[proofplan]
We prove the inclusion by working in a coordinate chart and using the local Fourier-transform definition of the wave front set. If a covector is absent from $\operatorname{WF}(u)$, then after localizing $u$ by a cutoff, its [Fourier transform](/page/Fourier%20Transform) decays rapidly in a conic neighbourhood of that covector. Multiplication by a smooth function becomes multiplication by a compactly supported smooth function after localization, and the Fourier transform of the product is convolution with a Schwartz function. A cone-shrinking convolution estimate preserves rapid decay, so the same covector is absent from $\operatorname{WF}(a u)$. If $a$ is nonzero on an [open set](/page/Open%20Set), then $1/a$ is smooth there and applying the inclusion to $u = (1/a)(a u)$ gives the reverse inclusion over that open set.
[/proofplan]
[step:Localize near a covector outside $\operatorname{WF}(u)$]
Let $(x_0,\xi_0) \in T^*M \setminus 0$ satisfy $(x_0,\xi_0) \notin \operatorname{WF}(u)$. Choose a coordinate chart
\begin{align*}
\kappa: U \to \Omega \subset \mathbb{R}^n
\end{align*}
with $x_0 \in U$. Let $\eta_0 \in \mathbb{R}^n \setminus \{0\}$ denote the coordinate representation of $\xi_0$ under the cotangent pullback map
\begin{align*}
d\kappa_{x_0}^{-\top}: T_{x_0}^*M \to \mathbb{R}^n.
\end{align*}
By the local definition of the wave front set, there exist a cutoff function $\psi \in C_c^\infty(U)$ with $\psi = 1$ on an open neighbourhood $U_0 \subset U$ of $x_0$, and an open conic neighbourhood $\Gamma_0 \subset \mathbb{R}^n \setminus \{0\}$ of $\eta_0$, such that the compactly supported distribution
\begin{align*}
v := (\kappa^{-1})^*(\psi u) \in \mathcal{E}'(\Omega)
\end{align*}
has rapidly decreasing Fourier transform on $\Gamma_0$: for every integer $N \ge 0$ there is a constant $C_N > 0$ such that
\begin{align*}
|\widehat v(\eta)| \le C_N(1+|\eta|)^{-N}
\end{align*}
for every $\eta \in \Gamma_0$.
[guided]
We begin with a point and direction which are already known not to be singular for $u$. The coordinate chart
\begin{align*}
\kappa: U \to \Omega \subset \mathbb{R}^n
\end{align*}
turns a neighbourhood of $x_0$ in the manifold into an open set in Euclidean space. Since wave front sets live in the cotangent bundle, the covector $\xi_0 \in T_{x_0}^*M$ must also be transported to Euclidean cotangent coordinates. We denote its coordinate representative by $\eta_0 \in \mathbb{R}^n \setminus \{0\}$, using the cotangent pullback identification
\begin{align*}
d\kappa_{x_0}^{-\top}: T_{x_0}^*M \to \mathbb{R}^n.
\end{align*}
The condition $(x_0,\xi_0) \notin \operatorname{WF}(u)$ means that there is some localization of $u$ near $x_0$ whose Fourier transform decays faster than every power in a conic neighbourhood of the frequency direction $\eta_0$. Concretely, we choose $\psi \in C_c^\infty(U)$ with $\psi = 1$ on an open neighbourhood $U_0 \subset U$ of $x_0$, and define the compactly supported distribution
\begin{align*}
v := (\kappa^{-1})^*(\psi u) \in \mathcal{E}'(\Omega).
\end{align*}
There is an open cone $\Gamma_0 \subset \mathbb{R}^n \setminus \{0\}$ containing $\eta_0$ such that for every integer $N \ge 0$ there is a constant $C_N > 0$ with
\begin{align*}
|\widehat v(\eta)| \le C_N(1+|\eta|)^{-N}
\end{align*}
for every $\eta \in \Gamma_0$. This is the precise Fourier decay that we must transfer from $u$ to $a u$.
[/guided]
[/step]
[step:Write the localized product as a smooth multiplier times the localized distribution]
Choose $\chi \in C_c^\infty(U_0)$ with $\chi = 1$ on an open neighbourhood $U_1 \subset U_0$ of $x_0$. Define
\begin{align*}
b: \Omega \to \mathbb{R}, \qquad b(y) := (\chi a)(\kappa^{-1}(y)).
