Level sets of fractional Sobolev functions

Abstract

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s,p}(\hat{I}\text{c◯})$ with $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, $0<s<1$, and $1\le p<\infty$. 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 $\mathrm{liminf}$ and the $\mathrm{limsup}$ (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 $n-s$.

26 pages