[proofplan]
We prove the inclusion by showing that every covector outside the displayed right-hand side is microlocally smooth for $u \otimes v$. The argument is local in product coordinate charts, where a product cutoff $\chi(x)\psi(y)$ turns the [tensor product](/page/Tensor%20Product) into a compactly supported distribution whose [Fourier transform](/page/Fourier%20Transform) factors as the product of the two localized Fourier transforms. If a covector component avoids the corresponding wave front set, that factor has rapid decay in a conic neighbourhood; the remaining factor is controlled by the standard polynomial growth estimate for compactly supported distributions. A conic separation argument then combines these estimates on a smaller cone around the given covector.
[/proofplan]
[step:Reduce the assertion to one product coordinate neighbourhood]
Let $((x_0,y_0),(\xi_0,\eta_0)) \in T^*(X \times Y)\setminus 0$ be outside the right-hand side in the statement. We prove that
\begin{align*}
((x_0,y_0),(\xi_0,\eta_0)) \notin \operatorname{WF}(u \otimes v).
\end{align*}
Choose coordinate charts $(U,\kappa)$ on $X$ and $(V,\lambda)$ on $Y$ with $x_0 \in U$ and $y_0 \in V$. Let $m = \dim X$ and $n = \dim Y$. The product chart
\begin{align*}
\kappa \times \lambda: U \times V \to \kappa(U) \times \lambda(V) \subset \mathbb{R}^{m+n}
\end{align*}
identifies the covector $(\xi_0,\eta_0)$ with a nonzero vector $(\xi_0',\eta_0') \in \mathbb{R}^m \times \mathbb{R}^n$. Since the wave front set is characterized locally by rapid decay of localized Fourier transforms in coordinate charts, it is enough to prove the corresponding Euclidean assertion for the coordinate representatives of $u$ and $v$ near $(x_0,y_0)$. We therefore write the local coordinates again as $x \in \mathbb{R}^m$ and $y \in \mathbb{R}^n$, and write the covector as $(\xi_0,\eta_0) \in \mathbb{R}^m \times \mathbb{R}^n$.
Let
\begin{align*}
\chi: \mathbb{R}^m \to \mathbb{R}
\end{align*}
be a function in $C_c^\infty(\kappa(U))$ with $\chi = 1$ on a neighbourhood of $x_0$, and let
\begin{align*}
\psi: \mathbb{R}^n \to \mathbb{R}
\end{align*}
be a function in $C_c^\infty(\lambda(V))$ with $\psi = 1$ on a neighbourhood of $y_0$. Define the product cutoff
\begin{align*}
\varphi: \mathbb{R}^{m+n} \to \mathbb{R}, \qquad \varphi(x,y) = \chi(x)\psi(y).
\end{align*}
Then $\varphi \in C_c^\infty(\kappa(U)\times\lambda(V))$ and $\varphi = 1$ near $(x_0,y_0)$.
It remains to find an open conic neighbourhood $\Gamma \subset \mathbb{R}^{m+n}\setminus\{0\}$ of $(\xi_0,\eta_0)$ such that, for every $N \in \mathbb{N}$, there exists $C_N > 0$ with
\begin{align*}
|\widehat{\varphi(u \otimes v)}(\xi,\eta)| \le C_N(1+|(\xi,\eta)|)^{-N}
\end{align*}
for all $(\xi,\eta) \in \Gamma$.
[/step]
[step:Factor the Fourier transform after product localization]
For compactly supported distributions $\chi u \in \mathcal{E}'(\mathbb{R}^m)$ and $\psi v \in \mathcal{E}'(\mathbb{R}^n)$, define their Fourier transforms by
\begin{align*}
\widehat{\chi u}(\xi) = (\chi u)(x \mapsto e^{-i x\cdot \xi})
\end{align*}
and
\begin{align*}
\widehat{\psi v}(\eta) = (\psi v)(y \mapsto e^{-i y\cdot \eta}).
\end{align*}
Likewise, define
\begin{align*}
\widehat{\varphi(u \otimes v)}(\xi,\eta) = \varphi(u \otimes v)((x,y) \mapsto e^{-i(x\cdot \xi + y\cdot \eta)}).
