A sharp monomial Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper establishes the optimal constant and classifies optimizers for a monomial Caffarelli-Kohn-Nirenberg inequality using geometric and regularity techniques, also revealing symmetry-breaking phenomena.

Contribution

It introduces a novel approach combining $ ext{Gamma}$-calculus and geometric methods to analyze the inequality and its optimizers, including symmetry-breaking results.

Findings

Optimal constant for the inequality determined

Classification of all optimizers provided

Symmetry-breaking phenomenon identified

Abstract

We consider a monomial Caffarelli-Kohn-Nirenberg inequality, find the optimal constant and classify the optimizers under an integrated curvature dimension condition. We take advantage of the $\Gamma$-calculus to exploit geometrical techniques to tackle the problem and regularity results to justify some integration by parts. A symmetry-breaking result is also provided.