  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.