\end{align*}
Then $b \in C_c^\infty(\Omega)$. Since $\psi = 1$ on $\operatorname{supp}\chi$, the localized product satisfies
\begin{align*}
(\kappa^{-1})^*(\chi a u) = b\,v
\end{align*}
as compactly supported distributions on $\Omega$.
The Fourier transform convention is
\begin{align*}
\widehat f(\zeta) := (2\pi)^{-n/2}\int_{\mathbb{R}^n} f(y)e^{-i y\cdot \zeta}\,d\mathcal{L}^n(y)
\end{align*}
for $f \in \mathcal{S}(\mathbb{R}^n)$. We regard $b$ and $v$ as compactly supported objects on $\mathbb{R}^n$ by extension by zero outside $\Omega$. Under this convention, the Fourier transform of the product of the compactly supported smooth function $b$ and the compactly supported distribution $v$ is
\begin{align*}
\widehat{b v}(\zeta) = (2\pi)^{-n/2}(\widehat b * \widehat v)(\zeta).
\end{align*}
Here $\widehat b \in \mathcal{S}(\mathbb{R}^n)$ because $b \in C_c^\infty(\Omega)$, and $\widehat v$ has at most polynomial growth because $v \in \mathcal{E}'(\Omega)$.
[guided]
We now localize $a u$ with a smaller cutoff so that the already chosen localization of $u$ is still valid. Choose $\chi \in C_c^\infty(U_0)$ with $\chi = 1$ on an open neighbourhood $U_1 \subset U_0$ of $x_0$. Define the smooth compactly supported multiplier in coordinates by
\begin{align*}
b: \Omega \to \mathbb{R}, \qquad b(y) := (\chi a)(\kappa^{-1}(y)).
\end{align*}
Since $\chi a \in C_c^\infty(U)$ and $\kappa^{-1}: \Omega \to U$ is smooth, we have $b \in C_c^\infty(\Omega)$.
The reason for inserting $\chi$ inside $U_0$ is that $\psi = 1$ on $U_0$, hence $\psi = 1$ on $\operatorname{supp}\chi$. Therefore multiplying by $\chi a$ does not distinguish $u$ from $\psi u$:
\begin{align*}
\chi a u = \chi a\,\psi u
\end{align*}
as distributions on $U$. Pulling this identity through the coordinate chart gives
\begin{align*}
(\kappa^{-1})^*(\chi a u) = b\,v
\end{align*}
as compactly supported distributions on $\Omega$.
To take Fourier transforms, we use the convention
\begin{align*}
\widehat f(\zeta) := (2\pi)^{-n/2}\int_{\mathbb{R}^n} f(y)e^{-i y\cdot \zeta}\,d\mathcal{L}^n(y)
\end{align*}
for $f \in \mathcal{S}(\mathbb{R}^n)$. Since $b$ and $v$ are compactly supported in $\Omega$, we regard them as compactly supported objects on $\mathbb{R}^n$ by extension by zero outside $\Omega$. The product formula for the Fourier transform of a smooth compactly supported multiplier times a compactly supported distribution then gives
\begin{align*}
\widehat{b v}(\zeta) = (2\pi)^{-n/2}(\widehat b * \widehat v)(\zeta).
\end{align*}
Here $\widehat b \in \mathcal{S}(\mathbb{R}^n)$ because $b$ is smooth and compactly supported, while $\widehat v$ has at most polynomial growth because $v$ is compactly supported as a distribution. This reduces the problem to showing that convolution with the Schwartz function $\widehat b$ preserves the conic rapid decay of $\widehat v$.
[/guided]
[/step]
[step:Use convolution with a Schwartz function to preserve conic rapid decay]
Choose an open conic neighbourhood $\Gamma_1$ of $\eta_0$ with $\overline{\Gamma_1} \subset \Gamma_0 \cup \{0\}$.
