Remainder Terms in the Fractional Sobolev Inequality

摘要

We show that the fractional Sobolev inequality for the embedding H~(S/2) (?~N)→L~(2N/(N-s))(?~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.