Some new and short proofs for a class of Caffarelli–Kohn–Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli–Kohn–Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L2 spaces. These may be useful in studying solvability of problems involving new singular PDEs.