[claim:Cone-shrinking convolution preserves rapid decay]
If $F: \mathbb{R}^n \to \mathbb{C}$ has polynomial growth and decays rapidly on $\Gamma_0$, and if $S \in \mathcal{S}(\mathbb{R}^n)$, then $S * F$ decays rapidly on every open cone $\Gamma_1$ whose closure is contained in $\Gamma_0 \cup \{0\}$.
[/claim]
[proof]
Let $m \ge 0$ and $B > 0$ satisfy $|F(\rho)| \le B(1+|\rho|)^m$ for every $\rho \in \mathbb{R}^n$. The strict conic containment gives a constant $\delta \in (0,1)$ such that, whenever $\zeta \in \Gamma_1$ and $\rho \notin \Gamma_0$, one has $|\zeta-\rho| \ge \delta(|\zeta|+|\rho|)$. Split the convolution integral into the regions $\rho \in \Gamma_0$ and $\rho \notin \Gamma_0$. On $\Gamma_0$, use the rapid decay of $F$ with exponent $N+n+1$ and the integrability of $S$. On the complement, use the polynomial growth bound for $F$ and the Schwartz bound
\begin{align*}
|S(\zeta-\rho)| \le C_{N,m}(1+|\zeta-\rho|)^{-N-m-n-1}.
\end{align*}
The angular separation implies
\begin{align*}
(1+|\zeta-\rho|)^{-N-m-n-1}(1+|\rho|)^m \le C'_{N,m}(1+|\zeta|)^{-N}(1+|\rho|)^{-n-1}.
\end{align*}
Since $(1+|\rho|)^{-n-1}$ is integrable with respect to $\mathcal{L}^n$ on $\mathbb{R}^n$, both pieces are bounded by a constant times $(1+|\zeta|)^{-N}$. Thus $S * F$ decays rapidly on $\Gamma_1$.
[/proof]
Applying the claim with $F = \widehat v$ and $S = \widehat b$, we obtain that for every integer $N \ge 0$ there is a constant $A_N > 0$ such that
\begin{align*}
|\widehat{b v}(\zeta)| \le A_N(1+|\zeta|)^{-N}
\end{align*}
for every $\zeta \in \Gamma_1$.
Thus the Fourier transform of $(\kappa^{-1})^*(\chi a u)$ decays rapidly in a conic neighbourhood of $\eta_0$. By the local definition of the wave front set, $(x_0,\xi_0) \notin \operatorname{WF}(a u)$. Since every point outside $\operatorname{WF}(u)$ is outside $\operatorname{WF}(a u)$, we have
\begin{align*}
\operatorname{WF}(a u) \subset \operatorname{WF}(u).
\end{align*}
[guided]
The only analytic issue is whether convolution with $\widehat b$ can destroy the rapid decay of $\widehat v$ in the cone $\Gamma_0$. It cannot, because $\widehat b$ is a Schwartz function, so it decays faster than every power in all directions.
We shrink the cone slightly: choose an open conic neighbourhood $\Gamma_1$ of $\eta_0$ with
\begin{align*}
\overline{\Gamma_1} \subset \Gamma_0 \cup \{0\}.
\end{align*}
This strict containment gives angular separation between points of $\Gamma_1$ and points outside $\Gamma_0$. More precisely, there is $\delta \in (0,1)$ such that, whenever $\zeta \in \Gamma_1$ and $\rho \notin \Gamma_0$, one has
\begin{align*}
|\zeta-\rho| \ge \delta(|\zeta|+|\rho|).
\end{align*}
This separation is what prevents frequencies outside the good cone from contributing significantly to the convolution. Since $v \in \mathcal{E}'(\Omega)$, its Fourier transform has polynomial growth: there are $m \ge 0$ and $B > 0$ such that $|\widehat v(\rho)| \le B(1+|\rho|)^m$ for every $\rho \in \mathbb{R}^n$. Since $\widehat v$ decays rapidly on $\Gamma_0$, the part of the convolution integral over $\Gamma_0$ has rapid decay. On the complementary region $\mathbb{R}^n \setminus \Gamma_0$, the polynomial growth of $\widehat v$ is dominated by the Schwartz decay of $\widehat b(\zeta-\rho)$ together with the separation estimate above. Therefore the cone-shrinking convolution estimate applies with $F = \widehat v$ and $S = \widehat b$.
