It is a standard exercise to show that if $X\subseteq\mathbb{R}^n$ has smooth boundary, then the characteristic function $1\_X$ has wavefront set $$\{(x,\xi)\in\partial X\times\mathbb{R}^n\setminus\{0\}:\xi\text{ is normal to }\partial X\}.$$ This is shown by locally "flattening" to reduce to the case of the upper half space.


My question is what happens when $X$ does not have smooth boundary, such as the interior of a fractal curve. For example, if $X$ is the interior of the Koch snowflake, how would one compute the wavefront set of $1\_X$?



Wavefront set of characteristic function of rough set
Let me first begin with an elementary example, taking $X=[0,1]^2$ in $\mathbb R^2$. It is then easy to see directly that the wave-front-set of $\mathbb 1\_X$ is everywhere the conormal bundle except at the four corners $c\_j$: 
$$
WF\mathbb 1\_X=\cup\_{1\le j\le 4}(c\_j;\mathbb R^2\backslash {(0,0)})\cup\text{conormal bundle at the smooth points}.
$$
To prove this, it is enough to check the wave-front-set of $H(x\_1)H(x\_2)$ with $H=\mathbb 1\_{[0,+\infty)}$ which amounts to check the wave-front-set of $\delta\_0(x\_1)\otimes\delta\_0(x\_2)$ which contains at $(0,0)$ every direction since the Fourier transform is 1.


There are of course (much) more refined results. If you take a look at Lars Hörmander's ALPDO first volume, Springer Grundlehren 256 on page 300, you will find the definition of the normal set to any closed subset of $\mathbb R^n$ and Theorem 8.5.6' asserts that the analytic wave-front-set of a distribution $u$ contains the normal set to the support.


For the example you give, I believe that the $C^\infty$ wave-front-set is simply the product of the boundary with $\mathbb R^2\backslash {(0,0)}$. However for $s$ real, we may define the $H^s$ wave-front-set and I guess that the Hausdorff dimension of the fractal set you consider should be linked to the $H^s$ wave-front-set, in the sense that it should be trivial at some threshold $s\_0$ and that $s\_0$ should be linked to the Hausdorff dimension. The paper MR1277392, by Falcone, is addressing a related question.


