Microlocal Parametrix Theorem for Elliptic Fourier Integral Operators

Theorem

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Let $X$ and $Y$ be smooth manifolds, let $\Omega_X^{1/2}$ and $\Omega_Y^{1/2}$ denote their half-density bundles, let $m \in \mathbb{R}$, and let $\Gamma_Y \subset T^*Y \setminus 0$ and $\Gamma_X \subset T^*X \setminus 0$ be conic open sets. Let \begin{align*} \kappa: \Gamma_Y \to \Gamma_X \end{align*} be a homogeneous canonical diffeomorphism, and let $C \subset (T^*X \setminus 0) \times (T^*Y \setminus 0)$ be the graph canonical relation \begin{align*} C = \{(\kappa(q), q) : q \in \Gamma_Y\}. \end{align*} Let $C^t \subset (T^*Y \setminus 0) \times (T^*X \setminus 0)$ denote the transpose relation \begin{align*} C^t = \{(q, \kappa(q)) : q \in \Gamma_Y\}. \end{align*} Fix $q_0 = (y_0,\eta_0) \in \Gamma_Y$, and write $p_0 := \kappa(q_0) \in \Gamma_X$. Let \begin{align*} A: C_c^\infty(Y;\Omega_Y^{1/2}) \to \mathcal{D}'(X;\Omega_X^{1/2}) \end{align*} be a properly supported microlocal Fourier integral operator $A \in I^m(X,Y;C)$ acting on half-densities near $(p_0,q_0)$. Suppose that $A$ is elliptic at $(p_0,q_0) \in C$, meaning that its principal symbol $\sigma_m(A)(p_0,q_0)$ is nonzero and hence invertible in the Maslov and half-density symbol line $L_C$ over $C$ under the standard half-density principal-symbol pairing for graph canonical relations. Use the convention that $\Psi^0(M)$ denotes the properly supported order-$0$ pseudodifferential operators on half-densities over a smooth manifold $M$, and that an operator is smoothing microlocally on a conic set $V \subset T^*M \setminus 0$ when, after applying pseudodifferential cutoffs elliptic on $V$, its Schwartz kernel is smooth in the corresponding microlocal region. Then, after inserting microlocal cutoffs supported in sufficiently small conic neighbourhoods of $q_0$ and $p_0$ so that the relevant compositions are properly supported in the microlocal region under consideration, there exist conic neighbourhoods $V_Y \subset \Gamma_Y$ of $q_0$ and $V_X \subset \Gamma_X$ of $p_0$, with $\kappa(V_Y) = V_X$, and a Fourier integral operator \begin{align*} B: C_c^\infty(X;\Omega_X^{1/2}) \to \mathcal{D}'(Y;\Omega_Y^{1/2}) \end{align*} such that $B \in I^{-m}(Y,X;C^t)$ and \begin{align*} BA - I_Y \end{align*} is smoothing microlocally on $V_Y$, while \begin{align*} AB - I_X \end{align*} is smoothing microlocally on $V_X$. Equivalently, $BA = I_Y$ microlocally near $q_0$ and $AB = I_X$ microlocally near $p_0$.