Consequently, for every integer $N \ge 0$, there is a constant $A_N > 0$ such that
\begin{align*}
|\widehat{b v}(\zeta)| \le A_N(1+|\zeta|)^{-N}
\end{align*}
for every $\zeta \in \Gamma_1$. Since
\begin{align*}
(\kappa^{-1})^*(\chi a u) = b v,
\end{align*}
this proves that the localization of $a u$ by $\chi$ has rapidly decreasing Fourier transform in a conic neighbourhood of $\eta_0$. The cutoff $\chi$ equals $1$ near $x_0$, so the local definition of the wave front set gives
\begin{align*}
(x_0,\xi_0) \notin \operatorname{WF}(a u).
\end{align*}
We started with an arbitrary covector outside $\operatorname{WF}(u)$ and proved that it is outside $\operatorname{WF}(a u)$. Equivalently,
\begin{align*}
\operatorname{WF}(a u) \subset \operatorname{WF}(u).
\end{align*}
[/guided]
[/step]
[step:Invert the multiplier on an open set where it has no zeros]
Let $V \subset M$ be an open set such that $a(x) \ne 0$ for every $x \in V$. Define
\begin{align*}
c: V \to \mathbb{R}, \qquad c(x) := \frac{1}{a(x)}.
\end{align*}
Then $c \in C^\infty(V)$ and, as distributions on $V$,
\begin{align*}
u|_V = c\,(a u)|_V.
\end{align*}
Applying the already proved inclusion on the manifold $V$ with smooth multiplier $c$ gives
\begin{align*}
\operatorname{WF}(u|_V) \subset \operatorname{WF}((a u)|_V).
\end{align*}
By locality of the wave front set, this is exactly
\begin{align*}
\operatorname{WF}(u) \cap \pi^{-1}(V) \subset \operatorname{WF}(a u) \cap \pi^{-1}(V).
\end{align*}
The first inclusion proved above gives the opposite containment after intersecting with $\pi^{-1}(V)$. Hence
\begin{align*}
\operatorname{WF}(a u) \cap \pi^{-1}(V) = \operatorname{WF}(u) \cap \pi^{-1}(V).
\end{align*}
This proves both assertions.
[guided]
Now assume $V \subset M$ is open and $a(x) \ne 0$ for every $x \in V$. Since $a: M \to \mathbb{R}$ is smooth and nowhere zero on $V$, the reciprocal map
\begin{align*}
c: V \to \mathbb{R}, \qquad c(x) := \frac{1}{a(x)}
\end{align*}
is smooth on $V$. Multiplication by $c$ is therefore an allowed smooth multiplication operation on the restricted distribution $(a u)|_V \in \mathcal{D}'(V)$.
On $V$, the equality $c a = 1$ gives the distributional identity
\begin{align*}
u|_V = c\,(a u)|_V.
\end{align*}
We may apply the inclusion already proved, but now on the smooth manifold $V$ and with the smooth multiplier $c$. This gives
\begin{align*}
\operatorname{WF}(u|_V) = \operatorname{WF}(c\,(a u)|_V) \subset \operatorname{WF}((a u)|_V).
\end{align*}
The wave front set is local under restriction to an open subset, so this inclusion is the same as
\begin{align*}
\operatorname{WF}(u) \cap \pi^{-1}(V) \subset \operatorname{WF}(a u) \cap \pi^{-1}(V).
\end{align*}
The inclusion proved earlier in the proof gives
\begin{align*}
\operatorname{WF}(a u) \subset \operatorname{WF}(u),
\end{align*}
and intersecting this inclusion with $\pi^{-1}(V)$ gives the reverse containment over $V$:
\begin{align*}
\operatorname{WF}(a u) \cap \pi^{-1}(V) \subset \operatorname{WF}(u) \cap \pi^{-1}(V).
\end{align*}
The two containments yield
\begin{align*}
\operatorname{WF}(a u) \cap \pi^{-1}(V) = \operatorname{WF}(u) \cap \pi^{-1}(V).
\end{align*}
This proves the equality of wave front sets over every open set on which $a$ has no zeros, and completes the proof.
[/guided]
[/step]
