Wavefront sets are invariant under smooth coordinate changes

In a chart, localize $u$ by a cutoff $\chi$ and write $\widehat{\chi u}(\xi )=⟨u,\chi (x)e^{-2\pi ix\cdot \xi }\mathrm{\mid }dx\mathrm{\mid }⟩$. The finite-order bound [F3] gives polynomial growth in $\xi$ by differentiating the test at most its fixed order; parameter derivatives multiply it by powers of $x$, which remain bounded on the fixed compact support. Difference quotients converge in each fixed-support test seminorm by the Taylor remainder bound, so pairing shows that $\widehat{\chi u}$ is smooth. This is also the localized Fourier transform in [F4].

Suppose $(p,{\xi }_0)$ is regular for $u$ in the $x$ chart, witnessed by a cutoff ${\chi }_x$ and an open cone ${\mathrm{\Gamma }}_x$. Let $y=\psi (x)$ be a smooth coordinate change, restrict to a compact patch about $p$, and put ${\eta }_0=(D\psi (p){)}^{-T}{\xi }_0$. Shrink the patch and choose nested cones ${\mathrm{\Gamma }}_0⋐{\mathrm{\Gamma }}_1⋐{\mathrm{\Gamma }}_x$ around ${\xi }_0$ and a target cone $V$ around ${\eta }_0$ so that $D\psi (x{)}^{T}V\subset {\mathrm{\Gamma }}_0$ throughout the patch. Choose a target cutoff ${\chi }_y$ equal to one near $p$ and supported where ${\chi }_x=1$, using [F8]. The target localized Fourier value is the source distribution paired with $a_{\eta }(x)={\chi }_y(\psi (x))\mathrm{\mid }\det D\psi (x)\mathrm{\mid }{e}^{-2\pi i\psi (x)\cdot \eta }$; because ${\chi }_x=1$ on its support this is also the pairing with ${\chi }_{x}u$. Fourier inversion [F5], passed through this compactly supported distribution by [F6], expresses the value as $\int \widehat{{\chi }_{x}u}(\xi )\widehat{a_{\eta }}(-\xi )d\xi$. The phase in $\widehat{a_{\eta }}(-\xi )$ is $2\pi (x\cdot \xi -\psi (x)\cdot \eta )$, with gradient $2\pi (\xi -D\psi (x{)}^T\eta )$. In the near-stationary region $\mathrm{\mid }\xi -D\psi (x{)}^T\eta \mathrm{\mid }\le c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$, for sufficiently small $c>0$, cone separation gives $\xi \in {\mathrm{\Gamma }}_x$ and $\mathrm{\mid }\xi \mathrm{\mid }\asymp \mathrm{\mid }\eta \mathrm{\mid }$; there $\widehat{{\chi }_{x}u}(\xi )$ decreases faster than every power, while $\mathrm{\mid }\widehat{a_{\eta }}(-\xi )\mathrm{\mid }\le \mathrm{\parallel }{a}_{\eta }{\mathrm{\parallel }}_1$ is uniformly bounded, and the region has only polynomial volume in $\mathrm{\mid }\eta \mathrm{\mid }$. On the complementary region the phase gradient is bounded below by $c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$; repeated integration by parts gives arbitrary powers of joint decay for $\widehat{a_{\eta }}(-\xi )$. The polynomial bound [F3] for $\widehat{{\chi }_{x}u}$ then makes this part rapidly decreasing after integration. This proves regularity in the target chart.

Applying step 2.1 also to ${\psi }^{-1}$ proves equivalence of regularity in both charts, and [F7] gives exactly $\eta =(D\psi {)}^{-T}\xi$. Regularity is open because the same cutoff and a smaller cone work for nearby base points; positive rescaling of a covector does not change rapid decay. Hence its complement is closed and conic.

