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](http://www.math.utoronto.ca/almut/rearrange.pdf)), 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.



What functions are equal to their symmetric decreasing rearrangement?
It should be straightforward to verify that $\mathcal A$ consists *exactly* of the lower semi-continuous radial and decreasing functions (which are 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 $n=1$).


