A clustering theorem in fractional Sobolev spaces

We prove a general clustering result for the fractional Sobolev spaceW_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.