Non-linear ground state representations and sharp Hardy inequalities

TL;DR

This paper establishes the optimal constants in Hardy inequalities for fractional Sobolev spaces using a novel non-linear, non-local ground state approach, and explores related Sobolev embeddings and equality cases.

Contribution

It introduces a new non-linear, non-local ground state representation to determine sharp Hardy constants and related embeddings in fractional Sobolev spaces.

Findings

Sharp Hardy inequality constants for fractional Sobolev spaces

Optimal Sobolev embedding constants in Lorentz scale

Characterization of equality cases in rearrangement inequalities

Abstract

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.