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 functionu 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 applicationsin 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