Local maximal operators on fractional Sobolev spaces
  In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0<s<1$ and $1<p<\infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set $G$ by a Maz'ya-type testing condition localized to Whitney cubes. 
Hannes Luiro and Antti V. V\"ah\"akangas
math.CA math.FA