Discussion

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Proof

[proofplan] The graph hypothesis turns the compositions $C^t \circ C$ and $C \circ C^t$ into diagonal canonical relations near the chosen covectors, so compositions of an operator on $C^t$ with $A$ become pseudodifferential operators. We first choose an initial inverse $B_0$ whose principal symbol is the inverse of the elliptic principal symbol of $A$. Then $B_0A$ and $AB_0$ are elliptic pseudodifferential operators with principal symbol $1$, so the pseudodifferential parametrix construction corrects them to the identity modulo microlocally smoothing errors. A short two-sidedness argument shows that the left and right corrected inverses differ only by a microlocally smoothing operator, giving a single Fourier integral operator $B$ with both inverse properties. [/proofplan] [step:Shrink to conic neighbourhoods where the canonical relation is a graph] Let $q_0 = (y_0,\eta_0) \in \Gamma_Y$ and $p_0 = \kappa(q_0) \in \Gamma_X$. Since $\kappa:\Gamma_Y \to \Gamma_X$ is a homogeneous canonical diffeomorphism, after replacing $\Gamma_Y$ and $\Gamma_X$ by smaller conic neighbourhoods $V_Y$ of $q_0$ and $V_X$ of $p_0$, we may assume that \begin{align*} \kappa: V_Y \to V_X \end{align*} is a homogeneous canonical diffeomorphism and that $C$ is exactly the graph of this map over $V_Y$. The transpose canonical relation is \begin{align*} C^t = \{(q,\kappa(q)) : q \in V_Y\} \subset (T^*Y \setminus 0) \times (T^*X \setminus 0). \end{align*} Because $\kappa$ is one-to-one on $V_Y$, the relational composition $C^t \circ C$ is the diagonal relation over $V_Y$: \begin{align*} C^t \circ C = \{(q,q) : q \in V_Y\}. \end{align*} Similarly, because $\kappa$ maps $V_Y$ onto $V_X$, the relational composition $C \circ C^t$ is the diagonal relation over $V_X$: \begin{align*} C \circ C^t = \{(p,p) : p \in V_X\}. \end{align*} Choose properly supported microlocal cutoffs whose wavefront supports lie in smaller conic neighbourhoods of $V_Y$ and $V_X$ and are equal to $1$ near $q_0$ and $p_0$. After inserting these cutoffs, all compositions below are properly supported in the microlocal regions under consideration. Thus, microlocally on these neighbourhoods, composing an operator in $I^{-m}(Y,X;C^t)$ with $A \in I^m(X,Y;C)$ gives a pseudodifferential operator on $Y$, and composing in the opposite order gives a pseudodifferential operator on $X$. This is the clean FIO composition theorem for properly supported Fourier integral operators whose canonical relations compose transversely; the hypotheses are satisfied here because both relations are graphs of the diffeomorphism $\kappa$ and its inverse. [/step] [step:Choose an initial inverse symbol on the transpose canonical relation] Let \begin{align*} a_0 := \sigma_m(A)(p_0,q_0) \end{align*} denote the principal symbol of $A$ at $(p_0,q_0)$ in the Maslov and half-density symbol line of $C$. By ellipticity, $a_0$ is invertible in that one-dimensional symbol line. After shrinking $V_Y$ and $V_X$ again if necessary, the principal symbol \begin{align*} \sigma_m(A): C \cap (V_X \times V_Y) \to L_C \end{align*} is nowhere zero, where $L_C$ denotes the relevant Maslov and half-density symbol line over $C$. Define a principal symbol \begin{align*} b_{0,-m}: C^t \cap (V_Y \times V_X) \to L_{C^t} \end{align*} as the dual inverse of $\sigma_m(A)$ under the half-density and Maslov pairing for graph canonical relations. This means that the principal-symbol composition pairing of $b_{0,-m}$ with $\sigma_m(A)$ gives the unit half-density symbol on $\Delta_{V_Y}$ for $C^t \circ C$, and the pairing of $\sigma_m(A)$ with $b_{0,-m}$ gives the unit half-density symbol on $\Delta_{V_X}$ for $C \circ C^t$. By the local existence theorem for Fourier integral operators with prescribed