A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality

TL;DR

This paper presents a novel proof of the sharp Hardy-Littlewood-Sobolev inequality that avoids rearrangement inequalities, extending the approach previously used for the Heisenberg group to the entire parameter range.

Contribution

It introduces the first rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality applicable across all parameters.

Findings

Proof method avoids rearrangement inequalities

Applicable to the entire parameter range

Extends previous approach from the Heisenberg group

Abstract

We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.