Remainder Terms in the Fractional Sobolev Inequality

We show that the fractional Sobolev inequality for the embedding H̊ s 2 (R ) →֒ L 2N N−s (R ), 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.