Suppose ${\chi }_1$ is a regularity witness for $u$ on an open cone $\mathrm{\Gamma }$, and let $\rho \in C_c^{\mathrm{\infty }}$ have support where ${\chi }_1=1$. If ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ is an open cone whose closure of directions on the unit sphere is contained in $\mathrm{\Gamma }$, then $\widehat{\rho u}$ is rapidly decreasing on ${\mathrm{\Gamma }}^{\mathrm{\prime }}$. Indeed, $\rho u=\rho ({\chi }_{1}u)$ and Fourier inversion gives $\widehat{\rho u}=\widehat{\rho }\ast \widehat{{\chi }_{1}u}$; passing the Schwartz integral through the compactly supported distribution uses [F6]. Repeated integration by parts makes $\widehat{\rho }$ rapidly decreasing, and [F3] bounds $\widehat{{\chi }_{1}u}$ polynomially. Choose an intermediate cone with closure of directions between those of ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ and $\mathrm{\Gamma }$. In the convolution, for frequencies $\eta$ in this intermediate cone with $\mathrm{\mid }\eta \mathrm{\mid }\ge \mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, the second factor is rapidly decreasing at frequency comparable to $\mathrm{\mid }\xi \mathrm{\mid }$. If $\mathrm{\mid }\eta \mathrm{\mid }<\mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, then $\mathrm{\mid }\xi -\eta \mathrm{\mid }>\mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$; for the remaining frequencies, $\eta$ lies outside the intermediate cone and $\mathrm{\mid }\eta \mathrm{\mid }\ge \mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, so angular separation gives $\mathrm{\mid }\xi -\eta \mathrm{\mid }\ge c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$ for some $c>0$. In these latter regions the first factor is rapidly decreasing, while [F3] bounds the other factor polynomially. These estimates give the arbitrary-power bound for $\mathrm{\mid }\xi \mathrm{\mid }\ge 1$ on ${\mathrm{\Gamma }}^{\mathrm{\prime }}$, and [F4] gives boundedness on bounded sets. The convolution is absolutely integrable by the same estimates, and [F9] justifies its Fourier-integral identity. Applying this cutoff transfer to a common $\rho$ for witnesses for $u$ and $v$, and intersecting their smaller cones, gives rapid decay for $\widehat{\rho (u+v)}=\widehat{\rho u}+\widehat{\rho v}$. This proves $\mathrm{WF}(u+v)\subseteq \mathrm{WF}(u)\cup \mathrm{WF}(v)$. For a diffeomorphism, step 2.1 gives the cotangent image inclusion for pullback; applying the same argument to ${\psi }^{-1}$ gives the reverse inclusion.

If $u$ is smooth, integration by parts in a compact chart cutoff gives rapid decay in every cone. Conversely, suppose $\mathrm{WF}(u)=\mathrm{\varnothing }$ and fix a point $p$ in a chart. In dimension zero every distribution is locally a scalar function, so assume the chart dimension is at least one. Form the set of all open cones ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ for which there exist a regularity witness $(\chi ,\mathrm{\Gamma })$ at $p$ with the closure of the directions of ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ on the unit sphere contained in $\mathrm{\Gamma }$. This set covers the unit covector sphere: every direction has a witness by emptiness of the local wavefront set, and its open cone contains a smaller cone about that direction. Compactness gives a finite subcover ${\mathrm{\Gamma }}_1^{\mathrm{\prime }},\dots ,{\mathrm{\Gamma }}_m^{\mathrm{\prime }}$. For each member of this finite family, take a corresponding witness ${\chi }_j,{\mathrm{\Gamma }}_j$; finite choice suffices. Their witness cutoffs all equal one on a neighborhood of $p$. By [F8], choose one cutoff $\rho$ equal to one near $p$ and supported in the intersection of those neighborhoods. Step 3.2 shows that the single localization $\rho u$ has rapidly decreasing Fourier transform on each ${\mathrm{\Gamma }}_j^{\mathrm{\prime }}$. Since these finitely many cones cover the unit sphere, $\widehat{\rho u}$ is rapidly decreasing in every direction: for each $N$, take the maximum of the finitely many cone bounds outside a bounded ball. Therefore every differentiated inverse Fourier integral is absolutely convergent and defines a smooth function. Fourier inversion [F5] and the pairing interchange [F6], applied after a cutoff equal to one near $\mathrm{supp}(\rho u)$, identify this function with the compactly supported distribution $\rho u$. Hence $u$ is smooth near $p$; as $p$ was arbitrary, it is smooth in every chart.