How to connect the functions in spaces $H^1$ and $H_r$?

\begin{align*} L^2 (\mathbb{R}^3)& {}=\{ u : \int_{\mathbb{R}^3} \lvert u\rvert^2 dx<+\infty \}. \\ H^1(\mathbb{R}^3) & {}=\{ u\in L^2 (\mathbb{R}^3):\, \lvert\nabla u\rvert\in L^2(\mathbb{R}^3) \}. \\ H_r(\mathbb{R}^3) & {}=\{ u\in H^1 (\mathbb{R}^3):\, \text{u is radial} \}. \end{align*}

How to connect the functions in spaces $$H^1$$ and $$H_r$$?

I saw a lemma like that, for every $$u\in H^1(\mathbb{R}^N)$$, $$u\geq 0$$, there results $$u^*\in H_r$$, $$u^*\geq 0$$,

\begin{align*} \int_{\mathbb{R}^N}|\nabla u^*|^2dx\leq \int_{\mathbb{R}^N}|\nabla u|^2dx\quad and \quad\quad\quad\\ \int_{\mathbb{R}^N}|u^*|^pdx = \int_{\mathbb{R}^N}|u|^pdx,\quad for\;all\;\, p>1. \end{align*}

So the functions in these two Spaces can be related by integration.

Is there some other relationship (e.g. inequality) between the functions in these two spaces $$H^1$$ and $$H_r$$?

e.g. (I guess) $$\,\forall\, u\in H^1$$, is there an $$u^*\in H_r$$ s.t. $$\lvert u(x)\rvert\leq \lvert u^*(x)\rvert\,$$?

This is not true since a radial majorant might not be in $$L^2$$. A counterexample is $$u(x,y)=(1+x^2+y^4)^{-\frac 12} \in H^1(\mathbb R^2)$$. If $$u^*$$ is radial and majorizes $$u$$, and $$r=\sqrt {x^2+y^2}$$, then $$u(r) \geq (1+r^2)^{-1} \not \in L^2(\mathbb R^2)$$ .