IUMJ

Title: Remainder terms in the fractional Sobolev inequality

Authors: Shibing Chen, Rupert Frank and Tobias Weth

Issue: Volume 62 (2013), Issue 4, 1381-1397

Abstract:

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