principal symbol on a conic graph canonical relation, whose hypotheses here are the smooth conic graph relation $C^t$, the chosen Maslov and half-density symbol line $L_{C^t}$, and the smooth homogeneous symbol $b_{0,-m}$, there exists \begin{align*} B_0: C_c^\infty(X;\Omega_X^{1/2}) \to \mathcal{D}'(Y;\Omega_Y^{1/2}) \end{align*} with $B_0 \in I^{-m}(Y,X;C^t)$ and principal symbol $\sigma_{-m}(B_0) = b_{0,-m}$ microlocally near $(q_0,p_0)$. We choose $B_0$ with the same properly supported microlocal cutoff convention fixed above, so that the later compositions with $A$ are defined in the indicated microlocal regions. [guided] The purpose of $B_0$ is to invert only the top-order part of $A$. The symbol $\sigma_m(A)$ lives in a complex line determined by the Maslov convention and the half-density factors. Ellipticity at $(p_0,q_0)$ means precisely that this symbol is nonzero, hence invertible in that line after choosing the dual line over the transpose relation. We shrink the conic neighbourhoods so that this nonvanishing persists on all of $C \cap (V_X \times V_Y)$. This is allowed because ellipticity is an open microlocal condition: the principal symbol is smooth on the conic relation and is nonzero at the chosen point. We then define \begin{align*} b_{0,-m}: C^t \cap (V_Y \times V_X) \to L_{C^t} \end{align*} to be the dual symbol satisfying the following condition: when $b_{0,-m}$ is paired with $\sigma_m(A)$ under the principal-symbol composition formula for $C^t \circ C$, the resulting principal symbol on the diagonal $\Delta_{V_Y}$ is $1$. The local existence theorem for FIOs with prescribed principal symbol on a graph canonical relation gives an operator \begin{align*} B_0: C_c^\infty(X;\Omega_X^{1/2}) \to \mathcal{D}'(Y;\Omega_Y^{1/2}) \end{align*} with $B_0 \in I^{-m}(Y,X;C^t)$ and $\sigma_{-m}(B_0)=b_{0,-m}$ near $(q_0,p_0)$. At this stage $B_0$ is not yet an inverse to $A$; it only makes the leading symbol of $B_0A$ equal to the leading symbol of the identity. [/guided] [/step] [step:Compute the leading symbols of the two initial compositions] By the FIO composition theorem for graph canonical relations, the composition \begin{align*} P_Y := B_0A \end{align*} is a pseudodifferential operator of order $0$ on $Y$ microlocally near $q_0$, because its canonical relation is $C^t \circ C = \Delta_{V_Y}$. The principal-symbol composition formula gives \begin{align*} \sigma_0(P_Y)(q) = b_{0,-m}(q,\kappa(q)) \, \sigma_m(A)(\kappa(q),q) \end{align*} for $q \in V_Y$, with the product interpreted through the half-density and Maslov pairing. By the definition of $b_{0,-m}$, this principal symbol is $1$ on $V_Y$. Similarly, \begin{align*} P_X := AB_0 \end{align*} is a pseudodifferential operator of order $0$ on $X$ microlocally near $p_0$, because its canonical relation is $C \circ C^t = \Delta_{V_X}$. For $p \in V_X$, write $q = \kappa^{-1}(p) \in V_Y$. The principal-symbol composition formula gives \begin{align*} \sigma_0(P_X)(p) = \sigma_m(A)(p,q)\, b_{0,-m}(q,p), \end{align*} again with the product interpreted through the half-density and Maslov pairing. The symbol line over $C^t$ was chosen as the dual symbol line to the symbol line over $C$ under this pairing. Since the scalar half-density pairing is commutative after identifying the two diagonal half-density factors with the identity half-density on $\Delta_{V_X}$ and $\Delta_{V_Y}$, the same inverse element $b_{0,-m}$ gives the value $1$ for both $C^t \circ C$ and $C \circ C^t$. Hence $\sigma_0(P_X)=1$ on $V_X$, and $P_Y$ and $P_X$ are elliptic order-$0$ pseudodifferential operators with principal symbol $1$ microlocally near $q_0$ and $p_0$, respectively. [/step] [step:Construct pseudodifferential parametrices for the two elliptic compositions] Let $\Psi^0(Y)$ denote the class of