A scaling approach to Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper introduces a new scaling approach to the Caffarelli-Kohn-Nirenberg inequality applicable in Euclidean and Riemannian contexts, utilizing a novel parameter to facilitate the proof.

Contribution

It presents a new method leveraging a parameter s to prove the Caffarelli-Kohn-Nirenberg inequality in different geometric settings.

Findings

Successful proof of the inequality using the new parameter s

Extension of the inequality to Riemannian manifolds

Simplification of existing proof techniques

Abstract

We mainly consider the general Caffarelli-Kohn-Nirenberg inequality in the Euclidean and Riemannian setting. In both cases, our proof relies mostly on a new parameter s conveniently introduced, see (2.7).