\end{align*}
Since $\varphi(x,y)=\chi(x)\psi(y)$ and the exponential factors as
\begin{align*}
e^{-i(x\cdot \xi + y\cdot \eta)} = e^{-i x\cdot \xi}e^{-i y\cdot \eta},
\end{align*}
the definition of the tensor product of distributions gives
\begin{align*}
\widehat{\varphi(u \otimes v)}(\xi,\eta) = \widehat{\chi u}(\xi)\widehat{\psi v}(\eta).
\end{align*}
We shall also use the standard polynomial growth estimate for compactly supported distributions: if $T \in \mathcal{E}'(\mathbb{R}^d)$, then there exist constants $A_T > 0$, $M_T \in \mathbb{N}$, and $C_T > 0$ such that
\begin{align*}
|\widehat{T}(\zeta)| \le C_T(1+|\zeta|)^{M_T}
\end{align*}
for all $\zeta \in \mathbb{R}^d$. This is the only external growth estimate used here; if it is not yet available in the wiki, it should be recorded as the polynomial growth of Fourier transforms of compactly supported distributions.
[/step]
[step:Separate the possible covector configurations outside the forbidden set]
Because $((x_0,y_0),(\xi_0,\eta_0))$ lies outside the right-hand side, one of the following alternatives holds.
First, if $\xi_0 \ne 0$ and $\eta_0 \ne 0$, then at least one of
\begin{align*}
(x_0,\xi_0) \notin \operatorname{WF}(u)
\end{align*}
or
\begin{align*}
(y_0,\eta_0) \notin \operatorname{WF}(v)
\end{align*}
holds.
Second, if $\xi_0 \ne 0$ and $\eta_0 = 0$, then the second set on the right-hand side excludes the point. Hence either
\begin{align*}
(x_0,\xi_0) \notin \operatorname{WF}(u)
\end{align*}
or
\begin{align*}
y_0 \notin \operatorname{supp}(v).
\end{align*}
Third, if $\xi_0 = 0$ and $\eta_0 \ne 0$, then the third set on the right-hand side excludes the point. Hence either
\begin{align*}
x_0 \notin \operatorname{supp}(u)
\end{align*}
or
\begin{align*}
(y_0,\eta_0) \notin \operatorname{WF}(v).
\end{align*}
The case $\xi_0=0$ and $\eta_0=0$ is impossible because $(\xi_0,\eta_0)$ is a nonzero covector in $T^*(X \times Y)\setminus 0$.
[/step]
[step:Obtain rapid decay when one factor is microlocally regular]
Assume first that $(x_0,\xi_0) \notin \operatorname{WF}(u)$ with $\xi_0 \ne 0$. By the local definition of the wave front set, after possibly shrinking the support of $\chi$ while keeping $\chi=1$ near $x_0$, there exists an open conic neighbourhood $\Gamma_X \subset \mathbb{R}^m\setminus\{0\}$ of $\xi_0$ such that, for every $N \in \mathbb{N}$, there is a constant $A_N > 0$ satisfying
\begin{align*}
|\widehat{\chi u}(\xi)| \le A_N(1+|\xi|)^{-N}
\end{align*}
for all $\xi \in \Gamma_X$.
Let $M_v \in \mathbb{N}$ and $B_v > 0$ be polynomial growth constants for $\psi v$, so that
\begin{align*}
|\widehat{\psi v}(\eta)| \le B_v(1+|\eta|)^{M_v}
\end{align*}
for all $\eta \in \mathbb{R}^n$.
Choose a conic neighbourhood $\Gamma \subset \mathbb{R}^{m+n}\setminus\{0\}$ of $(\xi_0,\eta_0)$ inside the product cone determined by the unit direction $(\xi_0,\eta_0)/|(\xi_0,\eta_0)|$. Because $\xi_0 \ne 0$, shrinking $\Gamma$ if necessary we may ensure that $\xi \in \Gamma_X$ and that the $\xi$-component stays uniformly away from $0$ relative to the full covector, so there is a constant $c>0$ with $|\xi| \ge c|(\xi,\eta)|$ for all $(\xi,\eta) \in \Gamma$.
For a prescribed $N \in \mathbb{N}$, apply the rapid-decay estimate for $\widehat{\chi u}$ with exponent $N+M_v+1$. For $(\xi,\eta)\in\Gamma$,
\begin{align*}
|\widehat{\varphi(u \otimes v)}(\xi,\eta)| \le A_{N+M_v+1}B_v(1+|\xi|)^{-N-M_v-1}(1+|\eta|)^{M_v}.
