What functions are equal to their symmetric decreasing rearrangement?

Question

I am trying to understand the set $$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$ where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if for all $t>0$ such that $\text{Vol}(\{x:f(x)>t\})<+\infty$ and $f^*$ denotes the symmetric decreasing rearrangement.

From elementary properties of $f^*$ (see Burchard - A short course on rearrangement inequalities), we know that $f^*$ satisfies

1. $f^*$ is a lower semi-continuous, radial, and decreasing function. Thus $\mathcal{A}$ consists of lower semi-continuous functions.
2. I also know that the operator which sends a function to its symmetric rearrangement is not always continuous on the space of $W^{1,p}$ functions.

I am curious to know if there are any results that characterize functions in the set $\mathcal{A}$ or in other words what are the possible functions that are equal to their rearrangement?

I understand that there can be many examples of such functions so perhaps it makes sense to ask if we can classify functions in the set $\mathcal{A}\cap L^p$, $\mathcal{A}\cap W^{k,p}$, etc.

Answer 1

It should be straightforward to verify that $\mathcal A$ consists exactly of the lower semi-continuous radial and decreasing functions (which are non-negative and vanish at $\infty$): You already know that every $f\in\mathcal A$ has these properties, so you only have to verify that if $f$ has these properties, then $f=f^*$. If the definition is of any value, this should be rather straightforward to do (start with the classical case $n=1$).