Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper presents concise, self-contained proofs of sharp weighted Caffarelli-Kohn-Nirenberg inequalities, improving understanding of minimizers and employing the expand of squares method for establishing these inequalities.

Contribution

It offers a simplified, self-contained proof of refined sharp CKN inequalities, highlighting the use of the expand of squares method and identifying minimizers in functional spaces.

Findings

Proofs are shorter and self-contained.

Results are sharp with identified minimizers.

Uses expand of squares method for inequalities.

Abstract

This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted $L^{2}$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments.