Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding), s ∈ (0, N ) can be sharpened by adding a remainder term proportional to the distance to the set of optimizers.As a corollary, we derive the existence of a remainder term in the weak L N N -s -norm for functions supported in a domain of finite measure.Our results generalize earlier work for the non-fractional case where s is an even integer.