Let $X$ and $Y$ be smooth second-countable Hausdorff manifolds of dimensions $m$ and $n$, respectively, and let $F: X \to Y$ be a smooth map. Let $\mathcal{D}'(X)$ and $\mathcal{D}'(Y)$ denote the spaces of complex-valued distributions on $X$ and $Y$. For each $x \in X$, let $dF_x: T_xX \to T_{F(x)}Y$ denote the differential and let $(dF_x)^*: T^*_{F(x)}Y \to T^*_xX$ denote its dual map. Define the normal set of $F$ by

\begin{align*} N_F := \{(F(x), \eta) \in T^*Y \setminus \{0\} : x \in X,\ \eta \in T^*_{F(x)}Y \setminus \{0\}, \text{ and } (dF_x)^*\eta = 0\}. \end{align*}

Let $\Gamma \subset T^*Y \setminus \{0\}$ be a closed conic subset, meaning a closed subset invariant under multiplication of cotangent vectors by positive real scalars, such that $\Gamma \cap N_F = \varnothing$. Define

\begin{align*} F^*\Gamma := \{(x, (dF_x)^*\eta) \in T^*X \setminus \{0\} : (F(x), \eta) \in \Gamma\}. \end{align*}

Then $F^*\Gamma$ is a closed conic subset of $T^*X \setminus \{0\}$, and the ordinary pullback on smooth functions extends uniquely to a continuous [linear map](/page/Linear%20Map)

\begin{align*} F^*: \mathcal{D}'_\Gamma(Y) \to \mathcal{D}'_{F^*\Gamma}(X), \end{align*}

where

\begin{align*} \mathcal{D}'_\Gamma(Y) := \{u \in \mathcal{D}'(Y) : \operatorname{WF}(u) \subset \Gamma\} \end{align*}

and

\begin{align*} \mathcal{D}'_{F^*\Gamma}(X) := \{v \in \mathcal{D}'(X) : \operatorname{WF}(v) \subset F^*\Gamma\} \end{align*}

are equipped with the Hörmander topologies generated in coordinate charts by ordinary distribution seminorms and by conic Fourier-decay seminorms away from the indicated closed conic set.

In particular, if $u \in \mathcal{D}'(Y)$ satisfies $\operatorname{WF}(u) \cap N_F = \varnothing$, then the pullback $F^*u \in \mathcal{D}'(X)$ is well-defined, agrees with ordinary composition $u \circ F$ when $u$ is smooth, and satisfies

\begin{align*} \operatorname{WF}(F^*u) \subset \{(x, (dF_x)^*\eta) : (F(x), \eta) \in \operatorname{WF}(u) \text{ and } (dF_x)^*\eta \neq 0\}. \end{align*}

All wave front sets are regarded as closed conic subsets of the corresponding cotangent bundle with the zero section removed. In local coordinates on open subsets of $\mathbb{R}^k$, $\mathcal{L}^k$ denotes $k$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure), $S^{k-1}:=\{\omega \in \mathbb{R}^k: |\omega|=1\}$ denotes the Euclidean unit sphere, and the [Fourier transform](/page/Fourier%20Transform) is taken with the convention $\hat{h}(\xi)=\int_{\mathbb{R}^k}e^{-ix\cdot\xi}h(x)\,d\mathcal{L}^k(x)$ for compactly supported smooth functions $h$.