Example of a Distribution where $(x, \xi) \in WF(u)$ but $(x, -\xi) \notin WF(u)$ I am searching for an example of a Distribution $u \in \mathcal{D'}(\mathbb{R}^2)$ where $(x, \xi) \in WF(u)$ but the opposite direction $(x, -\xi) \notin WF(u)$ is not. A quick Google-Search didn't bring anythin up and all "simple" distributions where the wavefront set doesn't contain all directions (e.g. the step function in the plane or $\delta$-Ridge along the y-Axis) always contain both directions (better: co-directions).
Also I don't really see, how I can picture such a distribution. To me it seems, if a distribution is singular at a point when approached from one direction, it also is when approached from the opposite direction.
 A: Let $H = \chi_{\mathbb R^+ \times \mathbb R}$ be the Heaviside function and define $f := \mathcal{F}^{-1}(H)$. I want to show that $((0, 0), (1,0))$ is an element of the wavefront set of $f$ and $((0, 0), (-1,0))$ is not.
Let $\phi \in C^\infty_c(\mathbb R^2)$ be a smooth cutoff function and $\phi(0) \neq 0$. 
Denoting by $*$ the convolution, we obtain by elementary computations that $\mathcal{F}(\phi \cdot f) = \mathcal{F}(\phi) * H$. 
Since $\phi \in C^\infty_c$, we can find for all $N\in \mathbb N$ a $C_N>0$ and a function $h_N \in L^1(\mathbb R^2)$ such that 
$$
|\mathcal{F}(\phi)(\xi)| \leq C_N h_N(\xi) (1+ |\xi_1|)^{-N}.
$$
Let $\xi \in \mathbb R^{-} \times \mathbb{R}$, then 
$$
|\mathcal{F}(\phi) * H(\xi)| = |\int_{\mathbb{R}^+ \times \mathbb{R}} \mathcal{F}(\phi)(\xi-y)dy| \leq C_N |\int_{\mathbb{R}^+ \times \mathbb{R}} h_N(\xi-y) (1+ |\xi_1-y_1|)^{-N} dy|.
$$
The last term is bounded by
$$
C_N \int |h_N(y)| dy (1+ |\xi_1|)^{-N}.
$$
In other words, for all smooth cutoff functions $\phi$ with $\phi(0) \neq 0$ and for all $N\in \mathbb N$ there extists a constant $C_N'>0$ so that $|\mathcal{F}(\phi \cdot f)(\xi)| \leq C_N' (1+ |\xi_1|)^{-N}$ for $\xi_1 \to -\infty$. This shows that $((0, 0), (-1,0))$ is not in the wavefront set of $f$.
Let now $\xi \in \mathbb R^{+} \times \mathbb{R}$, then we denote by $1$ the constant function taking only the value 1, and we get that 
$$
|\mathcal{F}(\phi) * H(\xi)| =  |\mathcal{F}(\phi) * 1(\xi) -  \mathcal{F}(\phi) * (1-H)(\xi)|.
$$
It follows from an elementary calculation that $|\mathcal{F}(\phi) * 1(\xi)| = |\phi(0)| =: c_0 >0$, where the positivity is by assumption.
Moreover, $(1-H) = \chi_{\mathbb R^{-} \times \mathbb R}$ and thus, by repeating the previous computation with flipped signs, we get that for all $N \in \mathbb N$ there exists $C_N''>0$ such that 
$|\mathcal{F}(\phi) * (1-H)(\xi)| \leq C_N'' (1+ |\xi_1|)^{-N}$ for $\xi_1 \to \infty$. This implies, that 
$$
|\mathcal{F}(\phi) * H(\xi)| \geq c_0 - C_N'' (1+ |\xi_1|)^{-N}
$$
for $\xi_1 \to \infty$. Since $c_0$ is positive, this shows that $((0,0), (1,0))$ is an element of the wavefront set of $f$.
