Level sets of fractional Sobolev functions

Abstract

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