Remainder terms in the fractional Sobolev inequality

TL;DR

This paper improves the fractional Sobolev inequality by adding a remainder term that measures the distance to optimal functions, leading to sharper bounds and new existence results in fractional Sobolev spaces.

Contribution

It introduces a remainder term in the fractional Sobolev inequality, generalizing previous results from the integer case to fractional orders.

Findings

Remainder term proportional to distance to optimizers derived

Sharper fractional Sobolev inequalities established

Existence of remainder term in weak L^{N/(N-s)} norm proved

Abstract

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