Local maximal operators on fractional Sobolev spaces

Abstract

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 ℝⁿ, 0 < s < 1 and 1 < p < ∞. As an application, we characterize the fractional (s,p)-Hardy inequality on a bounded open set by a Maz'ya-type testing condition localized to Whitney cubes.