Analytic incidence localization away from a data cotangent fiber

If $K=\mathrm{\varnothing }$, then $u=0$ and the conclusion is immediate; assume $K\mathrm{\ne }\mathrm{\varnothing }$. Fix a relatively compact data chart $V_0$ about $z_0$ and a compact object neighborhood $K_2$ with $K\subset \mathrm{int}{K}_2$. By [F1], the incidence portion over ${\bar{V}}_0$ and $K_2$ is compact. The portion of $C_Z$ over it with unit data covector is compact as well. Indeed, at each incidence point the projection of its conormal fiber $(\eta ,-\xi )\mapsto \eta$ is injective: if $\eta =0$, then $-\xi$ annihilates the image of $d{\pi }_X$, which is all of $T_{x}X$ by [F1], so $\xi =0$. On the compact incidence portion these injective linear maps have a uniform lower bound in fixed local metrics. Thus $\mathrm{\mid }\eta \mathrm{\mid }=1$ bounds $\xi$, and closedness of the normalized relation gives compactness. Write ${\omega }_0={\zeta }_0\mathrm{/}\mathrm{\mid }{\zeta }_0\mathrm{\mid }$ in the data chart. If no product neighborhood of $(z_0,{\omega }_0,K)$ were disjoint from the normalized relation, there would be relation points with data bases tending to $z_0$, unit data covectors tending to ${\omega }_0$, and object bases tending to $K$. Compactness would give a limiting relation point over $(z_0,{\omega }_0)$ and some $x\in K$, contradicting the hypothesis after rescaling by $\mathrm{\mid }{\zeta }_0\mathrm{\mid }$. Hence there are neighborhoods $V⋐V_0$, $\mathrm{\Gamma }\subset S^{\dim G-1}$ of ${\omega }_0$, and an open neighborhood $O$ of $K$, with $\bar{O}\subset \mathrm{int}{K}_2$, such that no relation point lies over $(q,\omega ,x)$ for $q\in V$, $\omega \in \mathrm{\Gamma }$, and $x\in O$. Choose a compact neighborhood $K_1$ with $K\subset \mathrm{int}{K}_1\subset K_1\subset O$. The incidence portion over $\bar{V}$ and $K_1$ is compact by [F1]. Use [F5] to cover it by finitely many analytic fiber charts for the projection to $X$, written $(t,x)\mapsto (z(t,x),x)$. In these coordinates the transpose data packet has integrals of the form $\int e^{i\lambda \mathrm{\Psi }(t,x;q,\omega )}b(t,x)\chi (z(t,x))dt,\mathrm{\Psi }=(q-z(t,x))\cdot \omega +\frac{i}{2}\mathrm{\mid }q-z(t,x){\mathrm{\mid }}^2,$ where $b$ is analytic and $\chi$ is a data cutoff. At incidence points $z(t,x)=q$, ${\mathrm{\partial }}_t\mathrm{\Psi }=-D_t{z}^T\omega$. If this vanished in every fiber direction, then $\omega$ would annihilate the vertical tangent of ${\pi }_X$ and extend to a nonzero conormal $(\omega ,-\xi )$ of $Z$; the submersion hypotheses ensure both components are nonzero. This would be a relation point over $(q,\omega ,x)$, excluded by the neighborhoods just chosen.

On the compact incidence portion over $\bar{V}$ and $K_1$, the no-relation property from step 1.1 gives a positive lower bound for ${\max }_j\mathrm{\mid }{\mathrm{\partial }}_{t_j}\mathrm{\Psi }\mathrm{\mid }$ when $z=q$ and $(q,\omega )$ ranges over smaller compactly contained neighborhoods of $(z_0,{\omega }_0)$. By continuity, shrink the data chart, direction neighborhood and fiber charts so this lower bound persists whenever $\mathrm{\mid }z-q\mathrm{\mid }$ is sufficiently small. A finite refinement therefore gives the uniform alternative on each piece, for $x$ in a neighborhood of $K$ and $(q,\omega )$ near the supplied data point: either $\mathrm{\mid }z-q\mathrm{\mid }\ge d>0$, or one fiber coordinate satisfies $\mathrm{\mid }{\mathrm{\partial }}_{t_j}\mathrm{\Psi }\mathrm{\mid }\ge d>0$. The first pieces have Gaussian exponential decay in every fixed object derivative. For the second pieces shrink the analytic neighborhoods so $1\mathrm{/}{\mathrm{\partial }}_{t_j}\mathrm{\Psi }$ is holomorphic and has uniform factorial derivative bounds by [F4]. The data cutoff can be chosen equal to one on the central data neighborhood. Its transition contributes only pieces of the first type, so it is never differentiated to an order growing with $\lambda$ in the second estimate.

