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 a in a cube has measure bounded from below by a multiple of the cube's volume, and the W^s,p-seminorm of a is bounded from above by a convenient power of the cube's side, then a 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.