Evans points out that sobolev embedding theorem does not imply hardy's inequality. I am having trouble figuring out how one would make this statement rigorous.
The reason why I am having trouble is that to prove hardy's inequality $\left( \int (u/|x|)^2 \leq \frac{2}{d-2}\int |Du|^2 \right)$, one does not need many manipulations anyway. So if in the course of manipulating $|u|_{H_1(R^n)} \leq C |u|_{L_p*(R^n)}, \frac{1}{p*}={\frac{1}{2}-\frac{1}{n}}$, one uses too many steps then one will end up proving hardy's inequality from scratch. Moreover, sobolev's inequality bounds a derivative by the function and hardy's seems to bound the function by a derivative so I am not sure why one would think that sobolev embedding theorem implies hardy's inequality.
Any insight is appreciated. Thanks.