Remainder terms in the fractional Sobolev inequality

Abstract

We show that the fractional Sobolev inequality for the embedding $\stackrel{˝}{\hookrightarrow }{L}^{\frac{2N}{N-s}}({\mathbb{R}}^N)$, $s\in (0,N)$ 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 $L^{\frac{N}{N-s}}$-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.

13 pages

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