Analytic wavefront set by Gaussian FBI packets

Let $u$ be a distribution, not necessarily compactly supported, or a distributional density (a continuous functional on compactly supported smooth scalar functions), on a real-analytic $n$-manifold $X$. Let $x_0$ lie in an analytic coordinate chart and let ${\xi }_0\in T_{x_0}^{\ast }X\setminus \{0\}$ have coordinate components in that chart. For $\chi \in C_c^{\mathrm{\infty }}$ supported in the chart and equal to $1$ near $x_0$, define the Gaussian FBI packet

$${\mathcal{F}}_{\chi }u(x,\lambda \omega )=⟨u(y),\chi (y)\exp \text{\hspace{0.17em}⁣}(i\lambda (x-y)\cdot \omega -\frac{\lambda }{2}\mathrm{\mid }x-y{\mathrm{\mid }}^2)⟩,\lambda \ge 1,\mathrm{\mid }\omega \mathrm{\mid }=1.$$

When $u$ is a scalar distribution, the pairing is with the displayed function times coordinate density $\mathrm{\mid }dy\mathrm{\mid }$; when $u$ is a distributional density, the pairing is with the displayed scalar function. In both cases $\lambda \omega$ is the cotangent frequency. The covector $(x_0,{\xi }_0)$ is analytically regular for $u$ in this chart if there are such a cutoff $\chi$, a neighborhood $U$ of $x_0$, an open neighborhood $\mathrm{\Gamma }$ of ${\xi }_0\mathrm{/}\mathrm{\mid }{\xi }_0\mathrm{\mid }$ in the unit sphere, and constants $c,C>0$ such that

$$\mathrm{\mid }{\mathcal{F}}_{\chi }u(x,\lambda \omega )\mathrm{\mid }\le C{e}^{-c\lambda }(x\in U,\omega \in \mathrm{\Gamma },\lambda \ge 1)\mathrm{.}$$

The analytic wavefront set ${\mathrm{WF}}_A(u)$ is the set of nonzero cotangent vectors for which this exponential estimate fails. This is a chart-local definition. The separate local FBI-invariance claim needed for a geometric interpretation is stated by the companion lemma.

Distribution and density pairing

Analytic coordinate charts

Power-series convention