Radial Sobolev inequality
= Radial Sobolev inequality
{c}
For a radial function $u\in H^1(\mathbb R^d)$,
$$
|u(r)|^2\lesssim_d r^{-(d-1)}\|u\|_2\|\nabla u\|_2.
$$
This supplies spatial decay without requiring a weighted moment.
= Radial Sobolev inequality
{c}
For a radial function $u\in H^1(\mathbb R^d)$,
$$
|u(r)|^2\lesssim_d r^{-(d-1)}\|u\|_2\|\nabla u\|_2.
$$
This supplies spatial decay without requiring a weighted moment.