For integration by parts choose finite-order localizers in the finitely many refined fiber charts. They can form an exact partition near the incidence compact without reciprocal derivatives: if ${\chi }_{j,N}$ are the cutoffs in [F3], take ${\rho }_{j,N}={\chi }_{j,N}{\prod }_{\mathrm{\ell }<j}(1-{\chi }_{\mathrm{\ell },N})$. At every point of the compact at least one ${\chi }_{j,N}$ is identically one on a fixed neighborhood. Hence ${\sum }_j{\rho }_{j,N}=1$ there. There are a fixed finite number of factors; the product rule gives $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\rho }_{j,N}\mathrm{\mid }\le C(CN{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le N$. Extend the construction to a fixed slightly larger compact in the incidence charts, so all object derivatives through any fixed order $m$ preserve the exact partition identity near $K$. Use the sequence indexed by $N+m$.

On a nonstationary piece use $L=(i\lambda {\mathrm{\partial }}_{t_j}\mathrm{\Psi }{)}^{-1}{\mathrm{\partial }}_{t_j}$, so $L{e}^{i\lambda \mathrm{\Psi }}=e^{i\lambda \mathrm{\Psi }}$. There are no boundary terms because the localizers have compact support in the chart. After $N$ integrations by parts the integrand is $e^{i\lambda \mathrm{\Psi }}(L^{\ast }{)}^N(b{\rho }_{j,N+m})$. The factorial coefficient bounds and the product rule give $\mathrm{\parallel }(L^{\ast }{)}^N(b{\rho }_{j,N+m}){\mathrm{\parallel }}_{\mathrm{\infty }}\le C^{N+1}{\lambda }^{-N}{\sum }_{k=0}^N\frac{N!}{k!}(C(N+m){)}^k\le C_m(C_{m}N\mathrm{/}\lambda {)}^N$ for $N\ge m+1$. To obtain the first bound, expand each adjoint as a first-order derivative plus its differentiated analytic coefficient; induction distributes $k$ derivatives to the localizer and bounds the remaining analytic derivatives by $C^{N+1}N!\mathrm{/}k!$. In the second bound use $N!\le N^N$ and ${\sum }_{k\le N}(C(N+m){)}^k\mathrm{/}k!\le e^{C(N+m)}$. The bounded integration volumes and $\mathrm{\mid }{e}^{i\lambda \mathrm{\Psi }}\mathrm{\mid }\le 1$ leave the same bound for the integral. Applying $m$ object derivatives first adds at most a fixed power of $\lambda$ and $N$; the same induction with $N+m$ gives $C_m{\lambda }^{M_m}{N}^{M_m}(C_{m}N\mathrm{/}\lambda {)}^N$. Fix this $m$ and take $N=\lfloor {ϵ}_m\lambda \rfloor$ with $C_m{ϵ}_m<1\mathrm{/}2$. The result is $O(e^{-c_m\lambda })$.

The transpose packet coefficient is independent of $N$ because the partition is exact near the support. Summing the finitely many estimates in steps 2.1 and 4.1 proves the asserted $C^m$ estimate on a fixed neighborhood of $K$. Take $m$ to be the finite distributional order in [F2] and pair with $u$. It follows that the data Gaussian FBI pairing of $Ru$ is uniformly exponentially small. The definition of analytic wavefront gives the conclusion. This proof uses neither an unproved wavefront pushforward theorem nor a compact analytic bump.