Let $U,V \subset \mathbb{R}^n$ be open sets, let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$, and let $\mathbb{R}^n_0:=\mathbb{R}^n\setminus\{0\}$. Let $\kappa:V \to U$ be a $C^\infty$ diffeomorphism, and let $u \in \mathcal{D}'(U)$. The pullback distribution $\kappa^*u\in\mathcal{D}'(V)$ is defined by

\begin{align*} (\kappa^*u)(\varphi)=u\left(x\mapsto \varphi(\kappa^{-1}(x))\,|\det d\kappa^{-1}_x|\right) \end{align*}

for every $\varphi\in C_c^\infty(V)$. Fourier transforms are taken with the convention

\begin{align*} \widehat{f}(\xi)=\int_{\mathbb{R}^n} f(x)e^{-ix\cdot\xi}\,d\mathcal{L}^n(x) \end{align*}

for $f\in C_c^\infty(\mathbb{R}^n)$, and by duality for compactly supported distributions. The wave front set $\operatorname{WF}(u)$ is defined by this Fourier-transform convention: $(x_0,\xi_0)\notin\operatorname{WF}(u)$ if and only if there are $\chi\in C_c^\infty(U)$ with $\chi(x_0)\neq0$ and an open conic neighbourhood $\Gamma\subset\mathbb{R}^n_0$ of $\xi_0$ such that $\widehat{\chi u}$ decays rapidly on $\Gamma$. For each $y \in V$, let

\begin{align*} (d\kappa_y)^\top:(\mathbb{R}^n)^* \to (\mathbb{R}^n)^* \end{align*}

denote the transpose of the differential $d\kappa_y:\mathbb{R}^n \to \mathbb{R}^n$ under the standard Euclidean coordinates. Then

\begin{align*} \operatorname{WF}(\kappa^*u)=\{(y,(d\kappa_y)^\top\xi):(\kappa(y),\xi)\in \operatorname{WF}(u)\}. \end{align*}