properly supported order-$0$ pseudodifferential operators on half-densities over $Y$. Since $P_Y$ is an elliptic pseudodifferential operator of order $0$ with principal symbol $1$ on $V_Y$, the elliptic pseudodifferential parametrix construction gives an operator \begin{align*} Q_Y \in \Psi^0(Y) \end{align*} such that \begin{align*} Q_Y P_Y - I_Y \end{align*} is smoothing microlocally on $V_Y$. This uses the usual asymptotic symbol expansion: if the full symbol of $P_Y$ is $1+r_{-1}+r_{-2}+\cdots$, one chooses the full symbol of $Q_Y$ recursively as $1+q_{-1}+q_{-2}+\cdots$ so that the product symbol is $1$ to all orders. Let $\Psi^0(X)$ denote the class of properly supported order-$0$ pseudodifferential operators on half-densities over $X$. Likewise, since $P_X$ is an elliptic pseudodifferential operator of order $0$ with principal symbol $1$ on $V_X$, the microlocal parametrix theorem for elliptic pseudodifferential operators gives \begin{align*} Q_X \in \Psi^0(X) \end{align*} such that \begin{align*} P_X Q_X - I_X \end{align*} is smoothing microlocally on $V_X$. The theorem applies because $P_Y$ and $P_X$ are properly supported microlocally by the cutoff convention above and elliptic on the conic sets $V_Y$ and $V_X$. [/step] [step:Form left and right FIO parametrices] Define \begin{align*} B_L := Q_YB_0. \end{align*} Because $Q_Y \in \Psi^0(Y)$ has diagonal canonical relation on $T^*Y \setminus 0$ and $B_0 \in I^{-m}(Y,X;C^t)$, the FIO composition theorem gives \begin{align*} B_L \in I^{-m}(Y,X;C^t). \end{align*} Moreover, \begin{align*} B_LA - I_Y = Q_YB_0A - I_Y = Q_YP_Y - I_Y, \end{align*} so $B_LA-I_Y$ is smoothing microlocally on $V_Y$. Define \begin{align*} B_R := B_0Q_X. \end{align*} Since $B_0 \in I^{-m}(Y,X;C^t)$ and $Q_X \in \Psi^0(X)$ has diagonal canonical relation on $T^*X \setminus 0$, again \begin{align*} B_R \in I^{-m}(Y,X;C^t). \end{align*} Also, \begin{align*} AB_R - I_X = AB_0Q_X - I_X = P_XQ_X - I_X, \end{align*} so $AB_R-I_X$ is smoothing microlocally on $V_X$. [/step] [step:Show the left and right parametrices agree modulo smoothing errors] We prove that $B_L-B_R$ is smoothing microlocally from $V_X$ to $V_Y$. Since $AB_R-I_X$ is smoothing microlocally on $V_X$, composition on the left by the properly supported microlocal FIO $B_L$ gives \begin{align*} B_L(AB_R)-B_L = B_L(AB_R-I_X) \end{align*} smoothing microlocally from $V_X$ to $V_Y$. Since $B_LA-I_Y$ is smoothing microlocally on $V_Y$, composition on the right by the properly supported microlocal FIO $B_R$ gives \begin{align*} (B_LA)B_R-B_R = (B_LA-I_Y)B_R \end{align*} smoothing microlocally from $V_X$ to $V_Y$. Associativity of properly supported microlocal composition gives \begin{align*} B_L(AB_R) = (B_LA)B_R \end{align*} modulo smoothing operators in the same microlocal region. Subtracting the first smoothing relation from the second, using this associativity identity, yields \begin{align*} B_L-B_R = \bigl[(B_LA)B_R-B_R\bigr] - \bigl[B_L(AB_R)-B_L\bigr] \end{align*} modulo a smoothing operator. Therefore $B_L-B_R$ is smoothing microlocally from $V_X$ to $V_Y$. [/step] [step:Choose a single microlocal inverse] Set \begin{align*} B := B_L. \end{align*} Then $B \in I^{-m}(Y,X;C^t)$ and, from the construction of $B_L$, \begin{align*} BA-I_Y \end{align*} is smoothing microlocally on $V_Y$. Since $B-B_R$ is smoothing microlocally from $V_X$ to $V_Y$ and $AB_R-I_X$ is smoothing microlocally on $V_X$, we have \begin{align*} AB-I_X = A(B-B_R) + (AB_R-I_X) \end{align*} smoothing microlocally on $V_X$. Thus \begin{align*} BA = I_Y \end{align*} microlocally near $q_0=(y_0,\eta_0)$ and \begin{align*} AB = I_X \end{align*} microlocally near $p_0=\kappa(y_0,\eta_0)$. This is the desired two-sided microlocal inverse. [/step]

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