Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding H^(s/2)(R^N),↪ L^(2N)/_(N-s) (R^N) 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.