Suppose $B_1$ is the unit ball centered at the origin in $ R^N$ with $N \ge 3$. Let $ q= 2^* = \frac{2N}{N-2}$. Does there exist some $C>0$ such that $$\int_{B_1} \frac{ u(x)^2}{|x|^2} dx \le C \| u \|_{L^q(B_1)}^2,$$ for all smooth compactly supported (in $B_1$) radial functions? I assume the answer is false but a counter example maybe isn't completely trivial since both sides scale the same (or maybe it is trivial). If it holds then is $C$ known?

I tried the obvious thing with H"older's inequality but of course you just miss; maybe one can use weak $L^p$ spaces? any comments greatly appreciated.

Suppose $B_1$ is the unit ball centered at the origin in $ R^N$ with $N \ge 3$. Let $ q= 2^* = \frac{2N}{N-2}$. Does there exist some $C>0$ such that $$\int_{B_1} \frac{ u(x)^2}{|x|^2} dx \le C \| u \|_{L^q(B_1)}^2,$$ for all smooth compactly supported (in $B_1$) radial functions? I assume the answer is false but a counter example maybe isn't completely trivial since both sides scale the same (or maybe it is trivial). If it holds then is $C$ known?

I tried the obvious thing with Hölder's inequality but of course you just miss; maybe one can use weak $L^p$ spaces? any comments greatly appreciated.