Remainder Terms in the Fractional Sobolev Inequality

We show that the fractional Sobolev inequality for the embedding H-s/2(R-N) hooked right arrow L2N/(N-s)(R-N), S is an element of (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 LN/(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.