A note on the fractional perimeter and interpolation

Ponce, Augusto [UCL] Spector, Daniel

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces W^{alpha, 1} of order 0 < alpha < 1.