\end{align*}
Since $|\eta| \le |(\xi,\eta)|$ and $|\xi| \ge c|(\xi,\eta)|$ on $\Gamma$, there is a constant $C_N>0$ such that
\begin{align*}
|\widehat{\varphi(u \otimes v)}(\xi,\eta)| \le C_N(1+|(\xi,\eta)|)^{-N}
\end{align*}
for all $(\xi,\eta)\in\Gamma$.
Assume instead that $(y_0,\eta_0)\notin \operatorname{WF}(v)$ with $\eta_0 \ne 0$. By the local definition of the wave front set, after possibly shrinking the support of $\psi$ while keeping $\psi=1$ near $y_0$, there exists an open conic neighbourhood $\Gamma_Y \subset \mathbb{R}^n\setminus\{0\}$ of $\eta_0$ such that, for every $N \in \mathbb{N}$, there is a constant $A'_N > 0$ satisfying
\begin{align*}
|\widehat{\psi v}(\eta)| \le A'_N(1+|\eta|)^{-N}
\end{align*}
for all $\eta \in \Gamma_Y$.
Let $M_u \in \mathbb{N}$ and $B_u > 0$ be polynomial growth constants for $\chi u$, so that
\begin{align*}
|\widehat{\chi u}(\xi)| \le B_u(1+|\xi|)^{M_u}
\end{align*}
for all $\xi \in \mathbb{R}^m$.
Choose a conic neighbourhood $\Gamma \subset \mathbb{R}^{m+n}\setminus\{0\}$ of $(\xi_0,\eta_0)$ and a constant $c>0$ such that $\eta \in \Gamma_Y$ and $|\eta| \ge c|(\xi,\eta)|$ for all $(\xi,\eta) \in \Gamma$. This is possible because $\eta_0 \ne 0$ and $\Gamma_Y$ is an open conic neighbourhood of $\eta_0$.
For a prescribed $N \in \mathbb{N}$, apply the rapid-decay estimate for $\widehat{\psi v}$ with exponent $N+M_u+1$. For $(\xi,\eta) \in \Gamma$,
\begin{align*}
|\widehat{\varphi(u \otimes v)}(\xi,\eta)| \le A'_{N+M_u+1}B_u(1+|\eta|)^{-N-M_u-1}(1+|\xi|)^{M_u}.
\end{align*}
Since $|\xi| \le |(\xi,\eta)|$ and $|\eta| \ge c|(\xi,\eta)|$ on $\Gamma$, there is a constant $C'_N>0$ such that
\begin{align*}
|\widehat{\varphi(u \otimes v)}(\xi,\eta)| \le C'_N(1+|(\xi,\eta)|)^{-N}
\end{align*}
for all $(\xi,\eta) \in \Gamma$.
[guided]
We explain the main estimate in the case where the $u$-factor is microlocally regular. The assumption $(x_0,\xi_0)\notin \operatorname{WF}(u)$ with $\xi_0\ne 0$ means exactly that we can localize $u$ near $x_0$ and obtain rapid decay of its Fourier transform in a conic neighbourhood of $\xi_0$. Thus, after choosing $\chi \in C_c^\infty(\mathbb{R}^m)$ with $\chi=1$ near $x_0$, there is an open conic set $\Gamma_X \subset \mathbb{R}^m\setminus\{0\}$ containing $\xi_0$ such that for every $N \in \mathbb{N}$ there exists $A_N>0$ with
\begin{align*}
|\widehat{\chi u}(\xi)| \le A_N(1+|\xi|)^{-N}
\end{align*}
for all $\xi \in \Gamma_X$.
The other factor need not decay. However, because $\psi v$ is compactly supported, its Fourier transform has at most polynomial growth. Thus there are constants $M_v \in \mathbb{N}$ and $B_v>0$ such that
\begin{align*}
|\widehat{\psi v}(\eta)| \le B_v(1+|\eta|)^{M_v}
\end{align*}
for every $\eta\in\mathbb{R}^n$. This is why rapid decay is strong enough: we may spend extra powers of decay from the good factor to absorb the polynomial growth of the other factor.
