A fractional Michael–Simon Sobolev inequality on convex hypersurfaces

The classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional Lp term on the right-hand side which is weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo seminorm of the function, as well as its Lp norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.