We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{ a,b}^(q,p)(R^N\{ 0}) with power weights |x|^a and |x|^b is continuously embedded into L^{ r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{ p,q}, are also obtained along the way.
Categories: math.AP math.FA