We now choose the cone in the product variables. Since $\xi_0\ne 0$, a sufficiently small conic neighbourhood $\Gamma$ of $(\xi_0,\eta_0)$ has two useful properties: first, its $\xi$-projection lies in $\Gamma_X$; second, the $\xi$-component remains a fixed positive fraction of the full covector. Thus there is $c>0$ such that
\begin{align*}
\xi \in \Gamma_X
\end{align*}
and
\begin{align*}
|\xi| \ge c|(\xi,\eta)|
\end{align*}
for all $(\xi,\eta)\in\Gamma$.
Using the factorization of the localized Fourier transform, for $(\xi,\eta)\in\Gamma$ we get
\begin{align*}
|\widehat{\varphi(u\otimes v)}(\xi,\eta)| = |\widehat{\chi u}(\xi)|\,|\widehat{\psi v}(\eta)|.
\end{align*}
Given the desired decay order $N$, we apply the rapid-decay estimate for $\widehat{\chi u}$ with the larger exponent $N+M_v+1$. Then
\begin{align*}
|\widehat{\varphi(u\otimes v)}(\xi,\eta)| \le A_{N+M_v+1}B_v(1+|\xi|)^{-N-M_v-1}(1+|\eta|)^{M_v}.
\end{align*}
Because $|\eta|\le |(\xi,\eta)|$ and $|\xi|\ge c|(\xi,\eta)|$ on $\Gamma$, the right-hand side is bounded by a constant times $(1+|(\xi,\eta)|)^{-N}$. Therefore there exists $C_N>0$ such that
\begin{align*}
|\widehat{\varphi(u\otimes v)}(\xi,\eta)| \le C_N(1+|(\xi,\eta)|)^{-N}
\end{align*}
for every $(\xi,\eta)\in\Gamma$.
The case in which the $v$-factor is microlocally regular is identical after exchanging the variables $x,\xi,u,\chi$ with $y,\eta,v,\psi$. The key point is that one rapidly decreasing factor always dominates one polynomially growing factor.
[/guided]
[/step]
[step:Remove the tensor product near points outside one support]
Suppose $y_0 \notin \operatorname{supp}(v)$. By definition of the support of a distribution, there is an open neighbourhood $W \subset \mathbb{R}^n$ of $y_0$ such that $v|_{C_c^\infty(W)} = 0$. Choose $\psi \in C_c^\infty(\mathbb{R}^n)$ with $\psi=1$ near $y_0$ and $\operatorname{supp}\psi \subset W$. Then $\psi v=0$, and hence
\begin{align*}
\widehat{\varphi(u\otimes v)}(\xi,\eta)=\widehat{\chi u}(\xi)\widehat{\psi v}(\eta)=0
\end{align*}
for all $(\xi,\eta)\in\mathbb{R}^{m+n}$. This gives rapid decay in every conic neighbourhood of $(\xi_0,\eta_0)$.
Likewise, suppose $x_0 \notin \operatorname{supp}(u)$. Then there is an open neighbourhood $U' \subset \mathbb{R}^m$ of $x_0$ such that $u|_{C_c^\infty(U')} = 0$. Choose $\chi \in C_c^\infty(\mathbb{R}^m)$ with $\chi=1$ near $x_0$ and $\operatorname{supp}\chi \subset U'$. Then $\chi u=0$, so the localized tensor product has identically zero Fourier transform.
[/step]
[step:Conclude the wave front inclusion from the local estimates]
Combining the alternatives from the conic separation step with the estimates above, every covector $((x_0,y_0),(\xi_0,\eta_0))$ outside the displayed right-hand side admits a product cutoff $\varphi=\chi\psi$ equal to $1$ near $(x_0,y_0)$ and an open conic neighbourhood $\Gamma$ of $(\xi_0,\eta_0)$ such that, for every $N\in\mathbb{N}$, there exists $C_N>0$ with
\begin{align*}
|\widehat{\varphi(u\otimes v)}(\xi,\eta)| \le C_N(1+|(\xi,\eta)|)^{-N}
\end{align*}
for all $(\xi,\eta)\in\Gamma$.
By the local coordinate characterization of the wave front set, this means
\begin{align*}
((x_0,y_0),(\xi_0,\eta_0)) \notin \operatorname{WF}(u\otimes v).
\end{align*}
Since the covector outside the right-hand side was arbitrary, the desired inclusion follows.
[/step]
