I am wondering if by replacing the Lebesgue measure in the definition of symmetric-decreasing function with a weighted Lebesgue measure, Riesz rearrangement inequality still holds. For clarity, I explain below.
If $w$ is a nice weight on $\mathbb{R}^n$, let $d\mu=w(x)dx$ and for any measurable set $A$ with $\mu(A)<\infty$, let $A^*$ be the open ball centred at the origin which satisfies $\mu(A)=\mu(A^*)$. Define the weighted symmetric-decreasing rearrangement of a Borel measurable function vanishing at infinity by
$$ f^*(x)=\displaystyle\int_0^\infty 1_{\left\{|f|>t\right\}^*}(x) dt,$$
where $1_A$ is just the characteristic function of $A$. As in the classical case, this is radially symmetric and non-decreasing.
Firstly, has this been studied in more detail? I came up with the definition because it seems to work in my problem, but I haven't found much about it. Secondly, as announced above, does the Riesz rearrangement inequality hold?
