A clustering theorem in fractional Sobolev spaces

We prove a general clustering result for the fractional Sobolev space \(W^{s,p}\): whenever the positivity set of a function \(u\) in a cube has measure bounded from below by a multiple of the cube's volume, and the \(W^{s,p}\)-seminorm of \(u\) is bounded from above by a convenient power of the cube's side, then \(u\) is positive in a universally reduced cube. Our result aims at applications in regularity theory for fractional elliptic and parabolic equations. Also, by means of suitable interpolation inequalities, we show that clustering results in \(W^{1,p}\) and \(BV\), respectively, can be deduced as special cases.