Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities

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 $\mathit{\text{expand}}\mathit{\text{of}}\mathit{\text{squares}}$ method to establish sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments.