operator from $Z$ to $X$, and

paragraph admin

\begin{align*} AB \in I^{m_1+m_2+e/2}(X,Z;C_1 \circ C_2) \end{align*}

latex_env admin

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.

paragraph admin

Verification Progress

31 Total Blocks

0 Verified

0% verified

Contributors

admin 31 blocks (0 verified)

Who Can Verify

Areas: Analysis

Viktor Miykov Admin

Max Vassiliev Global Reviewer

Horia Neagu Global Reviewer

강현욱 Global Reviewer

Demo Testing Global Reviewer

Archie Pennycook Global Reviewer

Quick Actions

Edit Theorem

Raw Attribution Data

Loading attribution data...

What brings you to Androma?

Start with a route through the knowledge graph.

Attributions & Verification

Statement

Verification Progress

Contributors

Who Can Verify

Quick Actions

Sign in to Androma

Check your inbox

One last step

Attributions & Verification

Statement

Verification Progress

Contributors

Who Can Verify

Quick Actions

Raw Attribution Data