A boxing inequality for the fractional perimeter

We prove the Boxing inequality (equation presented) for every α 2 (0, 1) and every bounded open subset U ⊂Rd , where Hd-α 1 (U) is the Hausdorff content of U of dimension d - α and the constant C >0 depends only on d. We then show how this estimate implies a trace inequality in the fractional Sobolev space Wα,1(Rd ) that includes Sobolev's L d d-α embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as α tends to 0 and 1, recovering in particular the classical inequalities of first order. Their counterparts in the full range α 2 (0, d) are also investigated.