Norm Constants in cases of the Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper simplifies the derivation of sharp constants in the Caffarelli-Kohn-Nirenberg inequality using elementary linear algebra and explores symmetry breaking and the non-existence of optimizers for positive alpha.

Contribution

It provides radical simplifications of existing proofs and extends analysis to the full parameter range of alpha, including cases not previously addressed.

Findings

Sharp constants obtained via simplified methods

Symmetry breaking occurs for alpha > 0

No optimizers exist when alpha > 0

Abstract

By methods based on elementary Linear Algebra we obtain sharp constants in cases of the Caffarelli-Kohn-Nirenberg inequality via quasi-conformal changes of variables. Some of our results were obtained earlier by Lam and Lu. Our proofs are radical simplifications of earlier proofs. In the case $\alpha>0$ we establish that we have symmetry breaking and optimizers to the CKN inequalities do not exist. This case was not treated by Lam and Lu. Our results thus are in the full parameter range of \alpha.