Level sets of fractional Sobolev functions

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s,p}(\mathrm{\Omega })$ with $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $0<s<1$, and $1\le p<\mathrm{\infty }$. Our theorem shows that a.e. level set has zero Hausdorff ${\mathcal{H}}^{n-s}$ measure, where the level set $f^{-1}(y)$ is defined as the set all points at which $y$ is between the $\mathrm{lim\hspace{0.17em}inf}$ and the $\mathrm{lim\hspace{0.17em}sup}$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r(y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.