A counterexample to the fractional Michael-Simon inequality

Abstract

We show that the fractional Michael-Simon inequality does not hold for general smooth hypersurfaces and for the range of 0<s<(n-1)/n in the case p=1. We construct a family of smooth hypersurfaces, whose fractional mean curvature is nonnegative everywhere, which violates the inequality in the limit. The same construction yields a counterexample for every 1 \leq p<\infty for the range s<(n-1)/(np).